Subgroups — C₂₃×#25978 #19173604

Order 5888.

1032 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#417	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2440	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2339	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2096	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2068	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2066	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2024	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1853	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1851	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1809	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1788	C₂	1	1	yes	maximal, characteristic
16	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#476	—	2	4	no	—
24	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂₃×#417	—	2	4	no	—
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×#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×#102	C₂²	1	1	yes	characteristic
39	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₂₃×#398	—	2	1	no	—
41	C₂₃×#395	—	2	1	no	—
42	C₂₃×#391	C₂²	1	1	yes	characteristic
43	C₂₃×#391	C₂²	1	1	yes	characteristic
44	C₂₃×#389	C₂²	1	1	yes	characteristic
45	C₂₃×#389	C₂²	1	1	yes	characteristic
46	C₂₃×#386	—	2	1	no	—
47	C₂₃×#386	—	2	1	no	—
48	C₂₃×#385	—	2	1	no	—
49	C₂₃×#384	C₂²	1	1	yes	characteristic
50	C₂₃×#381	—	2	1	no	—
51	C₂₃×#380	C₂²	1	1	yes	characteristic
52	C₂₃×#380	C₂²	1	1	yes	characteristic
53	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×#95	—	2	1	no	—
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	C₂×#41×C₂₃	—	2	2	no	—
62	C₂×#41×C₂₃	—	2	2	no	—
63	C₂×#41×C₂₃	—	2	2	no	—
64	C₂×#41×C₂₃	—	4	1	no	—
65	C₂×#41×C₂₃	—	4	1	no	—
66	C₂×#41×C₂₃	Q₈	1	4	yes	complemented
67	C₂²×Q₈×C₂₃	—	2	1	no	—
68	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₂₃	—	2	1	no	—
70	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	—	2	1	no	—
72	C₂³×C₄×C₂₃	—	2	1	no	—
73	C₂³×C₄×C₂₃	—	2	1	no	—
74	C₂³×C₄×C₂₃	—	2	1	no	—
75	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
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×C₄⋊Q₈	—	2	2	no	—
84	C₂₃×C₄⋊Q₈	—	4	1	no	—
85	C₂₃×C₂²⋊Q₈	—	4	4	no	—
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	—	4	1	no	—
89	C₄×D₄×C₂₃	—	4	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	2	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	—	4	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×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	D₄	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂₃×#107	—	4	1	no	—
113	C₂₃×#106	—	2	2	no	—
114	C₂₃×#106	—	2	2	no	—
115	C₂₃×#106	—	4	1	no	—
116	C₂₃×C₄⋊C₈	—	2	2	no	—
117	C₂₃×C₄⋊C₈	—	4	1	no	—
118	C₂₃×#103	—	2	2	no	—
119	C₂₃×#103	—	2	2	no	—
120	C₂₃×#103	—	4	1	no	—
121	C₂₃×#103	—	4	1	no	—
122	C₂₃×#102	—	2	2	no	—
123	C₂₃×#102	—	2	2	no	—
124	C₂₃×#102	—	2	2	no	—
125	C₂₃×#102	D₄	1	4	yes	—
126	C₂₃×C₂²⋊C₈	—	2	2	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	C₂₃×#95	—	2	1	no	—
134	C₂₃×#95	—	2	1	no	—
135	C₂₃×#95	D₄	1	1	yes	characteristic
136	C₂₃×#95	D₄	1	1	yes	characteristic
137	C₂⁴×C₂₃	—	2	1	no	—
138	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
139	#41×C₂₃	—	2	8	no	—
140	#41×C₂₃	—	4	4	no	—
141	#41×C₂₃	—	4	4	no	—
142	#41×C₂₃	—	4	4	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	—	4	2	no	—
149	C₂×Q₈×C₂₃	—	4	2	no	—
150	C₂×Q₈×C₂₃	—	8	1	no	—
151	C₂×Q₈×C₂₃	C₂×Q₈	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	—	4	2	no	—
157	C₂×D₄×C₂₃	—	4	2	no	—
158	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
159	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
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	4	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	—	4	2	no	—
174	C₂²×C₄×C₂₃	—	4	2	no	—
175	C₂²×C₄×C₂₃	—	4	2	no	—
176	C₂²×C₄×C₂₃	—	4	2	no	—
177	C₂²×C₄×C₂₃	—	4	2	no	—
178	C₂²×C₄×C₂₃	—	4	2	no	—
179	C₂²×C₄×C₂₃	—	4	2	no	—
180	C₂²×C₄×C₂₃	—	4	2	no	—
181	C₂²×C₄×C₂₃	—	8	1	no	—
182	C₂²×C₄×C₂₃	—	8	1	no	—
183	C₂²×C₄×C₂₃	—	8	1	no	—
184	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
190	C₂²×C₄×C₂₃	#41	1	2	yes	—
191	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
192	M₁₆×C₂₃	—	4	4	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	—	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	C₂×D₄	1	2	yes	—
205	C₄⋊C₄×C₂₃	—	2	1	no	—
206	C₄⋊C₄×C₂₃	—	2	1	no	—
207	C₄⋊C₄×C₂₃	—	2	1	no	—
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	—	4	2	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	C₂×D₄	1	2	yes	—
223	C₂²⋊C₄×C₂₃	—	2	2	no	—
224	C₂²⋊C₄×C₂₃	—	2	2	no	—
225	C₂²⋊C₄×C₂₃	—	4	2	no	—
226	C₂²⋊C₄×C₂₃	—	4	2	no	—
227	C₂²⋊C₄×C₂₃	—	4	2	no	—
228	C₂²⋊C₄×C₂₃	—	8	1	no	—
229	C₄²×C₂₃	—	2	1	no	—
230	C₄²×C₂₃	—	2	1	no	—
231	C₄²×C₂₃	—	2	1	no	—
232	C₄²×C₂₃	—	2	1	no	—
233	#25978	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂³×C₂₃	—	2	2	no	—
235	C₂³×C₂₃	—	2	2	no	—
236	C₂³×C₂₃	—	2	1	no	—
237	C₂³×C₂₃	—	2	2	no	—
238	C₂³×C₂₃	—	2	1	no	—
239	C₂³×C₂₃	—	2	1	no	—
240	C₂³×C₂₃	—	2	1	no	—
241	C₂³×C₂₃	—	2	1	no	—
242	C₂³×C₂₃	—	2	1	no	—
243	C₂³×C₂₃	—	4	1	no	—
244	C₂³×C₂₃	—	4	1	no	—
245	C₂³×C₂₃	—	4	1	no	—
246	C₂³×C₂₃	—	4	1	no	—
247	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
248	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
249	Q₈×C₂₃	—	2	2	no	—
250	Q₈×C₂₃	—	2	2	no	—
251	Q₈×C₂₃	—	4	1	no	—
252	Q₈×C₂₃	—	4	1	no	—
253	Q₈×C₂₃	—	8	4	no	—
254	D₄×C₂₃	—	2	2	no	—
255	D₄×C₂₃	—	2	2	no	—
256	D₄×C₂₃	—	2	2	no	—
257	D₄×C₂₃	—	4	8	no	—
258	D₄×C₂₃	C₂²⋊Q₈	1	4	yes	complemented
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	1	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	2	no	—
268	C₂×C₄×C₂₃	—	2	2	no	—
269	C₂×C₄×C₂₃	—	2	2	no	—
270	C₂×C₄×C₂₃	—	2	1	no	—
271	C₂×C₄×C₂₃	—	2	2	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	8	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	—	8	4	no	—
298	C₂×C₄×C₂₃	—	8	4	no	—
299	C₂×C₄×C₂₃	—	8	4	no	—
300	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
301	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
302	C₂×C₄×C₂₃	#137	1	2	yes	—
303	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
305	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
306	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
307	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
308	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
309	C₈×C₂₃	—	2	2	no	—
310	C₈×C₂₃	—	4	1	no	—
311	C₈×C₂₃	—	4	1	no	—
312	C₈×C₂₃	—	4	1	no	—
313	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
314	C₂×#417	C₂×C₂₃	1	1	yes	characteristic
315	#2440	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#2339	C₂×C₂₃	1	1	yes	characteristic
317	#2096	C₂×C₂₃	1	1	yes	characteristic, complemented
318	#2068	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#2066	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#2024	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1864	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1862	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1853	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1851	C₂×C₂₃	1	1	yes	characteristic, complemented
325	#1809	C₂×C₂₃	1	1	yes	characteristic, complemented
326	#1802	C₂×C₂₃	1	1	yes	characteristic, complemented
327	#1788	C₂×C₂₃	1	1	yes	characteristic
328	C₂²×C₂₃	—	2	2	no	—
329	C₂²×C₂₃	—	2	2	no	—
330	C₂²×C₂₃	—	2	2	no	—
331	C₂²×C₂₃	—	2	4	no	—
332	C₂²×C₂₃	—	4	1	no	—
333	C₂²×C₂₃	—	4	2	no	—
334	C₂²×C₂₃	—	4	2	no	—
335	C₂²×C₂₃	—	4	2	no	—
336	C₂²×C₂₃	—	4	2	no	—
337	C₂²×C₂₃	—	4	1	no	—
338	C₂²×C₂₃	—	4	1	no	—
339	C₂²×C₂₃	—	4	2	no	—
340	C₂²×C₂₃	—	4	2	no	—
341	C₂²×C₂₃	—	4	1	no	—
342	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
343	C₂²×C₂₃	D₄²	1	1	yes	characteristic
344	C₂²×C₂₃	#546	1	1	yes	characteristic
345	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
346	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
347	C₂²×C₂₃	#577	1	1	yes	characteristic
348	C₂²×C₂₃	#577	1	1	yes	characteristic
349	C₄×C₂₃	—	2	2	no	—
350	C₄×C₂₃	—	2	2	no	—
351	C₄×C₂₃	—	2	2	no	—
352	C₄×C₂₃	—	4	1	no	—
353	C₄×C₂₃	—	4	2	no	—
354	C₄×C₂₃	—	4	1	no	—
355	C₄×C₂₃	—	4	2	no	—
356	C₄×C₂₃	—	8	1	no	—
357	C₄×C₂₃	—	8	1	no	—
358	C₄×C₂₃	—	8	1	no	—
359	C₄×C₂₃	—	8	1	no	—
360	C₄×C₂₃	—	8	1	no	—
361	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
362	C₄×C₂₃	#543	1	1	yes	characteristic
363	C₄×C₂₃	#546	1	1	yes	characteristic
364	C₄×C₂₃	#551	1	1	yes	characteristic
365	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
366	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
368	C₂×C₂²⋊Q₈	—	2	1	no	—
369	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
371	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
372	#476	—	2	4	no	—
373	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
375	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
376	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
378	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
379	#417	—	2	4	no	—
380	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
381	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
382	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
383	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
384	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
387	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
388	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
389	#398	—	2	1	no	—
390	#395	—	2	1	no	—
391	#391	C₂²×C₂₃	1	1	yes	characteristic
392	#391	C₂²×C₂₃	1	1	yes	characteristic
393	#389	C₂²×C₂₃	1	1	yes	characteristic
394	#389	C₂²×C₂₃	1	1	yes	characteristic
395	#386	—	2	1	no	—
396	#386	—	2	1	no	—
397	#385	—	2	1	no	—
398	#384	C₂²×C₂₃	1	1	yes	characteristic
399	#381	—	2	1	no	—
400	#380	C₂²×C₂₃	1	1	yes	characteristic
401	#380	C₂²×C₂₃	1	1	yes	characteristic
402	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
403	C₂×#95	—	2	1	no	—
404	#340	C₂²×C₂₃	1	1	yes	characteristic
405	#340	C₂²×C₂₃	1	1	yes	characteristic
406	#340	C₂²×C₂₃	1	1	yes	characteristic
407	#340	C₂²×C₂₃	1	1	yes	characteristic
408	#336	C₂²×C₂₃	1	1	yes	characteristic
409	#336	C₂²×C₂₃	1	1	yes	characteristic
410	C₂×C₂₃	—	4	2	no	—
411	C₂×C₂₃	—	4	4	no	—
412	C₂×C₂₃	#2437	1	1	yes	characteristic
413	C₂×C₂₃	#3079	1	1	yes	characteristic
414	C₂×C₂₃	#3084	1	1	yes	characteristic
415	C₂×C₂₃	#3266	1	1	yes	characteristic
416	C₂×C₂₃	#3272	1	1	yes	characteristic
417	C₂×C₂₃	#3306	1	1	yes	characteristic
418	C₂×C₂₃	#3331	1	1	yes	characteristic
419	C₂×#41	—	2	2	no	—
420	C₂×#41	—	2	2	no	—
421	C₂×#41	—	2	2	no	—
422	C₂×#41	—	4	1	no	—
423	C₂×#41	—	4	1	no	—
424	C₂×#41	Q₈×C₂₃	1	4	yes	complemented
425	C₂²×Q₈	—	2	1	no	—
426	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₂²×D₄	—	2	1	no	—
428	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
429	C₂³×C₄	—	2	1	no	—
430	C₂³×C₄	—	2	1	no	—
431	C₂³×C₄	—	2	1	no	—
432	C₂³×C₄	—	2	1	no	—
433	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
434	C₂×M₁₆	—	2	2	no	—
435	C₂×M₁₆	—	2	2	no	—
436	C₂×M₁₆	—	4	1	no	—
437	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
438	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
439	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
440	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
441	C₄⋊Q₈	—	2	2	no	—
442	C₄⋊Q₈	—	4	1	no	—
443	C₂²⋊Q₈	—	4	4	no	—
444	C₄×D₄	—	2	2	no	—
445	C₄×D₄	—	2	2	no	—
446	C₄×D₄	—	4	1	no	—
447	C₄×D₄	—	4	1	no	—
448	C₂×C₄⋊C₄	—	2	2	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	—	2	1	no	—
454	C₂×C₄⋊C₄	—	2	1	no	—
455	C₂×C₄⋊C₄	—	4	1	no	—
456	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
458	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
459	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
460	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
461	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
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	—	2	1	no	—
466	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
467	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
468	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
469	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
470	#107	—	4	1	no	—
471	#106	—	2	2	no	—
472	#106	—	2	2	no	—
473	#106	—	4	1	no	—
474	C₄⋊C₈	—	2	2	no	—
475	C₄⋊C₈	—	4	1	no	—
476	#103	—	2	2	no	—
477	#103	—	2	2	no	—
478	#103	—	4	1	no	—
479	#103	—	4	1	no	—
480	#102	—	2	2	no	—
481	#102	—	2	2	no	—
482	#102	—	2	2	no	—
483	#102	D₄×C₂₃	1	4	yes	—
484	C₂²⋊C₈	—	2	2	no	—
485	#95	—	2	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	—	2	1	no	—
492	#95	—	2	1	no	—
493	#95	D₄×C₂₃	1	1	yes	characteristic
494	#95	D₄×C₂₃	1	1	yes	characteristic
495	C₂₃	#25978	1	1	yes	characteristic, direct product factor, Sylow
496	C₂⁴	—	2	1	no	—
497	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
498	#41	—	2	8	no	—
499	#41	—	4	4	no	—
500	#41	—	4	4	no	—
501	#41	—	4	4	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	—	4	2	no	—
508	C₂×Q₈	—	4	2	no	—
509	C₂×Q₈	—	8	1	no	—
510	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	—	4	2	no	—
517	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
518	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
519	C₂×D₄	#41×C₂₃	1	2	yes	—
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	4	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	1	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	4	2	no	—
533	C₂²×C₄	—	4	2	no	—
534	C₂²×C₄	—	4	2	no	—
535	C₂²×C₄	—	4	2	no	—
536	C₂²×C₄	—	4	2	no	—
537	C₂²×C₄	—	4	2	no	—
538	C₂²×C₄	—	4	2	no	—
539	C₂²×C₄	—	4	2	no	—
540	C₂²×C₄	—	8	1	no	—
541	C₂²×C₄	—	8	1	no	—
542	C₂²×C₄	—	8	1	no	—
543	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
544	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
545	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
546	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
547	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
548	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
549	C₂²×C₄	#41×C₂₃	1	2	yes	—
550	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
551	M₁₆	—	4	4	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	2	1	no	—
557	C₂×C₈	—	2	1	no	—
558	C₂×C₈	—	2	1	no	—
559	C₂×C₈	—	2	1	no	—
560	C₂×C₈	—	2	1	no	—
561	C₂×C₈	—	2	1	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	1	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	4	2	no	—
576	C₄⋊C₄	—	4	2	no	—
577	C₄⋊C₄	—	4	2	no	—
578	C₄⋊C₄	—	4	2	no	—
579	C₄⋊C₄	—	4	2	no	—
580	C₄⋊C₄	—	8	1	no	—
581	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
582	C₂²⋊C₄	—	2	2	no	—
583	C₂²⋊C₄	—	2	2	no	—
584	C₂²⋊C₄	—	4	2	no	—
585	C₂²⋊C₄	—	4	2	no	—
586	C₂²⋊C₄	—	4	2	no	—
587	C₂²⋊C₄	—	8	1	no	—
588	C₄²	—	2	1	no	—
589	C₄²	—	2	1	no	—
590	C₄²	—	2	1	no	—
591	C₄²	—	2	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	2	1	no	—
595	C₂³	—	2	1	no	—
596	C₂³	—	2	2	no	—
597	C₂³	—	2	2	no	—
598	C₂³	—	2	1	no	—
599	C₂³	—	2	2	no	—
600	C₂³	—	2	1	no	—
601	C₂³	—	4	1	no	—
602	C₂³	—	4	1	no	—
603	C₂³	—	4	1	no	—
604	C₂³	—	4	1	no	—
605	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
606	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
607	Q₈	—	2	2	no	—
608	Q₈	—	2	2	no	—
609	Q₈	—	4	1	no	—
610	Q₈	—	4	1	no	—
611	Q₈	—	8	4	no	—
612	D₄	—	2	2	no	—
613	D₄	—	2	2	no	—
614	D₄	—	2	2	no	—
615	D₄	—	4	8	no	—
616	D₄	C₂₃×C₂²⋊Q₈	1	4	yes	complemented
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	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	1	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	1	no	—
635	C₂×C₄	—	4	8	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	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	8	4	no	—
656	C₂×C₄	—	8	4	no	—
657	C₂×C₄	—	8	4	no	—
658	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
659	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
660	C₂×C₄	C₂₃×#137	1	2	yes	—
661	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
662	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
663	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
664	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
665	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
666	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
667	C₈	—	2	2	no	—
668	C₈	—	4	1	no	—
669	C₈	—	4	1	no	—
670	C₈	—	4	1	no	—
671	C₂²	—	2	2	no	—
672	C₂²	—	2	4	no	—
673	C₂²	—	2	2	no	—
674	C₂²	—	2	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	4	1	no	—
677	C₂²	—	4	1	no	—
678	C₂²	—	4	2	no	—
679	C₂²	—	4	2	no	—
680	C₂²	—	4	1	no	—
681	C₂²	—	4	1	no	—
682	C₂²	—	4	2	no	—
683	C₂²	—	4	2	no	—
684	C₂²	—	4	2	no	—
685	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
686	C₂²	D₄²×C₂₃	1	1	yes	characteristic
687	C₂²	C₂₃×#546	1	1	yes	characteristic
688	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
689	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
690	C₂²	C₂₃×#577	1	1	yes	characteristic
691	C₂²	C₂₃×#577	1	1	yes	characteristic
692	C₄	—	2	2	no	—
693	C₄	—	2	2	no	—
694	C₄	—	2	2	no	—
695	C₄	—	4	2	no	—
696	C₄	—	4	1	no	—
697	C₄	—	4	2	no	—
698	C₄	—	4	1	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	—	8	1	no	—
702	C₄	—	8	1	no	—
703	C₄	—	8	1	no	—
704	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
705	C₄	C₂₃×#543	1	1	yes	characteristic
706	C₄	C₂₃×#546	1	1	yes	characteristic
707	C₄	C₂₃×#551	1	1	yes	characteristic
708	C₂	—	4	2	no	—
709	C₂	—	4	4	no	—
710	C₂	C₂₃×#2437	1	1	yes	characteristic
711	C₂	C₂₃×#3079	1	1	yes	characteristic
712	C₂	C₂₃×#3084	1	1	yes	characteristic
713	C₂	C₂₃×#3266	1	1	yes	characteristic
714	C₂	C₂₃×#3272	1	1	yes	characteristic
715	C₂	C₂₃×#3306	1	1	yes	characteristic
716	C₂	C₂₃×#3331	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×#41×C₂₃	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×#476	C₂	semidirect
3. C₂₃×#2440	C₂	semidirect
5. C₂₃×#2096	C₂	semidirect
6. C₂₃×#2068	C₂	semidirect
7. C₂₃×#2066	C₂	semidirect
8. C₂₃×#2024	C₂	semidirect
9. C₂₃×#1864	C₂	semidirect
10. C₂₃×#1862	C₂	semidirect
11. C₂₃×#1853	C₂	semidirect
12. C₂₃×#1851	C₂	semidirect
13. C₂₃×#1809	C₂	semidirect
14. C₂₃×#1802	C₂	semidirect
66. C₂×#41×C₂₃	Q₈	semidirect
233. #25978	C₂₃	direct (N × H)
258. D₄×C₂₃	C₂²⋊Q₈	semidirect
313. C₂×#476	C₂×C₂₃	semidirect
315. #2440	C₂×C₂₃	semidirect
317. #2096	C₂×C₂₃	semidirect
318. #2068	C₂×C₂₃	semidirect
319. #2066	C₂×C₂₃	semidirect
320. #2024	C₂×C₂₃	semidirect
321. #1864	C₂×C₂₃	semidirect
322. #1862	C₂×C₂₃	semidirect
323. #1853	C₂×C₂₃	semidirect
324. #1851	C₂×C₂₃	semidirect
325. #1809	C₂×C₂₃	semidirect
326. #1802	C₂×C₂₃	semidirect
424. C₂×#41	Q₈×C₂₃	semidirect
495. C₂₃	#25978	direct (N × H)
616. D₄	C₂₃×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#476	2
C₂×C₂₃×#417	2
C₂₃×#2440	2
C₂₃×#2339	2
C₂₃×#2096	2
C₂₃×#2068	2
C₂₃×#2066	2
C₂₃×#2024	2
C₂₃×#1864	2
C₂₃×#1862	2
C₂₃×#1853	2
C₂₃×#1851	2
C₂₃×#1809	2
C₂₃×#1802	2
C₂₃×#1788	2
#25978	23

Subgroups — C23×#25978 #19173604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25978 #19173604