Subgroups — #13092458

Order 1472.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#67262	C₂	1	1	yes	maximal, characteristic
2	#10625063	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	#10624999	C₂	1	1	yes	maximal, characteristic, complemented
6	#10624990	C₂	1	1	yes	maximal, characteristic, complemented
7	#10624989	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₃	—	2	1	no	—
9	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
13	(C₂×D₂₃)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
16	Dic₂₃⋊C₄	—	2	1	no	—
17	C₄×Dic₂₃	—	2	1	no	—
18	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	#67262	C₄	1	4	yes	complemented
20	C₂²×D₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	—	2	1	no	—
22	C₂×C₄×C₂₃	—	2	1	no	—
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₃	—	2	1	no	—
25	C₂×Dic₂₃	—	2	1	no	—
26	C₂×Dic₂₃	D₄	1	1	yes	characteristic
27	C₄×D₂₃	—	4	2	no	—
28	C₄×D₂₃	C₂×C₄	1	2	yes	—
29	C₄×D₂₃	Q₈	1	2	yes	—
30	C₈×C₂₃	C₂×C₄	1	2	yes	complemented
31	C₂₃⋊C₈	C₂×C₄	1	2	yes	complemented
32	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₃	—	2	1	no	—
34	C₂×D₂₃	—	2	1	no	—
35	C₂×D₂₃	C₄⋊C₄	1	2	yes	—
36	C₄×C₂₃	—	4	1	no	—
37	C₄×C₂₃	—	4	1	no	—
38	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	Dic₂₃	—	4	1	no	—
41	Dic₂₃	—	4	1	no	—
42	Dic₂₃	C₄⋊C₄	1	2	yes	—
43	#428	—	23	1	no	maximal, Sylow
44	C₂×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
46	C₂×C₂₃	#137	1	1	yes	characteristic
47	D₂₃	—	2	2	no	—
48	C₂×M₁₆	—	23	1	no	—
49	#117	—	23	1	no	—
50	C₂×C₄⋊C₄	—	23	1	no	—
51	#107	—	23	1	no	—
52	#107	D₂₃	1	1	yes	characteristic, complemented
53	#106	—	23	1	no	—
54	#106	—	23	1	no	—
55	C₂₃	#428	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₄	—	23	1	no	—
57	C₂²×C₄	—	46	1	no	—
58	M₁₆	—	23	4	no	—
59	C₂×C₈	—	23	1	no	—
60	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
61	C₄⋊C₄	—	23	1	no	—
62	C₄⋊C₄	—	23	1	no	—
63	C₄⋊C₄	—	46	1	no	—
64	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
65	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
66	C₂²⋊C₄	—	46	1	no	—
67	C₄²	—	46	1	no	—
68	C₂³	—	23	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	23	1	no	—
72	C₂×C₄	—	23	2	no	—
73	C₂×C₄	—	23	2	no	—
74	C₂×C₄	—	46	1	no	—
75	C₂×C₄	—	46	1	no	—
76	C₂×C₄	—	92	2	no	—
77	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
78	C₈	—	23	2	no	—
79	C₈	C₄×D₂₃	1	2	yes	complemented
80	C₂²	—	23	2	no	—
81	C₂²	—	46	1	no	—
82	C₂²	—	46	1	no	—
83	C₂²	D₄×D₂₃	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	23	2	no	—
87	C₄	—	92	1	no	—
88	C₄	—	92	1	no	—
89	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
90	C₄	Q₈×D₂₃	1	1	yes	characteristic
91	C₂	—	46	2	no	—
92	C₂	C₄⋊C₄×D₂₃	1	1	yes	characteristic
93	C₂	#10625082	1	1	yes	characteristic
94	C₂	#10625089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#107	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂²×D₂₃	D₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	#107	D₂₃
hypocenter	C₂₃	#428
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
4. C₂₃×#107	C₂	semidirect
5. #10624999	C₂	semidirect
6. #10624990	C₂	semidirect
7. #10624989	C₂	semidirect
19. #67262	C₄	semidirect
30. C₈×C₂₃	C₂×C₄	semidirect
31. C₂₃⋊C₈	C₂×C₄	semidirect
52. #107	D₂₃	semidirect
55. C₂₃	#428	semidirect
79. C₈	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#67262	2
#10625063	2
C₄⋊C₄×D₂₃	2
C₂₃×#107	2
#10624999	2
#10624990	2
#10624989	2
#428	23