Subgroups — #15686887

Order 2944.

1648 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₂₃	C₂	1	2	yes	maximal, complemented
2	Q₈×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#13093074	C₂	1	2	yes	maximal, complemented
4	#13093072	C₂	1	3	yes	maximal, complemented
5	#13093064	C₂	1	6	yes	maximal, complemented
6	#13093052	C₂	1	3	yes	maximal, complemented
7	#13092965	C₂	1	3	yes	maximal, complemented
8	#13092927	C₂	1	6	yes	maximal, complemented
9	C₄×#67292	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₂₃	C₂	1	3	yes	maximal, complemented
11	C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	complemented
13	#41×D₂₃	—	2	16	no	—
14	C₂×#67292	C₂²	1	1	yes	characteristic
15	C₂×Q₈×D₂₃	C₂²	1	2	yes	complemented
16	C₂×#67290	C₂²	1	6	yes	complemented
17	C₂×D₄×D₂₃	C₂²	1	3	yes	—
18	C₂×D₄×D₂₃	C₂²	1	6	yes	complemented
19	C₂×#67288	C₂²	1	6	yes	complemented
20	C₂²×C₄×D₂₃	C₂²	1	6	yes	complemented
21	C₂₃×#127	C₂²	1	3	yes	complemented
22	C₂₃×#121	C₂²	1	6	yes	complemented
23	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₄×D₄×C₂₃	C₂²	1	3	yes	complemented
25	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
26	#10625109	C₂²	1	6	yes	complemented
27	#10625109	C₂²	1	6	yes	complemented
28	#10625109	C₂²	1	6	yes	complemented
29	#10625108	C₂²	1	12	yes	complemented
30	#10625107	C₂²	1	6	yes	complemented
31	D₄×Dic₂₃	C₂²	1	3	yes	complemented
32	D₄×Dic₂₃	C₂²	1	6	yes	complemented
33	C₄×C₂²⋊D₂₃	C₂²	1	6	yes	complemented
34	#10625095	C₂²	1	6	yes	complemented
35	#10625068	C₂²	1	6	yes	complemented
36	#10625064	C₂²	1	1	yes	characteristic
37	#10625064	C₂²	1	3	yes	complemented
38	#10625063	C₂²	1	3	yes	complemented
39	C₄⋊C₄×D₂₃	C₂²	1	3	yes	complemented
40	#10625054	C₂²	1	12	yes	complemented
41	(C₂²⋊D₂₃)⋊C₄	C₂²	1	6	yes	complemented
42	C₂²⋊C₄×D₂₃	C₂²	1	6	yes	complemented
43	C₄×D₉₂	C₂²	1	3	yes	—
44	C₄×D₉₂	C₂²	1	3	yes	complemented
45	C₄²×D₂₃	C₂²	1	3	yes	complemented
46	#10625040	C₂²	1	3	yes	complemented
47	C₂³×D₂₃	C₂³	1	6	yes	—
48	#41×C₂₃	—	2	8	no	—
49	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₃	C₂³	1	3	yes	—
51	C₂×D₄×C₂₃	C₂³	1	6	yes	—
52	C₂×D₄×C₂₃	C₂³	1	6	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₂₃	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₂₃	C₂³	1	12	yes	—
56	C₂²×Dic₂₃	C₂³	1	6	yes	—
57	#67292	D₄	1	8	yes	complemented
58	Q₈×D₂₃	—	2	8	no	—
59	#67290	—	2	24	no	—
60	D₄×D₂₃	—	2	24	no	—
61	D₄×D₂₃	—	2	12	no	—
62	#67288	—	2	24	no	—
63	C₂×D₉₂	C₂³	1	3	yes	—
64	C₂×D₉₂	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₂₃	—	2	24	no	—
66	C₂×C₄×D₂₃	C₂³	1	3	yes	—
67	C₂×C₄×D₂₃	C₂³	1	6	yes	—
68	C₂×C₄×D₂₃	C₂³	1	6	yes	—
69	C₂×C₄×D₂₃	C₂³	1	2	yes	—
70	C₂×C₄×D₂₃	C₂³	1	3	yes	—
71	C₂×C₄×D₂₃	C₂³	1	3	yes	—
72	C₂×Dic₄₆	C₂³	1	6	yes	—
73	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
75	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
76	C₄²×C₂₃	C₂³	1	3	yes	—
77	C₂²⋊Dic₂₃	C₂³	1	6	yes	—
78	C₂²⋊Dic₂₃	C₂³	1	12	yes	—
79	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
80	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
81	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
82	C₄⋊Dic₂₃	C₂³	1	3	yes	—
83	C₄⋊Dic₂₃	C₂³	1	6	yes	—
84	C₄⋊Dic₂₃	C₂³	1	3	yes	—
85	Dic₂₃⋊C₄	C₂³	1	6	yes	—
86	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
87	C₄×Dic₂₃	C₂³	1	3	yes	—
88	C₄×Dic₂₃	C₂³	1	3	yes	—
89	C₂³×C₂₃	C₂⁴	1	6	yes	—
90	C₂²×D₂₃	—	2	12	no	—
91	C₂²×D₂₃	—	2	12	no	—
92	C₂²×D₂₃	—	4	6	no	—
93	C₂²×D₂₃	C₂⁴	1	3	yes	—
94	C₂²×D₂₃	C₂⁴	1	2	yes	—
95	Q₈×C₂₃	C₂×D₄	1	4	yes	complemented
96	D₄×C₂₃	—	2	12	no	—
97	D₄×C₂₃	—	2	6	no	—
98	D₄×C₂₃	—	4	6	no	—
99	C₂×C₄×C₂₃	—	2	12	no	—
100	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
101	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
102	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₂₃	—	2	24	no	—
105	C₂²⋊D₂₃	—	2	12	no	—
106	C₂×Dic₂₃	—	2	12	no	—
107	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
108	C₂×Dic₂₃	C₂⁴	1	3	yes	—
109	C₂×Dic₂₃	C₂⁴	1	6	yes	—
110	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
111	D₉₂	C₂×D₄	1	12	yes	complemented
112	D₉₂	#41	1	4	yes	complemented
113	C₄×D₂₃	—	2	12	no	—
114	C₄×D₂₃	—	2	12	no	—
115	C₄×D₂₃	—	2	6	no	—
116	C₄×D₂₃	—	2	4	no	—
117	C₄×D₂₃	—	4	3	no	—
118	C₄×D₂₃	C₂×D₄	1	12	yes	complemented
119	Dic₄₆	—	2	12	no	—
120	D₄×#41	—	23	1	no	maximal, Sylow
121	C₂²×C₂₃	—	2	6	no	—
122	C₂²×C₂₃	—	2	6	no	—
123	C₂²×C₂₃	—	2	6	no	—
124	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
125	C₂×D₂₃	—	2	2	no	—
126	C₂×D₂₃	—	2	3	no	—
127	C₂×D₂₃	—	2	3	no	—
128	C₂×D₂₃	—	2	2	no	—
129	C₂×D₂₃	—	4	24	no	—
130	C₂×D₂₃	C₂²×D₄	1	6	yes	—
131	C₂×D₂₃	C₂×#41	1	4	yes	complemented
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	3	no	—
134	C₄×C₂₃	C₂²×D₄	1	6	yes	—
135	C₄×C₂₃	C₂×#41	1	2	yes	complemented
136	Dic₂₃	—	2	6	no	—
137	Dic₂₃	—	2	3	no	—
138	Dic₂₃	—	2	1	no	—
139	Dic₂₃	C₂²×D₄	1	2	yes	—
140	C₂²×#41	—	23	2	no	—
141	#550	D₂₃	1	1	yes	characteristic, complemented
142	D₄×Q₈	—	23	1	no	—
143	#548	—	23	2	no	—
144	#547	—	23	3	no	—
145	#546	—	23	6	no	—
146	D₄²	—	23	3	no	—
147	#532	—	23	3	no	—
148	#525	—	23	6	no	—
149	C₄×#41	—	23	1	no	—
150	C₂×C₄×D₄	—	23	3	no	—
151	C₂×C₂₃	—	4	6	no	—
152	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
153	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
154	C₂×C₂₃	#585	1	1	yes	characteristic
155	D₂₃	—	2	4	no	—
156	D₂₃	—	2	6	no	—
157	C₂×#41	—	23	1	no	—
158	C₂×#41	—	23	6	no	—
159	C₂×#41	—	23	6	no	—
160	C₂×#41	—	46	16	no	—
161	C₂×#41	C₂×D₂₃	1	2	yes	complemented
162	C₂²×Q₈	—	23	2	no	—
163	C₂²×D₄	—	23	3	no	—
164	C₂²×D₄	—	23	6	no	—
165	C₂³×C₄	—	23	6	no	—
166	C₄⋊Q₈	—	23	3	no	—
167	#127	C₂×D₂₃	1	3	yes	complemented
168	#124	—	23	6	no	—
169	#123	—	23	12	no	—
170	C₂²⋊Q₈	—	23	6	no	—
171	C₂²⋊Q₈	—	23	6	no	—
172	#121	—	23	6	no	—
173	#121	—	23	6	no	—
174	#121	—	23	6	no	—
175	#121	C₂×D₂₃	1	6	yes	complemented
176	#120	—	23	12	no	—
177	C₄×Q₈	—	23	1	no	—
178	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
179	C₄×D₄	—	23	3	no	—
180	C₄×D₄	—	23	6	no	—
181	C₄×D₄	—	23	6	no	—
182	C₄×D₄	—	23	1	no	—
183	C₄×D₄	—	23	3	no	—
184	C₄×D₄	—	23	6	no	—
185	C₄×D₄	—	23	3	no	—
186	C₄×D₄	—	23	3	no	—
187	C₄×D₄	C₂×D₂₃	1	3	yes	complemented
188	#117	—	23	3	no	—
189	C₂×C₄⋊C₄	—	23	3	no	—
190	C₂×C₂²⋊C₄	—	23	6	no	—
191	C₂×C₄²	—	23	3	no	—
192	C₂₃	D₄×#41	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	23	6	no	—
194	#41	—	2	8	no	—
195	#41	—	23	8	no	—
196	#41	—	46	24	no	—
197	#41	—	46	24	no	—
198	C₂×Q₈	—	23	6	no	—
199	C₂×Q₈	—	46	8	no	—
200	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
201	C₂×D₄	—	23	6	no	—
202	C₂×D₄	—	23	12	no	—
203	C₂×D₄	—	23	1	no	—
204	C₂×D₄	—	23	3	no	—
205	C₂×D₄	—	46	24	no	—
206	C₂×D₄	—	46	12	no	—
207	C₂×D₄	C₂²×D₂₃	1	3	yes	—
208	C₂×D₄	C₂²×D₂₃	1	6	yes	—
209	C₂×D₄	C₂²×D₂₃	1	6	yes	—
210	C₂²×C₄	—	23	3	no	—
211	C₂²×C₄	—	23	6	no	—
212	C₂²×C₄	—	23	3	no	—
213	C₂²×C₄	—	23	6	no	—
214	C₂²×C₄	—	23	2	no	—
215	C₂²×C₄	—	23	6	no	—
216	C₂²×C₄	—	23	3	no	—
217	C₂²×C₄	—	46	24	no	—
218	C₂²×C₄	C₂²×D₂₃	1	6	yes	—
219	C₄⋊C₄	—	23	3	no	—
220	C₄⋊C₄	—	23	6	no	—
221	C₄⋊C₄	—	23	3	no	—
222	C₄⋊C₄	—	23	6	no	—
223	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
224	C₄⋊C₄	C₂²×D₂₃	1	3	yes	—
225	C₂²⋊C₄	—	23	6	no	—
226	C₂²⋊C₄	—	23	12	no	—
227	C₂²⋊C₄	—	23	2	no	—
228	C₂²⋊C₄	—	23	6	no	—
229	C₂²⋊C₄	—	23	6	no	—
230	C₂²⋊C₄	C₂²×D₂₃	1	6	yes	—
231	C₄²	—	23	1	no	—
232	C₄²	—	23	3	no	—
233	C₄²	—	23	3	no	—
234	C₄²	C₂²×D₂₃	1	3	yes	—
235	C₂³	—	23	2	no	—
236	C₂³	—	23	3	no	—
237	C₂³	—	46	12	no	—
238	C₂³	—	46	12	no	—
239	C₂³	—	92	6	no	—
240	C₂³	C₂³×D₂₃	1	6	yes	—
241	Q₈	—	46	12	no	—
242	Q₈	D₄×D₂₃	1	4	yes	complemented
243	D₄	—	2	12	no	—
244	D₄	—	2	6	no	—
245	D₄	—	4	6	no	—
246	D₄	—	23	12	no	—
247	D₄	—	23	4	no	—
248	D₄	—	46	24	no	—
249	D₄	—	46	12	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	23	1	no	—
252	C₂×C₄	—	23	12	no	—
253	C₂×C₄	—	23	1	no	—
254	C₂×C₄	—	23	3	no	—
255	C₂×C₄	—	23	6	no	—
256	C₂×C₄	—	46	4	no	—
257	C₂×C₄	—	46	12	no	—
258	C₂×C₄	—	46	12	no	—
259	C₂×C₄	—	46	12	no	—
260	C₂×C₄	—	46	6	no	—
261	C₂×C₄	—	92	3	no	—
262	C₂×C₄	C₂³×D₂₃	1	3	yes	—
263	C₂×C₄	C₂³×D₂₃	1	2	yes	—
264	C₂×C₄	C₂³×D₂₃	1	3	yes	—
265	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
266	C₂²	—	2	6	no	—
267	C₂²	—	2	6	no	—
268	C₂²	—	2	6	no	—
269	C₂²	—	23	6	no	—
270	C₂²	—	23	4	no	—
271	C₂²	—	46	2	no	—
272	C₂²	—	46	3	no	—
273	C₂²	—	46	3	no	—
274	C₂²	—	46	2	no	—
275	C₂²	—	92	24	no	—
276	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	3	no	—
279	C₄	—	23	2	no	—
280	C₄	—	46	6	no	—
281	C₄	—	46	3	no	—
282	C₄	—	46	1	no	—
283	C₄	C₂×D₄×D₂₃	1	6	yes	—
284	C₄	#41×D₂₃	1	2	yes	complemented
285	C₂	—	4	6	no	—
286	C₂	—	46	4	no	—
287	C₂	—	46	6	no	—
288	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
289	C₂	C₂×#41×D₂₃	1	1	yes	characteristic
290	C₂	#13093252	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×#550	C₂
hypercenter	#550	D₂₃
hypocenter	C₂₃	D₄×#41
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₂₃	C₂	semidirect
2. Q₈×D₉₂	C₂	semidirect
3. #13093074	C₂	semidirect
4. #13093072	C₂	semidirect
5. #13093064	C₂	semidirect
6. #13093052	C₂	semidirect
7. #13092965	C₂	semidirect
8. #13092927	C₂	semidirect
9. C₄×#67292	C₂	semidirect
10. C₄×D₄×D₂₃	C₂	semidirect
11. C₂₃×#550	C₂	semidirect
12. C₂×#41×C₂₃	C₂²	semidirect
15. C₂×Q₈×D₂₃	C₂²	semidirect
16. C₂×#67290	C₂²	semidirect
18. C₂×D₄×D₂₃	C₂²	semidirect
19. C₂×#67288	C₂²	semidirect
20. C₂²×C₄×D₂₃	C₂²	semidirect
21. C₂₃×#127	C₂²	semidirect
22. C₂₃×#121	C₂²	semidirect
23. C₄×Q₈×C₂₃	C₂²	semidirect
24. C₄×D₄×C₂₃	C₂²	semidirect
25. Q₈×Dic₂₃	C₂²	semidirect
26. #10625109	C₂²	semidirect
27. #10625109	C₂²	semidirect
28. #10625109	C₂²	semidirect
29. #10625108	C₂²	semidirect
30. #10625107	C₂²	semidirect
31. D₄×Dic₂₃	C₂²	semidirect
32. D₄×Dic₂₃	C₂²	semidirect
33. C₄×C₂²⋊D₂₃	C₂²	semidirect
34. #10625095	C₂²	semidirect
35. #10625068	C₂²	semidirect
37. #10625064	C₂²	semidirect
38. #10625063	C₂²	semidirect
39. C₄⋊C₄×D₂₃	C₂²	semidirect
40. #10625054	C₂²	semidirect
41. (C₂²⋊D₂₃)⋊C₄	C₂²	semidirect
42. C₂²⋊C₄×D₂₃	C₂²	semidirect
44. C₄×D₉₂	C₂²	semidirect
45. C₄²×D₂₃	C₂²	semidirect
46. #10625040	C₂²	semidirect
57. #67292	D₄	semidirect
95. Q₈×C₂₃	C₂×D₄	semidirect
111. D₉₂	C₂×D₄	semidirect
112. D₉₂	#41	semidirect
118. C₄×D₂₃	C₂×D₄	semidirect
131. C₂×D₂₃	C₂×#41	semidirect
135. C₄×C₂₃	C₂×#41	semidirect
141. #550	D₂₃	semidirect
161. C₂×#41	C₂×D₂₃	semidirect
167. #127	C₂×D₂₃	semidirect
175. #121	C₂×D₂₃	semidirect
178. C₄×Q₈	C₂×D₂₃	semidirect
187. C₄×D₄	C₂×D₂₃	semidirect
192. C₂₃	D₄×#41	semidirect
242. Q₈	D₄×D₂₃	semidirect
284. C₄	#41×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₂₃	2
Q₈×D₉₂	2
#13093074	2
#13093072	2
#13093064	2
#13093052	2
#13092965	2
#13092927	2
C₄×#67292	2
C₄×D₄×D₂₃	2
C₂₃×#550	2
D₄×#41	23