Subgroups — #15671569

Order 2944.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092785	C₂	1	1	yes	maximal, characteristic
2	#13092216	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#356	C₂	1	1	yes	maximal, characteristic
4	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
5	#10625041	C₄	1	1	yes	characteristic
6	C₂₃×#101	—	2	1	no	—
7	#10625018	—	2	1	no	—
8	#10625017	—	2	1	no	—
9	#10625017	—	2	1	no	—
10	#10624985	C₄	1	1	yes	characteristic
11	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
12	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
13	M₁₆×C₂₃	—	4	1	no	—
14	C₄⋊C₄×C₂₃	—	4	1	no	—
15	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₄⋊Dic₂₃	—	2	1	no	—
17	Dic₂₃⋊C₄	—	4	1	no	—
18	#67267	—	4	1	no	—
19	C₂×C₂₃⋊C₈	—	2	1	no	—
20	Q₈×C₂₃	—	4	1	no	—
21	Q₈×C₂₃	—	4	1	no	—
22	C₂×C₄×C₂₃	—	2	1	no	—
23	C₂×C₄×C₂₃	—	2	1	no	—
24	C₂×C₄×C₂₃	—	2	1	no	—
25	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₂₃	—	4	1	no	—
27	C₈×C₂₃	—	8	1	no	—
28	C₂₃⋊C₈	—	4	1	no	—
29	C₂₃⋊C₈	—	8	1	no	—
30	#1413	—	23	1	no	maximal, Sylow
31	C₂²×C₂₃	#99	1	1	yes	characteristic
32	C₄×C₂₃	—	2	1	no	—
33	C₄×C₂₃	—	4	1	no	—
34	C₄×C₂₃	—	4	1	no	—
35	C₄×C₂₃	—	4	1	no	—
36	Dic₂₃	—	8	1	no	—
37	#491	—	23	1	no	—
38	#356	—	23	1	no	—
39	#356	D₂₃	1	1	yes	characteristic
40	C₂×C₂₃	—	2	1	no	—
41	C₂×C₂₃	#354	1	1	yes	characteristic
42	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
43	#125	—	23	1	no	—
44	#103	—	46	1	no	—
45	#103	—	46	1	no	—
46	#101	—	2	1	no	—
47	#101	—	46	1	no	—
48	#97	—	23	1	no	—
49	C₂₃	#1413	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	D₉₂	1	1	yes	characteristic
51	C₂×Q₈	C₂²⋊D₂₃	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	92	1	no	—
54	C₂×C₈	—	46	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	46	1	no	—
57	C₄⋊C₄	—	92	1	no	—
58	C₄²	C₄×D₂₃	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	92	1	no	—
65	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	92	1	no	—
68	C₈	—	184	1	no	—
69	C₂²	#10624988	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	184	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#13092211	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092211
commutator	C₄²×C₂₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₃
Fitting	C₂₃×#356	C₂
socle	C₂×C₂₃	#354
prime radical	C₂²×C₂₃	#99
cosocle	C₂₃×C₄⋊Q₈	C₂²
hypercenter	#356	D₂₃
hypocenter	C₂₃	#1413
LCS	G ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
49. C₂₃	#1413	semidirect

Maximal subgroups

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

subgroup	index
#13092785	2
#13092216	2
C₂₃×#356	2
#1413	23