Subgroups — #13093052

Order 1472.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₂₃	C₂	1	4	yes	maximal, complemented
2	C₂₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625109	C₂	1	4	yes	maximal, complemented
4	#10625108	C₂	1	4	yes	maximal, complemented
5	C₄×D₉₂	C₂	1	2	yes	maximal, complemented
6	C₂³×D₂₃	C₂²	1	4	yes	complemented
7	C₂×D₄×C₂₃	C₂²	1	4	yes	complemented
8	C₂×D₄×C₂₃	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₂₃	C₂²	1	8	yes	complemented
10	D₄×D₂₃	—	2	16	no	—
11	C₂×D₉₂	C₂²	1	2	yes	—
12	C₂×C₄×D₂₃	C₂²	1	4	yes	complemented
13	C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
14	C₂²⋊Dic₂₃	C₂²	1	4	yes	complemented
15	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	complemented
16	C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
17	C₂³×C₂₃	C₂³	1	4	yes	—
18	C₂²×D₂₃	—	2	16	no	—
19	C₂²×D₂₃	—	4	4	no	—
20	C₂²×D₂₃	C₂³	1	4	yes	—
21	D₄×C₂₃	—	2	8	no	—
22	D₄×C₂₃	—	4	2	no	—
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₃	C₂³	1	2	yes	—
25	C₂²⋊D₂₃	—	2	16	no	—
26	C₂×Dic₂₃	C₂³	1	4	yes	—
27	D₉₂	D₄	1	8	yes	complemented
28	C₄×D₂₃	—	2	8	no	—
29	C₂²×C₂₃	—	2	8	no	—
30	C₂²×C₂₃	—	2	4	no	—
31	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
32	C₂×D₂₃	—	2	4	no	—
33	C₂×D₂₃	—	2	4	no	—
34	C₂×D₂₃	—	4	16	no	—
35	C₂×D₂₃	C₂×D₄	1	8	yes	complemented
36	C₄×C₂₃	—	2	1	no	—
37	C₄×C₂₃	C₂×D₄	1	4	yes	complemented
38	Dic₂₃	—	2	4	no	—
39	D₄²	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	—	4	4	no	—
41	C₂×C₂₃	C₂²×D₄	1	2	yes	—
42	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
43	D₂₃	—	2	8	no	—
44	C₂²×D₄	—	23	4	no	—
45	#127	D₂₃	1	1	yes	characteristic, complemented
46	#121	—	23	4	no	—
47	#120	—	23	4	no	—
48	C₄×D₄	—	23	2	no	—
49	C₂₃	D₄²	1	1	yes	characteristic, complemented, Sylow
50	C₂⁴	—	23	4	no	—
51	C₂×D₄	—	23	8	no	—
52	C₂×D₄	—	23	2	no	—
53	C₂×D₄	—	46	16	no	—
54	C₂×D₄	C₂×D₂₃	1	4	yes	complemented
55	C₂×D₄	C₂×D₂₃	1	2	yes	complemented
56	C₂²×C₄	—	23	4	no	—
57	C₄⋊C₄	—	23	2	no	—
58	C₂²⋊C₄	—	23	4	no	—
59	C₂²⋊C₄	—	23	4	no	—
60	C₄²	C₂×D₂₃	1	1	yes	characteristic, complemented
61	C₂³	—	23	4	no	—
62	C₂³	—	46	16	no	—
63	C₂³	—	92	4	no	—
64	C₂³	C₂²×D₂₃	1	4	yes	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	—	23	8	no	—
68	D₄	—	46	16	no	—
69	C₂×C₄	—	23	4	no	—
70	C₂×C₄	—	46	8	no	—
71	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₂₃	1	2	yes	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	23	8	no	—
76	C₂²	—	46	4	no	—
77	C₂²	—	46	4	no	—
78	C₂²	—	92	16	no	—
79	C₂²	C₂³×D₂₃	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	46	4	no	—
82	C₄	D₄×D₂₃	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	46	8	no	—
85	C₂	C₂×D₄×D₂₃	1	2	yes	—
86	C₂	#10625155	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#127	C₂
hypercenter	#127	D₂₃
hypocenter	C₂₃	D₄²
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. C₂×D₄×D₂₃	C₂	semidirect
2. C₂₃×#127	C₂	semidirect
3. #10625109	C₂	semidirect
4. #10625108	C₂	semidirect
5. C₄×D₉₂	C₂	semidirect
6. C₂³×D₂₃	C₂²	semidirect
7. C₂×D₄×C₂₃	C₂²	semidirect
8. C₂×D₄×C₂₃	C₂²	semidirect
9. C₂×C₂²⋊D₂₃	C₂²	semidirect
12. C₂×C₄×D₂₃	C₂²	semidirect
13. C₄²×C₂₃	C₂²	semidirect
14. C₂²⋊Dic₂₃	C₂²	semidirect
15. (C₂×D₂₃)⋊C₄	C₂²	semidirect
16. C₄⋊Dic₂₃	C₂²	semidirect
27. D₉₂	D₄	semidirect
35. C₂×D₂₃	C₂×D₄	semidirect
37. C₄×C₂₃	C₂×D₄	semidirect
45. #127	D₂₃	semidirect
49. C₂₃	D₄²	semidirect
54. C₂×D₄	C₂×D₂₃	semidirect
55. C₂×D₄	C₂×D₂₃	semidirect
60. C₄²	C₂×D₂₃	semidirect
82. C₄	D₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₂₃	2
C₂₃×#127	2
#10625109	2
#10625108	2
C₄×D₉₂	2
D₄²	23