Subgroups — #15687005

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	#13093153	C₂	1	1	yes	maximal, characteristic, complemented
2	#13093128	C₂	1	2	yes	maximal, complemented
3	#13093114	C₂	1	4	yes	maximal, complemented
4	#13093097	C₂	1	4	yes	maximal, complemented
5	#13093064	C₂	1	4	yes	maximal, complemented
6	#13093052	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092949	C₂	1	4	yes	maximal, complemented
8	#13092924	C₂	1	4	yes	maximal, complemented
9	C₄×#67288	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₂₃	C₂	1	4	yes	maximal, complemented
11	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	complemented
13	C₂×D₄×D₂₃	C₂²	1	4	yes	—
14	C₂×#67288	C₂²	1	2	yes	—
15	C₂²×C₄×D₂₃	C₂²	1	4	yes	complemented
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₂₃×#124	C₂²	1	2	yes	complemented
19	C₂₃×#121	C₂²	1	4	yes	complemented
20	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
21	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
22	#10625117	C₂²	1	8	yes	complemented
23	#10625109	C₂²	1	4	yes	—
24	#10625109	C₂²	1	8	yes	complemented
25	#10625108	C₂²	1	4	yes	—
26	#10625108	C₂²	1	4	yes	complemented
27	#10625107	C₂²	1	4	yes	complemented
28	#10625106	C₂²	1	4	yes	complemented
29	#10625106	C₂²	1	4	yes	complemented
30	D₄×Dic₂₃	C₂²	1	4	yes	complemented
31	C₄×C₂²⋊D₂₃	C₂²	1	8	yes	complemented
32	C₄×C₂²⋊D₂₃	C₂²	1	4	yes	complemented
33	C₄×C₂²⋊D₂₃	C₂²	1	4	yes	complemented
34	#10625097	C₂²	1	2	yes	complemented
35	#10625097	C₂²	1	4	yes	complemented
36	#10625068	C₂²	1	4	yes	complemented
37	#10625064	C₂²	1	4	yes	—
38	C₄⋊C₄×D₂₃	C₂²	1	4	yes	complemented
39	#10625061	C₂²	1	4	yes	complemented
40	#10625054	C₂²	1	8	yes	complemented
41	(C₂²⋊D₂₃)⋊C₄	C₂²	1	8	yes	complemented
42	C₂²⋊C₄×D₂₃	C₂²	1	8	yes	complemented
43	#10625050	C₂²	1	8	yes	complemented
44	C₄×D₉₂	C₂²	1	2	yes	—
45	C₄×D₉₂	C₂²	1	2	yes	—
46	#10625043	C₂²	1	4	yes	complemented
47	C₄²×D₂₃	C₂²	1	4	yes	complemented
48	C₄×Dic₄₆	C₂²	1	2	yes	complemented
49	C₄×Dic₄₆	C₂²	1	2	yes	complemented
50	C₂³×D₂₃	C₂³	1	4	yes	—
51	#41×C₂₃	—	4	4	no	—
52	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
53	C₂×D₄×C₂₃	C₂³	1	4	yes	—
54	C₂×D₄×C₂₃	C₂³	1	2	yes	—
55	C₂×D₄×C₂₃	C₂³	1	4	yes	—
56	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
58	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
60	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
61	C₂²×Dic₂₃	C₂³	1	4	yes	—
62	D₄×D₂₃	—	2	16	no	—
63	#67288	—	2	8	no	—
64	C₂×D₉₂	C₂³	1	2	yes	—
65	C₂×C₄×D₂₃	—	4	8	no	—
66	C₂×C₄×D₂₃	C₂³	1	4	yes	—
67	C₂×C₄×D₂₃	C₂³	1	4	yes	—
68	C₂×C₄×D₂₃	C₂³	1	4	yes	—
69	C₂×C₄×D₂₃	C₂³	1	8	yes	—
70	C₂×Dic₄₆	C₂³	1	2	yes	—
71	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
72	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
73	C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₄²×C₂₃	C₂³	1	2	yes	—
75	C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
77	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
78	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
79	C₂²⋊Dic₂₃	C₂³	1	4	yes	—
80	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
81	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
82	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
83	C₄⋊Dic₂₃	C₂³	1	2	yes	—
84	C₄⋊Dic₂₃	C₂³	1	2	yes	—
85	C₄⋊Dic₂₃	C₂³	1	8	yes	—
86	Dic₂₃⋊C₄	C₂³	1	4	yes	—
87	Dic₂₃⋊C₄	C₂³	1	4	yes	—
88	Dic₂₃⋊C₄	C₂³	1	8	yes	—
89	C₄×Dic₂₃	C₂³	1	4	yes	—
90	C₄×Dic₂₃	C₂³	1	4	yes	—
91	C₄×Dic₂₃	C₂³	1	4	yes	—
92	C₂³×C₂₃	C₂⁴	1	4	yes	—
93	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₂²×D₂₃	—	2	16	no	—
95	C₂²×D₂₃	—	4	4	no	—
96	C₂²×D₂₃	C₂⁴	1	4	yes	—
97	Q₈×C₂₃	—	4	2	no	—
98	D₄×C₂₃	—	2	8	no	—
99	D₄×C₂₃	—	4	4	no	—
100	D₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	2	8	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
107	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
109	C₂²⋊D₂₃	—	2	8	no	—
110	C₂²⋊D₂₃	—	2	16	no	—
111	C₂×Dic₂₃	—	4	4	no	—
112	C₂×Dic₂₃	C₂⁴	1	4	yes	—
113	C₂×Dic₂₃	C₂⁴	1	4	yes	—
114	C₂×Dic₂₃	C₂⁴	1	4	yes	—
115	D₉₂	#41	1	8	yes	complemented
116	C₄×D₂₃	—	2	8	no	—
117	C₄×D₂₃	—	2	8	no	—
118	C₄×D₂₃	—	2	8	no	—
119	C₄×D₂₃	—	4	8	no	—
120	Dic₄₆	—	2	4	no	—
121	#3481	—	23	1	no	maximal, Sylow
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	8	no	—
124	C₂²×C₂₃	—	2	4	no	—
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
127	C₂×D₂₃	—	2	4	no	—
128	C₂×D₂₃	—	2	4	no	—
129	C₂×D₂₃	—	4	16	no	—
130	C₂×D₂₃	C₂×#41	1	8	yes	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	4	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	C₂×#41	1	4	yes	—
135	C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
136	Dic₂₃	—	2	4	no	—
137	Dic₂₃	—	2	4	no	—
138	Dic₂₃	—	2	4	no	—
139	#559	—	23	1	no	—
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	D₂₃	1	1	yes	characteristic, complemented
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	—	4	1	no	—
152	C₂×C₂₃	C₂²×#41	1	2	yes	—
153	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
154	D₂₃	—	2	8	no	—
155	C₂×#41	—	23	2	no	—
156	C₂×#41	C₂×D₂₃	1	2	yes	complemented
157	C₂²×D₄	—	23	4	no	—
158	C₂³×C₄	—	23	4	no	—
159	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
160	#127	C₂×D₂₃	1	1	yes	characteristic
161	#126	—	23	8	no	—
162	#125	—	23	4	no	—
163	#124	—	23	4	no	—
164	#124	C₂×D₂₃	1	2	yes	complemented
165	#123	—	23	4	no	—
166	#123	—	23	4	no	—
167	#123	—	23	8	no	—
168	C₂²⋊Q₈	—	23	8	no	—
169	C₂²⋊Q₈	—	23	4	no	—
170	#121	—	23	8	no	—
171	#121	—	23	4	no	—
172	#121	C₂×D₂₃	1	4	yes	complemented
173	#120	—	23	4	no	—
174	#120	—	23	4	no	—
175	C₄×Q₈	—	23	2	no	—
176	C₄×Q₈	—	23	2	no	—
177	C₄×D₄	—	23	4	no	—
178	C₄×D₄	—	23	8	no	—
179	C₄×D₄	—	23	4	no	—
180	C₄×D₄	—	23	4	no	—
181	C₄×D₄	—	23	4	no	—
182	C₄×D₄	—	23	8	no	—
183	C₄×D₄	—	23	2	no	—
184	C₄×D₄	—	23	2	no	—
185	C₄×D₄	C₂×D₂₃	1	4	yes	complemented
186	#117	—	23	2	no	—
187	#117	—	23	4	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	C₂×D₂₃	1	1	yes	characteristic, complemented
193	C₂₃	#3481	1	1	yes	characteristic, complemented, Sylow
194	C₂⁴	—	23	4	no	—
195	#41	—	4	4	no	—
196	#41	—	46	8	no	—
197	C₂×Q₈	—	23	2	no	—
198	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
199	C₂×D₄	—	23	4	no	—
200	C₂×D₄	—	23	8	no	—
201	C₂×D₄	—	23	2	no	—
202	C₂×D₄	—	46	16	no	—
203	C₂×D₄	C₂²×D₂₃	1	4	yes	—
204	C₂×D₄	C₂²×D₂₃	1	2	yes	—
205	C₂×D₄	C₂²×D₂₃	1	4	yes	—
206	C₂²×C₄	—	23	4	no	—
207	C₂²×C₄	—	23	4	no	—
208	C₂²×C₄	—	23	4	no	—
209	C₂²×C₄	—	23	4	no	—
210	C₂²×C₄	—	23	8	no	—
211	C₂²×C₄	—	92	8	no	—
212	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
213	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
215	C₄⋊C₄	—	23	2	no	—
216	C₄⋊C₄	—	23	8	no	—
217	C₄⋊C₄	—	23	2	no	—
218	C₄⋊C₄	—	23	4	no	—
219	C₄⋊C₄	—	23	4	no	—
220	C₄⋊C₄	—	23	8	no	—
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	4	no	—
226	C₂²⋊C₄	—	23	4	no	—
227	C₂²⋊C₄	—	23	8	no	—
228	C₂²⋊C₄	—	23	4	no	—
229	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
230	C₄²	—	23	4	no	—
231	C₄²	—	23	4	no	—
232	C₄²	—	23	4	no	—
233	C₄²	C₂²×D₂₃	1	1	yes	characteristic
234	C₄²	C₂²×D₂₃	1	1	yes	characteristic
235	C₄²	C₂²×D₂₃	1	2	yes	—
236	C₂³	—	23	4	no	—
237	C₂³	—	46	16	no	—
238	C₂³	—	92	4	no	—
239	C₂³	C₂³×D₂₃	1	4	yes	—
240	C₂³	C₂³×D₂₃	1	1	yes	characteristic
241	Q₈	—	4	2	no	—
242	Q₈	—	46	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	D₄	—	23	8	no	—
247	D₄	—	46	16	no	—
248	D₄	—	46	8	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	23	4	no	—
253	C₂×C₄	—	23	4	no	—
254	C₂×C₄	—	23	4	no	—
255	C₂×C₄	—	46	8	no	—
256	C₂×C₄	—	46	8	no	—
257	C₂×C₄	—	46	8	no	—
258	C₂×C₄	—	92	8	no	—
259	C₂×C₄	—	92	4	no	—
260	C₂×C₄	C₂³×D₂₃	1	2	yes	—
261	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂³×D₂₃	1	4	yes	—
263	C₂×C₄	C₂³×D₂₃	1	2	yes	—
264	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	23	8	no	—
270	C₂²	—	46	4	no	—
271	C₂²	—	46	4	no	—
272	C₂²	—	92	16	no	—
273	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	46	4	no	—
278	C₄	—	46	4	no	—
279	C₄	—	46	4	no	—
280	C₄	#10625155	1	2	yes	—
281	C₄	#41×D₂₃	1	4	yes	—
282	C₂	—	4	4	no	—
283	C₂	—	4	1	no	—
284	C₂	—	46	8	no	—
285	C₂	C₂×#41×D₂₃	1	2	yes	—
286	C₂	C₂×#10625155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂₃×#532	C₂
hypercenter	#532	D₂₃
hypocenter	C₂₃	#3481
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #13093153	C₂	semidirect
2. #13093128	C₂	semidirect
3. #13093114	C₂	semidirect
4. #13093097	C₂	semidirect
5. #13093064	C₂	semidirect
6. #13093052	C₂	semidirect
7. #13092949	C₂	semidirect
8. #13092924	C₂	semidirect
9. C₄×#67288	C₂	semidirect
10. C₄×D₄×D₂₃	C₂	semidirect
11. C₂₃×#532	C₂	semidirect
12. C₂×#41×C₂₃	C₂²	semidirect
15. C₂²×C₄×D₂₃	C₂²	semidirect
16. C₂₃×C₄⋊Q₈	C₂²	semidirect
18. C₂₃×#124	C₂²	semidirect
19. C₂₃×#121	C₂²	semidirect
20. C₄×D₄×C₂₃	C₂²	semidirect
21. C₂×C₄²×C₂₃	C₂²	semidirect
22. #10625117	C₂²	semidirect
24. #10625109	C₂²	semidirect
26. #10625108	C₂²	semidirect
27. #10625107	C₂²	semidirect
28. #10625106	C₂²	semidirect
29. #10625106	C₂²	semidirect
30. D₄×Dic₂₃	C₂²	semidirect
31. C₄×C₂²⋊D₂₃	C₂²	semidirect
32. C₄×C₂²⋊D₂₃	C₂²	semidirect
33. C₄×C₂²⋊D₂₃	C₂²	semidirect
34. #10625097	C₂²	semidirect
35. #10625097	C₂²	semidirect
36. #10625068	C₂²	semidirect
38. C₄⋊C₄×D₂₃	C₂²	semidirect
39. #10625061	C₂²	semidirect
40. #10625054	C₂²	semidirect
41. (C₂²⋊D₂₃)⋊C₄	C₂²	semidirect
42. C₂²⋊C₄×D₂₃	C₂²	semidirect
43. #10625050	C₂²	semidirect
46. #10625043	C₂²	semidirect
47. C₄²×D₂₃	C₂²	semidirect
48. C₄×Dic₄₆	C₂²	semidirect
49. C₄×Dic₄₆	C₂²	semidirect
115. D₉₂	#41	semidirect
145. #532	D₂₃	semidirect
156. C₂×#41	C₂×D₂₃	semidirect
159. C₄⋊Q₈	C₂×D₂₃	semidirect
164. #124	C₂×D₂₃	semidirect
172. #121	C₂×D₂₃	semidirect
185. C₄×D₄	C₂×D₂₃	semidirect
192. C₂×C₄²	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
#13093153	2
#13093128	2
#13093114	2
#13093097	2
#13093064	2
#13093052	2
#13092949	2
#13092924	2
C₄×#67288	2
C₄×D₄×D₂₃	2
C₂₃×#532	2
#3481	23