Group #15673477 — subgroups

Subgroups — #15673477

Order 2944.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#13092381	C₂	1	2	yes	maximal
2	#13092186	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₂₃	C₄	1	2	yes	—
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×#67267	C₂²	1	4	yes	—
11	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
12	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
13	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₃	—	2	4	no	—
15	C₂²×C₄×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	—	4	4	no	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₃	—	2	4	no	—
24	C₄⋊C₄×C₂₃	—	2	4	no	—
25	#67267	—	2	8	no	—
26	C₂×C₂₃⋊C₈	—	2	4	no	—
27	C₂³×C₂₃	—	2	2	no	—
28	C₂³×C₂₃	—	2	2	no	—
29	C₂³×C₂₃	—	2	2	no	—
30	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	2	4	no	—
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	4	4	no	—
38	C₂×C₄×C₂₃	—	4	4	no	—
39	C₂×C₄×C₂₃	—	4	4	no	—
40	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₂₃	#41	1	4	yes	—
43	C₂₃⋊C₈	—	4	4	no	—
44	#1821	—	23	1	no	maximal, Sylow
45	C₂²×C₂₃	—	2	8	no	—
46	C₂²×C₂₃	—	2	1	no	—
47	C₂²×C₂₃	—	2	1	no	—
48	C₂²×C₂₃	—	2	1	no	—
49	C₂²×C₂₃	—	2	1	no	—
50	C₂²×C₂₃	—	2	1	no	—
51	C₂²×C₂₃	—	2	1	no	—
52	C₂²×C₂₃	#100	1	2	yes	—
53	C₂²×C₂₃	#101	1	2	yes	—
54	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₃	#117	1	2	yes	—
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	4	4	no	—
58	C₄×C₂₃	#123	1	4	yes	—
59	C₂²×C₄⋊C₄	D₂₃	1	1	yes	characteristic
60	#407	—	23	2	no	—
61	#343	—	23	4	no	—
62	C₂×C₂₃	—	2	2	no	—
63	C₂×C₂₃	—	2	2	no	—
64	C₂×C₂₃	—	2	2	no	—
65	C₂×C₂₃	#381	1	1	yes	characteristic
66	C₂×C₂₃	C₂×#100	1	1	yes	characteristic
67	C₂×C₂₃	C₂×#101	1	1	yes	characteristic
68	C₂³×C₄	Dic₂₃	1	2	yes	—
69	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
70	C₂×M₁₆	—	23	4	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	Dic₂₃	1	2	yes	—
74	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
75	C₂²⋊C₈	—	46	2	no	—
76	C₂²⋊C₈	—	46	2	no	—
77	C₂₃	#1821	1	1	yes	characteristic, complemented, Sylow
78	C₂⁴	C₂×Dic₂₃	1	1	yes	characteristic
79	C₂²×C₄	—	2	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	4	4	no	—
83	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
84	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
85	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
86	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
88	M₁₆	—	46	8	no	—
89	C₂×C₈	—	46	4	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₂³	—	2	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	2	no	—
95	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
96	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
97	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
106	C₂×C₄	#67288	1	4	yes	—
107	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
108	C₈	—	92	4	no	—
109	C₂²	—	2	8	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#10625015	1	2	yes	—
117	C₂²	#10625018	1	2	yes	—
118	C₂²	#10625097	1	2	yes	—
119	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	4	4	no	—
122	C₄	#10625100	1	4	yes	—
123	C₂	—	2	2	no	—
124	C₂	—	2	2	no	—
125	C₂	—	2	2	no	—
126	C₂	#13092280	1	1	yes	characteristic
127	C₂	C₂×#10625015	1	1	yes	characteristic
128	C₂	C₂×#10625018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092381

C₂

yes

maximal

#13092186

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₂₃

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₄

yes

—

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

—

C₂²⋊(C₂₃⋊C₈)

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

C₂³×C₂₃

C₂²⋊C₄

yes

characteristic

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

#1821

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#100

yes

—

C₂²×C₂₃

#101

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#123

yes

—

C₂²×C₄⋊C₄

D₂₃

yes

characteristic

#407

—

#343

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂³×C₄

Dic₂₃

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₂₃

yes

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1821

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₂₃

yes

characteristic

C₂³

C₂²⋊Dic₂₃

yes

characteristic

C₂³

C₂²×Dic₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₂₃

yes

—

106

C₂×C₄

#67288

yes

—

107

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

108

C₈

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#10625015

yes

—

117

C₂²

#10625018

yes

—

118

C₂²

#10625097

yes

—

119

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

#10625100

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#13092280

yes

characteristic

127

C₂

C₂×#10625015

yes

characteristic

128

C₂

C₂×#10625018

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂²×C₄⋊C₄×C₂₃	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂²×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#1821
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂²×C₄⋊C₄×C₂₃

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂²×C₄⋊C₄

D₂₃

hypocenter

C₂₃

#1821

LCS

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #13092186	C₂	semidirect
77. C₂₃	#1821	semidirect

H ≅ G/N

type

2. #13092186

C₂

semidirect

77. C₂₃

#1821

semidirect

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

subgroup	index
#13092381	2
#13092186	2
C₂²×C₄⋊C₄×C₂₃	2
#1821	23

subgroup

index

#13092381

#13092186

C₂²×C₄⋊C₄×C₂₃

#1821

Subgroups — #15673477

Conjugacy classes of subgroups

Complements

Maximal subgroups