Subgroups — #15686694

Order 2944.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092972	C₂	1	2	yes	maximal, complemented
2	#13092968	C₂	1	2	yes	maximal, complemented
3	#13092967	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092954	C₂	1	4	yes	maximal, complemented
5	#13092953	C₂	1	4	yes	maximal, complemented
6	#13092952	C₂	1	4	yes	maximal, complemented
7	#13092930	C₂	1	4	yes	maximal, complemented
8	#13092929	C₂	1	2	yes	maximal, complemented
9	C₄×#67292	C₂	1	4	yes	maximal, complemented
10	#13092885	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#67292	C₂²	1	4	yes	—
14	C₂×#67288	C₂²	1	2	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
17	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
18	C₂₃×#125	C₂²	1	2	yes	complemented
19	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
21	C₂₃×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	#10625118	C₂²	1	4	yes	complemented
24	#10625117	C₂²	1	4	yes	—
25	Q₈×Dic₂₃	C₂²	1	4	yes	complemented
26	#10625101	C₂²	1	4	yes	—
27	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
29	#10625069	C₂²	1	8	yes	complemented
30	#10625068	C₂²	1	4	yes	—
31	#10625066	C₂²	1	4	yes	—
32	#10625066	C₂²	1	4	yes	—
33	#10625065	C₂²	1	8	yes	—
34	#10625064	C₂²	1	8	yes	—
35	#10625064	C₂²	1	4	yes	complemented
36	#10625064	C₂²	1	4	yes	complemented
37	#10625063	C₂²	1	4	yes	—
38	#10625063	C₂²	1	8	yes	complemented
39	#10625061	C₂²	1	4	yes	complemented
40	#10625058	C₂²	1	4	yes	complemented
41	#10625057	C₂²	1	4	yes	—
42	#10625053	C₂²	1	4	yes	—
43	(C₂²⋊D₂₃)⋊C₄	C₂²	1	8	yes	—
44	#10625046	C₂²	1	2	yes	complemented
45	#10625045	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₉₂	C₂²	1	8	yes	—
47	C₄×D₉₂	C₂²	1	4	yes	complemented
48	#10625043	C₂²	1	4	yes	complemented
49	C₄²×D₂₃	C₂²	1	2	yes	—
50	C₄²×D₂₃	C₂²	1	2	yes	—
51	C₄²×D₂₃	C₂²	1	2	yes	—
52	C₄²×D₂₃	C₂²	1	2	yes	—
53	C₄²×D₂₃	C₂²	1	4	yes	complemented
54	#10625040	C₂²	1	1	yes	characteristic, complemented
55	C₂³×D₂₃	C₂³	1	1	yes	characteristic
56	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
57	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
60	C₂²×Dic₂₃	C₂³	1	2	yes	—
61	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
62	#67292	—	2	16	no	—
63	#67288	—	2	8	no	—
64	C₂×D₉₂	C₂³	1	8	yes	—
65	C₂×D₉₂	C₂³	1	4	yes	—
66	C₂×D₉₂	C₂³	1	2	yes	—
67	C₂×C₄×D₂₃	—	2	4	no	—
68	C₂×C₄×D₂₃	—	2	8	no	—
69	C₂×C₄×D₂₃	C₂³	1	2	yes	—
70	C₂×C₄×D₂₃	C₂³	1	2	yes	—
71	C₂×C₄×D₂₃	C₂³	1	2	yes	—
72	C₂×C₄×D₂₃	C₂³	1	2	yes	—
73	C₂×C₄×D₂₃	C₂³	1	2	yes	—
74	C₂×C₄×D₂₃	C₂³	1	4	yes	—
75	C₂×C₄×D₂₃	C₂³	1	8	yes	—
76	C₂×C₄×D₂₃	C₂³	1	2	yes	—
77	C₂×Dic₄₆	C₂³	1	2	yes	—
78	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
79	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
81	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
82	C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₄²×C₂₃	C₂³	1	2	yes	—
84	C₄²×C₂₃	C₂³	1	1	yes	characteristic
85	C₄²×C₂₃	C₂³	1	4	yes	—
86	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
87	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
88	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
89	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
90	C₄⋊Dic₂₃	C₂³	1	4	yes	—
91	C₄⋊Dic₂₃	C₂³	1	8	yes	—
92	Dic₂₃⋊C₄	C₂³	1	8	yes	—
93	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
94	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₂₃	C₂³	1	2	yes	—
96	C₄×Dic₂₃	C₂³	1	2	yes	—
97	C₄×Dic₂₃	C₂³	1	2	yes	—
98	C₄×Dic₂₃	C₂³	1	2	yes	—
99	C₄×Dic₂₃	C₂³	1	2	yes	—
100	C₄×Dic₂₃	C₂³	1	8	yes	—
101	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
102	C₂²×D₂₃	—	2	2	no	—
103	C₂²×D₂₃	—	2	4	no	—
104	C₂²×D₂₃	C₂⁴	1	4	yes	—
105	C₂²×D₂₃	C₂⁴	1	2	yes	—
106	Q₈×C₂₃	—	2	8	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	#41	1	4	yes	complemented
109	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
111	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₂₃	—	2	8	no	—
115	C₂×Dic₂₃	—	2	2	no	—
116	C₂×Dic₂₃	—	2	4	no	—
117	C₂×Dic₂₃	C₂⁴	1	2	yes	—
118	C₂×Dic₂₃	C₂⁴	1	2	yes	—
119	C₂×Dic₂₃	C₂⁴	1	4	yes	—
120	C₂×Dic₂₃	C₂⁴	1	2	yes	—
121	D₉₂	—	2	16	no	—
122	D₉₂	—	2	4	no	—
123	D₉₂	—	2	8	no	—
124	C₄×D₂₃	—	2	4	no	—
125	C₄×D₂₃	—	2	4	no	—
126	C₄×D₂₃	—	2	4	no	—
127	C₄×D₂₃	—	2	4	no	—
128	C₄×D₂₃	—	2	8	no	—
129	C₄×D₂₃	—	2	4	no	—
130	C₄×D₂₃	—	2	16	no	—
131	C₄×D₂₃	#41	1	8	yes	complemented
132	Dic₄₆	—	2	4	no	—
133	#3444	—	23	1	no	maximal, Sylow
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	C₂×#41	1	2	yes	—
136	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
137	C₂×D₂₃	—	2	8	no	—
138	C₂×D₂₃	—	2	2	no	—
139	C₂×D₂₃	—	2	2	no	—
140	C₂×D₂₃	—	2	8	no	—
141	C₂×D₂₃	—	2	4	no	—
142	C₂×D₂₃	C₂×#41	1	4	yes	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	2	8	no	—
145	C₄×C₂₃	C₂×#41	1	2	yes	—
146	C₄×C₂₃	C₂×#41	1	2	yes	—
147	C₄×C₂₃	C₂×#41	1	4	yes	—
148	Dic₂₃	—	2	4	no	—
149	Dic₂₃	—	2	2	no	—
150	Dic₂₃	—	2	2	no	—
151	Dic₂₃	C₂×#41	1	4	yes	—
152	#533	—	23	2	no	—
153	#533	D₂₃	1	1	yes	characteristic, complemented
154	#532	—	23	2	no	—
155	#532	—	23	1	no	—
156	#529	—	23	4	no	—
157	#529	—	23	4	no	—
158	#529	—	23	4	no	—
159	#525	—	23	4	no	—
160	#525	—	23	2	no	—
161	C₄×#41	—	23	4	no	—
162	C₄×#41	—	23	2	no	—
163	C₂²×C₄²	—	23	1	no	—
164	C₂×C₂₃	—	2	2	no	—
165	C₂×C₂₃	C₂²×#41	1	2	yes	—
166	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
167	D₂₃	—	2	4	no	—
168	D₂₃	—	4	4	no	—
169	C₂×#41	—	23	4	no	—
170	C₂×#41	—	23	2	no	—
171	C₂³×C₄	—	23	2	no	—
172	C₂³×C₄	—	23	1	no	—
173	C₄⋊Q₈	—	23	1	no	—
174	C₄⋊Q₈	C₂×D₂₃	1	2	yes	complemented
175	#127	—	23	1	no	—
176	#126	—	23	8	no	—
177	#125	—	23	4	no	—
178	#125	C₂×D₂₃	1	2	yes	complemented
179	#124	—	23	4	no	—
180	#124	—	23	2	no	—
181	#123	—	23	8	no	—
182	#123	—	23	4	no	—
183	C₂²⋊Q₈	—	23	4	no	—
184	C₂²⋊Q₈	—	23	4	no	—
185	C₂²⋊Q₈	C₂×D₂₃	1	4	yes	—
186	#121	—	23	4	no	—
187	#121	—	23	4	no	—
188	#121	—	23	4	no	—
189	#120	—	23	4	no	—
190	C₄×Q₈	—	23	4	no	—
191	C₄×Q₈	—	23	4	no	—
192	C₄×Q₈	C₂×D₂₃	1	4	yes	complemented
193	C₄×D₄	—	23	8	no	—
194	C₄×D₄	—	23	4	no	—
195	C₄×D₄	—	23	4	no	—
196	C₄×D₄	—	23	8	no	—
197	C₄×D₄	—	23	8	no	—
198	C₄×D₄	—	23	4	no	—
199	#117	—	23	4	no	—
200	#117	—	23	8	no	—
201	#117	—	23	4	no	—
202	#117	C₂×D₂₃	1	2	yes	—
203	C₂×C₄²	—	23	1	no	—
204	C₂×C₄²	—	23	2	no	—
205	C₂×C₄²	—	23	4	no	—
206	C₂×C₄²	—	23	2	no	—
207	C₂×C₄²	—	23	2	no	—
208	C₂×C₄²	—	23	2	no	—
209	C₂×C₄²	—	23	2	no	—
210	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
211	C₂₃	#3444	1	1	yes	characteristic, complemented, Sylow
212	C₂⁴	—	23	1	no	—
213	#41	—	46	16	no	—
214	#41	—	46	8	no	—
215	C₂×Q₈	—	23	2	no	—
216	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
217	C₂×D₄	—	23	4	no	—
218	C₂×D₄	—	23	8	no	—
219	C₂×D₄	—	23	2	no	—
220	C₂×D₄	—	23	4	no	—
221	C₂²×C₄	—	23	2	no	—
222	C₂²×C₄	—	23	2	no	—
223	C₂²×C₄	—	23	1	no	—
224	C₂²×C₄	—	23	2	no	—
225	C₂²×C₄	—	23	2	no	—
226	C₂²×C₄	—	23	2	no	—
227	C₂²×C₄	—	23	2	no	—
228	C₂²×C₄	—	23	8	no	—
229	C₂²×C₄	—	23	2	no	—
230	C₂²×C₄	—	23	4	no	—
231	C₂²×C₄	—	46	8	no	—
232	C₂²×C₄	—	46	4	no	—
233	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
234	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
235	C₄⋊C₄	—	23	4	no	—
236	C₄⋊C₄	—	23	8	no	—
237	C₄⋊C₄	—	23	8	no	—
238	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
239	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
240	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
241	C₂²⋊C₄	—	23	8	no	—
242	C₂²⋊C₄	—	23	8	no	—
243	C₂²⋊C₄	—	23	8	no	—
244	C₂²⋊C₄	—	23	8	no	—
245	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
246	C₄²	—	23	2	no	—
247	C₄²	—	23	2	no	—
248	C₄²	—	23	2	no	—
249	C₄²	—	23	2	no	—
250	C₄²	—	23	2	no	—
251	C₄²	—	23	8	no	—
252	C₄²	—	23	1	no	—
253	C₄²	—	23	1	no	—
254	C₄²	C₂²×D₂₃	1	1	yes	characteristic
255	C₄²	C₂²×D₂₃	1	1	yes	characteristic
256	C₄²	C₂²×D₂₃	1	2	yes	—
257	C₄²	C₂²×D₂₃	1	4	yes	—
258	C₂³	—	23	2	no	—
259	C₂³	—	23	4	no	—
260	C₂³	—	46	4	no	—
261	C₂³	—	46	2	no	—
262	C₂³	C₂³×D₂₃	1	1	yes	characteristic
263	Q₈	—	2	8	no	—
264	Q₈	—	46	4	no	—
265	D₄	—	46	16	no	—
266	D₄	—	46	8	no	—
267	D₄	—	46	4	no	—
268	D₄	—	46	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	23	2	no	—
271	C₂×C₄	—	23	8	no	—
272	C₂×C₄	—	23	2	no	—
273	C₂×C₄	—	23	2	no	—
274	C₂×C₄	—	23	4	no	—
275	C₂×C₄	—	46	4	no	—
276	C₂×C₄	—	46	16	no	—
277	C₂×C₄	—	46	4	no	—
278	C₂×C₄	—	46	4	no	—
279	C₂×C₄	—	46	2	no	—
280	C₂×C₄	—	46	4	no	—
281	C₂×C₄	—	46	4	no	—
282	C₂×C₄	—	46	8	no	—
283	C₂×C₄	—	46	4	no	—
284	C₂×C₄	#67292	1	4	yes	complemented
285	C₂×C₄	C₂³×D₂₃	1	2	yes	—
286	C₂×C₄	C₂³×D₂₃	1	8	yes	—
287	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₂₃	1	2	yes	—
289	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	23	4	no	—
292	C₂²	—	46	8	no	—
293	C₂²	—	46	2	no	—
294	C₂²	—	46	2	no	—
295	C₂²	—	46	8	no	—
296	C₂²	—	46	4	no	—
297	C₂²	C₂×#67292	1	2	yes	—
298	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	8	no	—
301	C₄	—	23	4	no	—
302	C₄	—	46	2	no	—
303	C₄	—	46	2	no	—
304	C₄	—	46	4	no	—
305	C₄	C₂×#67292	1	2	yes	—
306	C₄	C₂×#67292	1	2	yes	—
307	C₄	#41×D₂₃	1	4	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	46	4	no	—
310	C₂	—	92	4	no	—
311	C₂	C₂×#41×D₂₃	1	2	yes	—
312	C₂	C₂²×#67292	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂₃×#533	C₂
hypercenter	#533	D₂₃
hypocenter	C₂₃	#3444
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #13092972	C₂	semidirect
2. #13092968	C₂	semidirect
3. #13092967	C₂	semidirect
4. #13092954	C₂	semidirect
5. #13092953	C₂	semidirect
6. #13092952	C₂	semidirect
7. #13092930	C₂	semidirect
8. #13092929	C₂	semidirect
9. C₄×#67292	C₂	semidirect
10. #13092885	C₂	semidirect
11. C₂×C₄²×D₂₃	C₂	semidirect
12. C₂₃×#533	C₂	semidirect
17. C₂₃×C₄⋊Q₈	C₂²	semidirect
18. C₂₃×#125	C₂²	semidirect
20. C₄×Q₈×C₂₃	C₂²	semidirect
23. #10625118	C₂²	semidirect
25. Q₈×Dic₂₃	C₂²	semidirect
29. #10625069	C₂²	semidirect
35. #10625064	C₂²	semidirect
36. #10625064	C₂²	semidirect
38. #10625063	C₂²	semidirect
39. #10625061	C₂²	semidirect
40. #10625058	C₂²	semidirect
44. #10625046	C₂²	semidirect
45. #10625045	C₂²	semidirect
47. C₄×D₉₂	C₂²	semidirect
48. #10625043	C₂²	semidirect
53. C₄²×D₂₃	C₂²	semidirect
54. #10625040	C₂²	semidirect
108. C₂×C₄×C₂₃	#41	semidirect
131. C₄×D₂₃	#41	semidirect
153. #533	D₂₃	semidirect
174. C₄⋊Q₈	C₂×D₂₃	semidirect
178. #125	C₂×D₂₃	semidirect
192. C₄×Q₈	C₂×D₂₃	semidirect
211. C₂₃	#3444	semidirect
284. C₂×C₄	#67292	semidirect

Maximal subgroups

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

subgroup	index
#13092972	2
#13092968	2
#13092967	2
#13092954	2
#13092953	2
#13092952	2
#13092930	2
#13092929	2
C₄×#67292	2
#13092885	2
C₂×C₄²×D₂₃	2
C₂₃×#533	2
#3444	23