Subgroups — #13092979

Order 1472.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625101	C₂	1	4	yes	maximal, complemented
4	#10625057	C₂	1	4	yes	maximal, complemented
5	#10625053	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×D₂₃	C₂²	1	2	yes	complemented
8	C₂²×C₄×C₂₃	C₂²	1	2	yes	complemented
9	C₂×C₂²⋊D₂₃	—	2	8	no	—
10	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	—
11	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	complemented
12	C₂²×Dic₂₃	C₂²	1	2	yes	complemented
13	C₂×D₉₂	C₂²	1	4	yes	complemented
14	C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
15	(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	complemented
16	C₄⋊Dic₂₃	C₂²	1	4	yes	complemented
17	C₂³×C₂₃	—	4	2	no	—
18	C₂³×C₂₃	D₄	1	4	yes	complemented
19	C₂³×C₂₃	C₂³	1	2	yes	—
20	C₂³×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×D₂₃	—	2	4	no	—
22	C₂²×D₂₃	—	4	4	no	—
23	C₂²×D₂₃	C₂³	1	4	yes	—
24	C₂×C₄×C₂₃	—	4	2	no	—
25	C₂×C₄×C₂₃	C₂³	1	4	yes	—
26	C₂²⋊D₂₃	—	2	8	no	—
27	C₂²⋊D₂₃	—	2	8	no	—
28	C₂×Dic₂₃	—	2	4	no	—
29	C₂×Dic₂₃	C₂³	1	4	yes	—
30	D₉₂	—	4	4	no	—
31	C₂²×C₂₃	—	2	2	no	—
32	C₂²×C₂₃	—	2	4	no	—
33	C₂²×C₂₃	—	4	4	no	—
34	C₂²×C₂₃	C₂×D₄	1	4	yes	—
35	C₂²×C₂₃	C₂×D₄	1	2	yes	—
36	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
37	C₂×D₂₃	—	2	4	no	—
38	C₂×D₂₃	—	2	8	no	—
39	C₂×D₂₃	—	4	8	no	—
40	C₄×C₂₃	—	2	4	no	—
41	Dic₂₃	—	2	4	no	—
42	#534	—	23	1	no	maximal, Sylow
43	C₂×C₂₃	—	2	4	no	—
44	C₂×C₂₃	—	2	2	no	—
45	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
46	C₂×C₂₃	H(5,𝔽₂)	1	2	yes	—
47	D₂₃	—	4	4	no	—
48	C₂²×D₄	—	23	2	no	—
49	#123	—	23	4	no	—
50	#121	—	23	4	no	—
51	#120	—	23	4	no	—
52	C₂×C₂²⋊C₄	D₂₃	1	1	yes	characteristic, complemented
53	C₂₃	#534	1	1	yes	characteristic, complemented, Sylow
54	C₂⁴	—	23	2	no	—
55	C₂⁴	C₂×D₂₃	1	1	yes	characteristic
56	C₂×D₄	—	23	4	no	—
57	C₂×D₄	—	23	4	no	—
58	C₂×D₄	—	23	4	no	—
59	C₂×D₄	—	46	8	no	—
60	C₂²×C₄	—	23	2	no	—
61	C₂²×C₄	C₂×D₂₃	1	2	yes	complemented
62	C₄⋊C₄	—	23	4	no	—
63	C₂²⋊C₄	—	23	8	no	—
64	C₂²⋊C₄	C₂×D₂₃	1	4	yes	complemented
65	C₂³	—	4	2	no	—
66	C₂³	—	23	4	no	—
67	C₂³	—	46	4	no	—
68	C₂³	—	92	4	no	—
69	C₂³	D₉₂	1	4	yes	complemented
70	C₂³	C₂²×D₂₃	1	2	yes	—
71	C₂³	C₂²×D₂₃	1	1	yes	characteristic
72	D₄	—	46	8	no	—
73	D₄	—	46	8	no	—
74	D₄	—	92	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	23	4	no	—
77	C₂×C₄	—	46	4	no	—
78	C₂×C₄	C₂²×D₂₃	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	4	4	no	—
82	C₂²	—	46	4	no	—
83	C₂²	—	46	8	no	—
84	C₂²	—	92	8	no	—
85	C₂²	C₂×D₉₂	1	4	yes	—
86	C₂²	C₂×D₉₂	1	2	yes	—
87	C₂²	C₂³×D₂₃	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	46	4	no	—
90	C₂	—	2	4	no	—
91	C₂	—	2	2	no	—
92	C₂	—	92	4	no	—
93	C₂	C₂²×D₉₂	1	1	yes	characteristic
94	C₂	#10625155	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₂₃	C₂
hypercenter	C₂×C₂²⋊C₄	D₂₃
hypocenter	C₂₃	#534
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₂₃	C₂	semidirect
2. C₂×C₂²⋊C₄×C₂₃	C₂	semidirect
3. #10625101	C₂	semidirect
4. #10625057	C₂	semidirect
5. #10625053	C₂	semidirect
7. C₂³×D₂₃	C₂²	semidirect
8. C₂²×C₄×C₂₃	C₂²	semidirect
11. C₂×C₂²⋊D₂₃	C₂²	semidirect
12. C₂²×Dic₂₃	C₂²	semidirect
13. C₂×D₉₂	C₂²	semidirect
14. C₂²⋊C₄×C₂₃	C₂²	semidirect
15. (C₂×D₂₃)⋊C₄	C₂²	semidirect
16. C₄⋊Dic₂₃	C₂²	semidirect
18. C₂³×C₂₃	D₄	semidirect
52. C₂×C₂²⋊C₄	D₂₃	semidirect
53. C₂₃	#534	semidirect
61. C₂²×C₄	C₂×D₂₃	semidirect
64. C₂²⋊C₄	C₂×D₂₃	semidirect
69. C₂³	D₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊D₂₃	2
C₂×C₂²⋊C₄×C₂₃	2
#10625101	2
#10625057	2
#10625053	2
#534	23