Subgroups — #13092717

Order 1472.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	#10625041	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625014	C₂	1	1	yes	maximal, characteristic
4	#10625014	C₂	1	1	yes	maximal, characteristic
5	#10624990	C₂	1	1	yes	maximal, characteristic, complemented
6	#10624989	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₂₃	—	2	1	no	—
10	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊C₄×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	C₂³×C₂₃	—	2	1	no	—
20	D₄×C₂₃	—	2	1	no	—
21	D₄×C₂₃	Q₈	1	2	yes	—
22	C₂×C₄×C₂₃	—	2	1	no	—
23	C₂×C₄×C₂₃	—	4	2	no	—
24	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₂₃	—	2	1	no	—
28	C₂×Dic₂₃	—	2	1	no	—
29	C₂₃⋊C₈	—	2	1	no	—
30	C₂₃⋊C₈	—	2	1	no	—
31	C₂²×C₂₃	—	2	2	no	—
32	C₂²×C₂₃	—	4	1	no	—
33	C₂²×C₂₃	—	4	1	no	—
34	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₃	—	2	1	no	—
36	C₄×C₂₃	—	2	1	no	—
37	C₄×C₂₃	—	4	1	no	—
38	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₄×C₂₃	#41	1	1	yes	characteristic
40	Dic₂₃	—	4	1	no	—
41	Dic₂₃	—	4	1	no	—
42	#478	—	23	1	no	maximal, Sylow
43	C₂×C₂₃	—	4	2	no	—
44	C₂×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
45	C₂×C₂₃	#135	1	1	yes	characteristic
46	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
47	#125	—	23	1	no	—
48	C₄×D₄	D₂₃	1	1	yes	characteristic
49	#107	—	23	1	no	—
50	#106	—	23	1	no	—
51	C₄⋊C₈	—	23	1	no	—
52	#102	—	23	1	no	—
53	#102	—	23	1	no	—
54	C₂₃	#478	1	1	yes	characteristic, complemented, Sylow
55	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
56	C₂²×C₄	—	2	1	no	—
57	C₂×C₈	—	23	1	no	—
58	C₂×C₈	—	23	1	no	—
59	C₄⋊C₄	—	23	1	no	—
60	C₄⋊C₄	—	23	1	no	—
61	C₄⋊C₄	—	46	1	no	—
62	C₄⋊C₄	—	46	1	no	—
63	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×D₂₃	1	1	yes	characteristic
66	C₂³	—	2	1	no	—
67	D₄	—	2	1	no	—
68	D₄	Dic₄₆	1	2	yes	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	46	1	no	—
72	C₂×C₄	—	46	1	no	—
73	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
76	C₈	—	46	1	no	—
77	C₈	—	46	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	92	1	no	—
86	C₄	—	92	1	no	—
87	C₄	C₂×Dic₄₆	1	1	yes	characteristic
88	C₄	#67288	1	1	yes	characteristic
89	C₂	—	4	2	no	—
90	C₂	#10625095	1	1	yes	characteristic
91	C₂	#10625103	1	1	yes	characteristic
92	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×D₄×C₂₃	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂×D₄×C₂₃	C₂²
hypercenter	C₄×D₄	D₂₃
hypocenter	C₂₃	#478
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #10625041	C₂	semidirect
5. #10624990	C₂	semidirect
6. #10624989	C₂	semidirect
7. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
54. C₂₃	#478	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×C₂₃	2
#10625041	2
#10625014	2
#10624990	2
#10624989	2
C₄⋊(C₂₃⋊C₈)	2
#478	23