Group #15683777 — subgroups

Subgroups — #15683777

Order 2944.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	#13092785	C₂	1	4	yes	maximal, complemented
3	#13092767	C₂	1	4	yes	maximal, complemented
4	#13092426	C₂	1	4	yes	maximal
5	C₄×#67267	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂₃×C₄⋊Q₈	—	2	2	no	—
9	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
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	#10625017	C₂²	1	16	yes	—
20	#10624985	C₂²	1	2	yes	—
21	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
22	C₂×Q₈×C₂₃	—	2	2	no	—
23	C₂×Q₈×C₂₃	—	2	8	no	—
24	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	D₄	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₄₆	—	2	2	no	—
29	C₄⋊C₄×C₂₃	—	2	4	no	—
30	C₄⋊C₄×C₂₃	—	2	4	no	—
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	—	2	4	no	—
40	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
41	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₂₃	—	2	8	no	—
43	Q₈×C₂₃	—	4	4	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	—	4	4	no	—
47	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₃	#41	1	4	yes	—
51	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₂₃	—	2	4	no	—
53	Dic₄₆	—	4	4	no	—
54	C₂₃⋊C₈	—	2	4	no	—
55	#3139	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	#124	1	2	yes	—
58	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	—	4	4	no	—
61	C₄×C₂₃	#124	1	2	yes	—
62	C₄×C₂₃	#137	1	4	yes	—
63	C₄×C₂₃	C₂×#41	1	2	yes	—
64	Dic₂₃	—	4	4	no	—
65	#533	—	23	1	no	—
66	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic
67	#491	—	23	4	no	—
68	#487	—	23	4	no	—
69	#419	—	23	4	no	—
70	C₄×M₁₆	—	23	1	no	—
71	C₂×C₂₃	—	2	2	no	—
72	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
73	C₂×C₂₃	C₂×#137	1	2	yes	—
74	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
75	C₂×M₁₆	—	23	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	—	23	2	no	—
78	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
79	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
80	#125	—	23	2	no	—
81	C₂²⋊Q₈	—	46	2	no	—
82	C₄×Q₈	—	46	2	no	—
83	#117	—	23	2	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
86	#103	—	23	16	no	—
87	#97	—	23	2	no	—
88	C₄×C₈	—	23	2	no	—
89	C₂₃	#3139	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	46	2	no	—
93	C₂×Q₈	C₂²×D₂₃	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
97	M₁₆	—	46	4	no	—
98	C₂×C₈	—	23	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	23	4	no	—
102	C₄⋊C₄	—	46	4	no	—
103	C₄⋊C₄	—	46	2	no	—
104	C₂²⋊C₄	—	46	2	no	—
105	C₄²	—	46	2	no	—
106	C₄²	C₂²⋊D₂₃	1	2	yes	—
107	C₄²	C₂²×D₂₃	1	1	yes	characteristic
108	C₄²	C₂²×D₂₃	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	92	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	46	4	no	—
117	C₂×C₄	#67290	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
121	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
122	C₈	—	46	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#10625107	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	92	4	no	—
129	C₄	#10625107	1	2	yes	—
130	C₄	#10625113	1	4	yes	—
131	C₄	C₂×#67290	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#10625107	1	1	yes	characteristic
134	C₂	C₂×#10625113	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic

#13092785

C₂

yes

maximal, complemented

#13092767

C₂

yes

maximal, complemented

#13092426

C₂

yes

maximal

C₄×#67267

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625095

—

C₂×#67267

C₂²

yes

—

#10625041

C₂²

yes

—

#10625040

C₂²

yes

—

C₄×Dic₄₆

—

#10625017

C₂²

yes

—

#10624985

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₄₆

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

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

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

#3139

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

#137

yes

—

C₄×C₂₃

C₂×#41

yes

—

Dic₂₃

—

#533

—

C₂×C₄⋊Q₈

D₂₃

yes

—

—

—

C₄×M₁₆

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

C₂×#137

yes

—

C₂²×Q₈

C₂×D₂₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

—

C₄⋊Q₈

C₂×D₂₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂₃

#3139

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₂₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂²⋊D₂₃

yes

—

107

C₄²

C₂²×D₂₃

yes

characteristic

108

C₄²

C₂²×D₂₃

yes

characteristic

109

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#67290

yes

—

118

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

119

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

121

C₂×C₄

C₂³×D₂₃

yes

characteristic

122

C₈

—

123

C₂²

—

124

C₂²

#10625107

yes

—

125

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#10625107

yes

—

130

C₄

#10625113

yes

—

131

C₄

C₂×#67290

yes

—

132

C₂

—

133

C₂

C₂×#10625107

yes

characteristic

134

C₂

C₂×#10625113

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#3139
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₂₃×C₄⋊Q₈

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂³×C₂₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₂₃

hypocenter

C₂₃

#3139

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #13092785	C₂	semidirect
3. #13092767	C₂	semidirect
89. C₂₃	#3139	semidirect

H ≅ G/N

type

2. #13092785

C₂

semidirect

3. #13092767

C₂

semidirect

89. C₂₃

#3139

semidirect

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

subgroup	index
#13092974	2
#13092785	2
#13092767	2
#13092426	2
C₄×#67267	2
C₂×C₂₃×C₄⋊Q₈	2
#3139	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

Subgroups — #15683777

Conjugacy classes of subgroups

Complements

Maximal subgroups