Subgroups — #13092891

Order 1472.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625097	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625064	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₄×D₂₃	C₂	1	4	yes	maximal, complemented
6	C₄×D₉₂	C₂	1	4	yes	maximal, complemented
7	C₂³×D₂₃	C₂²	1	2	yes	—
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×D₉₂	C₄	1	8	yes	complemented
10	C₂×D₉₂	C₂²	1	2	yes	—
11	C₂×D₉₂	C₂²	1	2	yes	—
12	C₂×C₄×D₂₃	C₂²	1	8	yes	complemented
13	C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
14	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₄²×C₂₃	C₂²	1	2	yes	complemented
16	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
17	(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	complemented
18	C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
19	C₄×Dic₂₃	C₂²	1	2	yes	complemented
20	C₂³×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×D₂₃	—	4	4	no	—
22	C₂²×D₂₃	C₂×C₄	1	8	yes	complemented
23	C₂²×D₂₃	C₂³	1	4	yes	—
24	C₂×C₄×C₂₃	C₂×C₄	1	4	yes	complemented
25	C₂×C₄×C₂₃	C₂³	1	4	yes	—
26	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₂₃	C₂³	1	4	yes	—
29	D₉₂	C₂×C₄	1	8	yes	complemented
30	D₉₂	C₂×C₄	1	8	yes	complemented
31	C₄×D₂₃	—	4	8	no	—
32	C₂²×C₂₃	—	2	2	no	—
33	C₂²×C₂₃	C₂²×C₄	1	2	yes	—
34	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
35	C₂×D₂₃	—	2	8	no	—
36	C₂×D₂₃	—	4	8	no	—
37	C₂×D₂₃	C₂²×C₄	1	8	yes	—
38	C₄×C₂₃	—	2	4	no	—
39	C₄×C₂₃	C₂²×C₄	1	2	yes	—
40	C₄×C₂₃	C₂²×C₄	1	2	yes	—
41	Dic₂₃	—	2	4	no	—
42	#518	—	23	1	no	maximal, Sylow
43	C₂×C₂₃	—	2	2	no	—
44	C₂×C₂₃	C₂³×C₄	1	1	yes	characteristic
45	C₂×C₂₃	H(5,𝔽₂)	1	2	yes	—
46	D₂₃	—	2	8	no	—
47	C₂²×D₄	—	23	1	no	—
48	C₄×D₄	—	23	4	no	—
49	C₄×D₄	—	23	4	no	—
50	#117	—	23	1	no	—
51	#117	D₂₃	1	1	yes	characteristic, complemented
52	C₂×C₂²⋊C₄	—	23	4	no	—
53	C₂₃	#518	1	1	yes	characteristic, complemented, Sylow
54	C₂⁴	—	23	2	no	—
55	C₂×D₄	—	23	8	no	—
56	C₂×D₄	—	23	2	no	—
57	C₂×D₄	—	23	2	no	—
58	C₂²×C₄	—	23	8	no	—
59	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
60	C₄⋊C₄	—	23	2	no	—
61	C₄⋊C₄	C₂×D₂₃	1	2	yes	complemented
62	C₂²⋊C₄	—	23	2	no	—
63	C₂²⋊C₄	—	23	8	no	—
64	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
65	C₄²	—	23	2	no	—
66	C₄²	C₂×D₂₃	1	2	yes	complemented
67	C₂³	—	23	4	no	—
68	C₂³	—	23	8	no	—
69	C₂³	—	92	4	no	—
70	C₂³	C₂²×D₂₃	1	1	yes	characteristic
71	D₄	—	23	8	no	—
72	D₄	—	23	8	no	—
73	C₂×C₄	—	23	4	no	—
74	C₂×C₄	—	92	8	no	—
75	C₂×C₄	C₄×D₂₃	1	4	yes	complemented
76	C₂×C₄	C₂²×D₂₃	1	4	yes	—
77	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	—	23	8	no	—
81	C₂²	—	46	8	no	—
82	C₂²	—	92	8	no	—
83	C₂²	C₂×C₄×D₂₃	1	2	yes	—
84	C₂²	C₂³×D₂₃	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	46	4	no	—
87	C₄	C₂×C₄×D₂₃	1	2	yes	—
88	C₄	C₂×C₄×D₂₃	1	2	yes	—
89	C₂	—	2	2	no	—
90	C₂	—	46	8	no	—
91	C₂	C₂²×C₄×D₂₃	1	1	yes	characteristic
92	C₂	#10625160	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator	C₂×C₂₃	C₂³×C₄
Fitting	C₂₃×#117	C₂
socle, cosocle	C₂²×C₂₃	C₂⁴
prime radical	C₂²×D₉₂	C₂
hypercenter	#117	D₂₃
hypocenter	C₂₃	#518
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₂₃×#117	C₂	semidirect
3. #10625097	C₂	semidirect
4. #10625064	C₂	semidirect
5. C₂²⋊C₄×D₂₃	C₂	semidirect
6. C₄×D₉₂	C₂	semidirect
9. C₂×D₉₂	C₄	semidirect
12. C₂×C₄×D₂₃	C₂²	semidirect
13. C₄⋊C₄×C₂₃	C₂²	semidirect
15. C₄²×C₂₃	C₂²	semidirect
17. (C₂×D₂₃)⋊C₄	C₂²	semidirect
18. C₄⋊Dic₂₃	C₂²	semidirect
19. C₄×Dic₂₃	C₂²	semidirect
22. C₂²×D₂₃	C₂×C₄	semidirect
24. C₂×C₄×C₂₃	C₂×C₄	semidirect
29. D₉₂	C₂×C₄	semidirect
30. D₉₂	C₂×C₄	semidirect
51. #117	D₂₃	semidirect
53. C₂₃	#518	semidirect
61. C₄⋊C₄	C₂×D₂₃	semidirect
66. C₄²	C₂×D₂₃	semidirect
75. C₂×C₄	C₄×D₂₃	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
C₂₃×#117	2
#10625097	2
#10625064	2
C₂²⋊C₄×D₂₃	2
C₄×D₉₂	2
#518	23