Subgroups — C₂₃×#26956 #19174582

Order 5888.

892 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#485	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2429	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2363	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2087	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2000	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#2000	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1973	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1927	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1924	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1880	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1873	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1763	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	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
25	C₂₃×#485	—	2	4	no	—
26	C₂₃×#461	—	2	4	no	—
27	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₄⋊C₈	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×#103	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×#103	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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
41	C₂₃×#398	—	2	1	no	—
42	C₂₃×#396	—	2	1	no	—
43	C₂₃×#391	C₂²	1	1	yes	characteristic
44	C₂₃×#390	—	2	1	no	—
45	C₂₃×#390	C₂²	1	1	yes	characteristic
46	C₂₃×#389	—	2	1	no	—
47	C₂₃×#389	C₂²	1	1	yes	characteristic
48	C₂₃×#389	C₂²	1	1	yes	characteristic
49	C₂₃×#388	—	2	1	no	—
50	C₂₃×#388	—	2	1	no	—
51	C₂₃×#386	—	2	1	no	—
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
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×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	—	2	1	no	—
64	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×SD₁₆×C₂₃	—	2	2	no	—
68	C₂×SD₁₆×C₂₃	—	2	2	no	—
69	C₂×SD₁₆×C₂₃	—	2	2	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	—	4	1	no	—
72	C₂×SD₁₆×C₂₃	—	4	1	no	—
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	—	4	1	no	—
79	C₂₃×C₂²⋊Q₈	—	2	2	no	—
80	C₂₃×C₂²⋊Q₈	—	2	2	no	—
81	C₂₃×C₂²⋊Q₈	—	4	1	no	—
82	C₂₃×C₂²⋊Q₈	—	4	1	no	—
83	C₂₃×#121	—	2	2	no	—
84	C₂₃×#121	—	2	2	no	—
85	C₄×Q₈×C₂₃	—	2	2	no	—
86	C₄×Q₈×C₂₃	—	4	1	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	—	4	1	no	—
89	C₂₃×#117	—	2	2	no	—
90	C₂₃×#117	—	4	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	—	2	1	no	—
113	C₂×C₄²×C₂₃	—	2	1	no	—
114	C₂×C₄²×C₂₃	—	2	1	no	—
115	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
117	C₂₃×#107	—	2	2	no	—
118	C₂₃×#106	—	2	2	no	—
119	C₂₃×C₄⋊C₈	—	2	2	no	—
120	C₂₃×#103	—	2	2	no	—
121	C₂₃×#103	—	2	2	no	—
122	C₂₃×#103	—	2	2	no	—
123	C₂₃×#103	—	2	2	no	—
124	C₂₃×#103	—	4	1	no	—
125	C₂₃×#103	D₄	1	4	yes	complemented
126	C₂₃×#102	—	2	2	no	—
127	C₂₃×#102	—	4	1	no	—
128	C₂₃×C₂²⋊C₈	—	2	2	no	—
129	C₂₃×C₂²⋊C₈	—	4	1	no	—
130	C₂₃×#97	—	4	1	no	—
131	C₂₃×#95	—	2	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	D₄	1	1	yes	characteristic
137	C₂₃×#95	D₄	1	1	yes	characteristic
138	C₂⁴×C₂₃	—	2	1	no	—
139	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
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	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₂₃	—	2	4	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	—	8	1	no	—
159	C₂×D₄×C₂₃	—	8	1	no	—
160	C₂×D₄×C₂₃	#41	1	2	yes	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	1	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
170	C₂²×C₄×C₂₃	—	2	1	no	—
171	C₂²×C₄×C₂₃	—	2	1	no	—
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	1	no	—
174	C₂²×C₄×C₂₃	—	4	2	no	—
175	C₂²×C₄×C₂₃	—	4	2	no	—
176	C₂²×C₄×C₂₃	—	8	1	no	—
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
183	SD₁₆×C₂₃	—	4	4	no	—
184	SD₁₆×C₂₃	—	4	4	no	—
185	C₂×C₈×C₂₃	—	2	1	no	—
186	C₂×C₈×C₂₃	—	2	1	no	—
187	C₂×C₈×C₂₃	—	2	1	no	—
188	C₂×C₈×C₂₃	—	2	1	no	—
189	C₂×C₈×C₂₃	—	2	1	no	—
190	C₂×C₈×C₂₃	—	2	1	no	—
191	C₂×C₈×C₂₃	—	2	1	no	—
192	C₂×C₈×C₂₃	—	2	1	no	—
193	C₂×C₈×C₂₃	—	2	1	no	—
194	C₂×C₈×C₂₃	—	2	1	no	—
195	C₂×C₈×C₂₃	—	2	1	no	—
196	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	2	1	no	—
200	C₄⋊C₄×C₂₃	—	2	1	no	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	1	no	—
203	C₄⋊C₄×C₂₃	—	2	1	no	—
204	C₄⋊C₄×C₂₃	—	4	2	no	—
205	C₄⋊C₄×C₂₃	—	4	2	no	—
206	C₄⋊C₄×C₂₃	—	4	2	no	—
207	C₄⋊C₄×C₂₃	—	4	2	no	—
208	C₄⋊C₄×C₂₃	—	4	2	no	—
209	C₄⋊C₄×C₂₃	—	4	2	no	—
210	C₄⋊C₄×C₂₃	—	8	1	no	—
211	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
212	C₄⋊C₄×C₂₃	#41	1	2	yes	—
213	C₄⋊C₄×C₂₃	#41	1	2	yes	—
214	C₂²⋊C₄×C₂₃	—	2	4	no	—
215	C₂²⋊C₄×C₂₃	—	2	2	no	—
216	C₂²⋊C₄×C₂₃	—	4	1	no	—
217	C₂²⋊C₄×C₂₃	—	4	2	no	—
218	C₂²⋊C₄×C₂₃	—	4	2	no	—
219	C₂²⋊C₄×C₂₃	—	4	2	no	—
220	C₂²⋊C₄×C₂₃	—	4	2	no	—
221	C₂²⋊C₄×C₂₃	—	4	2	no	—
222	C₂²⋊C₄×C₂₃	—	8	1	no	—
223	C₄²×C₂₃	—	2	1	no	—
224	C₄²×C₂₃	—	2	1	no	—
225	C₄²×C₂₃	—	2	1	no	—
226	C₄²×C₂₃	—	2	1	no	—
227	C₄²×C₂₃	—	4	2	no	—
228	C₄²×C₂₃	—	4	2	no	—
229	C₄²×C₂₃	—	8	1	no	—
230	#26956	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂³×C₂₃	—	2	2	no	—
232	C₂³×C₂₃	—	2	1	no	—
233	C₂³×C₂₃	—	2	1	no	—
234	C₂³×C₂₃	—	2	1	no	—
235	C₂³×C₂₃	—	2	1	no	—
236	C₂³×C₂₃	—	2	1	no	—
237	C₂³×C₂₃	—	2	1	no	—
238	C₂³×C₂₃	—	2	1	no	—
239	C₂³×C₂₃	—	4	1	no	—
240	C₂³×C₂₃	—	4	1	no	—
241	C₂³×C₂₃	—	4	1	no	—
242	C₂³×C₂₃	—	4	1	no	—
243	C₂³×C₂₃	—	4	1	no	—
244	C₂³×C₂₃	—	4	1	no	—
245	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
246	Q₈×C₂₃	—	2	2	no	—
247	Q₈×C₂₃	—	2	2	no	—
248	Q₈×C₂₃	—	4	1	no	—
249	Q₈×C₂₃	—	4	1	no	—
250	Q₈×C₂₃	—	4	1	no	—
251	Q₈×C₂₃	—	4	1	no	—
252	Q₈×C₂₃	—	4	1	no	—
253	Q₈×C₂₃	—	4	1	no	—
254	D₄×C₂₃	—	2	2	no	—
255	D₄×C₂₃	—	2	2	no	—
256	D₄×C₂₃	—	4	1	no	—
257	D₄×C₂₃	—	4	1	no	—
258	D₄×C₂₃	—	8	4	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	2	no	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	1	no	—
265	C₂×C₄×C₂₃	—	2	2	no	—
266	C₂×C₄×C₂₃	—	2	2	no	—
267	C₂×C₄×C₂₃	—	2	1	no	—
268	C₂×C₄×C₂₃	—	2	1	no	—
269	C₂×C₄×C₂₃	—	2	2	no	—
270	C₂×C₄×C₂₃	—	2	1	no	—
271	C₂×C₄×C₂₃	—	2	1	no	—
272	C₂×C₄×C₂₃	—	2	1	no	—
273	C₂×C₄×C₂₃	—	2	1	no	—
274	C₂×C₄×C₂₃	—	2	1	no	—
275	C₂×C₄×C₂₃	—	2	1	no	—
276	C₂×C₄×C₂₃	—	2	1	no	—
277	C₂×C₄×C₂₃	—	4	1	no	—
278	C₂×C₄×C₂₃	—	4	1	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	4	1	no	—
281	C₂×C₄×C₂₃	—	4	1	no	—
282	C₂×C₄×C₂₃	—	4	1	no	—
283	C₂×C₄×C₂₃	—	4	1	no	—
284	C₂×C₄×C₂₃	—	4	1	no	—
285	C₂×C₄×C₂₃	—	4	1	no	—
286	C₂×C₄×C₂₃	—	4	1	no	—
287	C₂×C₄×C₂₃	—	4	1	no	—
288	C₂×C₄×C₂₃	—	4	1	no	—
289	C₂×C₄×C₂₃	—	4	1	no	—
290	C₂×C₄×C₂₃	—	4	1	no	—
291	C₂×C₄×C₂₃	—	4	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	—	8	4	no	—
299	C₂×C₄×C₂₃	#137	1	2	yes	—
300	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
302	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
303	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
304	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
305	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
306	C₈×C₂₃	—	2	2	no	—
307	C₈×C₂₃	—	4	1	no	—
308	C₈×C₂₃	—	4	1	no	—
309	C₈×C₂₃	—	4	1	no	—
310	C₂×#485	C₂×C₂₃	1	1	yes	characteristic, complemented
311	C₂×#461	C₂×C₂₃	1	1	yes	characteristic, complemented
312	#2429	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#2363	C₂×C₂₃	1	1	yes	characteristic
314	#2087	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#2043	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#2000	C₂×C₂₃	1	1	yes	characteristic
317	#2000	C₂×C₂₃	1	1	yes	characteristic
318	#1973	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#1927	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#1924	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1880	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1873	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1864	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1763	C₂×C₂₃	1	1	yes	characteristic, complemented
325	C₂²×C₂₃	—	4	1	no	—
326	C₂²×C₂₃	—	4	1	no	—
327	C₂²×C₂₃	—	4	2	no	—
328	C₂²×C₂₃	—	4	1	no	—
329	C₂²×C₂₃	—	4	2	no	—
330	C₂²×C₂₃	—	4	1	no	—
331	C₂²×C₂₃	—	4	2	no	—
332	C₂²×C₂₃	—	4	1	no	—
333	C₂²×C₂₃	—	4	1	no	—
334	C₂²×C₂₃	—	4	1	no	—
335	C₂²×C₂₃	—	8	1	no	—
336	C₂²×C₂₃	—	8	1	no	—
337	C₂²×C₂₃	—	8	1	no	—
338	C₂²×C₂₃	—	8	1	no	—
339	C₂²×C₂₃	#525	1	1	yes	characteristic
340	C₂²×C₂₃	D₄²	1	1	yes	characteristic
341	C₂²×C₂₃	#546	1	1	yes	characteristic
342	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
343	C₂²×C₂₃	#575	1	1	yes	characteristic
344	C₂²×C₂₃	#577	1	1	yes	characteristic
345	C₂²×C₂₃	#577	1	1	yes	characteristic
346	C₄×C₂₃	—	2	2	no	—
347	C₄×C₂₃	—	4	1	no	—
348	C₄×C₂₃	—	4	2	no	—
349	C₄×C₂₃	—	4	1	no	—
350	C₄×C₂₃	—	4	2	no	—
351	C₄×C₂₃	—	4	2	no	—
352	C₄×C₂₃	—	4	1	no	—
353	C₄×C₂₃	—	4	1	no	—
354	C₄×C₂₃	—	4	2	no	—
355	C₄×C₂₃	—	8	1	no	—
356	C₄×C₂₃	—	8	1	no	—
357	C₄×C₂₃	—	8	1	no	—
358	C₄×C₂₃	#532	1	1	yes	characteristic
359	C₄×C₂₃	#540	1	1	yes	characteristic
360	C₄×C₂₃	#546	1	1	yes	characteristic
361	C₄×C₂₃	#553	1	1	yes	characteristic
362	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
363	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
364	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
371	#485	—	2	4	no	—
372	#461	—	2	4	no	—
373	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
375	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
380	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
381	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
383	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
387	#398	—	2	1	no	—
388	#396	—	2	1	no	—
389	#391	C₂²×C₂₃	1	1	yes	characteristic
390	#390	—	2	1	no	—
391	#390	C₂²×C₂₃	1	1	yes	characteristic
392	#389	—	2	1	no	—
393	#389	C₂²×C₂₃	1	1	yes	characteristic
394	#389	C₂²×C₂₃	1	1	yes	characteristic
395	#388	—	2	1	no	—
396	#388	—	2	1	no	—
397	#386	—	2	1	no	—
398	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
399	C₄×C₂²⋊C₄	—	2	1	no	—
400	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
401	#340	C₂²×C₂₃	1	1	yes	characteristic
402	#340	C₂²×C₂₃	1	1	yes	characteristic
403	#340	C₂²×C₂₃	1	1	yes	characteristic
404	#340	C₂²×C₂₃	1	1	yes	characteristic
405	#336	C₂²×C₂₃	1	1	yes	characteristic
406	#336	C₂²×C₂₃	1	1	yes	characteristic
407	C₂×C₂₃	—	8	1	no	—
408	C₂×C₂₃	—	8	2	no	—
409	C₂×C₂₃	#2464	1	1	yes	characteristic
410	C₂×C₂₃	#3116	1	1	yes	characteristic
411	C₂×C₂₃	#3123	1	1	yes	characteristic
412	C₂×C₂₃	#3266	1	1	yes	characteristic
413	C₂×C₂₃	#3271	1	1	yes	characteristic
414	C₂×C₂₃	#3307	1	1	yes	characteristic
415	C₂×C₂₃	#3329	1	1	yes	characteristic
416	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
417	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
418	C₂²×D₄	—	2	1	no	—
419	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
420	C₂³×C₄	—	2	1	no	—
421	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
422	C₂×SD₁₆	—	2	2	no	—
423	C₂×SD₁₆	—	2	2	no	—
424	C₂×SD₁₆	—	2	2	no	—
425	C₂×SD₁₆	—	2	2	no	—
426	C₂×SD₁₆	—	4	1	no	—
427	C₂×SD₁₆	—	4	1	no	—
428	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
430	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₄⋊Q₈	—	2	2	no	—
433	C₄⋊Q₈	—	4	1	no	—
434	C₂²⋊Q₈	—	2	2	no	—
435	C₂²⋊Q₈	—	2	2	no	—
436	C₂²⋊Q₈	—	4	1	no	—
437	C₂²⋊Q₈	—	4	1	no	—
438	#121	—	2	2	no	—
439	#121	—	2	2	no	—
440	C₄×Q₈	—	2	2	no	—
441	C₄×Q₈	—	4	1	no	—
442	C₄×D₄	—	2	2	no	—
443	C₄×D₄	—	4	1	no	—
444	#117	—	2	2	no	—
445	#117	—	4	1	no	—
446	C₂×C₄⋊C₄	—	2	1	no	—
447	C₂×C₄⋊C₄	—	2	1	no	—
448	C₂×C₄⋊C₄	—	2	1	no	—
449	C₂×C₄⋊C₄	—	2	1	no	—
450	C₂×C₄⋊C₄	—	2	1	no	—
451	C₂×C₄⋊C₄	—	2	1	no	—
452	C₂×C₄⋊C₄	—	2	1	no	—
453	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
454	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
455	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₂²⋊C₄	—	2	1	no	—
460	C₂×C₂²⋊C₄	—	2	1	no	—
461	C₂×C₂²⋊C₄	—	2	1	no	—
462	C₂×C₂²⋊C₄	—	2	1	no	—
463	C₂×C₂²⋊C₄	—	2	1	no	—
464	C₂×C₂²⋊C₄	—	2	1	no	—
465	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
466	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
467	C₂×C₄²	—	2	1	no	—
468	C₂×C₄²	—	2	1	no	—
469	C₂×C₄²	—	2	1	no	—
470	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
471	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
472	#107	—	2	2	no	—
473	#106	—	2	2	no	—
474	C₄⋊C₈	—	2	2	no	—
475	#103	—	2	2	no	—
476	#103	—	2	2	no	—
477	#103	—	2	2	no	—
478	#103	—	2	2	no	—
479	#103	—	4	1	no	—
480	#103	D₄×C₂₃	1	4	yes	complemented
481	#102	—	2	2	no	—
482	#102	—	4	1	no	—
483	C₂²⋊C₈	—	2	2	no	—
484	C₂²⋊C₈	—	4	1	no	—
485	#97	—	4	1	no	—
486	#95	—	2	1	no	—
487	#95	—	2	1	no	—
488	#95	—	2	1	no	—
489	#95	—	2	1	no	—
490	#95	—	2	1	no	—
491	#95	D₄×C₂₃	1	1	yes	characteristic
492	#95	D₄×C₂₃	1	1	yes	characteristic
493	C₂₃	#26956	1	1	yes	characteristic, direct product factor, Sylow
494	C₂⁴	—	2	1	no	—
495	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
496	C₂×Q₈	—	2	1	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	2	1	no	—
502	C₂×Q₈	—	2	1	no	—
503	C₂×Q₈	—	2	1	no	—
504	C₂×Q₈	—	2	1	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	—	2	1	no	—
507	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	2	1	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	8	1	no	—
515	C₂×D₄	—	8	1	no	—
516	C₂×D₄	#41×C₂₃	1	2	yes	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	1	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	4	2	no	—
531	C₂²×C₄	—	4	2	no	—
532	C₂²×C₄	—	8	1	no	—
533	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
534	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
535	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
536	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
537	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
538	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
539	SD₁₆	—	4	4	no	—
540	SD₁₆	—	4	4	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	1	no	—
558	C₄⋊C₄	—	2	1	no	—
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	4	2	no	—
561	C₄⋊C₄	—	4	2	no	—
562	C₄⋊C₄	—	4	2	no	—
563	C₄⋊C₄	—	4	2	no	—
564	C₄⋊C₄	—	4	2	no	—
565	C₄⋊C₄	—	4	2	no	—
566	C₄⋊C₄	—	8	1	no	—
567	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
568	C₄⋊C₄	#41×C₂₃	1	2	yes	—
569	C₄⋊C₄	#41×C₂₃	1	2	yes	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	2	no	—
572	C₂²⋊C₄	—	4	1	no	—
573	C₂²⋊C₄	—	4	2	no	—
574	C₂²⋊C₄	—	4	2	no	—
575	C₂²⋊C₄	—	4	2	no	—
576	C₂²⋊C₄	—	4	2	no	—
577	C₂²⋊C₄	—	4	2	no	—
578	C₂²⋊C₄	—	8	1	no	—
579	C₄²	—	2	1	no	—
580	C₄²	—	2	1	no	—
581	C₄²	—	2	1	no	—
582	C₄²	—	2	1	no	—
583	C₄²	—	4	2	no	—
584	C₄²	—	4	2	no	—
585	C₄²	—	8	1	no	—
586	C₂³	—	2	1	no	—
587	C₂³	—	2	1	no	—
588	C₂³	—	2	1	no	—
589	C₂³	—	2	1	no	—
590	C₂³	—	2	2	no	—
591	C₂³	—	2	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	4	1	no	—
595	C₂³	—	4	1	no	—
596	C₂³	—	4	1	no	—
597	C₂³	—	4	1	no	—
598	C₂³	—	4	1	no	—
599	C₂³	—	4	1	no	—
600	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
601	Q₈	—	2	2	no	—
602	Q₈	—	2	2	no	—
603	Q₈	—	4	1	no	—
604	Q₈	—	4	1	no	—
605	Q₈	—	4	1	no	—
606	Q₈	—	4	1	no	—
607	Q₈	—	4	1	no	—
608	Q₈	—	4	1	no	—
609	D₄	—	2	2	no	—
610	D₄	—	2	2	no	—
611	D₄	—	4	1	no	—
612	D₄	—	4	1	no	—
613	D₄	—	8	4	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	1	no	—
653	C₂×C₄	—	8	4	no	—
654	C₂×C₄	C₂₃×#137	1	2	yes	—
655	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
656	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
657	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
658	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
659	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
660	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
661	C₈	—	2	2	no	—
662	C₈	—	4	1	no	—
663	C₈	—	4	1	no	—
664	C₈	—	4	1	no	—
665	C₂²	—	4	1	no	—
666	C₂²	—	4	2	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	1	no	—
669	C₂²	—	4	1	no	—
670	C₂²	—	4	1	no	—
671	C₂²	—	4	1	no	—
672	C₂²	—	4	2	no	—
673	C₂²	—	4	1	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	8	1	no	—
676	C₂²	—	8	1	no	—
677	C₂²	—	8	1	no	—
678	C₂²	—	8	1	no	—
679	C₂²	C₂₃×#525	1	1	yes	characteristic
680	C₂²	D₄²×C₂₃	1	1	yes	characteristic
681	C₂²	C₂₃×#546	1	1	yes	characteristic
682	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
683	C₂²	C₂₃×#575	1	1	yes	characteristic
684	C₂²	C₂₃×#577	1	1	yes	characteristic
685	C₂²	C₂₃×#577	1	1	yes	characteristic
686	C₄	—	2	2	no	—
687	C₄	—	4	1	no	—
688	C₄	—	4	2	no	—
689	C₄	—	4	2	no	—
690	C₄	—	4	1	no	—
691	C₄	—	4	1	no	—
692	C₄	—	4	2	no	—
693	C₄	—	4	1	no	—
694	C₄	—	4	2	no	—
695	C₄	—	8	1	no	—
696	C₄	—	8	1	no	—
697	C₄	—	8	1	no	—
698	C₄	C₂₃×#532	1	1	yes	characteristic
699	C₄	C₂₃×#540	1	1	yes	characteristic
700	C₄	C₂₃×#546	1	1	yes	characteristic
701	C₄	C₂₃×#553	1	1	yes	characteristic
702	C₂	—	8	1	no	—
703	C₂	—	8	2	no	—
704	C₂	C₂₃×#2464	1	1	yes	characteristic
705	C₂	C₂₃×#3116	1	1	yes	characteristic
706	C₂	C₂₃×#3123	1	1	yes	characteristic
707	C₂	C₂₃×#3266	1	1	yes	characteristic
708	C₂	C₂₃×#3271	1	1	yes	characteristic
709	C₂	C₂₃×#3307	1	1	yes	characteristic
710	C₂	C₂₃×#3329	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
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×#485	C₂	semidirect
2. C₂×C₂₃×#461	C₂	semidirect
3. C₂₃×#2429	C₂	semidirect
5. C₂₃×#2087	C₂	semidirect
6. C₂₃×#2043	C₂	semidirect
9. C₂₃×#1973	C₂	semidirect
10. C₂₃×#1927	C₂	semidirect
11. C₂₃×#1924	C₂	semidirect
12. C₂₃×#1880	C₂	semidirect
13. C₂₃×#1873	C₂	semidirect
14. C₂₃×#1864	C₂	semidirect
15. C₂₃×#1763	C₂	semidirect
125. C₂₃×#103	D₄	semidirect
230. #26956	C₂₃	direct (N × H)
310. C₂×#485	C₂×C₂₃	semidirect
311. C₂×#461	C₂×C₂₃	semidirect
312. #2429	C₂×C₂₃	semidirect
314. #2087	C₂×C₂₃	semidirect
315. #2043	C₂×C₂₃	semidirect
318. #1973	C₂×C₂₃	semidirect
319. #1927	C₂×C₂₃	semidirect
320. #1924	C₂×C₂₃	semidirect
321. #1880	C₂×C₂₃	semidirect
322. #1873	C₂×C₂₃	semidirect
323. #1864	C₂×C₂₃	semidirect
324. #1763	C₂×C₂₃	semidirect
480. #103	D₄×C₂₃	semidirect
493. C₂₃	#26956	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×#485	2
C₂×C₂₃×#461	2
C₂₃×#2429	2
C₂₃×#2363	2
C₂₃×#2087	2
C₂₃×#2043	2
C₂₃×#2000	2
C₂₃×#1973	2
C₂₃×#1927	2
C₂₃×#1924	2
C₂₃×#1880	2
C₂₃×#1873	2
C₂₃×#1864	2
C₂₃×#1763	2
#26956	23

Subgroups — C23×#26956 #19174582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26956 #19174582