Group #13093144 — subgroups

Subgroups — #13093144

Order 1472.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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
2	#10625061	C₂	1	1	yes	maximal, characteristic
3	#10625060	C₂	1	2	yes	maximal
4	#10625060	C₂	1	2	yes	maximal
5	#10625059	C₂	1	1	yes	maximal, characteristic
6	#10625059	C₂	1	2	yes	maximal
7	#10625058	C₂	1	1	yes	maximal, characteristic
8	#10625058	C₂	1	2	yes	maximal
9	#10625058	C₂	1	1	yes	maximal, characteristic
10	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₆	C₂²	1	2	yes	—
14	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
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	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₂₃	C₂²	1	2	yes	—
23	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
24	Dic₂₃⋊C₄	C₂²	1	2	yes	—
25	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
26	Dic₂₃⋊C₄	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	1	yes	characteristic
31	C₄×Dic₂₃	C₂²	1	2	yes	—
32	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₃	C₂²	1	2	yes	—
34	C₄×Dic₂₃	C₂²	1	2	yes	—
35	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
41	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₂₃	C₂³	1	2	yes	—
43	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
44	C₂×Dic₂₃	C₂³	1	2	yes	—
45	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
46	Dic₄₆	—	2	2	no	—
47	Dic₄₆	—	2	4	no	—
48	Dic₄₆	Q₈	1	4	yes	complemented
49	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₄×C₂₃	—	2	2	no	—
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	—	2	1	no	—
53	C₄×C₂₃	—	2	2	no	—
54	C₄×C₂₃	C₂×Q₈	1	2	yes	—
55	Dic₂₃	—	2	1	no	—
56	Dic₂₃	—	2	2	no	—
57	Dic₂₃	—	2	1	no	—
58	Dic₂₃	—	2	1	no	—
59	Dic₂₃	C₂×Q₈	1	4	yes	—
60	Dic₂₃	#41	1	2	yes	—
61	#557	—	23	1	no	maximal, Sylow
62	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
63	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
64	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
65	C₄⋊Q₈	—	23	2	no	—
66	C₄⋊Q₈	—	23	1	no	—
67	#125	—	23	2	no	—
68	#125	—	23	2	no	—
69	#125	—	23	1	no	—
70	#125	D₂₃	1	1	yes	characteristic
71	C₄×Q₈	—	23	1	no	—
72	C₄×Q₈	—	23	2	no	—
73	C₄×Q₈	—	23	1	no	—
74	C₄×Q₈	—	23	1	no	—
75	C₄×Q₈	—	23	1	no	—
76	C₂₃	#557	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₈	—	23	1	no	—
78	C₂×Q₈	—	23	2	no	—
79	C₂×Q₈	—	23	1	no	—
80	C₄⋊C₄	—	23	1	no	—
81	C₄⋊C₄	—	23	2	no	—
82	C₄⋊C₄	—	23	2	no	—
83	C₄⋊C₄	—	23	1	no	—
84	C₄⋊C₄	—	23	1	no	—
85	C₄⋊C₄	—	23	2	no	—
86	C₄⋊C₄	—	23	2	no	—
87	C₄⋊C₄	—	23	1	no	—
88	C₄⋊C₄	—	23	2	no	—
89	C₄⋊C₄	—	23	2	no	—
90	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
91	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
92	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
93	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
94	C₄²	—	23	2	no	—
95	C₄²	—	23	2	no	—
96	C₄²	—	23	1	no	—
97	C₄²	—	23	2	no	—
98	C₄²	—	23	1	no	—
99	C₄²	C₂×D₂₃	1	1	yes	characteristic
100	Q₈	—	23	4	no	—
101	Q₈	—	46	4	no	—
102	Q₈	—	46	2	no	—
103	C₂×C₄	—	23	2	no	—
104	C₂×C₄	—	23	1	no	—
105	C₂×C₄	—	23	1	no	—
106	C₂×C₄	—	23	1	no	—
107	C₂×C₄	—	23	1	no	—
108	C₂×C₄	—	23	2	no	—
109	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
110	C₂×C₄	C₂²×D₂₃	1	2	yes	—
111	C₂×C₄	C₂²×D₂₃	1	2	yes	—
112	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
114	C₂²	C₂³×D₂₃	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	23	4	no	—
120	C₄	—	23	2	no	—
121	C₄	—	46	1	no	—
122	C₄	—	46	2	no	—
123	C₄	—	46	1	no	—
124	C₄	—	46	1	no	—
125	C₄	Q₈×D₂₃	1	2	yes	—
126	C₂	C₂×Q₈×D₂₃	1	1	yes	characteristic
127	C₂	#41×D₂₃	1	1	yes	characteristic
128	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#125

C₂

yes

maximal, characteristic

#10625061

C₂

yes

maximal, characteristic

#10625060

C₂

yes

maximal

#10625060

C₂

yes

maximal

#10625059

C₂

yes

maximal, characteristic

#10625059

C₂

yes

maximal

#10625058

C₂

yes

maximal, characteristic

#10625058

C₂

yes

maximal

#10625058

C₂

yes

maximal, characteristic

C₄×Dic₄₆

C₂

yes

maximal, characteristic

C₄×Dic₄₆

C₂

yes

maximal, characteristic

C₂×Dic₄₆

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄²×C₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

Dic₄₆

—

Dic₄₆

—

Dic₄₆

Q₈

yes

complemented

C₂²×C₂₃

C₂⁴

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₂×Q₈

yes

—

Dic₂₃

#41

yes

—

#557

—

maximal, Sylow

C₂×C₂₃

C₂²×Q₈

yes

characteristic

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

D₂₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂₃

#557

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₂₃

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²×D₂₃

yes

characteristic

110

C₂×C₄

C₂²×D₂₃

yes

—

111

C₂×C₄

C₂²×D₂₃

yes

—

112

C₂×C₄

C₂²×D₂₃

yes

characteristic

113

C₂×C₄

C₂²×D₂₃

yes

characteristic

114

C₂²

C₂³×D₂₃

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

Q₈×D₂₃

yes

—

126

C₂

C₂×Q₈×D₂₃

yes

characteristic

127

C₂

#41×D₂₃

yes

characteristic

128

C₂

#10625161

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, prime radical, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#125	C₂
hypercenter	#125	D₂₃
hypocenter	C₂₃	#557
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, prime radical, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#125

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
48. Dic₄₆	Q₈	semidirect
76. C₂₃	#557	semidirect

H ≅ G/N

type

48. Dic₄₆

Q₈

semidirect

76. C₂₃

#557

semidirect

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

subgroup	index
C₂₃×#125	2
#10625061	2
#10625060	2
#10625059	2
#10625058	2
C₄×Dic₄₆	2
#557	23

subgroup

index

C₂₃×#125

C₄×Dic₄₆

Subgroups — #13093144

Conjugacy classes of subgroups

Complements

Maximal subgroups