Subgroups — #13092704

Order 1472.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₉₂	C₂	1	1	yes	maximal, characteristic
3	#10624991	C₂	1	2	yes	maximal
4	#10624990	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₂₃	—	2	1	no	—
7	C₂×D₉₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₃	—	2	1	no	—
9	C₄⋊C₄×C₂₃	—	2	1	no	—
10	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	(C₂×D₂₃)⋊C₄	—	2	1	no	—
13	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×D₂₃	—	2	1	no	—
16	Q₈×C₂₃	—	4	2	no	—
17	C₂×C₄×C₂₃	—	2	2	no	—
18	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₂₃	—	2	1	no	—
22	D₉₂	—	2	1	no	—
23	D₉₂	Q₈	1	2	yes	complemented
24	C₄×D₂₃	—	4	2	no	—
25	C₂₃⋊C₈	—	2	2	no	—
26	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₃	—	2	2	no	—
28	C₂×D₂₃	—	4	1	no	—
29	C₂×D₂₃	—	4	1	no	—
30	C₄×C₂₃	—	2	1	no	—
31	C₄×C₂₃	—	4	2	no	—
32	C₄×C₂₃	SD₁₆	1	2	yes	—
33	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₂₃	#41	1	1	yes	characteristic
35	Dic₂₃	—	4	1	no	—
36	#476	—	23	1	no	maximal, Sylow
37	C₂×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
40	D₂₃	—	4	2	no	—
41	C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
42	C₄×D₄	—	23	1	no	—
43	#106	—	23	2	no	—
44	C₄⋊C₈	—	23	1	no	—
45	#102	—	23	2	no	—
46	C₂₃	#476	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	23	1	no	—
49	C₂²×C₄	—	46	1	no	—
50	C₂×C₈	—	23	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	23	1	no	—
53	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
54	C₂²⋊C₄	—	46	1	no	—
55	C₄²	C₂×D₂₃	1	1	yes	characteristic
56	C₂³	—	46	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	23	2	no	—
59	D₄	—	46	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	46	1	no	—
62	C₂×C₄	—	92	2	no	—
63	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
66	C₈	—	46	2	no	—
67	C₂²	—	46	2	no	—
68	C₂²	—	92	1	no	—
69	C₂²	—	92	1	no	—
70	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	92	1	no	—
74	C₄	#67274	1	2	yes	—
75	C₄	Q₈×D₂₃	1	1	yes	characteristic
76	C₄	#67292	1	1	yes	characteristic
77	C₂	—	92	2	no	—
78	C₂	#10625103	1	1	yes	characteristic
79	C₂	C₂×#67274	1	1	yes	characteristic
80	C₂	#10625117	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×C₄⋊Q₈	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂×D₉₂	C₂²
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#476
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
1. C₂₃×C₄⋊Q₈	C₂	semidirect
4. #10624990	C₂	semidirect
5. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
23. D₉₂	Q₈	semidirect
41. C₄⋊Q₈	D₂₃	semidirect
46. C₂₃	#476	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×C₄⋊Q₈	2
C₄×D₉₂	2
#10624991	2
#10624990	2
C₄⋊(C₂₃⋊C₈)	2
#476	23