Subgroups — #13092812

Order 1472.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×#67275	C₂	1	2	yes	maximal, complemented
3	C₂×#67273	C₂	1	2	yes	maximal
4	#10625040	C₂	1	1	yes	maximal, characteristic, complemented
5	#10624985	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×Dic₄₆	C₂²	1	2	yes	—
9	C₂²⋊C₄×C₂₃	—	2	2	no	—
10	C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	#67275	—	2	4	no	—
12	#67273	—	2	4	no	—
13	C₄⋊Dic₂₃	—	2	2	no	—
14	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂³×C₂₃	—	2	1	no	—
16	Q₈×C₂₃	—	2	2	no	—
17	D₄×C₂₃	—	2	2	no	—
18	C₂×C₄×C₂₃	—	2	1	no	—
19	C₂×C₄×C₂₃	D₄	1	2	yes	—
20	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₂₃	—	2	2	no	—
22	Dic₄₆	—	2	2	no	—
23	Dic₄₆	—	2	2	no	—
24	C₂₃⋊C₈	D₄	1	4	yes	—
25	C₂²×C₂₃	—	4	1	no	—
26	C₂²×C₂₃	—	4	2	no	—
27	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₃	—	2	2	no	—
29	C₄×C₂₃	—	4	1	no	—
30	C₄×C₂₃	C₂×D₄	1	2	yes	—
31	Dic₂₃	—	4	2	no	—
32	#497	—	23	1	no	maximal, Sylow
33	C₂×C₂₃	—	8	1	no	—
34	C₂×C₂₃	#127	1	1	yes	characteristic
35	C₂×C₂₃	#137	1	2	yes	—
36	C₂×Q₁₆	—	23	2	no	—
37	C₂×SD₁₆	—	23	2	no	—
38	C₄⋊Q₈	—	23	1	no	—
39	#124	D₂₃	1	1	yes	characteristic
40	#97	—	23	1	no	—
41	C₂₃	#497	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₈	—	23	2	no	—
43	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
44	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
45	Q₁₆	—	46	4	no	—
46	SD₁₆	—	46	4	no	—
47	C₂×C₈	—	23	2	no	—
48	C₄⋊C₄	—	46	2	no	—
49	C₂²⋊C₄	—	2	2	no	—
50	C₄²	C₂×D₂₃	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	46	2	no	—
54	Q₈	—	46	2	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	46	2	no	—
58	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
59	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
60	C₈	—	23	4	no	—
61	C₂²	—	4	1	no	—
62	C₂²	—	4	2	no	—
63	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	92	2	no	—
67	C₄	D₄×D₂₃	1	2	yes	—
68	C₂	—	8	1	no	—
69	C₂	#10625111	1	1	yes	characteristic
70	C₂	#10625122	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#124	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂×D₄×C₂₃	C₂²
hypercenter	#124	D₂₃
hypocenter	C₂₃	#497
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
2. C₂×#67275	C₂	semidirect
4. #10625040	C₂	semidirect
5. #10624985	C₂	semidirect
41. C₂₃	#497	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#124	2
C₂×#67275	2
C₂×#67273	2
#10625040	2
#10624985	2
#497	23