Subgroups — #13092796

Order 1472.

122 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
2	C₂×#67275	C₂	1	4	yes	maximal
3	#10625040	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₂₃	C₂²	1	2	yes	—
6	C₂×Dic₄₆	C₂²	1	2	yes	—
7	C₄⋊C₄×C₂₃	—	2	2	no	—
8	C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	#67275	—	2	8	no	—
10	C₄⋊Dic₂₃	—	2	2	no	—
11	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
12	Q₈×C₂₃	—	2	4	no	—
13	C₂×C₄×C₂₃	—	2	2	no	—
14	C₂×C₄×C₂₃	D₄	1	2	yes	—
15	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×Dic₂₃	—	2	2	no	—
17	Dic₄₆	—	2	4	no	—
18	C₂₃⋊C₈	D₄	1	4	yes	—
19	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
20	C₄×C₂₃	—	4	2	no	—
21	C₄×C₂₃	Q₁₆	1	4	yes	complemented
22	C₄×C₂₃	C₂×D₄	1	2	yes	—
23	Dic₂₃	—	4	2	no	—
24	#494	—	23	1	no	maximal, Sylow
25	C₂×C₂₃	#127	1	1	yes	characteristic
26	C₂×C₂₃	C₂×Q₁₆	1	2	yes	—
27	C₂×Q₁₆	—	23	4	no	—
28	C₄⋊Q₈	—	23	1	no	—
29	C₄⋊Q₈	D₂₃	1	1	yes	characteristic
30	C₄×C₈	—	23	1	no	—
31	C₂₃	#494	1	1	yes	characteristic, complemented, Sylow
32	C₂×Q₈	—	23	2	no	—
33	C₂×Q₈	C₂×D₂₃	1	2	yes	—
34	Q₁₆	—	46	8	no	—
35	C₂×C₈	—	23	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	46	2	no	—
38	C₄²	C₂×D₂₃	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	46	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	46	2	no	—
43	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
44	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
45	C₈	—	23	4	no	—
46	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
47	C₄	—	4	2	no	—
48	C₄	—	92	2	no	—
49	C₄	#67275	1	4	yes	complemented
50	C₄	D₄×D₂₃	1	2	yes	—
51	C₂	#10625111	1	1	yes	characteristic
52	C₂	C₂×#67275	1	2	yes	—

	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, prime radical	C₂²×C₂₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#494
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
21. C₄×C₂₃	Q₁₆	semidirect
31. C₂₃	#494	semidirect
49. C₄	#67275	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
C₂×#67275	2
#10625040	2
C₄×C₂₃⋊C₈	2
#494	23