Subgroups — #10625113

Order 736.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#67288	C₂	1	1	yes	maximal, characteristic
3	#67275	C₂	1	2	yes	maximal, complemented
4	#67274	C₂	1	2	yes	maximal, complemented
5	#67267	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₂₃	—	2	1	no	—
7	Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄×C₂₃	—	2	1	no	—
9	C₂×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²⋊D₂₃	—	2	1	no	—
11	D₉₂	C₂²	1	1	yes	characteristic
12	C₄×D₂₃	—	2	1	no	—
13	Dic₄₆	C₂²	1	1	yes	characteristic
14	C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₂₃	D₄	1	1	yes	characteristic
16	C₂×D₂₃	—	2	1	no	—
17	C₄×C₂₃	—	2	2	no	—
18	C₄×C₂₃	D₄	1	1	yes	characteristic
19	C₄×C₂₃	C₂³	1	1	yes	characteristic
20	Dic₂₃	—	2	1	no	—
21	C₂×C₂₃	—	2	1	no	—
22	C₂×C₂₃	C₂×D₄	1	1	yes	characteristic
23	D₂₃	—	4	1	no	—
24	#137	—	23	1	no	maximal, Sylow
25	C₂₃	#137	1	1	yes	characteristic, complemented, Sylow
26	#41	—	23	1	no	—
27	C₂×Q₈	D₂₃	1	1	yes	characteristic, complemented
28	Q₁₆	—	23	2	no	—
29	SD₁₆	—	23	2	no	—
30	M₁₆	—	23	1	no	—
31	Q₈	—	2	1	no	—
32	Q₈	—	23	1	no	—
33	Q₈	C₂×D₂₃	1	2	yes	—
34	D₄	—	23	1	no	—
35	D₄	—	46	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	46	1	no	—
38	C₂×C₄	C₂×D₂₃	1	1	yes	characteristic
39	C₈	—	23	2	no	—
40	C₂²	—	46	1	no	—
41	C₂²	C₂²⋊D₂₃	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	46	1	no	—
44	C₄	C₂²⋊D₂₃	1	1	yes	characteristic
45	C₄	C₂²×D₂₃	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	92	1	no	—
48	C₂	C₂×C₂²⋊D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₂₃
commutator, cosocle	C₄×C₂₃	C₂³
Frattini	C₄	C₂²×D₂₃
Fitting	C₂×Q₈×C₂₃	C₂
socle	C₂×C₂₃	C₂×D₄
prime radical	#67288	C₂
hypercenter	C₂×Q₈	D₂₃
hypocenter	C₂₃	#137
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₂₃	C₂	semidirect
3. #67275	C₂	semidirect
4. #67274	C₂	semidirect
5. #67267	C₂	semidirect
25. C₂₃	#137	semidirect
27. C₂×Q₈	D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₂₃	2
#67288	2
#67275	2
#67274	2
#67267	2
#137	23