Subgroups — #13092617

Order 1472.

128 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, complemented
2	C₂×#67274	C₂	1	2	yes	maximal, complemented
3	#10625045	C₂	1	1	yes	maximal, characteristic
4	#10624991	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×D₉₂	—	2	1	no	—
8	C₂×D₉₂	—	2	1	no	—
9	C₂×D₉₂	C₂²	1	2	yes	—
10	C₄⋊C₄×C₂₃	—	2	1	no	—
11	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₄²×C₂₃	—	2	1	no	—
13	C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	#67274	—	2	4	no	—
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
16	C₂²×D₂₃	—	2	2	no	—
17	Q₈×C₂₃	—	2	1	no	—
18	Q₈×C₂₃	D₄	1	2	yes	complemented
19	C₂×C₄×C₂₃	—	2	1	no	—
20	C₂×C₄×C₂₃	—	2	1	no	—
21	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	D₉₂	—	2	2	no	—
25	D₉₂	—	2	2	no	—
26	D₉₂	—	4	2	no	—
27	D₉₂	—	4	2	no	—
28	C₂₃⋊C₈	—	2	2	no	—
29	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂×D₂₃	—	4	2	no	—
31	C₂×D₂₃	—	4	2	no	—
32	C₂×D₂₃	—	4	2	no	—
33	C₄×C₂₃	—	2	2	no	—
34	C₄×C₂₃	—	2	1	no	—
35	C₄×C₂₃	—	4	1	no	—
36	C₄×C₂₃	SD₁₆	1	2	yes	complemented
37	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₄×C₂₃	#41	1	1	yes	characteristic
39	#460	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	#121	1	1	yes	characteristic
41	C₂×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
43	D₂₃	—	8	2	no	—
44	C₂×SD₁₆	—	23	2	no	—
45	#127	—	23	1	no	—
46	C₄×Q₈	D₂₃	1	1	yes	characteristic, complemented
47	C₄⋊C₈	—	23	1	no	—
48	#102	—	23	2	no	—
49	C₂₃	#460	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
51	C₂×D₄	—	23	2	no	—
52	C₂×D₄	—	46	1	no	—
53	C₂×D₄	—	46	1	no	—
54	SD₁₆	—	46	4	no	—
55	C₂×C₈	—	23	2	no	—
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
58	C₄²	—	2	1	no	—
59	C₄²	C₂×D₂₃	1	1	yes	characteristic
60	C₂³	—	46	2	no	—
61	Q₈	—	2	1	no	—
62	Q₈	D₉₂	1	2	yes	complemented
63	D₄	—	46	2	no	—
64	D₄	—	46	2	no	—
65	D₄	—	92	2	no	—
66	D₄	—	92	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
72	C₈	—	46	2	no	—
73	C₂²	—	92	2	no	—
74	C₂²	—	92	2	no	—
75	C₂²	—	92	2	no	—
76	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	#67274	1	2	yes	complemented
81	C₄	C₂×D₉₂	1	1	yes	characteristic
82	C₄	#67288	1	1	yes	characteristic
83	C₂	—	184	2	no	—
84	C₂	#10625101	1	1	yes	characteristic
85	C₂	C₂×#67274	1	1	yes	characteristic
86	C₂	#10625120	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	C₂²×C₂₃	C₂×D₄
prime radical	#10625045	C₂
hypercenter	C₄×Q₈	D₂₃
hypocenter	C₂₃	#460
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. C₄×Q₈×C₂₃	C₂	semidirect
2. C₂×#67274	C₂	semidirect
4. #10624991	C₂	semidirect
5. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
18. Q₈×C₂₃	D₄	semidirect
36. C₄×C₂₃	SD₁₆	semidirect
46. C₄×Q₈	D₂₃	semidirect
49. C₂₃	#460	semidirect
62. Q₈	D₉₂	semidirect
80. C₄	#67274	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
C₂×#67274	2
#10625045	2
#10624991	2
C₄⋊(C₂₃⋊C₈)	2
#460	23