Subgroups — C₂₃×#26132 #19173758

Order 5888.

938 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2417	C₂	1	1	yes	maximal, characteristic
4	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2095	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2076	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2036	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1928	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1921	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1892	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1872	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1869	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1852	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1852	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	—	2	1	no	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#484	—	2	4	no	—
25	C₂×C₂₃×#107	—	2	4	no	—
26	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#107	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×C₂²⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂₃×#398	—	2	1	no	—
40	C₂₃×#396	—	2	1	no	—
41	C₂₃×#391	C₂²	1	1	yes	characteristic
42	C₂₃×#389	—	2	1	no	—
43	C₂₃×#389	C₂²	1	1	yes	characteristic
44	C₂₃×#386	—	2	1	no	—
45	C₂₃×#385	—	2	1	no	—
46	C₂₃×#385	—	2	1	no	—
47	C₂₃×#384	C₂²	1	1	yes	characteristic
48	C₂₃×#380	—	2	1	no	—
49	C₂₃×#380	C₂²	1	1	yes	characteristic
50	C₂₃×#380	C₂²	1	1	yes	characteristic
51	C₂₃×#380	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	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	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	—	2	1	no	—
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂³×C₄×C₂₃	—	2	1	no	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₂₃	—	2	2	no	—
69	C₂²×C₈×C₂₃	—	2	2	no	—
70	C₂²×C₈×C₂₃	—	2	2	no	—
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×C₂²⋊Q₈	—	2	2	no	—
76	C₂₃×C₂²⋊Q₈	—	2	2	no	—
77	C₂₃×C₂²⋊Q₈	—	2	2	no	—
78	C₂₃×C₂²⋊Q₈	—	4	1	no	—
79	C₄×D₄×C₂₃	—	2	2	no	—
80	C₄×D₄×C₂₃	—	2	2	no	—
81	C₄×D₄×C₂₃	—	4	4	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
96	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
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×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	—	2	1	no	—
112	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
113	C₂₃×#107	—	2	2	no	—
114	C₂₃×#107	—	2	2	no	—
115	C₂₃×#107	—	2	2	no	—
116	C₂₃×#107	—	2	2	no	—
117	C₂₃×#107	—	4	1	no	—
118	C₂₃×#107	D₄	1	4	yes	complemented
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#103	—	2	2	no	—
121	C₂₃×#103	—	2	2	no	—
122	C₂₃×#103	—	4	1	no	—
123	C₂₃×C₂²⋊C₈	—	2	2	no	—
124	C₂₃×C₂²⋊C₈	—	2	2	no	—
125	C₂₃×C₂²⋊C₈	—	2	2	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	—	2	1	no	—
130	C₂₃×#95	—	2	1	no	—
131	C₂₃×#95	—	2	1	no	—
132	C₂₃×#95	—	2	1	no	—
133	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	2	no	—
142	C₂×D₄×C₂₃	—	2	2	no	—
143	C₂×D₄×C₂₃	Q₁₆	1	4	yes	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	4	2	no	—
163	C₂²×C₄×C₂₃	—	4	2	no	—
164	C₂²×C₄×C₂₃	—	4	2	no	—
165	C₂²×C₄×C₂₃	—	4	2	no	—
166	C₂²×C₄×C₂₃	—	4	2	no	—
167	C₂²×C₄×C₂₃	—	4	2	no	—
168	C₂²×C₄×C₂₃	—	8	1	no	—
169	C₂²×C₄×C₂₃	—	8	1	no	—
170	C₂²×C₄×C₂₃	—	8	1	no	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₂₃	#41	1	2	yes	—
176	C₂²×C₄×C₂₃	#41	1	2	yes	—
177	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂×C₈×C₂₃	—	2	8	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	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	C₂×D₄	1	2	yes	—
190	C₂×C₈×C₂₃	#41	1	2	yes	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
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	—	8	1	no	—
212	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
213	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
214	C₂²⋊C₄×C₂₃	—	2	2	no	—
215	C₂²⋊C₄×C₂₃	—	2	2	no	—
216	C₂²⋊C₄×C₂₃	—	4	2	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	—	8	1	no	—
221	C₄²×C₂₃	—	2	1	no	—
222	C₄²×C₂₃	—	2	1	no	—
223	C₄²×C₂₃	—	4	2	no	—
224	#26132	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₂₃	—	2	1	no	—
226	C₂³×C₂₃	—	2	1	no	—
227	C₂³×C₂₃	—	2	1	no	—
228	C₂³×C₂₃	—	2	1	no	—
229	C₂³×C₂₃	—	2	1	no	—
230	C₂³×C₂₃	—	2	1	no	—
231	C₂³×C₂₃	—	2	1	no	—
232	C₂³×C₂₃	—	2	1	no	—
233	C₂³×C₂₃	—	2	1	no	—
234	C₂³×C₂₃	—	2	1	no	—
235	C₂³×C₂₃	#123	1	2	yes	—
236	C₂³×C₂₃	C₂×Q₁₆	1	2	yes	—
237	C₂³×C₂₃	C₂×Q₁₆	1	2	yes	—
238	C₂³×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
239	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
240	Q₈×C₂₃	—	4	1	no	—
241	Q₈×C₂₃	—	4	1	no	—
242	Q₈×C₂₃	—	4	1	no	—
243	Q₈×C₂₃	—	4	1	no	—
244	D₄×C₂₃	—	2	8	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	2	no	—
257	C₂×C₄×C₂₃	—	2	1	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	1	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	4	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	—	8	4	no	—
288	C₂×C₄×C₂₃	—	8	4	no	—
289	C₂×C₄×C₂₃	—	8	4	no	—
290	C₂×C₄×C₂₃	#123	1	2	yes	—
291	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
292	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
294	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
295	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
296	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
297	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
298	C₈×C₂₃	—	2	2	no	—
299	C₈×C₂₃	—	2	2	no	—
300	C₈×C₂₃	—	4	1	no	—
301	C₈×C₂₃	—	4	1	no	—
302	C₂×#484	C₂×C₂₃	1	1	yes	characteristic, complemented
303	C₂²×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#2417	C₂×C₂₃	1	1	yes	characteristic
305	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
306	#2095	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#2083	C₂×C₂₃	1	1	yes	characteristic, complemented
308	#2076	C₂×C₂₃	1	1	yes	characteristic, complemented
309	#2036	C₂×C₂₃	1	1	yes	characteristic, complemented
310	#1928	C₂×C₂₃	1	1	yes	characteristic, complemented
311	#1921	C₂×C₂₃	1	1	yes	characteristic, complemented
312	#1892	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#1872	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#1869	C₂×C₂₃	1	1	yes	characteristic
315	#1852	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#1852	C₂×C₂₃	1	1	yes	characteristic, complemented
317	C₂²×C₂₃	—	2	2	no	—
318	C₂²×C₂₃	—	2	2	no	—
319	C₂²×C₂₃	—	2	2	no	—
320	C₂²×C₂₃	—	2	2	no	—
321	C₂²×C₂₃	—	2	2	no	—
322	C₂²×C₂₃	—	2	2	no	—
323	C₂²×C₂₃	—	2	2	no	—
324	C₂²×C₂₃	—	2	2	no	—
325	C₂²×C₂₃	—	2	2	no	—
326	C₂²×C₂₃	—	4	1	no	—
327	C₂²×C₂₃	—	4	1	no	—
328	C₂²×C₂₃	—	4	1	no	—
329	C₂²×C₂₃	—	4	1	no	—
330	C₂²×C₂₃	#484	1	4	yes	complemented
331	C₂²×C₂₃	C₂×#123	1	1	yes	characteristic
332	C₂²×C₂₃	D₄²	1	1	yes	characteristic
333	C₂²×C₂₃	#547	1	1	yes	characteristic
334	C₂²×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
335	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
336	C₂²×C₂₃	#576	1	1	yes	characteristic
337	C₂²×C₂₃	#578	1	1	yes	characteristic
338	C₄×C₂₃	—	2	2	no	—
339	C₄×C₂₃	—	2	2	no	—
340	C₄×C₂₃	—	4	1	no	—
341	C₄×C₂₃	—	4	1	no	—
342	C₄×C₂₃	—	4	2	no	—
343	C₄×C₂₃	—	4	2	no	—
344	C₄×C₂₃	—	8	1	no	—
345	C₄×C₂₃	—	8	1	no	—
346	C₄×C₂₃	—	8	1	no	—
347	C₄×C₂₃	—	8	1	no	—
348	C₄×C₂₃	—	8	1	no	—
349	C₄×C₂₃	—	8	1	no	—
350	C₄×C₂₃	#525	1	1	yes	characteristic
351	C₄×C₂₃	#539	1	1	yes	characteristic
352	C₄×C₂₃	#548	1	1	yes	characteristic
353	C₄×C₂₃	#553	1	1	yes	characteristic
354	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
357	C₂×C₄×D₄	—	2	1	no	—
358	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
359	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
360	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
361	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
362	#484	—	2	4	no	—
363	C₂×#107	—	2	4	no	—
364	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
365	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
371	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
372	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
375	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
377	#398	—	2	1	no	—
378	#396	—	2	1	no	—
379	#391	C₂²×C₂₃	1	1	yes	characteristic
380	#389	—	2	1	no	—
381	#389	C₂²×C₂₃	1	1	yes	characteristic
382	#386	—	2	1	no	—
383	#385	—	2	1	no	—
384	#385	—	2	1	no	—
385	#384	C₂²×C₂₃	1	1	yes	characteristic
386	#380	—	2	1	no	—
387	#380	C₂²×C₂₃	1	1	yes	characteristic
388	#380	C₂²×C₂₃	1	1	yes	characteristic
389	#380	C₂²×C₂₃	1	1	yes	characteristic
390	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
391	#340	C₂²×C₂₃	1	1	yes	characteristic
392	#340	C₂²×C₂₃	1	1	yes	characteristic
393	#340	C₂²×C₂₃	1	1	yes	characteristic
394	#340	C₂²×C₂₃	1	1	yes	characteristic
395	#340	C₂²×C₂₃	1	1	yes	characteristic
396	#340	C₂²×C₂₃	1	1	yes	characteristic
397	#336	C₂²×C₂₃	1	1	yes	characteristic
398	#336	C₂²×C₂₃	1	1	yes	characteristic
399	C₂×C₂₃	—	2	4	no	—
400	C₂×C₂₃	—	4	2	no	—
401	C₂×C₂₃	#2443	1	1	yes	characteristic
402	C₂×C₂₃	C₂×#484	1	1	yes	characteristic
403	C₂×C₂₃	#3098	1	1	yes	characteristic
404	C₂×C₂₃	#3264	1	1	yes	characteristic
405	C₂×C₂₃	D₄×Q₁₆	1	1	yes	characteristic
406	C₂×C₂₃	#3338	1	1	yes	characteristic
407	C₂×C₂₃	#3343	1	1	yes	characteristic
408	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
410	C₂³×C₄	—	2	1	no	—
411	C₂³×C₄	—	2	1	no	—
412	C₂³×C₄	—	2	1	no	—
413	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
415	C₂²×C₈	—	2	2	no	—
416	C₂²×C₈	—	2	2	no	—
417	C₂²×C₈	—	2	2	no	—
418	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
419	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
420	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₂²⋊Q₈	—	2	2	no	—
423	C₂²⋊Q₈	—	2	2	no	—
424	C₂²⋊Q₈	—	2	2	no	—
425	C₂²⋊Q₈	—	4	1	no	—
426	C₄×D₄	—	2	2	no	—
427	C₄×D₄	—	2	2	no	—
428	C₄×D₄	—	4	4	no	—
429	C₂×C₄⋊C₄	—	2	2	no	—
430	C₂×C₄⋊C₄	—	2	2	no	—
431	C₂×C₄⋊C₄	—	2	2	no	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	2	1	no	—
434	C₂×C₄⋊C₄	—	2	1	no	—
435	C₂×C₄⋊C₄	—	2	1	no	—
436	C₂×C₄⋊C₄	—	2	1	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	—	2	1	no	—
440	C₂×C₄⋊C₄	—	4	1	no	—
441	C₂×C₄⋊C₄	—	4	1	no	—
442	C₂×C₄⋊C₄	—	4	1	no	—
443	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
447	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
448	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
449	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
451	C₂×C₂²⋊C₄	—	2	1	no	—
452	C₂×C₂²⋊C₄	—	2	1	no	—
453	C₂×C₂²⋊C₄	—	2	1	no	—
454	C₂×C₂²⋊C₄	—	2	1	no	—
455	C₂×C₂²⋊C₄	—	2	1	no	—
456	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
457	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
458	C₂×C₄²	—	2	1	no	—
459	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
460	#107	—	2	2	no	—
461	#107	—	2	2	no	—
462	#107	—	2	2	no	—
463	#107	—	2	2	no	—
464	#107	—	4	1	no	—
465	#107	D₄×C₂₃	1	4	yes	complemented
466	#103	—	2	2	no	—
467	#103	—	2	2	no	—
468	#103	—	2	2	no	—
469	#103	—	4	1	no	—
470	C₂²⋊C₈	—	2	2	no	—
471	C₂²⋊C₈	—	2	2	no	—
472	C₂²⋊C₈	—	2	2	no	—
473	#95	—	2	1	no	—
474	#95	—	2	1	no	—
475	#95	—	2	1	no	—
476	#95	—	2	1	no	—
477	#95	—	2	1	no	—
478	#95	—	2	1	no	—
479	#95	—	2	1	no	—
480	C₂₃	#26132	1	1	yes	characteristic, direct product factor, Sylow
481	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×D₄	—	2	2	no	—
490	C₂×D₄	—	2	2	no	—
491	C₂×D₄	Q₁₆×C₂₃	1	4	yes	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	2	1	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	4	2	no	—
511	C₂²×C₄	—	4	2	no	—
512	C₂²×C₄	—	4	2	no	—
513	C₂²×C₄	—	4	2	no	—
514	C₂²×C₄	—	4	2	no	—
515	C₂²×C₄	—	4	2	no	—
516	C₂²×C₄	—	8	1	no	—
517	C₂²×C₄	—	8	1	no	—
518	C₂²×C₄	—	8	1	no	—
519	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
520	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
521	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
522	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
523	C₂²×C₄	#41×C₂₃	1	2	yes	—
524	C₂²×C₄	#41×C₂₃	1	2	yes	—
525	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
526	C₂×C₈	—	2	8	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	1	no	—
533	C₂×C₈	—	2	1	no	—
534	C₂×C₈	—	2	1	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
538	C₂×C₈	#41×C₂₃	1	2	yes	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	4	2	no	—
553	C₄⋊C₄	—	4	2	no	—
554	C₄⋊C₄	—	4	2	no	—
555	C₄⋊C₄	—	4	2	no	—
556	C₄⋊C₄	—	4	2	no	—
557	C₄⋊C₄	—	4	2	no	—
558	C₄⋊C₄	—	8	1	no	—
559	C₄⋊C₄	—	8	1	no	—
560	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
561	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
562	C₂²⋊C₄	—	2	2	no	—
563	C₂²⋊C₄	—	2	2	no	—
564	C₂²⋊C₄	—	4	2	no	—
565	C₂²⋊C₄	—	4	2	no	—
566	C₂²⋊C₄	—	4	2	no	—
567	C₂²⋊C₄	—	4	2	no	—
568	C₂²⋊C₄	—	8	1	no	—
569	C₄²	—	2	1	no	—
570	C₄²	—	2	1	no	—
571	C₄²	—	4	2	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	—	2	1	no	—
577	C₂³	—	2	1	no	—
578	C₂³	—	2	1	no	—
579	C₂³	—	2	1	no	—
580	C₂³	—	2	1	no	—
581	C₂³	—	2	1	no	—
582	C₂³	C₂₃×#123	1	2	yes	—
583	C₂³	C₂×Q₁₆×C₂₃	1	2	yes	—
584	C₂³	C₂×Q₁₆×C₂₃	1	2	yes	—
585	C₂³	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
586	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
587	Q₈	—	4	1	no	—
588	Q₈	—	4	1	no	—
589	Q₈	—	4	1	no	—
590	Q₈	—	4	1	no	—
591	D₄	—	2	8	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	8	4	no	—
635	C₂×C₄	—	8	4	no	—
636	C₂×C₄	—	8	4	no	—
637	C₂×C₄	C₂₃×#123	1	2	yes	—
638	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
639	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
640	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
641	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
642	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
643	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
644	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
645	C₈	—	2	2	no	—
646	C₈	—	2	2	no	—
647	C₈	—	4	1	no	—
648	C₈	—	4	1	no	—
649	C₂²	—	2	2	no	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	2	2	no	—
653	C₂²	—	2	2	no	—
654	C₂²	—	2	2	no	—
655	C₂²	—	2	2	no	—
656	C₂²	—	2	2	no	—
657	C₂²	—	2	2	no	—
658	C₂²	—	4	1	no	—
659	C₂²	—	4	1	no	—
660	C₂²	—	4	1	no	—
661	C₂²	—	4	1	no	—
662	C₂²	C₂₃×#484	1	4	yes	complemented
663	C₂²	C₂×C₂₃×#123	1	1	yes	characteristic
664	C₂²	D₄²×C₂₃	1	1	yes	characteristic
665	C₂²	C₂₃×#547	1	1	yes	characteristic
666	C₂²	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
667	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
668	C₂²	C₂₃×#576	1	1	yes	characteristic
669	C₂²	C₂₃×#578	1	1	yes	characteristic
670	C₄	—	2	2	no	—
671	C₄	—	2	2	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	2	no	—
675	C₄	—	4	1	no	—
676	C₄	—	8	1	no	—
677	C₄	—	8	1	no	—
678	C₄	—	8	1	no	—
679	C₄	—	8	1	no	—
680	C₄	—	8	1	no	—
681	C₄	—	8	1	no	—
682	C₄	C₂₃×#525	1	1	yes	characteristic
683	C₄	C₂₃×#539	1	1	yes	characteristic
684	C₄	C₂₃×#548	1	1	yes	characteristic
685	C₄	C₂₃×#553	1	1	yes	characteristic
686	C₂	—	2	4	no	—
687	C₂	—	4	2	no	—
688	C₂	C₂₃×#2443	1	1	yes	characteristic
689	C₂	C₂×C₂₃×#484	1	1	yes	characteristic
690	C₂	C₂₃×#3098	1	1	yes	characteristic
691	C₂	C₂₃×#3264	1	1	yes	characteristic
692	C₂	D₄×Q₁₆×C₂₃	1	1	yes	characteristic
693	C₂	C₂₃×#3338	1	1	yes	characteristic
694	C₂	C₂₃×#3343	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#484	C₂	semidirect
2. C₂²×C₂₃×#107	C₂	semidirect
5. C₂₃×#2095	C₂	semidirect
6. C₂₃×#2083	C₂	semidirect
7. C₂₃×#2076	C₂	semidirect
8. C₂₃×#2036	C₂	semidirect
9. C₂₃×#1928	C₂	semidirect
10. C₂₃×#1921	C₂	semidirect
11. C₂₃×#1892	C₂	semidirect
12. C₂₃×#1872	C₂	semidirect
14. C₂₃×#1852	C₂	semidirect
15. C₂₃×#1852	C₂	semidirect
118. C₂₃×#107	D₄	semidirect
224. #26132	C₂₃	direct (N × H)
302. C₂×#484	C₂×C₂₃	semidirect
303. C₂²×#107	C₂×C₂₃	semidirect
306. #2095	C₂×C₂₃	semidirect
307. #2083	C₂×C₂₃	semidirect
308. #2076	C₂×C₂₃	semidirect
309. #2036	C₂×C₂₃	semidirect
310. #1928	C₂×C₂₃	semidirect
311. #1921	C₂×C₂₃	semidirect
312. #1892	C₂×C₂₃	semidirect
313. #1872	C₂×C₂₃	semidirect
315. #1852	C₂×C₂₃	semidirect
316. #1852	C₂×C₂₃	semidirect
330. C₂²×C₂₃	#484	semidirect
465. #107	D₄×C₂₃	semidirect
480. C₂₃	#26132	direct (N × H)
662. C₂²	C₂₃×#484	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#484	2
C₂²×C₂₃×#107	2
C₂₃×#2417	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#2095	2
C₂₃×#2083	2
C₂₃×#2076	2
C₂₃×#2036	2
C₂₃×#1928	2
C₂₃×#1921	2
C₂₃×#1892	2
C₂₃×#1872	2
C₂₃×#1869	2
C₂₃×#1852	2
#26132	23

Subgroups — C23×#26132 #19173758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26132 #19173758