Subgroups — #15686789

Order 2944.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093041	C₂	1	4	yes	maximal, complemented
2	#13093030	C₂	1	2	yes	maximal, complemented
3	#13093023	C₂	1	2	yes	maximal, complemented
4	#13093016	C₂	1	4	yes	maximal, complemented
5	#13093007	C₂	1	4	yes	maximal, complemented
6	#13092967	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092954	C₂	1	8	yes	maximal, complemented
8	#13092929	C₂	1	2	yes	maximal, complemented
9	#13092885	C₂	1	2	yes	maximal, complemented
10	C₂×#10625043	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	2	yes	—
13	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
15	C₂₃×#125	C₂²	1	2	yes	complemented
16	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
17	C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
18	C₂₃×#117	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	#10625118	C₂²	1	8	yes	—
21	#10625117	C₂²	1	8	yes	complemented
22	#10625101	C₂²	1	4	yes	—
23	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
26	#10625069	C₂²	1	8	yes	—
27	#10625069	C₂²	1	8	yes	complemented
28	#10625068	C₂²	1	4	yes	—
29	#10625066	C₂²	1	4	yes	—
30	#10625066	C₂²	1	8	yes	—
31	#10625065	C₂²	1	8	yes	—
32	#10625065	C₂²	1	8	yes	complemented
33	#10625064	C₂²	1	4	yes	complemented
34	#10625061	C₂²	1	8	yes	complemented
35	#10625058	C₂²	1	4	yes	complemented
36	#10625057	C₂²	1	4	yes	—
37	#10625053	C₂²	1	4	yes	—
38	(C₂²⋊D₂₃)⋊C₄	C₂²	1	8	yes	—
39	#10625046	C₂²	1	2	yes	complemented
40	#10625045	C₂²	1	1	yes	characteristic, complemented
41	C₄×D₉₂	C₂²	1	8	yes	—
42	C₄×D₉₂	C₂²	1	4	yes	complemented
43	#10625043	C₂²	1	4	yes	—
44	#10625043	C₂²	1	2	yes	—
45	#10625043	C₂²	1	2	yes	—
46	#10625043	C₂²	1	4	yes	complemented
47	C₄²×D₂₃	C₂²	1	4	yes	complemented
48	#10625040	C₂²	1	1	yes	characteristic, complemented
49	C₂³×D₂₃	C₂³	1	1	yes	characteristic
50	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₂₃	C₂³	1	4	yes	—
54	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₂₃	C₂³	1	2	yes	—
56	#67288	—	2	8	no	—
57	C₂×D₉₂	C₂³	1	2	yes	—
58	C₂×D₉₂	C₂³	1	8	yes	—
59	C₂×D₉₂	C₂³	1	4	yes	—
60	C₂×C₄×D₂₃	—	2	4	no	—
61	C₂×C₄×D₂₃	C₂³	1	4	yes	—
62	C₂×C₄×D₂₃	C₂³	1	2	yes	—
63	C₂×C₄×D₂₃	C₂³	1	2	yes	—
64	C₂×C₄×D₂₃	C₂³	1	8	yes	—
65	C₂×Dic₄₆	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
67	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
68	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
70	C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₄²×C₂₃	C₂³	1	4	yes	—
72	C₄²×C₂₃	C₂³	1	2	yes	—
73	C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
75	(C₂×D₂₃)⋊C₄	C₂³	1	4	yes	—
76	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₂₃)⋊C₄	C₂³	1	8	yes	—
80	C₄⋊Dic₂₃	C₂³	1	4	yes	—
81	C₄⋊Dic₂₃	C₂³	1	8	yes	—
82	Dic₂₃⋊C₄	C₂³	1	4	yes	—
83	Dic₂₃⋊C₄	C₂³	1	4	yes	—
84	Dic₂₃⋊C₄	C₂³	1	8	yes	—
85	C₄×Dic₂₃	C₂³	1	2	yes	—
86	C₄×Dic₂₃	C₂³	1	8	yes	—
87	C₄×Dic₂₃	C₂³	1	2	yes	—
88	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
89	C₂²×D₂₃	—	2	2	no	—
90	C₂²×D₂₃	—	4	2	no	—
91	C₂²×D₂₃	C₂⁴	1	2	yes	—
92	C₂²×D₂₃	C₂⁴	1	4	yes	—
93	Q₈×C₂₃	—	4	4	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	#41	1	4	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
97	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₂₃	—	2	8	no	—
102	C₂×Dic₂₃	—	2	2	no	—
103	C₂×Dic₂₃	—	4	2	no	—
104	C₂×Dic₂₃	C₂⁴	1	4	yes	—
105	C₂×Dic₂₃	C₂⁴	1	4	yes	—
106	C₂×Dic₂₃	C₂⁴	1	2	yes	—
107	D₉₂	—	2	4	no	—
108	D₉₂	—	4	8	no	—
109	D₉₂	—	4	4	no	—
110	C₄×D₂₃	—	2	4	no	—
111	C₄×D₂₃	—	2	4	no	—
112	C₄×D₂₃	—	2	8	no	—
113	C₄×D₂₃	—	4	8	no	—
114	Dic₄₆	—	2	4	no	—
115	#3455	—	23	1	no	maximal, Sylow
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	C₂×#41	1	2	yes	—
118	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
119	C₂×D₂₃	—	2	2	no	—
120	C₂×D₂₃	—	2	4	no	—
121	C₂×D₂₃	—	2	8	no	—
122	C₂×D₂₃	—	2	4	no	—
123	C₂×D₂₃	—	4	4	no	—
124	C₄×C₂₃	—	2	8	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	C₂×#41	1	2	yes	—
128	C₄×C₂₃	C₂×#41	1	2	yes	—
129	Dic₂₃	—	2	4	no	—
130	Dic₂₃	—	2	2	no	—
131	Dic₂₃	—	2	4	no	—
132	#542	—	23	4	no	—
133	#541	—	23	2	no	—
134	#540	—	23	2	no	—
135	#539	—	23	4	no	—
136	#538	—	23	4	no	—
137	#533	D₂₃	1	1	yes	characteristic, complemented
138	#532	—	23	1	no	—
139	#529	—	23	8	no	—
140	#525	—	23	2	no	—
141	C₄×#41	—	23	2	no	—
142	C₂×#117	—	23	1	no	—
143	C₂×C₂₃	—	2	2	no	—
144	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
145	C₂×C₂₃	#585	1	2	yes	—
146	D₂₃	—	4	4	no	—
147	D₂₃	—	4	2	no	—
148	C₂×#41	—	23	2	no	—
149	C₂³×C₄	—	23	1	no	—
150	C₄⋊Q₈	—	23	1	no	—
151	C₄⋊Q₈	C₂×D₂₃	1	2	yes	complemented
152	#127	—	23	1	no	—
153	#126	—	23	8	no	—
154	#126	—	23	8	no	—
155	#125	—	23	8	no	—
156	#125	C₂×D₂₃	1	2	yes	complemented
157	#124	—	23	8	no	—
158	#124	—	23	2	no	—
159	#123	—	23	8	no	—
160	#123	—	23	8	no	—
161	#123	—	23	4	no	—
162	C₂²⋊Q₈	—	23	8	no	—
163	C₂²⋊Q₈	—	23	4	no	—
164	C₂²⋊Q₈	C₂×D₂₃	1	4	yes	—
165	#121	—	23	4	no	—
166	#121	—	23	4	no	—
167	#121	—	23	8	no	—
168	#120	—	23	4	no	—
169	C₄×Q₈	—	23	4	no	—
170	C₄×Q₈	C₂×D₂₃	1	4	yes	complemented
171	C₄×D₄	—	23	4	no	—
172	C₄×D₄	—	23	8	no	—
173	C₄×D₄	—	23	8	no	—
174	C₄×D₄	—	23	4	no	—
175	#117	—	23	4	no	—
176	#117	—	23	2	no	—
177	#117	—	23	2	no	—
178	#117	—	23	4	no	—
179	#117	C₂×D₂₃	1	2	yes	—
180	C₂×C₄⋊C₄	—	23	2	no	—
181	C₂×C₂²⋊C₄	—	23	2	no	—
182	C₂×C₄²	—	23	1	no	—
183	C₂×C₄²	—	23	4	no	—
184	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
185	C₂₃	#3455	1	1	yes	characteristic, complemented, Sylow
186	C₂⁴	—	23	1	no	—
187	#41	—	46	8	no	—
188	C₂×Q₈	—	23	2	no	—
189	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
190	C₂×D₄	—	23	4	no	—
191	C₂×D₄	—	23	2	no	—
192	C₂×D₄	—	23	8	no	—
193	C₂×D₄	—	23	4	no	—
194	C₂²×C₄	—	23	1	no	—
195	C₂²×C₄	—	23	2	no	—
196	C₂²×C₄	—	23	2	no	—
197	C₂²×C₄	—	23	4	no	—
198	C₂²×C₄	—	23	2	no	—
199	C₂²×C₄	—	23	8	no	—
200	C₂²×C₄	—	46	4	no	—
201	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
203	C₄⋊C₄	—	23	4	no	—
204	C₄⋊C₄	—	23	4	no	—
205	C₄⋊C₄	—	23	8	no	—
206	C₄⋊C₄	—	23	4	no	—
207	C₄⋊C₄	—	23	8	no	—
208	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
209	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
210	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
211	C₂²⋊C₄	—	23	4	no	—
212	C₂²⋊C₄	—	23	4	no	—
213	C₂²⋊C₄	—	23	8	no	—
214	C₂²⋊C₄	—	23	8	no	—
215	C₂²⋊C₄	—	23	8	no	—
216	C₂²⋊C₄	—	23	8	no	—
217	C₂²⋊C₄	C₂²×D₂₃	1	4	yes	—
218	C₄²	—	23	2	no	—
219	C₄²	—	23	8	no	—
220	C₄²	—	23	2	no	—
221	C₄²	C₂²×D₂₃	1	1	yes	characteristic
222	C₄²	C₂²×D₂₃	1	1	yes	characteristic
223	C₄²	C₂²×D₂₃	1	4	yes	—
224	C₄²	C₂²×D₂₃	1	2	yes	—
225	C₂³	—	23	2	no	—
226	C₂³	—	23	4	no	—
227	C₂³	—	46	2	no	—
228	C₂³	—	92	2	no	—
229	C₂³	C₂³×D₂₃	1	1	yes	characteristic
230	Q₈	—	4	4	no	—
231	Q₈	—	46	4	no	—
232	D₄	—	46	8	no	—
233	D₄	—	46	4	no	—
234	D₄	—	92	4	no	—
235	D₄	—	92	8	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	23	4	no	—
238	C₂×C₄	—	23	4	no	—
239	C₂×C₄	—	23	2	no	—
240	C₂×C₄	—	46	2	no	—
241	C₂×C₄	—	46	4	no	—
242	C₂×C₄	—	46	4	no	—
243	C₂×C₄	—	46	8	no	—
244	C₂×C₄	—	92	8	no	—
245	C₂×C₄	—	92	2	no	—
246	C₂×C₄	#67292	1	4	yes	—
247	C₂×C₄	C₂³×D₂₃	1	8	yes	—
248	C₂×C₄	C₂³×D₂₃	1	2	yes	—
249	C₂×C₄	C₂³×D₂₃	1	2	yes	—
250	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	46	2	no	—
254	C₂²	—	46	4	no	—
255	C₂²	—	46	8	no	—
256	C₂²	—	46	4	no	—
257	C₂²	—	92	4	no	—
258	C₂²	C₂×#67292	1	2	yes	—
259	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	8	no	—
263	C₄	—	46	2	no	—
264	C₄	—	46	4	no	—
265	C₄	—	46	4	no	—
266	C₄	C₂×#67292	1	2	yes	—
267	C₄	C₂×#67292	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	92	2	no	—
270	C₂	—	92	4	no	—
271	C₂	C₂²×#67292	1	1	yes	characteristic
272	C₂	#13093253	1	2	yes	—

	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	#3455
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. #13093041	C₂	semidirect
2. #13093030	C₂	semidirect
3. #13093023	C₂	semidirect
4. #13093016	C₂	semidirect
5. #13093007	C₂	semidirect
6. #13092967	C₂	semidirect
7. #13092954	C₂	semidirect
8. #13092929	C₂	semidirect
9. #13092885	C₂	semidirect
10. C₂×#10625043	C₂	semidirect
11. C₂₃×#533	C₂	semidirect
14. C₂₃×C₄⋊Q₈	C₂²	semidirect
15. C₂₃×#125	C₂²	semidirect
17. C₄×Q₈×C₂₃	C₂²	semidirect
21. #10625117	C₂²	semidirect
27. #10625069	C₂²	semidirect
32. #10625065	C₂²	semidirect
33. #10625064	C₂²	semidirect
34. #10625061	C₂²	semidirect
35. #10625058	C₂²	semidirect
39. #10625046	C₂²	semidirect
40. #10625045	C₂²	semidirect
42. C₄×D₉₂	C₂²	semidirect
46. #10625043	C₂²	semidirect
47. C₄²×D₂₃	C₂²	semidirect
48. #10625040	C₂²	semidirect
137. #533	D₂₃	semidirect
151. C₄⋊Q₈	C₂×D₂₃	semidirect
156. #125	C₂×D₂₃	semidirect
170. C₄×Q₈	C₂×D₂₃	semidirect
185. C₂₃	#3455	semidirect

Maximal subgroups

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

subgroup	index
#13093041	2
#13093030	2
#13093023	2
#13093016	2
#13093007	2
#13092967	2
#13092954	2
#13092929	2
#13092885	2
C₂×#10625043	2
C₂₃×#533	2
#3455	23