Subgroups — #15687006

Order 2944.

1242 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093127	C₂	1	2	yes	maximal, complemented
2	#13093117	C₂	1	4	yes	maximal, complemented
3	#13093095	C₂	1	4	yes	maximal, complemented
4	#13093060	C₂	1	4	yes	maximal, complemented
5	#13093052	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092965	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092946	C₂	1	4	yes	maximal, complemented
8	#13092925	C₂	1	4	yes	maximal, complemented
9	C₄×#67292	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₂₃	C₂	1	4	yes	maximal, complemented
11	C₂₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67292	C₂²	1	2	yes	—
13	C₂×#67290	C₂²	1	2	yes	complemented
14	C₂×D₄×D₂₃	C₂²	1	4	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	4	yes	complemented
16	C₂₃×#127	C₂²	1	1	yes	characteristic
17	C₂₃×#124	C₂²	1	4	yes	complemented
18	C₂₃×#123	C₂²	1	4	yes	complemented
19	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
20	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
21	#10625117	C₂²	1	4	yes	complemented
22	Q₈×Dic₂₃	C₂²	1	2	yes	complemented
23	#10625109	C₂²	1	4	yes	—
24	#10625109	C₂²	1	4	yes	complemented
25	#10625108	C₂²	1	4	yes	—
26	#10625107	C₂²	1	2	yes	complemented
27	D₄×Dic₂₃	C₂²	1	4	yes	complemented
28	#10625100	C₂²	1	4	yes	complemented
29	C₄×C₂²⋊D₂₃	C₂²	1	8	yes	complemented
30	#10625067	C₂²	1	8	yes	complemented
31	#10625064	C₂²	1	2	yes	—
32	#10625064	C₂²	1	4	yes	complemented
33	#10625063	C₂²	1	4	yes	complemented
34	#10625063	C₂²	1	2	yes	complemented
35	C₄⋊C₄×D₂₃	C₂²	1	4	yes	complemented
36	#10625055	C₂²	1	8	yes	complemented
37	#10625054	C₂²	1	8	yes	complemented
38	#10625053	C₂²	1	4	yes	complemented
39	(C₂²⋊D₂₃)⋊C₄	C₂²	1	8	yes	complemented
40	C₂²⋊C₄×D₂₃	C₂²	1	8	yes	complemented
41	#10625050	C₂²	1	8	yes	complemented
42	#10625048	C₂²	1	4	yes	complemented
43	C₄×D₉₂	C₂²	1	4	yes	—
44	C₄×D₉₂	C₂²	1	2	yes	—
45	C₄×D₉₂	C₂²	1	4	yes	complemented
46	C₄²×D₂₃	C₂²	1	4	yes	complemented
47	C₄²×D₂₃	C₂²	1	1	yes	characteristic, complemented
48	#10625041	C₂²	1	4	yes	complemented
49	#10625040	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₂₃	C₂³	1	4	yes	—
51	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂×D₄×C₂₃	C₂³	1	4	yes	—
53	C₂×D₄×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
56	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
57	C₂²×Dic₂₃	C₂³	1	4	yes	—
58	#67292	—	2	8	no	—
59	#67290	—	4	4	no	—
60	D₄×D₂₃	—	2	16	no	—
61	C₂×D₉₂	C₂³	1	4	yes	—
62	C₂×D₉₂	C₂³	1	2	yes	—
63	C₂×C₄×D₂₃	—	4	8	no	—
64	C₂×C₄×D₂₃	C₂³	1	4	yes	—
65	C₂×C₄×D₂₃	C₂³	1	2	yes	—
66	C₂×C₄×D₂₃	C₂³	1	4	yes	—
67	C₂×C₄×D₂₃	C₂³	1	4	yes	—
68	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₂₃	C₂³	1	8	yes	—
70	C₂×Dic₄₆	C₂³	1	2	yes	—
71	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
72	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
74	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
75	C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₄²×C₂₃	C₂³	1	4	yes	—
77	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
78	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
79	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
81	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
82	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
83	C₄⋊Dic₂₃	C₂³	1	4	yes	—
84	C₄⋊Dic₂₃	C₂³	1	2	yes	—
85	C₄⋊Dic₂₃	C₂³	1	4	yes	—
86	Dic₂₃⋊C₄	C₂³	1	8	yes	—
87	Dic₂₃⋊C₄	C₂³	1	8	yes	—
88	C₄×Dic₂₃	C₂³	1	2	yes	—
89	C₄×Dic₂₃	C₂³	1	4	yes	—
90	C₄×Dic₂₃	C₂³	1	4	yes	—
91	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
92	C₂³×C₂₃	C₂⁴	1	4	yes	—
93	C₂²×D₂₃	—	2	16	no	—
94	C₂²×D₂₃	—	4	4	no	—
95	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
96	C₂²×D₂₃	C₂⁴	1	4	yes	—
97	Q₈×C₂₃	—	2	4	no	—
98	D₄×C₂₃	—	2	8	no	—
99	D₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
102	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
105	C₂²⋊D₂₃	—	2	16	no	—
106	C₂²⋊D₂₃	—	4	4	no	—
107	C₂×Dic₂₃	—	2	8	no	—
108	C₂×Dic₂₃	C₂⁴	1	2	yes	—
109	C₂×Dic₂₃	C₂⁴	1	4	yes	—
110	C₂×Dic₂₃	C₂⁴	1	4	yes	—
111	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
112	D₉₂	—	2	8	no	—
113	D₉₂	#41	1	8	yes	complemented
114	C₄×D₂₃	—	2	8	no	—
115	C₄×D₂₃	—	2	8	no	—
116	C₄×D₂₃	—	2	4	no	—
117	C₄×D₂₃	—	2	8	no	—
118	C₄×D₂₃	—	2	2	no	—
119	C₄×D₂₃	—	4	8	no	—
120	Dic₄₆	—	4	2	no	—
121	#3481	—	23	1	no	maximal, Sylow
122	C₂²×C₂₃	—	2	8	no	—
123	C₂²×C₂₃	—	2	4	no	—
124	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
125	C₂×D₂₃	—	2	4	no	—
126	C₂×D₂₃	—	2	4	no	—
127	C₂×D₂₃	—	2	2	no	—
128	C₂×D₂₃	—	2	1	no	—
129	C₂×D₂₃	—	4	16	no	—
130	C₂×D₂₃	C₂×#41	1	8	yes	—
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	4	no	—
134	C₄×C₂₃	C₂×#41	1	4	yes	—
135	Dic₂₃	—	2	2	no	—
136	Dic₂₃	—	2	4	no	—
137	Dic₂₃	—	2	4	no	—
138	Dic₂₃	H(5,𝔽₂)	1	2	yes	—
139	#559	D₂₃	1	1	yes	characteristic, complemented
140	#554	—	23	2	no	—
141	#553	—	23	4	no	—
142	#551	—	23	4	no	—
143	#546	—	23	4	no	—
144	D₄²	—	23	1	no	—
145	#532	—	23	1	no	—
146	#529	—	23	4	no	—
147	#525	—	23	4	no	—
148	C₄×#41	—	23	2	no	—
149	C₂×C₄×D₄	—	23	4	no	—
150	C₂×C₂₃	—	4	4	no	—
151	C₂×C₂₃	C₂²×#41	1	2	yes	—
152	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
153	D₂₃	—	2	8	no	—
154	D₂₃	—	4	1	no	—
155	C₂×#41	—	23	2	no	—
156	C₂×#41	—	23	2	no	—
157	C₂²×D₄	—	23	4	no	—
158	C₂³×C₄	—	23	4	no	—
159	C₄⋊Q₈	—	23	1	no	—
160	#127	C₂×D₂₃	1	1	yes	characteristic
161	#126	—	23	8	no	—
162	#125	—	23	4	no	—
163	#124	—	23	2	no	—
164	#124	C₂×D₂₃	1	4	yes	complemented
165	#123	—	23	4	no	—
166	#123	—	23	8	no	—
167	#123	C₂×D₂₃	1	4	yes	complemented
168	C₂²⋊Q₈	—	23	4	no	—
169	C₂²⋊Q₈	—	23	8	no	—
170	#121	—	23	4	no	—
171	#121	—	23	4	no	—
172	#121	—	23	8	no	—
173	#120	—	23	4	no	—
174	#120	—	23	4	no	—
175	C₄×Q₈	—	23	2	no	—
176	C₄×Q₈	C₂×D₂₃	1	2	yes	complemented
177	C₄×D₄	—	23	4	no	—
178	C₄×D₄	—	23	8	no	—
179	C₄×D₄	—	23	2	no	—
180	C₄×D₄	—	23	4	no	—
181	C₄×D₄	—	23	8	no	—
182	C₄×D₄	—	23	4	no	—
183	C₄×D₄	—	23	2	no	—
184	C₄×D₄	—	23	4	no	—
185	C₄×D₄	C₂×D₂₃	1	4	yes	complemented
186	#117	—	23	4	no	—
187	#117	—	23	2	no	—
188	#117	—	23	4	no	—
189	C₂×C₄⋊C₄	—	23	4	no	—
190	C₂×C₂²⋊C₄	—	23	8	no	—
191	C₂×C₄²	—	23	4	no	—
192	C₂×C₄²	—	23	1	no	—
193	C₂₃	#3481	1	1	yes	characteristic, complemented, Sylow
194	C₂⁴	—	23	4	no	—
195	#41	—	46	8	no	—
196	#41	—	92	4	no	—
197	C₂×Q₈	—	23	2	no	—
198	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
199	C₂×D₄	—	23	8	no	—
200	C₂×D₄	—	23	4	no	—
201	C₂×D₄	—	23	2	no	—
202	C₂×D₄	—	23	4	no	—
203	C₂×D₄	—	46	16	no	—
204	C₂×D₄	C₂²×D₂₃	1	4	yes	—
205	C₂×D₄	C₂²×D₂₃	1	2	yes	—
206	C₂²×C₄	—	23	4	no	—
207	C₂²×C₄	—	23	4	no	—
208	C₂²×C₄	—	23	4	no	—
209	C₂²×C₄	—	23	2	no	—
210	C₂²×C₄	—	23	4	no	—
211	C₂²×C₄	—	23	1	no	—
212	C₂²×C₄	—	23	8	no	—
213	C₂²×C₄	—	92	8	no	—
214	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
215	C₄⋊C₄	—	23	4	no	—
216	C₄⋊C₄	—	23	8	no	—
217	C₄⋊C₄	—	23	2	no	—
218	C₄⋊C₄	—	23	4	no	—
219	C₄⋊C₄	—	23	8	no	—
220	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
221	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
222	C₂²⋊C₄	—	23	8	no	—
223	C₂²⋊C₄	—	23	4	no	—
224	C₂²⋊C₄	—	23	4	no	—
225	C₂²⋊C₄	—	23	8	no	—
226	C₂²⋊C₄	—	23	4	no	—
227	C₂²⋊C₄	—	23	4	no	—
228	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
229	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
230	C₄²	—	23	2	no	—
231	C₄²	—	23	1	no	—
232	C₄²	—	23	4	no	—
233	C₄²	—	23	4	no	—
234	C₄²	C₂²×D₂₃	1	4	yes	—
235	C₄²	C₂²×D₂₃	1	1	yes	characteristic
236	C₂³	—	23	4	no	—
237	C₂³	—	23	1	no	—
238	C₂³	—	46	16	no	—
239	C₂³	—	92	4	no	—
240	C₂³	C₂³×D₂₃	1	4	yes	—
241	Q₈	—	2	4	no	—
242	Q₈	—	92	2	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	2	no	—
245	D₄	—	23	8	no	—
246	D₄	—	46	8	no	—
247	D₄	—	46	16	no	—
248	D₄	—	92	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	23	1	no	—
251	C₂×C₄	—	23	2	no	—
252	C₂×C₄	—	23	4	no	—
253	C₂×C₄	—	23	4	no	—
254	C₂×C₄	—	46	8	no	—
255	C₂×C₄	—	46	2	no	—
256	C₂×C₄	—	46	8	no	—
257	C₂×C₄	—	46	8	no	—
258	C₂×C₄	—	46	8	no	—
259	C₂×C₄	—	46	4	no	—
260	C₂×C₄	—	92	8	no	—
261	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂³×D₂₃	1	4	yes	—
263	C₂×C₄	C₂³×D₂₃	1	4	yes	—
264	C₂×C₄	C₂³×D₂₃	1	2	yes	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	23	8	no	—
268	C₂²	—	46	4	no	—
269	C₂²	—	46	4	no	—
270	C₂²	—	46	2	no	—
271	C₂²	—	46	1	no	—
272	C₂²	—	92	16	no	—
273	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	23	2	no	—
278	C₄	—	46	2	no	—
279	C₄	—	46	4	no	—
280	C₄	—	46	4	no	—
281	C₄	#41×D₂₃	1	4	yes	—
282	C₂	—	4	4	no	—
283	C₂	—	46	8	no	—
284	C₂	—	92	1	no	—
285	C₂	C₂×#41×D₂₃	1	2	yes	—
286	C₂	H(5,𝔽₂)×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×#559	C₂
hypercenter	#559	D₂₃
hypocenter	C₂₃	#3481
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #13093127	C₂	semidirect
2. #13093117	C₂	semidirect
3. #13093095	C₂	semidirect
4. #13093060	C₂	semidirect
5. #13093052	C₂	semidirect
6. #13092965	C₂	semidirect
7. #13092946	C₂	semidirect
8. #13092925	C₂	semidirect
9. C₄×#67292	C₂	semidirect
10. C₄×D₄×D₂₃	C₂	semidirect
11. C₂₃×#559	C₂	semidirect
13. C₂×#67290	C₂²	semidirect
15. C₂²×C₄×D₂₃	C₂²	semidirect
17. C₂₃×#124	C₂²	semidirect
18. C₂₃×#123	C₂²	semidirect
19. C₄×Q₈×C₂₃	C₂²	semidirect
20. C₄×D₄×C₂₃	C₂²	semidirect
21. #10625117	C₂²	semidirect
22. Q₈×Dic₂₃	C₂²	semidirect
24. #10625109	C₂²	semidirect
26. #10625107	C₂²	semidirect
27. D₄×Dic₂₃	C₂²	semidirect
28. #10625100	C₂²	semidirect
29. C₄×C₂²⋊D₂₃	C₂²	semidirect
30. #10625067	C₂²	semidirect
32. #10625064	C₂²	semidirect
33. #10625063	C₂²	semidirect
34. #10625063	C₂²	semidirect
35. C₄⋊C₄×D₂₃	C₂²	semidirect
36. #10625055	C₂²	semidirect
37. #10625054	C₂²	semidirect
38. #10625053	C₂²	semidirect
39. (C₂²⋊D₂₃)⋊C₄	C₂²	semidirect
40. C₂²⋊C₄×D₂₃	C₂²	semidirect
41. #10625050	C₂²	semidirect
42. #10625048	C₂²	semidirect
45. C₄×D₉₂	C₂²	semidirect
46. C₄²×D₂₃	C₂²	semidirect
47. C₄²×D₂₃	C₂²	semidirect
48. #10625041	C₂²	semidirect
49. #10625040	C₂²	semidirect
113. D₉₂	#41	semidirect
139. #559	D₂₃	semidirect
164. #124	C₂×D₂₃	semidirect
167. #123	C₂×D₂₃	semidirect
176. C₄×Q₈	C₂×D₂₃	semidirect
185. C₄×D₄	C₂×D₂₃	semidirect
193. C₂₃	#3481	semidirect

Maximal subgroups

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

subgroup	index
#13093127	2
#13093117	2
#13093095	2
#13093060	2
#13093052	2
#13092965	2
#13092946	2
#13092925	2
C₄×#67292	2
C₄×D₄×D₂₃	2
C₂₃×#559	2
#3481	23