Subgroups — #13092209

Order 1472.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10625111	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#99	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625016	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊D₂₃	—	2	1	no	—
6	C₂×C₂²⋊D₂₃	—	2	1	no	—
7	C₂×D₉₂	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊C₄×C₂₃	—	2	1	no	—
9	C₂²⋊Dic₂₃	—	2	1	no	—
10	C₄×Dic₂₃	C₄	1	1	yes	characteristic, complemented
11	C₂³×C₂₃	D₄	1	1	yes	characteristic
12	C₂³×C₂₃	D₄	1	1	yes	characteristic
13	C₂²×D₂₃	—	2	1	no	—
14	D₄×C₂₃	—	4	1	no	—
15	C₂×C₄×C₂₃	—	4	1	no	—
16	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²⋊D₂₃	—	4	2	no	—
18	C₂²⋊D₂₃	—	4	2	no	—
19	C₂×Dic₂₃	—	2	1	no	—
20	C₂×Dic₂₃	—	4	1	no	—
21	D₉₂	—	4	1	no	—
22	C₂²×C₂₃	—	2	1	no	—
23	C₂²×C₂₃	—	2	1	no	—
24	C₂²×C₂₃	—	4	1	no	—
25	C₂²×C₂₃	—	4	1	no	—
26	C₂²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₂₃	—	4	1	no	—
28	C₂×D₂₃	—	4	1	no	—
29	C₂×D₂₃	—	4	1	no	—
30	C₄×C₂₃	—	2	1	no	—
31	C₄×C₂₃	—	8	1	no	—
32	Dic₂₃	—	2	2	no	—
33	Dic₂₃	—	8	1	no	—
34	#353	—	23	1	no	maximal, Sylow
35	C₂×C₂₃	—	2	1	no	—
36	C₂×C₂₃	—	4	1	no	—
37	C₂×C₂₃	—	4	1	no	—
38	C₂×C₂₃	#99	1	1	yes	characteristic
39	D₂₃	—	8	1	no	—
40	#127	—	23	1	no	—
41	#99	—	23	1	no	—
42	#99	D₂₃	1	1	yes	characteristic, complemented
43	C₂₃	#353	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₄	—	23	1	no	—
45	C₂×D₄	—	46	1	no	—
46	C₂×D₄	—	46	1	no	—
47	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₂²⋊C₄	—	46	1	no	—
50	C₄²	—	23	1	no	—
51	C₂³	—	46	1	no	—
52	C₂³	D₉₂	1	1	yes	characteristic
53	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
54	D₄	—	4	1	no	—
55	D₄	—	92	1	no	—
56	D₄	—	92	2	no	—
57	D₄	—	92	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	46	1	no	—
60	C₂×C₄	—	92	1	no	—
61	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	92	1	no	—
67	C₂²	—	92	1	no	—
68	C₂²	—	92	1	no	—
69	C₂²	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	8	1	no	—
72	C₄	—	46	2	no	—
73	C₄	—	184	1	no	—
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	184	1	no	—
78	C₂	#10624988	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10624988
commutator	C₂×C₄×C₂₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₃
Fitting	C₂₃×#99	C₂
socle	C₂×C₂₃	#99
prime radical	#10625111	C₂
cosocle	C₂×D₄×C₂₃	C₂²
hypercenter	#99	D₂₃
hypocenter	C₂₃	#353
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
2. C₂₃×#99	C₂	semidirect
3. #10625016	C₂	semidirect
7. C₂×D₉₂	C₄	semidirect
10. C₄×Dic₂₃	C₄	semidirect
42. #99	D₂₃	semidirect
43. C₂₃	#353	semidirect

Maximal subgroups

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

subgroup	index
#10625111	2
C₂₃×#99	2
#10625016	2
#353	23