Subgroups — #13092336

Order 1472.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₃⋊C₄	C₂	1	3	yes	maximal
3	C₂₃⋊#95	C₂	1	3	yes	maximal
4	C₂×Q₈×C₂₃	—	2	6	no	—
5	C₂²×C₄×C₂₃	C₂²	1	3	yes	—
6	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₃	C₂²	1	3	yes	—
8	Dic₂₃⋊C₄	—	2	6	no	—
9	C₂³×C₂₃	C₂³	1	1	yes	characteristic
10	Q₈×C₂₃	—	2	8	no	—
11	C₂×C₄×C₂₃	—	2	6	no	—
12	C₂×C₄×C₂₃	D₄	1	6	yes	—
13	C₂×Dic₂₃	—	2	3	no	—
14	C₂×Dic₂₃	—	2	3	no	—
15	C₂×Dic₂₃	—	2	3	no	—
16	C₂×Dic₂₃	Q₈	1	6	yes	complemented
17	C₂²×C₂₃	C₂×D₄	1	3	yes	—
18	C₂²×C₂₃	C₂×Q₈	1	3	yes	—
19	C₂²×C₂₃	#41	1	1	yes	characteristic
20	C₄×C₂₃	—	2	6	no	—
21	Dic₂₃	—	2	6	no	—
22	Dic₂₃	—	4	1	no	—
23	#395	—	23	1	no	maximal, Sylow
24	C₂×C₂₃	#120	1	1	yes	characteristic
25	C₂×C₂₃	C₂²⋊Q₈	1	3	yes	—
26	C₂×C₂₃	C₄⋊Q₈	1	3	yes	—
27	C₂²×Q₈	D₂₃	1	1	yes	characteristic
28	C₂×C₄⋊C₄	—	23	3	no	—
29	#95	—	23	3	no	—
30	C₂₃	#395	1	1	yes	characteristic, complemented, Sylow
31	C₂×Q₈	—	2	6	no	—
32	C₂²×C₄	—	23	3	no	—
33	C₂²×C₄	—	23	1	no	—
34	C₂²×C₄	C₂×D₂₃	1	3	yes	—
35	C₄⋊C₄	—	46	6	no	—
36	C₂³	C₂²×D₂₃	1	1	yes	characteristic
37	Q₈	—	2	8	no	—
38	C₂×C₄	—	2	6	no	—
39	C₂×C₄	—	23	6	no	—
40	C₂×C₄	—	46	3	no	—
41	C₂×C₄	—	46	3	no	—
42	C₂×C₄	—	46	3	no	—
43	C₂×C₄	C₂²⋊D₂₃	1	6	yes	—
44	C₂²	Q₈×D₂₃	1	3	yes	—
45	C₂²	#67292	1	1	yes	characteristic
46	C₂²	C₂×C₂²⋊D₂₃	1	3	yes	—
47	C₄	—	2	6	no	—
48	C₄	—	46	6	no	—
49	C₄	—	92	1	no	—
50	C₂	#10625115	1	3	yes	—
51	C₂	#10625117	1	3	yes	—
52	C₂	#10625124	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃
commutator, socle, prime radical, cosocle	C₂³×C₂₃	C₂³
Fitting	C₂²×Q₈×C₂₃	C₂
hypercenter	C₂²×Q₈	D₂₃
hypocenter	C₂₃	#395
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
16. C₂×Dic₂₃	Q₈	semidirect
30. C₂₃	#395	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₂₃	2
C₂×Dic₂₃⋊C₄	2
C₂₃⋊#95	2
#395	23