Subgroups — C₂₃×#3006 #15670625

Order 2944.

806 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#449	C₂	1	4	yes	maximal, complemented
5	C₂₃×#448	C₂	1	2	yes	maximal, complemented
6	C₂₃×#447	C₂	1	2	yes	maximal, complemented
7	C₂₃×#417	C₂	1	2	yes	maximal, complemented
8	C₂₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	—	2	8	no	—
10	C₂×#41×C₂₃	—	2	1	no	—
11	C₂×#41×C₂₃	—	2	1	no	—
12	C₂×#41×C₂₃	C₂²	1	2	yes	—
13	C₂×#41×C₂₃	C₂²	1	2	yes	complemented
14	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂²×D₄×C₂₃	—	2	1	no	—
16	C₂²×D₄×C₂₃	—	2	1	no	—
17	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
19	C₂³×C₄×C₂₃	—	2	1	no	—
20	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×AGL(1,ℤ/8)	—	2	8	no	—
22	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
23	C₂×D₈×C₂₃	C₂²	1	4	yes	complemented
24	C₂×M₁₆×C₂₃	C₂²	1	2	yes	complemented
25	C₂₃×#121	—	2	1	no	—
26	C₂₃×#121	—	2	1	no	—
27	C₂₃×#121	C₂²	1	2	yes	—
28	C₂₃×#121	C₂²	1	2	yes	complemented
29	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
31	C₂₃×#103	C₂²	1	4	yes	complemented
32	C₂₃×#102	C₂²	1	4	yes	complemented
33	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
34	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
35	C₂⁴×C₂₃	—	2	1	no	—
36	C₂⁴×C₂₃	—	2	1	no	—
37	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
38	#41×C₂₃	—	2	8	no	—
39	#41×C₂₃	—	2	8	no	—
40	#41×C₂₃	—	2	4	no	—
41	#41×C₂₃	—	2	4	no	—
42	#41×C₂₃	—	2	4	no	—
43	#41×C₂₃	D₄	1	8	yes	complemented
44	C₂×Q₈×C₂₃	—	2	4	no	—
45	C₂×Q₈×C₂₃	—	2	1	no	—
46	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
47	C₂×D₄×C₂₃	—	2	8	no	—
48	C₂×D₄×C₂₃	—	2	2	no	—
49	C₂×D₄×C₂₃	—	2	2	no	—
50	C₂×D₄×C₂₃	—	2	4	no	—
51	C₂×D₄×C₂₃	—	2	2	no	—
52	C₂×D₄×C₂₃	—	2	2	no	—
53	C₂×D₄×C₂₃	—	2	1	no	—
54	C₂×D₄×C₂₃	—	2	4	no	—
55	C₂×D₄×C₂₃	—	2	1	no	—
56	C₂×D₄×C₂₃	—	4	4	no	—
57	C₂×D₄×C₂₃	C₂³	1	2	yes	—
58	C₂×D₄×C₂₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₂₃	—	2	8	no	—
60	C₂²×C₄×C₂₃	—	2	1	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	—	2	4	no	—
64	C₂²×C₄×C₂₃	—	2	1	no	—
65	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	SD₁₆×C₂₃	—	2	8	no	—
72	D₈×C₂₃	—	2	8	no	—
73	M₁₆×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	C₂³	1	4	yes	—
75	C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₂₃	—	2	2	no	—
78	C₂²⋊C₄×C₂₃	—	2	2	no	—
79	C₂³×C₂₃	—	2	4	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	—	4	1	no	—
85	C₂³×C₂₃	—	4	2	no	—
86	C₂³×C₂₃	—	4	2	no	—
87	C₂³×C₂₃	—	4	2	no	—
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	—	4	2	no	—
90	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
93	Q₈×C₂₃	—	2	4	no	—
94	Q₈×C₂₃	—	2	2	no	—
95	Q₈×C₂₃	C₂×D₄	1	4	yes	complemented
96	D₄×C₂₃	—	2	8	no	—
97	D₄×C₂₃	—	2	4	no	—
98	D₄×C₂₃	—	2	8	no	—
99	D₄×C₂₃	—	2	2	no	—
100	D₄×C₂₃	—	2	4	no	—
101	D₄×C₂₃	—	4	4	no	—
102	D₄×C₂₃	—	4	4	no	—
103	D₄×C₂₃	—	4	2	no	—
104	D₄×C₂₃	C₂×D₄	1	4	yes	complemented
105	C₂×C₄×C₂₃	—	2	4	no	—
106	C₂×C₄×C₂₃	—	2	8	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	8	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
120	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
121	C₈×C₂₃	—	2	4	no	—
122	#3006	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₂₃	—	2	1	no	—
124	C₂²×C₂₃	—	2	1	no	—
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	2	1	no	—
128	C₂²×C₂₃	—	2	4	no	—
129	C₂²×C₂₃	—	2	1	no	—
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	4	4	no	—
132	C₂²×C₂₃	—	4	8	no	—
133	C₂²×C₂₃	—	4	2	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	—	8	4	no	—
139	C₂²×C₂₃	#120	1	2	yes	—
140	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
141	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	4	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	#120	1	2	yes	—
147	C₄×C₂₃	C₂²×D₄	1	2	yes	—
148	C₂²×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	2	yes	complemented
150	C₂×#121	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#449	C₂×C₂₃	1	4	yes	complemented
152	#448	C₂×C₂₃	1	2	yes	complemented
153	#447	C₂×C₂₃	1	2	yes	complemented
154	#417	C₂×C₂₃	1	2	yes	complemented
155	#407	C₂×C₂₃	1	1	yes	characteristic, complemented
156	C₂×C₂₃	—	2	2	no	—
157	C₂×C₂₃	—	2	2	no	—
158	C₂×C₂₃	—	4	1	no	—
159	C₂×C₂₃	—	4	4	no	—
160	C₂×C₂₃	—	8	2	no	—
161	C₂×C₂₃	C₂×#120	1	1	yes	characteristic
162	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₂₃	#575	1	1	yes	characteristic
164	C₂×#41	—	2	8	no	—
165	C₂×#41	—	2	1	no	—
166	C₂×#41	—	2	1	no	—
167	C₂×#41	C₂²×C₂₃	1	2	yes	—
168	C₂×#41	C₂²×C₂₃	1	2	yes	complemented
169	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	2	1	no	—
172	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
173	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
174	C₂³×C₄	—	2	1	no	—
175	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
176	AGL(1,ℤ/8)	—	2	8	no	—
177	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
178	C₂×D₈	C₂²×C₂₃	1	4	yes	complemented
179	C₂×M₁₆	C₂²×C₂₃	1	2	yes	complemented
180	#121	—	2	1	no	—
181	#121	—	2	1	no	—
182	#121	C₂²×C₂₃	1	2	yes	—
183	#121	C₂²×C₂₃	1	2	yes	complemented
184	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
185	C₂×C₂²⋊C₄	—	2	1	no	—
186	#103	C₂²×C₂₃	1	4	yes	complemented
187	#102	C₂²×C₂₃	1	4	yes	complemented
188	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
189	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
190	C₂₃	#3006	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	—	2	1	no	—
192	C₂⁴	—	2	1	no	—
193	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
194	#41	—	2	8	no	—
195	#41	—	2	8	no	—
196	#41	—	2	4	no	—
197	#41	—	2	4	no	—
198	#41	—	2	4	no	—
199	#41	D₄×C₂₃	1	8	yes	complemented
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	1	no	—
202	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
203	C₂×D₄	—	2	8	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	C₂³×C₂₃	1	2	yes	—
214	C₂×D₄	C₂³×C₂₃	1	2	yes	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
227	SD₁₆	—	2	8	no	—
228	D₈	—	2	8	no	—
229	M₁₆	—	2	4	no	—
230	C₂×C₈	C₂³×C₂₃	1	4	yes	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	4	1	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
247	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
248	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	2	2	no	—
251	Q₈	C₂×D₄×C₂₃	1	4	yes	complemented
252	D₄	—	2	8	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	8	no	—
255	D₄	—	2	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	4	no	—
259	D₄	—	4	2	no	—
260	D₄	C₂×D₄×C₂₃	1	4	yes	complemented
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
276	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	8	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	—	8	4	no	—
294	C₂²	C₂₃×#120	1	2	yes	—
295	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	complemented
296	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	2	no	—
301	C₄	C₂₃×#120	1	2	yes	—
302	C₄	C₂²×D₄×C₂₃	1	2	yes	—
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	—	4	1	no	—
306	C₂	—	4	4	no	—
307	C₂	—	8	2	no	—
308	C₂	C₂×C₂₃×#120	1	1	yes	characteristic
309	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂	C₂₃×#575	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₂₃	C₂	semidirect
2. C₂×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₂₃×#121	C₂	semidirect
4. C₂₃×#449	C₂	semidirect
5. C₂₃×#448	C₂	semidirect
6. C₂₃×#447	C₂	semidirect
7. C₂₃×#417	C₂	semidirect
8. C₂₃×#407	C₂	semidirect
13. C₂×#41×C₂₃	C₂²	semidirect
14. C₂²×Q₈×C₂₃	C₂²	semidirect
17. C₂²×D₄×C₂₃	C₂²	semidirect
18. C₂²×D₄×C₂₃	C₂²	semidirect
22. C₂×SD₁₆×C₂₃	C₂²	semidirect
23. C₂×D₈×C₂₃	C₂²	semidirect
24. C₂×M₁₆×C₂₃	C₂²	semidirect
28. C₂₃×#121	C₂²	semidirect
29. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
31. C₂₃×#103	C₂²	semidirect
32. C₂₃×#102	C₂²	semidirect
33. C₂₃×C₂²⋊C₈	C₂²	semidirect
34. C₂₃×C₂²⋊C₈	C₂²	semidirect
43. #41×C₂₃	D₄	semidirect
95. Q₈×C₂₃	C₂×D₄	semidirect
104. D₄×C₂₃	C₂×D₄	semidirect
122. #3006	C₂₃	direct (N × H)
140. C₂²×C₂₃	AGL(1,ℤ/8)	semidirect
148. C₂²×#41	C₂×C₂₃	semidirect
149. C₂×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
150. C₂×#121	C₂×C₂₃	semidirect
151. #449	C₂×C₂₃	semidirect
152. #448	C₂×C₂₃	semidirect
153. #447	C₂×C₂₃	semidirect
154. #417	C₂×C₂₃	semidirect
155. #407	C₂×C₂₃	semidirect
168. C₂×#41	C₂²×C₂₃	semidirect
169. C₂²×Q₈	C₂²×C₂₃	semidirect
172. C₂²×D₄	C₂²×C₂₃	semidirect
173. C₂²×D₄	C₂²×C₂₃	semidirect
177. C₂×SD₁₆	C₂²×C₂₃	semidirect
178. C₂×D₈	C₂²×C₂₃	semidirect
179. C₂×M₁₆	C₂²×C₂₃	semidirect
183. #121	C₂²×C₂₃	semidirect
184. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
186. #103	C₂²×C₂₃	semidirect
187. #102	C₂²×C₂₃	semidirect
188. C₂²⋊C₈	C₂²×C₂₃	semidirect
189. C₂²⋊C₈	C₂²×C₂₃	semidirect
190. C₂₃	#3006	direct (N × H)
199. #41	D₄×C₂₃	semidirect
251. Q₈	C₂×D₄×C₂₃	semidirect
260. D₄	C₂×D₄×C₂₃	semidirect
295. C₂²	C₂₃×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₃	2
C₂×C₂₃×AGL(1,ℤ/8)	2
C₂×C₂₃×#121	2
C₂₃×#449	2
C₂₃×#448	2
C₂₃×#447	2
C₂₃×#417	2
C₂₃×#407	2
#3006	23

Subgroups — C23×#3006 #15670625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3006 #15670625