Subgroups — C₂₃×#2668 #15670287

Order 2944.

816 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#121	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#120	C₂	1	2	yes	maximal, complemented
3	C₂₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#397	C₂	1	2	yes	maximal, complemented
5	C₂₃×#394	C₂	1	2	yes	maximal, complemented
6	C₂₃×#392	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#386	C₂	1	2	yes	maximal, complemented
8	C₂₃×#386	C₂	1	2	yes	maximal, complemented
9	C₂₃×#379	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₄×C₂₃	C₂²	1	2	yes	complemented
16	C₂₃×#121	—	2	8	no	—
17	C₂₃×#120	—	2	8	no	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	complemented
29	C₂₃×#95	C₂²	1	2	yes	complemented
30	C₂₃×#95	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#95	C₂²	1	2	yes	complemented
32	C₂₃×#95	C₂²	1	1	yes	characteristic, complemented
33	C₂⁴×C₂₃	—	2	4	no	—
34	C₂⁴×C₂₃	—	2	2	no	—
35	C₂⁴×C₂₃	—	4	1	no	—
36	C₂⁴×C₂₃	—	4	2	no	—
37	C₂⁴×C₂₃	—	4	1	no	—
38	C₂⁴×C₂₃	C₂³	1	2	yes	—
39	C₂⁴×C₂₃	C₂³	1	2	yes	—
40	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₂₃	—	2	4	no	—
42	C₂×D₄×C₂₃	—	2	4	no	—
43	C₂×D₄×C₂₃	—	2	4	no	—
44	C₂×D₄×C₂₃	—	2	4	no	—
45	C₂×D₄×C₂₃	—	2	4	no	—
46	C₂×D₄×C₂₃	—	4	2	no	—
47	C₂×D₄×C₂₃	—	4	2	no	—
48	C₂×D₄×C₂₃	—	4	2	no	—
49	C₂×D₄×C₂₃	—	4	2	no	—
50	C₂×D₄×C₂₃	—	4	1	no	—
51	C₂×D₄×C₂₃	D₄	1	8	yes	complemented
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	—	4	2	no	—
54	C₂²×C₄×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
61	C₄⋊C₄×C₂₃	—	2	4	no	—
62	C₄⋊C₄×C₂₃	—	4	1	no	—
63	C₂²⋊C₄×C₂₃	—	2	4	no	—
64	C₂²⋊C₄×C₂₃	—	2	4	no	—
65	C₂²⋊C₄×C₂₃	—	2	4	no	—
66	C₂²⋊C₄×C₂₃	—	2	4	no	—
67	C₂²⋊C₄×C₂₃	—	2	4	no	—
68	C₂²⋊C₄×C₂₃	—	4	2	no	—
69	C₂²⋊C₄×C₂₃	—	4	1	no	—
70	C₂²⋊C₄×C₂₃	—	4	1	no	—
71	C₂²⋊C₄×C₂₃	—	4	2	no	—
72	C₂³×C₂₃	—	2	2	no	—
73	C₂³×C₂₃	—	2	2	no	—
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	—	2	2	no	—
77	C₂³×C₂₃	—	2	2	no	—
78	C₂³×C₂₃	—	2	1	no	—
79	C₂³×C₂₃	—	2	1	no	—
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	2	2	no	—
84	C₂³×C₂₃	—	2	2	no	—
85	C₂³×C₂₃	—	2	2	no	—
86	C₂³×C₂₃	—	2	2	no	—
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	4	4	no	—
89	C₂³×C₂₃	—	4	8	no	—
90	C₂³×C₂₃	—	4	8	no	—
91	C₂³×C₂₃	—	4	4	no	—
92	C₂³×C₂₃	—	4	4	no	—
93	C₂³×C₂₃	C₂×D₄	1	4	yes	—
94	C₂³×C₂₃	C₂×D₄	1	4	yes	—
95	C₂³×C₂₃	#41	1	2	yes	—
96	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
97	D₄×C₂₃	—	4	8	no	—
98	D₄×C₂₃	—	4	4	no	—
99	D₄×C₂₃	—	4	8	no	—
100	D₄×C₂₃	—	4	8	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	4	8	no	—
117	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
118	#2668	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₂₃	—	2	4	no	—
120	C₂²×C₂₃	—	2	4	no	—
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	2	4	no	—
123	C₂²×C₂₃	—	2	4	no	—
124	C₂²×C₂₃	—	2	4	no	—
125	C₂²×C₂₃	—	2	4	no	—
126	C₂²×C₂₃	—	2	4	no	—
127	C₂²×C₂₃	—	4	16	no	—
128	C₂²×C₂₃	—	4	2	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	4	2	no	—
132	C₂²×C₂₃	—	4	2	no	—
133	C₂²×C₂₃	—	4	1	no	—
134	C₂²×C₂₃	—	4	2	no	—
135	C₂²×C₂₃	—	4	2	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	—	4	2	no	—
138	C₂²×C₂₃	—	4	1	no	—
139	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
140	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
141	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
143	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
144	C₄×C₂₃	—	2	4	no	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	—	4	1	no	—
148	C₄×C₂₃	—	4	2	no	—
149	C₄×C₂₃	—	4	1	no	—
150	C₂×#121	C₂×C₂₃	1	2	yes	complemented
151	C₂×#120	C₂×C₂₃	1	2	yes	complemented
152	#399	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#397	C₂×C₂₃	1	2	yes	complemented
154	#394	C₂×C₂₃	1	2	yes	complemented
155	#392	C₂×C₂₃	1	1	yes	characteristic, complemented
156	#386	C₂×C₂₃	1	2	yes	complemented
157	#386	C₂×C₂₃	1	2	yes	complemented
158	#379	C₂×C₂₃	1	1	yes	characteristic, complemented
159	C₂×C₂₃	—	4	4	no	—
160	C₂×C₂₃	—	4	4	no	—
161	C₂×C₂₃	—	4	2	no	—
162	C₂×C₂₃	#534	1	2	yes	—
163	C₂×C₂₃	#538	1	1	yes	characteristic
164	C₂×C₂₃	D₄²	1	1	yes	characteristic
165	C₂×C₂₃	#546	1	2	yes	—
166	C₂×C₂₃	#560	1	1	yes	characteristic
167	C₂⁵	C₂²×C₂₃	1	1	yes	characteristic
168	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
169	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
170	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
171	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
172	C₂³×C₄	C₂²×C₂₃	1	2	yes	complemented
173	#121	—	2	8	no	—
174	#120	—	2	8	no	—
175	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
176	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
177	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
179	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
183	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	complemented
185	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	complemented
186	#95	C₂²×C₂₃	1	2	yes	complemented
187	#95	C₂²×C₂₃	1	1	yes	characteristic, complemented
188	#95	C₂²×C₂₃	1	2	yes	complemented
189	#95	C₂²×C₂₃	1	1	yes	characteristic, complemented
190	C₂₃	#2668	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	—	2	4	no	—
192	C₂⁴	—	2	2	no	—
193	C₂⁴	—	4	1	no	—
194	C₂⁴	—	4	2	no	—
195	C₂⁴	—	4	1	no	—
196	C₂⁴	C₂³×C₂₃	1	2	yes	—
197	C₂⁴	C₂³×C₂₃	1	2	yes	—
198	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	D₄×C₂₃	1	8	yes	complemented
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
214	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
215	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
218	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	1	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	8	no	—
248	C₂³	—	4	8	no	—
249	C₂³	—	4	4	no	—
250	C₂³	—	4	4	no	—
251	C₂³	C₂×D₄×C₂₃	1	4	yes	—
252	C₂³	C₂×D₄×C₂₃	1	4	yes	—
253	C₂³	#41×C₂₃	1	2	yes	—
254	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
255	D₄	—	4	8	no	—
256	D₄	—	4	4	no	—
257	D₄	—	4	8	no	—
258	D₄	—	4	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	16	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
297	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
298	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
299	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
301	C₄	—	2	4	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	2	no	—
307	C₂	—	4	4	no	—
308	C₂	—	4	2	no	—
309	C₂	—	4	4	no	—
310	C₂	C₂₃×#534	1	2	yes	—
311	C₂	C₂₃×#538	1	1	yes	characteristic
312	C₂	D₄²×C₂₃	1	1	yes	characteristic
313	C₂	C₂₃×#546	1	2	yes	—
314	C₂	C₂₃×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#121	C₂	semidirect
2. C₂×C₂₃×#120	C₂	semidirect
3. C₂₃×#399	C₂	semidirect
4. C₂₃×#397	C₂	semidirect
5. C₂₃×#394	C₂	semidirect
6. C₂₃×#392	C₂	semidirect
7. C₂₃×#386	C₂	semidirect
8. C₂₃×#386	C₂	semidirect
9. C₂₃×#379	C₂	semidirect
14. C₂²×D₄×C₂₃	C₂²	semidirect
15. C₂³×C₄×C₂₃	C₂²	semidirect
18. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
19. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
27. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
28. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
29. C₂₃×#95	C₂²	semidirect
30. C₂₃×#95	C₂²	semidirect
31. C₂₃×#95	C₂²	semidirect
32. C₂₃×#95	C₂²	semidirect
51. C₂×D₄×C₂₃	D₄	semidirect
118. #2668	C₂₃	direct (N × H)
150. C₂×#121	C₂×C₂₃	semidirect
151. C₂×#120	C₂×C₂₃	semidirect
152. #399	C₂×C₂₃	semidirect
153. #397	C₂×C₂₃	semidirect
154. #394	C₂×C₂₃	semidirect
155. #392	C₂×C₂₃	semidirect
156. #386	C₂×C₂₃	semidirect
157. #386	C₂×C₂₃	semidirect
158. #379	C₂×C₂₃	semidirect
171. C₂²×D₄	C₂²×C₂₃	semidirect
172. C₂³×C₄	C₂²×C₂₃	semidirect
175. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
176. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
184. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
185. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
186. #95	C₂²×C₂₃	semidirect
187. #95	C₂²×C₂₃	semidirect
188. #95	C₂²×C₂₃	semidirect
189. #95	C₂²×C₂₃	semidirect
190. C₂₃	#2668	direct (N × H)
209. C₂×D₄	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#121	2
C₂×C₂₃×#120	2
C₂₃×#399	2
C₂₃×#397	2
C₂₃×#394	2
C₂₃×#392	2
C₂₃×#386	2
C₂₃×#379	2
#2668	23

Subgroups — C23×#2668 #15670287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2668 #15670287