Subgroups — #13093154

Order 1472.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625110	C₂	1	6	yes	maximal, complemented
3	#10625108	C₂	1	3	yes	maximal, complemented
4	#10625106	C₂	1	3	yes	maximal, complemented
5	#10625047	C₂	1	2	yes	maximal, complemented
6	C₂³×D₂₃	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₂₃	C₂²	1	3	yes	—
8	C₂×D₄×C₂₃	C₂²	1	3	yes	complemented
9	C₂×C₂²⋊D₂₃	C₂²	1	6	yes	—
10	C₂²×Dic₂₃	C₂²	1	3	yes	complemented
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
12	C₂²⋊Dic₂₃	C₂²	1	3	yes	—
13	C₂²⋊Dic₂₃	C₂²	1	3	yes	complemented
14	(C₂×D₂₃)⋊C₄	C₂²	1	6	yes	—
15	Dic₂₃⋊C₄	C₂²	1	6	yes	complemented
16	C₂³×C₂₃	C₂³	1	1	yes	characteristic
17	C₂³×C₂₃	C₂³	1	3	yes	—
18	C₂²×D₂₃	—	4	3	no	—
19	C₂²×D₂₃	C₂³	1	2	yes	—
20	D₄×C₂₃	—	4	3	no	—
21	D₄×C₂₃	—	4	3	no	—
22	C₂×C₄×C₂₃	C₂³	1	3	yes	—
23	C₂²⋊D₂₃	—	4	6	no	—
24	C₂×Dic₂₃	—	4	3	no	—
25	C₂×Dic₂₃	C₂³	1	6	yes	—
26	C₂²×C₂₃	—	2	3	no	—
27	C₂²×C₂₃	—	2	6	no	—
28	C₂²×C₂₃	—	2	3	no	—
29	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
30	C₂×D₂₃	—	2	6	no	—
31	C₂×D₂₃	—	4	4	no	—
32	C₄×C₂₃	—	2	3	no	—
33	Dic₂₃	—	2	6	no	—
34	#560	—	23	1	no	maximal, Sylow
35	C₂×C₂₃	—	4	1	no	—
36	C₂×C₂₃	—	4	3	no	—
37	C₂×C₂₃	H(5,𝔽₂)	1	3	yes	—
38	D₂₃	—	4	2	no	—
39	#127	D₂₃	1	1	yes	characteristic, complemented
40	#126	—	23	2	no	—
41	#123	—	23	3	no	—
42	#121	—	23	6	no	—
43	#120	—	23	3	no	—
44	C₂₃	#560	1	1	yes	characteristic, complemented, Sylow
45	C₂⁴	—	23	1	no	—
46	C₂×D₄	—	23	6	no	—
47	C₂×D₄	C₂×D₂₃	1	3	yes	—
48	C₂×D₄	C₂×D₂₃	1	3	yes	complemented
49	C₂²×C₄	—	23	3	no	—
50	C₄⋊C₄	—	23	6	no	—
51	C₂²⋊C₄	—	23	3	no	—
52	C₂²⋊C₄	—	23	3	no	—
53	C₂²⋊C₄	—	23	6	no	—
54	C₄²	C₂×D₂₃	1	1	yes	characteristic, complemented
55	C₂³	—	23	2	no	—
56	C₂³	—	92	3	no	—
57	C₂³	C₂²×D₂₃	1	1	yes	characteristic
58	C₂³	C₂²×D₂₃	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	92	6	no	—
62	C₂×C₄	—	23	6	no	—
63	C₂×C₄	—	92	3	no	—
64	C₂×C₄	C₂²×D₂₃	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	46	6	no	—
69	C₂²	—	92	4	no	—
70	C₂²	C₂³×D₂₃	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	46	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	92	2	no	—
76	C₂	#10625155	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#127	C₂
hypercenter	#127	D₂₃
hypocenter	C₂₃	#560
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂₃×#127	C₂	semidirect
2. #10625110	C₂	semidirect
3. #10625108	C₂	semidirect
4. #10625106	C₂	semidirect
5. #10625047	C₂	semidirect
8. C₂×D₄×C₂₃	C₂²	semidirect
10. C₂²×Dic₂₃	C₂²	semidirect
11. C₄²×C₂₃	C₂²	semidirect
13. C₂²⋊Dic₂₃	C₂²	semidirect
15. Dic₂₃⋊C₄	C₂²	semidirect
39. #127	D₂₃	semidirect
44. C₂₃	#560	semidirect
48. C₂×D₄	C₂×D₂₃	semidirect
54. C₄²	C₂×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#127	2
#10625110	2
#10625108	2
#10625106	2
#10625047	2
#560	23