Group #13093171 — subgroups

Subgroups — #13093171

Order 1472.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625069	C₂	1	2	yes	maximal, complemented
3	#10625069	C₂	1	2	yes	maximal, complemented
4	#10625068	C₂	1	2	yes	maximal, complemented
5	#10625067	C₂	1	2	yes	maximal, complemented
6	#10625065	C₂	1	2	yes	maximal
7	#10625059	C₂	1	2	yes	maximal, complemented
8	#10625046	C₂	1	1	yes	maximal, characteristic
9	#10625041	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₉₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₂₃	C₂²	1	2	yes	—
12	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₆	C₂²	1	2	yes	—
14	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
24	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₂₃	C₂²	1	2	yes	—
27	Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	Dic₂₃⋊C₄	C₂²	1	2	yes	—
29	Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	C₄×Dic₂₃	C₂²	1	2	yes	—
31	C₂²×D₂₃	C₂³	1	2	yes	—
32	C₂×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×Dic₂₃	C₂³	1	2	yes	—
38	C₂×Dic₂₃	C₂³	1	2	yes	—
39	C₂×Dic₂₃	C₂³	1	2	yes	—
40	D₉₂	—	4	1	no	—
41	C₄×D₂₃	—	4	2	no	—
42	Dic₄₆	—	4	1	no	—
43	Dic₄₆	—	4	2	no	—
44	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₂₃	—	2	2	no	—
46	C₂×D₂₃	—	2	2	no	—
47	C₂×D₂₃	—	2	2	no	—
48	C₄×C₂₃	—	2	2	no	—
49	C₄×C₂₃	—	2	1	no	—
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	—	2	2	no	—
52	C₄×C₂₃	—	2	1	no	—
53	Dic₂₃	—	2	2	no	—
54	Dic₂₃	—	2	2	no	—
55	Dic₂₃	—	2	2	no	—
56	#563	—	23	1	no	maximal, Sylow
57	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
58	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
59	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
60	D₂₃	—	4	2	no	—
61	C₄⋊Q₈	—	23	2	no	—
62	#126	—	23	2	no	—
63	#126	—	23	2	no	—
64	#125	—	23	1	no	—
65	#125	D₂₃	1	1	yes	characteristic, complemented
66	#124	—	23	1	no	—
67	#123	—	23	2	no	—
68	C₂²⋊Q₈	—	23	2	no	—
69	C₂²⋊Q₈	—	23	2	no	—
70	C₂₃	#563	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₈	—	23	2	no	—
72	C₂×Q₈	—	23	1	no	—
73	C₂×D₄	—	23	1	no	—
74	C₂²×C₄	—	23	2	no	—
75	C₄⋊C₄	—	23	1	no	—
76	C₄⋊C₄	—	23	2	no	—
77	C₄⋊C₄	—	23	2	no	—
78	C₄⋊C₄	—	23	2	no	—
79	C₄⋊C₄	—	23	2	no	—
80	C₄⋊C₄	—	23	1	no	—
81	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
83	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
84	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
85	C₂²⋊C₄	—	23	2	no	—
86	C₂²⋊C₄	—	23	2	no	—
87	C₂²⋊C₄	—	23	2	no	—
88	C₂²⋊C₄	—	23	2	no	—
89	C₂²⋊C₄	—	23	2	no	—
90	C₄²	—	23	2	no	—
91	C₄²	C₂×D₂₃	1	1	yes	characteristic
92	C₂³	—	23	2	no	—
93	Q₈	—	92	2	no	—
94	Q₈	—	92	1	no	—
95	D₄	—	92	1	no	—
96	C₂×C₄	—	23	2	no	—
97	C₂×C₄	—	23	2	no	—
98	C₂×C₄	—	23	2	no	—
99	C₂×C₄	—	92	2	no	—
100	C₂×C₄	C₂²×D₂₃	1	2	yes	—
101	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₂₃	1	2	yes	—
105	C₂²	—	46	2	no	—
106	C₂²	—	46	2	no	—
107	C₂²	—	46	2	no	—
108	C₂²	C₂³×D₂₃	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	2	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	46	2	no	—
115	C₄	—	46	2	no	—
116	C₄	—	46	2	no	—
117	C₂	—	92	2	no	—
118	C₂	#10625158	1	1	yes	characteristic
119	C₂	#10625160	1	1	yes	characteristic
120	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#125

C₂

yes

maximal, characteristic, complemented

#10625069

C₂

yes

maximal, complemented

#10625069

C₂

yes

maximal, complemented

#10625068

C₂

yes

maximal, complemented

#10625067

C₂

yes

maximal, complemented

#10625065

C₂

yes

maximal

#10625059

C₂

yes

maximal, complemented

#10625046

C₂

yes

maximal, characteristic

#10625041

C₂

yes

maximal, characteristic, complemented

C₂×D₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₄²×C₂₃

C₂²

yes

characteristic

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₂²×D₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

—

Dic₄₆

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

C₂×C₂₃

yes

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

D₂₃

—

C₄⋊Q₈

—

—

—

—

D₂₃

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂₃

#563

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

—

C₄⋊C₄

C₂×D₂₃

yes

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₂₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²×D₂₃

yes

—

101

C₂×C₄

C₂²×D₂₃

yes

characteristic

102

C₂×C₄

C₂²×D₂₃

yes

characteristic

103

C₂×C₄

C₂²×D₂₃

yes

characteristic

104

C₂×C₄

C₂²×D₂₃

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂³×D₂₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₂

—

118

C₂

#10625158

yes

characteristic

119

C₂

#10625160

yes

characteristic

120

C₂

#10625161

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#125	C₂
prime radical	C₂×D₉₂	C₂²
hypercenter	#125	D₂₃
hypocenter	C₂₃	#563
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#125

C₂

prime radical

C₂×D₉₂

C₂²

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₂₃×#125	C₂	semidirect
2. #10625069	C₂	semidirect
3. #10625069	C₂	semidirect
4. #10625068	C₂	semidirect
5. #10625067	C₂	semidirect
7. #10625059	C₂	semidirect
9. #10625041	C₂	semidirect
65. #125	D₂₃	semidirect
70. C₂₃	#563	semidirect

H ≅ G/N

type

1. C₂₃×#125

C₂

semidirect

2. #10625069

C₂

semidirect

3. #10625069

C₂

semidirect

4. #10625068

C₂

semidirect

5. #10625067

C₂

semidirect

7. #10625059

C₂

semidirect

9. #10625041

C₂

semidirect

65. #125

D₂₃

semidirect

70. C₂₃

#563

semidirect

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

subgroup	index
C₂₃×#125	2
#10625069	2
#10625068	2
#10625067	2
#10625065	2
#10625059	2
#10625046	2
#10625041	2
#563	23

subgroup

index

C₂₃×#125

Subgroups — #13093171

Conjugacy classes of subgroups

Complements

Maximal subgroups