Subgroups — #13092724

Order 1472.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625041	C₂	1	1	yes	maximal, characteristic
3	#10625017	C₂	1	1	yes	maximal, characteristic
4	#10625017	C₂	1	1	yes	maximal, characteristic
5	#10624990	C₂	1	1	yes	maximal, characteristic
6	#10624989	C₂	1	1	yes	maximal, characteristic
7	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₃	—	2	1	no	—
10	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₄²×C₂₃	—	2	1	no	—
12	C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	Dic₂₃⋊C₄	—	2	1	no	—
16	Dic₂₃⋊C₄	—	2	1	no	—
17	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	Q₈×C₂₃	—	2	1	no	—
20	Q₈×C₂₃	Q₈	1	2	yes	—
21	C₂×C₄×C₂₃	—	2	1	no	—
22	C₂×C₄×C₂₃	—	2	1	no	—
23	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₂₃	—	2	1	no	—
27	C₂×Dic₂₃	—	2	1	no	—
28	C₂₃⋊C₈	—	2	1	no	—
29	C₂₃⋊C₈	—	2	1	no	—
30	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₃	—	2	2	no	—
32	C₄×C₂₃	—	2	1	no	—
33	C₄×C₂₃	—	2	1	no	—
34	C₄×C₂₃	—	4	1	no	—
35	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₂₃	#41	1	1	yes	characteristic
37	Dic₂₃	—	4	1	no	—
38	Dic₂₃	—	4	1	no	—
39	#479	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₂₃	#135	1	1	yes	characteristic
42	C₂×C₂₃	#137	1	1	yes	characteristic
43	#125	—	23	1	no	—
44	C₄×Q₈	D₂₃	1	1	yes	characteristic
45	#107	—	23	1	no	—
46	#106	—	23	1	no	—
47	C₄⋊C₈	—	23	1	no	—
48	#103	—	23	1	no	—
49	#103	—	23	1	no	—
50	C₂₃	#479	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
52	C₂×C₈	—	23	1	no	—
53	C₂×C₈	—	23	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	23	1	no	—
56	C₄⋊C₄	—	23	1	no	—
57	C₄⋊C₄	—	46	1	no	—
58	C₄⋊C₄	—	46	1	no	—
59	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
60	C₄²	—	2	1	no	—
61	C₄²	C₂×D₂₃	1	1	yes	characteristic
62	Q₈	—	2	1	no	—
63	Q₈	Dic₄₆	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	46	1	no	—
67	C₂×C₄	—	46	1	no	—
68	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
71	C₈	—	46	1	no	—
72	C₈	—	46	1	no	—
73	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	92	1	no	—
79	C₄	—	92	1	no	—
80	C₄	C₂×Dic₄₆	1	1	yes	characteristic
81	C₄	#67288	1	1	yes	characteristic
82	C₂	#10625095	1	1	yes	characteristic
83	C₂	#10625113	1	1	yes	characteristic
84	C₂	#10625121	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₄×Q₈×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₄
hypercenter	C₄×Q₈	D₂₃
hypocenter	C₂₃	#479
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
50. C₂₃	#479	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₂₃	2
#10625041	2
#10625017	2
#10624990	2
#10624989	2
C₄⋊(C₂₃⋊C₈)	2
#479	23