Subgroups — C₂₃×#26987 #19174613

Order 5888.

974 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#468	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#440	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2428	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2083	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2033	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2014	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2000	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1957	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1936	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1933	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1895	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1880	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1753	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#121	—	2	1	no	—
22	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#468	—	2	4	no	—
27	C₂₃×#440	—	2	4	no	—
28	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#399	—	2	1	no	—
42	C₂₃×#398	—	2	1	no	—
43	C₂₃×#394	—	2	1	no	—
44	C₂₃×#392	—	2	1	no	—
45	C₂₃×#391	C₂²	1	1	yes	characteristic
46	C₂₃×#390	C₂²	1	1	yes	characteristic
47	C₂₃×#389	C₂²	1	1	yes	characteristic
48	C₂₃×#389	C₂²	1	1	yes	characteristic
49	C₂₃×#388	—	2	1	no	—
50	C₂₃×#387	—	2	1	no	—
51	C₂₃×#386	—	2	1	no	—
52	C₂₃×#384	C₂²	1	1	yes	characteristic
53	C₂₃×#380	C₂²	1	1	yes	characteristic
54	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	—	2	1	no	—
63	C₂²×D₄×C₂₃	—	2	1	no	—
64	C₂²×D₄×C₂₃	—	2	1	no	—
65	C₂²×D₄×C₂₃	—	2	1	no	—
66	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	—	2	2	no	—
72	C₂×SD₁₆×C₂₃	—	2	2	no	—
73	C₂×SD₁₆×C₂₃	—	2	2	no	—
74	C₂×SD₁₆×C₂₃	—	2	2	no	—
75	C₂×SD₁₆×C₂₃	—	4	1	no	—
76	C₂×M₁₆×C₂₃	—	2	2	no	—
77	C₂×M₁₆×C₂₃	—	2	2	no	—
78	C₂×M₁₆×C₂₃	—	4	1	no	—
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₂₃×#127	—	2	2	no	—
84	C₂₃×#127	—	4	1	no	—
85	C₂₃×C₂²⋊Q₈	—	2	2	no	—
86	C₂₃×C₂²⋊Q₈	—	4	1	no	—
87	C₂₃×#121	—	2	2	no	—
88	C₂₃×#121	—	2	2	no	—
89	C₂₃×#121	—	4	4	no	—
90	C₂₃×#121	—	4	1	no	—
91	C₂₃×#121	—	4	1	no	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	4	1	no	—
94	C₄×D₄×C₂₃	—	2	2	no	—
95	C₄×D₄×C₂₃	—	4	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₂₃	—	2	1	no	—
117	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
119	C₂₃×#107	—	2	2	no	—
120	C₂₃×#107	—	2	2	no	—
121	C₂₃×#103	—	4	1	no	—
122	C₂₃×#103	D₄	1	4	yes	complemented
123	C₂₃×#102	—	2	2	no	—
124	C₂₃×#102	—	2	2	no	—
125	C₂₃×#102	—	2	2	no	—
126	C₂₃×#102	—	2	2	no	—
127	C₂₃×#102	—	4	1	no	—
128	C₂₃×#102	—	4	1	no	—
129	C₂₃×C₂²⋊C₈	—	2	2	no	—
130	C₂₃×#97	—	2	2	no	—
131	C₄×C₈×C₂₃	—	4	1	no	—
132	C₂₃×#95	—	2	1	no	—
133	C₂₃×#95	—	2	1	no	—
134	C₂₃×#95	—	2	1	no	—
135	C₂₃×#95	—	2	1	no	—
136	C₂₃×#95	—	2	1	no	—
137	C₂₃×#95	D₄	1	1	yes	characteristic
138	C₂₃×#95	D₄	1	1	yes	characteristic
139	C₂⁴×C₂₃	—	2	1	no	—
140	C₂⁴×C₂₃	—	2	1	no	—
141	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	—	2	1	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	—	4	2	no	—
160	C₂×D₄×C₂₃	—	4	2	no	—
161	C₂×D₄×C₂₃	—	4	2	no	—
162	C₂×D₄×C₂₃	—	4	2	no	—
163	C₂×D₄×C₂₃	—	4	2	no	—
164	C₂×D₄×C₂₃	—	4	2	no	—
165	C₂×D₄×C₂₃	—	4	2	no	—
166	C₂×D₄×C₂₃	—	4	2	no	—
167	C₂×D₄×C₂₃	—	8	1	no	—
168	C₂×D₄×C₂₃	—	8	1	no	—
169	C₂×D₄×C₂₃	—	8	1	no	—
170	C₂×D₄×C₂₃	—	8	1	no	—
171	C₂×D₄×C₂₃	#41	1	2	yes	—
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	1	no	—
174	C₂²×C₄×C₂₃	—	2	1	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	1	no	—
177	C₂²×C₄×C₂₃	—	2	1	no	—
178	C₂²×C₄×C₂₃	—	2	1	no	—
179	C₂²×C₄×C₂₃	—	2	1	no	—
180	C₂²×C₄×C₂₃	—	2	1	no	—
181	C₂²×C₄×C₂₃	—	2	1	no	—
182	C₂²×C₄×C₂₃	—	2	1	no	—
183	C₂²×C₄×C₂₃	—	2	1	no	—
184	C₂²×C₄×C₂₃	—	4	2	no	—
185	C₂²×C₄×C₂₃	—	4	2	no	—
186	C₂²×C₄×C₂₃	—	8	1	no	—
187	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
193	SD₁₆×C₂₃	—	2	8	no	—
194	SD₁₆×C₂₃	—	4	4	no	—
195	M₁₆×C₂₃	—	4	4	no	—
196	C₂×C₈×C₂₃	—	2	1	no	—
197	C₂×C₈×C₂₃	—	2	1	no	—
198	C₂×C₈×C₂₃	—	2	1	no	—
199	C₂×C₈×C₂₃	—	2	1	no	—
200	C₂×C₈×C₂₃	—	2	1	no	—
201	C₂×C₈×C₂₃	—	2	1	no	—
202	C₂×C₈×C₂₃	—	2	1	no	—
203	C₂×C₈×C₂₃	—	2	1	no	—
204	C₂×C₈×C₂₃	—	2	1	no	—
205	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
206	C₂×C₈×C₂₃	#41	1	2	yes	—
207	C₂×C₈×C₂₃	#41	1	2	yes	—
208	C₄⋊C₄×C₂₃	—	2	1	no	—
209	C₄⋊C₄×C₂₃	—	2	1	no	—
210	C₄⋊C₄×C₂₃	—	2	1	no	—
211	C₄⋊C₄×C₂₃	—	2	1	no	—
212	C₄⋊C₄×C₂₃	—	2	1	no	—
213	C₄⋊C₄×C₂₃	—	2	1	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	4	2	no	—
218	C₄⋊C₄×C₂₃	—	4	2	no	—
219	C₄⋊C₄×C₂₃	—	4	2	no	—
220	C₄⋊C₄×C₂₃	—	4	2	no	—
221	C₄⋊C₄×C₂₃	—	8	1	no	—
222	C₄⋊C₄×C₂₃	—	8	1	no	—
223	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
224	C₂²⋊C₄×C₂₃	—	2	2	no	—
225	C₂²⋊C₄×C₂₃	—	2	2	no	—
226	C₂²⋊C₄×C₂₃	—	4	2	no	—
227	C₂²⋊C₄×C₂₃	—	4	2	no	—
228	C₂²⋊C₄×C₂₃	—	4	2	no	—
229	C₂²⋊C₄×C₂₃	—	4	2	no	—
230	C₂²⋊C₄×C₂₃	—	4	2	no	—
231	C₂²⋊C₄×C₂₃	—	8	1	no	—
232	C₂²⋊C₄×C₂₃	—	8	1	no	—
233	C₄²×C₂₃	—	2	1	no	—
234	C₄²×C₂₃	—	2	1	no	—
235	C₄²×C₂₃	—	2	1	no	—
236	C₄²×C₂₃	—	2	1	no	—
237	C₄²×C₂₃	—	4	2	no	—
238	#26987	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂³×C₂₃	—	2	1	no	—
240	C₂³×C₂₃	—	2	1	no	—
241	C₂³×C₂₃	—	2	2	no	—
242	C₂³×C₂₃	—	2	1	no	—
243	C₂³×C₂₃	—	2	1	no	—
244	C₂³×C₂₃	—	2	1	no	—
245	C₂³×C₂₃	—	2	1	no	—
246	C₂³×C₂₃	—	2	1	no	—
247	C₂³×C₂₃	—	4	1	no	—
248	C₂³×C₂₃	—	4	1	no	—
249	C₂³×C₂₃	—	4	1	no	—
250	C₂³×C₂₃	—	4	1	no	—
251	C₂³×C₂₃	—	4	1	no	—
252	C₂³×C₂₃	—	4	1	no	—
253	C₂³×C₂₃	—	4	1	no	—
254	C₂³×C₂₃	—	4	1	no	—
255	C₂³×C₂₃	—	4	1	no	—
256	C₂³×C₂₃	—	4	1	no	—
257	C₂³×C₂₃	—	4	1	no	—
258	C₂³×C₂₃	—	4	1	no	—
259	C₂³×C₂₃	—	4	1	no	—
260	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
261	Q₈×C₂₃	—	2	2	no	—
262	Q₈×C₂₃	—	2	2	no	—
263	Q₈×C₂₃	—	4	1	no	—
264	Q₈×C₂₃	—	4	1	no	—
265	D₄×C₂₃	—	2	2	no	—
266	D₄×C₂₃	—	2	2	no	—
267	D₄×C₂₃	—	4	1	no	—
268	D₄×C₂₃	—	4	1	no	—
269	D₄×C₂₃	—	4	1	no	—
270	D₄×C₂₃	—	4	1	no	—
271	D₄×C₂₃	—	4	1	no	—
272	D₄×C₂₃	—	4	1	no	—
273	D₄×C₂₃	—	8	4	no	—
274	D₄×C₂₃	—	8	4	no	—
275	D₄×C₂₃	—	8	4	no	—
276	D₄×C₂₃	—	8	4	no	—
277	C₂×C₄×C₂₃	—	2	1	no	—
278	C₂×C₄×C₂₃	—	2	1	no	—
279	C₂×C₄×C₂₃	—	2	1	no	—
280	C₂×C₄×C₂₃	—	2	1	no	—
281	C₂×C₄×C₂₃	—	2	1	no	—
282	C₂×C₄×C₂₃	—	2	1	no	—
283	C₂×C₄×C₂₃	—	2	1	no	—
284	C₂×C₄×C₂₃	—	2	2	no	—
285	C₂×C₄×C₂₃	—	2	2	no	—
286	C₂×C₄×C₂₃	—	2	1	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	2	1	no	—
289	C₂×C₄×C₂₃	—	2	1	no	—
290	C₂×C₄×C₂₃	—	2	1	no	—
291	C₂×C₄×C₂₃	—	2	1	no	—
292	C₂×C₄×C₂₃	—	4	1	no	—
293	C₂×C₄×C₂₃	—	4	1	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	4	1	no	—
298	C₂×C₄×C₂₃	—	4	1	no	—
299	C₂×C₄×C₂₃	—	4	1	no	—
300	C₂×C₄×C₂₃	—	4	1	no	—
301	C₂×C₄×C₂₃	—	4	1	no	—
302	C₂×C₄×C₂₃	—	4	1	no	—
303	C₂×C₄×C₂₃	—	4	1	no	—
304	C₂×C₄×C₂₃	—	4	1	no	—
305	C₂×C₄×C₂₃	—	4	1	no	—
306	C₂×C₄×C₂₃	—	4	1	no	—
307	C₂×C₄×C₂₃	—	4	1	no	—
308	C₂×C₄×C₂₃	—	4	1	no	—
309	C₂×C₄×C₂₃	—	4	1	no	—
310	C₂×C₄×C₂₃	—	4	1	no	—
311	C₂×C₄×C₂₃	—	4	1	no	—
312	C₂×C₄×C₂₃	—	8	4	no	—
313	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
314	C₂×C₄×C₂₃	#137	1	2	yes	—
315	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
317	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
318	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
319	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
320	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
321	C₈×C₂₃	—	2	2	no	—
322	C₈×C₂₃	—	2	2	no	—
323	C₈×C₂₃	—	2	2	no	—
324	C₈×C₂₃	—	4	1	no	—
325	C₂×#468	C₂×C₂₃	1	1	yes	characteristic, complemented
326	C₂×#440	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#2428	C₂×C₂₃	1	1	yes	characteristic
328	#2423	C₂×C₂₃	1	1	yes	characteristic, complemented
329	#2083	C₂×C₂₃	1	1	yes	characteristic, complemented
330	#2033	C₂×C₂₃	1	1	yes	characteristic, complemented
331	#2014	C₂×C₂₃	1	1	yes	characteristic, complemented
332	#2000	C₂×C₂₃	1	1	yes	characteristic, complemented
333	#1957	C₂×C₂₃	1	1	yes	characteristic, complemented
334	#1936	C₂×C₂₃	1	1	yes	characteristic, complemented
335	#1933	C₂×C₂₃	1	1	yes	characteristic, complemented
336	#1895	C₂×C₂₃	1	1	yes	characteristic, complemented
337	#1880	C₂×C₂₃	1	1	yes	characteristic, complemented
338	#1862	C₂×C₂₃	1	1	yes	characteristic, complemented
339	#1753	C₂×C₂₃	1	1	yes	characteristic, complemented
340	C₂²×C₂₃	—	4	1	no	—
341	C₂²×C₂₃	—	4	1	no	—
342	C₂²×C₂₃	—	4	1	no	—
343	C₂²×C₂₃	—	4	2	no	—
344	C₂²×C₂₃	—	4	1	no	—
345	C₂²×C₂₃	—	4	1	no	—
346	C₂²×C₂₃	—	4	1	no	—
347	C₂²×C₂₃	—	4	2	no	—
348	C₂²×C₂₃	—	4	2	no	—
349	C₂²×C₂₃	—	4	1	no	—
350	C₂²×C₂₃	—	8	1	no	—
351	C₂²×C₂₃	—	8	1	no	—
352	C₂²×C₂₃	—	8	1	no	—
353	C₂²×C₂₃	—	8	1	no	—
354	C₂²×C₂₃	—	8	1	no	—
355	C₂²×C₂₃	—	8	1	no	—
356	C₂²×C₂₃	—	8	1	no	—
357	C₂²×C₂₃	—	8	1	no	—
358	C₂²×C₂₃	—	8	1	no	—
359	C₂²×C₂₃	—	8	1	no	—
360	C₂²×C₂₃	—	8	1	no	—
361	C₂²×C₂₃	#532	1	1	yes	characteristic
362	C₂²×C₂₃	D₄²	1	1	yes	characteristic
363	C₂²×C₂₃	#546	1	1	yes	characteristic
364	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
365	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
366	C₂²×C₂₃	#576	1	1	yes	characteristic
367	C₂²×C₂₃	#577	1	1	yes	characteristic
368	C₄×C₂₃	—	2	2	no	—
369	C₄×C₂₃	—	2	2	no	—
370	C₄×C₂₃	—	4	1	no	—
371	C₄×C₂₃	—	4	2	no	—
372	C₄×C₂₃	—	4	1	no	—
373	C₄×C₂₃	—	4	2	no	—
374	C₄×C₂₃	—	4	1	no	—
375	C₄×C₂₃	—	8	1	no	—
376	C₄×C₂₃	—	8	1	no	—
377	C₄×C₂₃	—	8	1	no	—
378	C₄×C₂₃	—	8	1	no	—
379	C₄×C₂₃	#525	1	1	yes	characteristic
380	C₄×C₂₃	#540	1	1	yes	characteristic
381	C₄×C₂₃	#546	1	1	yes	characteristic
382	C₄×C₂₃	#553	1	1	yes	characteristic
383	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
384	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
385	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×#121	—	2	1	no	—
389	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
390	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
391	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
392	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
393	#468	—	2	4	no	—
394	#440	—	2	4	no	—
395	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
396	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
397	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
398	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
399	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
400	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
401	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
402	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
403	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
404	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
405	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
406	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
407	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
408	#399	—	2	1	no	—
409	#398	—	2	1	no	—
410	#394	—	2	1	no	—
411	#392	—	2	1	no	—
412	#391	C₂²×C₂₃	1	1	yes	characteristic
413	#390	C₂²×C₂₃	1	1	yes	characteristic
414	#389	C₂²×C₂₃	1	1	yes	characteristic
415	#389	C₂²×C₂₃	1	1	yes	characteristic
416	#388	—	2	1	no	—
417	#387	—	2	1	no	—
418	#386	—	2	1	no	—
419	#384	C₂²×C₂₃	1	1	yes	characteristic
420	#380	C₂²×C₂₃	1	1	yes	characteristic
421	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
422	#340	C₂²×C₂₃	1	1	yes	characteristic
423	#340	C₂²×C₂₃	1	1	yes	characteristic
424	#340	C₂²×C₂₃	1	1	yes	characteristic
425	#340	C₂²×C₂₃	1	1	yes	characteristic
426	#336	C₂²×C₂₃	1	1	yes	characteristic
427	#336	C₂²×C₂₃	1	1	yes	characteristic
428	C₂×C₂₃	—	8	1	no	—
429	C₂×C₂₃	—	8	2	no	—
430	C₂×C₂₃	—	16	1	no	—
431	C₂×C₂₃	#2464	1	1	yes	characteristic
432	C₂×C₂₃	#3179	1	1	yes	characteristic
433	C₂×C₂₃	#3182	1	1	yes	characteristic
434	C₂×C₂₃	#3274	1	1	yes	characteristic
435	C₂×C₂₃	#3278	1	1	yes	characteristic
436	C₂×C₂₃	#3310	1	1	yes	characteristic
437	C₂×C₂₃	#3314	1	1	yes	characteristic
438	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
439	C₂²×D₄	—	2	1	no	—
440	C₂²×D₄	—	2	1	no	—
441	C₂²×D₄	—	2	1	no	—
442	C₂²×D₄	—	2	1	no	—
443	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
444	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
445	C₂³×C₄	—	2	1	no	—
446	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
447	C₂×SD₁₆	—	2	2	no	—
448	C₂×SD₁₆	—	2	2	no	—
449	C₂×SD₁₆	—	2	2	no	—
450	C₂×SD₁₆	—	2	2	no	—
451	C₂×SD₁₆	—	2	2	no	—
452	C₂×SD₁₆	—	4	1	no	—
453	C₂×M₁₆	—	2	2	no	—
454	C₂×M₁₆	—	2	2	no	—
455	C₂×M₁₆	—	4	1	no	—
456	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
457	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
458	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
459	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
460	#127	—	2	2	no	—
461	#127	—	4	1	no	—
462	C₂²⋊Q₈	—	2	2	no	—
463	C₂²⋊Q₈	—	4	1	no	—
464	#121	—	2	2	no	—
465	#121	—	2	2	no	—
466	#121	—	4	4	no	—
467	#121	—	4	1	no	—
468	#121	—	4	1	no	—
469	C₄×Q₈	—	2	2	no	—
470	C₄×Q₈	—	4	1	no	—
471	C₄×D₄	—	2	2	no	—
472	C₄×D₄	—	4	1	no	—
473	C₂×C₄⋊C₄	—	2	1	no	—
474	C₂×C₄⋊C₄	—	2	1	no	—
475	C₂×C₄⋊C₄	—	2	1	no	—
476	C₂×C₄⋊C₄	—	2	1	no	—
477	C₂×C₄⋊C₄	—	2	1	no	—
478	C₂×C₄⋊C₄	—	2	1	no	—
479	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
480	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
481	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
482	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
483	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
484	C₂×C₂²⋊C₄	—	2	1	no	—
485	C₂×C₂²⋊C₄	—	2	1	no	—
486	C₂×C₂²⋊C₄	—	2	1	no	—
487	C₂×C₂²⋊C₄	—	2	1	no	—
488	C₂×C₂²⋊C₄	—	2	1	no	—
489	C₂×C₂²⋊C₄	—	2	1	no	—
490	C₂×C₂²⋊C₄	—	2	1	no	—
491	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
492	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
493	C₂×C₄²	—	2	1	no	—
494	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
495	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
496	#107	—	2	2	no	—
497	#107	—	2	2	no	—
498	#103	—	4	1	no	—
499	#103	D₄×C₂₃	1	4	yes	complemented
500	#102	—	2	2	no	—
501	#102	—	2	2	no	—
502	#102	—	2	2	no	—
503	#102	—	2	2	no	—
504	#102	—	4	1	no	—
505	#102	—	4	1	no	—
506	C₂²⋊C₈	—	2	2	no	—
507	#97	—	2	2	no	—
508	C₄×C₈	—	4	1	no	—
509	#95	—	2	1	no	—
510	#95	—	2	1	no	—
511	#95	—	2	1	no	—
512	#95	—	2	1	no	—
513	#95	—	2	1	no	—
514	#95	D₄×C₂₃	1	1	yes	characteristic
515	#95	D₄×C₂₃	1	1	yes	characteristic
516	C₂₃	#26987	1	1	yes	characteristic, direct product factor, Sylow
517	C₂⁴	—	2	1	no	—
518	C₂⁴	—	2	1	no	—
519	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
520	C₂×Q₈	—	2	1	no	—
521	C₂×Q₈	—	2	1	no	—
522	C₂×Q₈	—	2	1	no	—
523	C₂×Q₈	—	2	1	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
526	C₂×D₄	—	2	1	no	—
527	C₂×D₄	—	2	1	no	—
528	C₂×D₄	—	2	1	no	—
529	C₂×D₄	—	2	1	no	—
530	C₂×D₄	—	2	1	no	—
531	C₂×D₄	—	2	1	no	—
532	C₂×D₄	—	2	1	no	—
533	C₂×D₄	—	2	1	no	—
534	C₂×D₄	—	2	1	no	—
535	C₂×D₄	—	2	1	no	—
536	C₂×D₄	—	2	1	no	—
537	C₂×D₄	—	4	2	no	—
538	C₂×D₄	—	4	2	no	—
539	C₂×D₄	—	4	2	no	—
540	C₂×D₄	—	4	2	no	—
541	C₂×D₄	—	4	2	no	—
542	C₂×D₄	—	4	2	no	—
543	C₂×D₄	—	4	2	no	—
544	C₂×D₄	—	4	2	no	—
545	C₂×D₄	—	8	1	no	—
546	C₂×D₄	—	8	1	no	—
547	C₂×D₄	—	8	1	no	—
548	C₂×D₄	—	8	1	no	—
549	C₂×D₄	#41×C₂₃	1	2	yes	—
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	1	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	2	1	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	2	1	no	—
557	C₂²×C₄	—	2	1	no	—
558	C₂²×C₄	—	2	1	no	—
559	C₂²×C₄	—	2	1	no	—
560	C₂²×C₄	—	2	1	no	—
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	4	2	no	—
563	C₂²×C₄	—	4	2	no	—
564	C₂²×C₄	—	8	1	no	—
565	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
566	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
567	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
568	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
569	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
570	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
571	SD₁₆	—	2	8	no	—
572	SD₁₆	—	4	4	no	—
573	M₁₆	—	4	4	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	1	no	—
579	C₂×C₈	—	2	1	no	—
580	C₂×C₈	—	2	1	no	—
581	C₂×C₈	—	2	1	no	—
582	C₂×C₈	—	2	1	no	—
583	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
584	C₂×C₈	#41×C₂₃	1	2	yes	—
585	C₂×C₈	#41×C₂₃	1	2	yes	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	4	2	no	—
596	C₄⋊C₄	—	4	2	no	—
597	C₄⋊C₄	—	4	2	no	—
598	C₄⋊C₄	—	4	2	no	—
599	C₄⋊C₄	—	8	1	no	—
600	C₄⋊C₄	—	8	1	no	—
601	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
602	C₂²⋊C₄	—	2	2	no	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	4	2	no	—
605	C₂²⋊C₄	—	4	2	no	—
606	C₂²⋊C₄	—	4	2	no	—
607	C₂²⋊C₄	—	4	2	no	—
608	C₂²⋊C₄	—	4	2	no	—
609	C₂²⋊C₄	—	8	1	no	—
610	C₂²⋊C₄	—	8	1	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	2	1	no	—
613	C₄²	—	2	1	no	—
614	C₄²	—	2	1	no	—
615	C₄²	—	4	2	no	—
616	C₂³	—	2	1	no	—
617	C₂³	—	2	1	no	—
618	C₂³	—	2	1	no	—
619	C₂³	—	2	1	no	—
620	C₂³	—	2	1	no	—
621	C₂³	—	2	1	no	—
622	C₂³	—	2	2	no	—
623	C₂³	—	2	1	no	—
624	C₂³	—	4	1	no	—
625	C₂³	—	4	1	no	—
626	C₂³	—	4	1	no	—
627	C₂³	—	4	1	no	—
628	C₂³	—	4	1	no	—
629	C₂³	—	4	1	no	—
630	C₂³	—	4	1	no	—
631	C₂³	—	4	1	no	—
632	C₂³	—	4	1	no	—
633	C₂³	—	4	1	no	—
634	C₂³	—	4	1	no	—
635	C₂³	—	4	1	no	—
636	C₂³	—	4	1	no	—
637	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
638	Q₈	—	2	2	no	—
639	Q₈	—	2	2	no	—
640	Q₈	—	4	1	no	—
641	Q₈	—	4	1	no	—
642	D₄	—	2	2	no	—
643	D₄	—	2	2	no	—
644	D₄	—	4	1	no	—
645	D₄	—	4	1	no	—
646	D₄	—	4	1	no	—
647	D₄	—	4	1	no	—
648	D₄	—	4	1	no	—
649	D₄	—	4	1	no	—
650	D₄	—	8	4	no	—
651	D₄	—	8	4	no	—
652	D₄	—	8	4	no	—
653	D₄	—	8	4	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	1	no	—
657	C₂×C₄	—	2	1	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	2	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	1	no	—
668	C₂×C₄	—	2	1	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	1	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	4	1	no	—
677	C₂×C₄	—	4	1	no	—
678	C₂×C₄	—	4	1	no	—
679	C₂×C₄	—	4	1	no	—
680	C₂×C₄	—	4	1	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	4	1	no	—
683	C₂×C₄	—	4	1	no	—
684	C₂×C₄	—	4	1	no	—
685	C₂×C₄	—	4	1	no	—
686	C₂×C₄	—	4	1	no	—
687	C₂×C₄	—	4	1	no	—
688	C₂×C₄	—	4	1	no	—
689	C₂×C₄	—	8	4	no	—
690	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
691	C₂×C₄	C₂₃×#137	1	2	yes	—
692	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
693	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
694	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
695	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
696	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
697	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
698	C₈	—	2	2	no	—
699	C₈	—	2	2	no	—
700	C₈	—	2	2	no	—
701	C₈	—	4	1	no	—
702	C₂²	—	4	1	no	—
703	C₂²	—	4	1	no	—
704	C₂²	—	4	1	no	—
705	C₂²	—	4	2	no	—
706	C₂²	—	4	1	no	—
707	C₂²	—	4	1	no	—
708	C₂²	—	4	1	no	—
709	C₂²	—	4	2	no	—
710	C₂²	—	4	2	no	—
711	C₂²	—	4	1	no	—
712	C₂²	—	8	1	no	—
713	C₂²	—	8	1	no	—
714	C₂²	—	8	1	no	—
715	C₂²	—	8	1	no	—
716	C₂²	—	8	1	no	—
717	C₂²	—	8	1	no	—
718	C₂²	—	8	1	no	—
719	C₂²	—	8	1	no	—
720	C₂²	—	8	1	no	—
721	C₂²	—	8	1	no	—
722	C₂²	—	8	1	no	—
723	C₂²	C₂₃×#532	1	1	yes	characteristic
724	C₂²	D₄²×C₂₃	1	1	yes	characteristic
725	C₂²	C₂₃×#546	1	1	yes	characteristic
726	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
727	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
728	C₂²	C₂₃×#576	1	1	yes	characteristic
729	C₂²	C₂₃×#577	1	1	yes	characteristic
730	C₄	—	2	2	no	—
731	C₄	—	2	2	no	—
732	C₄	—	4	1	no	—
733	C₄	—	4	2	no	—
734	C₄	—	4	1	no	—
735	C₄	—	4	2	no	—
736	C₄	—	4	1	no	—
737	C₄	—	8	1	no	—
738	C₄	—	8	1	no	—
739	C₄	—	8	1	no	—
740	C₄	—	8	1	no	—
741	C₄	C₂₃×#525	1	1	yes	characteristic
742	C₄	C₂₃×#540	1	1	yes	characteristic
743	C₄	C₂₃×#546	1	1	yes	characteristic
744	C₄	C₂₃×#553	1	1	yes	characteristic
745	C₂	—	8	1	no	—
746	C₂	—	8	2	no	—
747	C₂	—	16	1	no	—
748	C₂	C₂₃×#2464	1	1	yes	characteristic
749	C₂	C₂₃×#3179	1	1	yes	characteristic
750	C₂	C₂₃×#3182	1	1	yes	characteristic
751	C₂	C₂₃×#3274	1	1	yes	characteristic
752	C₂	C₂₃×#3278	1	1	yes	characteristic
753	C₂	C₂₃×#3310	1	1	yes	characteristic
754	C₂	C₂₃×#3314	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#2428	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₂₃×#468	C₂	semidirect
2. C₂×C₂₃×#440	C₂	semidirect
4. C₂₃×#2423	C₂	semidirect
5. C₂₃×#2083	C₂	semidirect
6. C₂₃×#2033	C₂	semidirect
7. C₂₃×#2014	C₂	semidirect
8. C₂₃×#2000	C₂	semidirect
9. C₂₃×#1957	C₂	semidirect
10. C₂₃×#1936	C₂	semidirect
11. C₂₃×#1933	C₂	semidirect
12. C₂₃×#1895	C₂	semidirect
13. C₂₃×#1880	C₂	semidirect
14. C₂₃×#1862	C₂	semidirect
15. C₂₃×#1753	C₂	semidirect
122. C₂₃×#103	D₄	semidirect
238. #26987	C₂₃	direct (N × H)
325. C₂×#468	C₂×C₂₃	semidirect
326. C₂×#440	C₂×C₂₃	semidirect
328. #2423	C₂×C₂₃	semidirect
329. #2083	C₂×C₂₃	semidirect
330. #2033	C₂×C₂₃	semidirect
331. #2014	C₂×C₂₃	semidirect
332. #2000	C₂×C₂₃	semidirect
333. #1957	C₂×C₂₃	semidirect
334. #1936	C₂×C₂₃	semidirect
335. #1933	C₂×C₂₃	semidirect
336. #1895	C₂×C₂₃	semidirect
337. #1880	C₂×C₂₃	semidirect
338. #1862	C₂×C₂₃	semidirect
339. #1753	C₂×C₂₃	semidirect
499. #103	D₄×C₂₃	semidirect
516. C₂₃	#26987	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#468	2
C₂×C₂₃×#440	2
C₂₃×#2428	2
C₂₃×#2423	2
C₂₃×#2083	2
C₂₃×#2033	2
C₂₃×#2014	2
C₂₃×#2000	2
C₂₃×#1957	2
C₂₃×#1936	2
C₂₃×#1933	2
C₂₃×#1895	2
C₂₃×#1880	2
C₂₃×#1862	2
C₂₃×#1753	2
#26987	23

Subgroups — C23×#26987 #19174613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26987 #19174613