Subgroups — #13092609

Order 1472.

140 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	C₂×#67272	C₂	1	2	yes	maximal, complemented
3	C₄×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625014	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₂₃	—	2	1	no	—
7	C₂×D₄×C₂₃	—	2	1	no	—
8	C₂×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂×D₉₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₃	—	2	1	no	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	#67272	—	2	4	no	—
13	(C₂×D₂₃)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
16	C₂³×C₂₃	—	2	2	no	—
17	C₂²×D₂₃	—	2	1	no	—
18	D₄×C₂₃	—	2	2	no	—
19	D₄×C₂₃	—	2	2	no	—
20	D₄×C₂₃	—	4	2	no	—
21	D₄×C₂₃	—	4	2	no	—
22	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₂₃	—	2	1	no	—
26	D₉₂	—	2	1	no	—
27	D₉₂	D₄	1	2	yes	complemented
28	C₄×D₂₃	—	4	2	no	—
29	C₂₃⋊C₈	—	2	2	no	—
30	C₂²×C₂₃	—	4	2	no	—
31	C₂²×C₂₃	—	4	2	no	—
32	C₂²×C₂₃	—	4	2	no	—
33	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×D₂₃	—	2	2	no	—
35	C₂×D₂₃	—	4	1	no	—
36	C₂×D₂₃	—	4	1	no	—
37	C₄×C₂₃	—	2	1	no	—
38	C₄×C₂₃	D₈	1	2	yes	complemented
39	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₄×C₂₃	#41	1	1	yes	characteristic
41	Dic₂₃	—	4	1	no	—
42	#459	—	23	1	no	maximal, Sylow
43	C₂×C₂₃	—	8	2	no	—
44	C₂×C₂₃	#121	1	1	yes	characteristic
45	C₂×C₂₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₂₃	—	4	2	no	—
48	C₂×D₈	—	23	2	no	—
49	#127	D₂₃	1	1	yes	characteristic, complemented
50	C₄×D₄	—	23	1	no	—
51	C₄⋊C₈	—	23	1	no	—
52	#102	—	23	2	no	—
53	C₂₃	#459	1	1	yes	characteristic, complemented, Sylow
54	C₂×D₄	—	2	1	no	—
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	—	23	1	no	—
57	C₂×D₄	C₂×D₂₃	1	2	yes	—
58	C₂²×C₄	—	46	1	no	—
59	D₈	—	46	4	no	—
60	C₂×C₈	—	23	2	no	—
61	C₄⋊C₄	—	23	1	no	—
62	C₂²⋊C₄	—	46	1	no	—
63	C₄²	C₂×D₂₃	1	1	yes	characteristic
64	C₂³	—	2	2	no	—
65	C₂³	—	46	1	no	—
66	D₄	—	2	2	no	—
67	D₄	—	2	2	no	—
68	D₄	—	4	2	no	—
69	D₄	—	4	2	no	—
70	D₄	—	23	2	no	—
71	D₄	—	46	1	no	—
72	C₂×C₄	—	46	1	no	—
73	C₂×C₄	—	92	2	no	—
74	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
77	C₈	—	46	2	no	—
78	C₂²	—	4	2	no	—
79	C₂²	—	4	2	no	—
80	C₂²	—	4	2	no	—
81	C₂²	—	46	2	no	—
82	C₂²	—	92	1	no	—
83	C₂²	—	92	1	no	—
84	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	92	1	no	—
87	C₄	#67272	1	2	yes	complemented
88	C₄	D₄×D₂₃	1	1	yes	characteristic
89	C₄	#67290	1	1	yes	characteristic
90	C₂	—	8	2	no	—
91	C₂	—	92	2	no	—
92	C₂	C₂×#67272	1	1	yes	characteristic
93	C₂	#10625103	1	1	yes	characteristic
94	C₂	#10625109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#127	C₂
socle	C₂²×C₂₃	C₂×D₄
hypercenter	#127	D₂₃
hypocenter	C₂₃	#459
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂₃×#127	C₂	semidirect
2. C₂×#67272	C₂	semidirect
3. C₄×D₉₂	C₂	semidirect
4. #10625014	C₂	semidirect
5. C₄⋊(C₂₃⋊C₈)	C₂	semidirect
27. D₉₂	D₄	semidirect
38. C₄×C₂₃	D₈	semidirect
49. #127	D₂₃	semidirect
53. C₂₃	#459	semidirect
87. C₄	#67272	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
C₂×#67272	2
C₄×D₉₂	2
#10625014	2
C₄⋊(C₂₃⋊C₈)	2
#459	23