Subgroups — #13092327

Order 1472.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₉₂	C₂	1	3	yes	maximal, complemented
2	C₂×(C₂×D₂₃)⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₃	C₂²	1	3	yes	—
5	C₂²×C₄×C₂₃	C₂²	1	3	yes	—
6	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
7	C₂×D₉₂	—	2	12	no	—
8	C₂×D₉₂	—	2	6	no	—
9	(C₂×D₂₃)⋊C₄	—	2	6	no	—
10	C₂³×C₂₃	C₂³	1	1	yes	characteristic
11	C₂²×D₂₃	—	2	3	no	—
12	C₂²×D₂₃	—	2	6	no	—
13	C₂²×D₂₃	—	2	6	no	—
14	C₂²×D₂₃	—	2	3	no	—
15	C₂²×D₂₃	D₄	1	6	yes	complemented
16	C₂×C₄×C₂₃	—	2	6	no	—
17	C₂×C₄×C₂₃	D₄	1	6	yes	complemented
18	C₂×Dic₂₃	—	2	3	no	—
19	D₉₂	—	2	24	no	—
20	C₂²×C₂₃	C₂×D₄	1	3	yes	—
21	C₂²×C₂₃	C₂×D₄	1	3	yes	—
22	C₂²×C₂₃	#41	1	1	yes	characteristic
23	C₂×D₂₃	—	2	12	no	—
24	C₂×D₂₃	—	2	6	no	—
25	C₂×D₂₃	—	4	3	no	—
26	C₂×D₂₃	—	4	6	no	—
27	C₂×D₂₃	—	4	3	no	—
28	C₄×C₂₃	—	2	6	no	—
29	Dic₂₃	—	4	1	no	—
30	#392	—	23	1	no	maximal, Sylow
31	C₂×C₂₃	#120	1	3	yes	—
32	C₂×C₂₃	#121	1	3	yes	—
33	C₂×C₂₃	#127	1	1	yes	characteristic
34	D₂₃	—	4	6	no	—
35	C₂²×D₄	—	23	3	no	—
36	C₂×C₂²⋊C₄	—	23	3	no	—
37	#95	D₂₃	1	1	yes	characteristic, complemented
38	C₂₃	#392	1	1	yes	characteristic, complemented, Sylow
39	C₂⁴	—	23	3	no	—
40	C₂×D₄	—	46	12	no	—
41	C₂×D₄	—	46	6	no	—
42	C₂²×C₄	—	23	1	no	—
43	C₂²×C₄	C₂×D₂₃	1	3	yes	—
44	C₂²⋊C₄	—	46	6	no	—
45	C₂³	—	23	6	no	—
46	C₂³	—	46	3	no	—
47	C₂³	—	46	6	no	—
48	C₂³	—	46	6	no	—
49	C₂³	—	46	3	no	—
50	C₂³	C₂²×D₂₃	1	1	yes	characteristic
51	D₄	—	46	24	no	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	46	3	no	—
54	C₂×C₄	D₉₂	1	6	yes	complemented
55	C₂²	—	46	12	no	—
56	C₂²	—	46	6	no	—
57	C₂²	—	92	3	no	—
58	C₂²	—	92	3	no	—
59	C₂²	—	92	6	no	—
60	C₂²	C₂×D₉₂	1	3	yes	—
61	C₂²	D₄×D₂₃	1	3	yes	—
62	C₂²	#67292	1	1	yes	characteristic
63	C₄	—	2	6	no	—
64	C₄	—	92	1	no	—
65	C₂	—	92	6	no	—
66	C₂	#10625045	1	1	yes	characteristic
67	C₂	#10625053	1	3	yes	—
68	C₂	#10625066	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃
commutator, socle, cosocle	C₂³×C₂₃	C₂³
Fitting	C₂₃×#95	C₂
hypercenter	#95	D₂₃
hypocenter	C₂₃	#392
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂²×D₉₂	C₂	semidirect
2. C₂×(C₂×D₂₃)⋊C₄	C₂	semidirect
3. C₂₃×#95	C₂	semidirect
15. C₂²×D₂₃	D₄	semidirect
17. C₂×C₄×C₂₃	D₄	semidirect
37. #95	D₂₃	semidirect
38. C₂₃	#392	semidirect
54. C₂×C₄	D₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₉₂	2
C₂×(C₂×D₂₃)⋊C₄	2
C₂₃×#95	2
#392	23