Group #15683946 — subgroups

Subgroups — #15683946

Order 2944.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#13092974	C₂	1	1	yes	maximal, characteristic
2	#13092827	C₂	1	4	yes	maximal, complemented
3	#13092816	C₂	1	4	yes	maximal, complemented
4	#13092457	C₂	1	4	yes	maximal
5	C₄×#67267	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#125	—	2	2	no	—
8	C₂₃×#125	C₂²	1	2	yes	—
9	C₂₃×#125	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	#10625097	C₂²	1	2	yes	—
14	#10625095	—	2	2	no	—
15	C₂×#67267	C₂²	1	2	yes	—
16	#10625041	C₂²	1	2	yes	—
17	#10625040	C₂²	1	2	yes	—
18	C₄×Dic₄₆	—	2	2	no	—
19	#10624990	C₂²	1	8	yes	—
20	#10624989	C₂²	1	8	yes	—
21	#10624985	C₂²	1	2	yes	—
22	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
23	C₂²×C₄×C₂₃	—	2	2	no	—
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₄₆	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₄⋊C₄×C₂₃	—	2	4	no	—
29	C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
31	C₄²×C₂₃	D₄	1	2	yes	—
32	C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₂₃	—	2	2	no	—
35	C₄⋊Dic₂₃	—	2	2	no	—
36	C₄⋊Dic₂₃	C₂³	1	4	yes	—
37	Dic₂₃⋊C₄	—	2	4	no	—
38	C₄×Dic₂₃	—	2	2	no	—
39	#67267	Q₈	1	8	yes	complemented
40	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
41	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	4	2	no	—
44	C₂×C₄×C₂₃	—	4	4	no	—
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₂₃	—	2	4	no	—
51	Dic₄₆	—	4	4	no	—
52	C₂₃⋊C₈	C₂×Q₈	1	8	yes	—
53	#3166	—	23	1	no	maximal, Sylow
54	C₂²×C₂₃	—	2	2	no	—
55	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₂₃	—	2	2	no	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	—	4	4	no	—
60	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
61	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
62	Dic₂₃	—	4	4	no	—
63	#533	—	23	1	no	—
64	C₂×#125	D₂₃	1	1	yes	characteristic
65	#501	—	23	4	no	—
66	#499	—	23	4	no	—
67	#428	—	23	4	no	—
68	C₄×M₁₆	—	23	1	no	—
69	C₂×C₂₃	—	2	2	no	—
70	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₂₃	#575	1	2	yes	—
72	C₂×M₁₆	—	23	2	no	—
73	C₄⋊Q₈	—	23	2	no	—
74	#125	—	2	2	no	—
75	#125	—	23	2	no	—
76	#125	C₂×D₂₃	1	2	yes	—
77	#125	C₂×D₂₃	1	2	yes	—
78	C₂²⋊Q₈	—	46	2	no	—
79	C₄×Q₈	—	46	2	no	—
80	#117	—	23	2	no	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
83	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
84	#107	—	23	8	no	—
85	#106	—	23	8	no	—
86	#97	—	23	2	no	—
87	C₄×C₈	—	23	2	no	—
88	C₂₃	#3166	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	46	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
93	M₁₆	—	23	8	no	—
94	C₂×C₈	—	23	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	23	4	no	—
99	C₄⋊C₄	—	46	4	no	—
100	C₄⋊C₄	—	46	2	no	—
101	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
102	C₂²⋊C₄	—	46	2	no	—
103	C₄²	—	46	2	no	—
104	C₄²	C₂²⋊D₂₃	1	2	yes	—
105	C₄²	C₂²×D₂₃	1	1	yes	characteristic
106	C₄²	C₂²×D₂₃	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
108	Q₈	—	92	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	46	4	no	—
113	C₂×C₄	Q₈×D₂₃	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
118	C₈	—	23	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#10625115	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	92	4	no	—
126	C₄	#10625115	1	2	yes	—
127	C₄	C₂×Q₈×D₂₃	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#10625115	1	1	yes	characteristic
130	C₂	#13093215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic

#13092827

C₂

yes

maximal, complemented

#13092816

C₂

yes

maximal, complemented

#13092457

C₂

yes

maximal

C₄×#67267

C₂

yes

maximal, characteristic

C₂×C₂₃×#125

C₂

yes

maximal, characteristic

C₂₃×#125

—

C₂₃×#125

C₂²

yes

—

C₂₃×#125

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625095

—

C₂×#67267

C₂²

yes

—

#10625041

C₂²

yes

—

#10625040

C₂²

yes

—

C₄×Dic₄₆

—

#10624990

C₂²

yes

—

#10624989

C₂²

yes

—

#10624985

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

#67267

Q₈

yes

complemented

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

C₂×Q₈

yes

—

#3166

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

C₂²×Q₈

yes

—

Dic₂₃

—

#533

—

C₂×#125

D₂₃

yes

—

—

—

C₄×M₁₆

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

#575

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

—

#125

C₂×D₂₃

yes

—

#125

C₂×D₂₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

C₄×C₈

—

C₂₃

#3166

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂²×D₂₃

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₂²⋊D₂₃

yes

—

105

C₄²

C₂²×D₂₃

yes

characteristic

106

C₄²

C₂²×D₂₃

yes

characteristic

107

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

Q₈×D₂₃

yes

—

114

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

115

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

116

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

117

C₂×C₄

C₂³×D₂₃

yes

characteristic

118

C₈

—

119

C₂²

—

120

C₂²

#10625115

yes

—

121

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#10625115

yes

—

127

C₄

C₂×Q₈×D₂₃

yes

—

128

C₂

—

129

C₂

C₂×#10625115

yes

characteristic

130

C₂

#13093215

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#125	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	#3166
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₂₃×#125

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #13092827	C₂	semidirect
3. #13092816	C₂	semidirect
39. #67267	Q₈	semidirect
88. C₂₃	#3166	semidirect

H ≅ G/N

type

2. #13092827

C₂

semidirect

3. #13092816

C₂

semidirect

39. #67267

Q₈

semidirect

88. C₂₃

#3166

semidirect

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

subgroup	index
#13092974	2
#13092827	2
#13092816	2
#13092457	2
C₄×#67267	2
C₂×C₂₃×#125	2
#3166	23

subgroup

index

C₂×C₂₃×#125

Subgroups — #15683946

Conjugacy classes of subgroups

Complements

Maximal subgroups