Group #15674320 — subgroups

Subgroups — #15674320

Order 2944.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×#10625040	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#10624992	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₆	C₂²	1	2	yes	—
6	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	—	2	1	no	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	—	2	1	no	—
10	C₂×C₄⋊Dic₂₃	—	2	1	no	—
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	#10625040	—	2	2	no	—
13	#10625040	C₄	1	4	yes	complemented
14	C₂₃×#95	—	2	1	no	—
15	#10624992	—	2	8	no	—
16	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
17	C₂²×C₄×C₂₃	—	2	2	no	—
18	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₂₃	—	2	2	no	—
22	C₂×Dic₄₆	—	2	4	no	—
23	C₂×Dic₄₆	—	2	4	no	—
24	C₂×Dic₄₆	—	2	2	no	—
25	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
28	C₄²×C₂₃	—	4	2	no	—
29	C₄²×C₂₃	C₂×C₄	1	2	yes	—
30	C₄²×C₂₃	D₄	1	2	yes	—
31	C₄⋊Dic₂₃	—	2	4	no	—
32	C₄⋊Dic₂₃	—	2	4	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	—	2	4	no	—
35	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	—	2	4	no	—
38	C₂×C₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	—	4	2	no	—
40	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₂₃	Q₁₆	1	2	yes	—
44	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#41	1	2	yes	—
48	C₂×Dic₂₃	—	2	4	no	—
49	C₂×Dic₂₃	—	4	4	no	—
50	Dic₄₆	—	2	8	no	—
51	Dic₄₆	—	4	2	no	—
52	Dic₄₆	—	4	2	no	—
53	C₂₃⋊C₈	—	4	2	no	—
54	#1981	—	23	1	no	maximal, Sylow
55	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₃	#121	1	2	yes	—
57	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×C₂₃	#137	1	2	yes	—
60	C₄×C₂₃	—	2	2	no	—
61	C₄×C₂₃	—	4	4	no	—
62	C₄×C₂₃	#103	1	4	yes	complemented
63	C₄×C₂₃	C₄×D₄	1	2	yes	—
64	C₄×C₂₃	#124	1	1	yes	characteristic
65	C₄×C₂₃	#127	1	1	yes	characteristic
66	Dic₂₃	—	4	4	no	—
67	C₂×C₄⋊Q₈	—	23	1	no	—
68	C₂×C₄⋊C₈	—	23	1	no	—
69	C₂×#103	—	23	4	no	—
70	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic
71	C₂×C₂₃	#390	1	1	yes	characteristic
72	C₂×C₂₃	C₂×#103	1	1	yes	characteristic
73	C₂×C₂₃	#419	1	1	yes	characteristic
74	C₂×C₂₃	#461	1	2	yes	—
75	C₂×C₂₃	#462	1	2	yes	—
76	C₂²×Q₈	—	23	2	no	—
77	C₂²×C₈	—	23	2	no	—
78	C₄⋊Q₈	—	23	4	no	—
79	C₄⋊Q₈	—	46	2	no	—
80	C₂×C₄⋊C₄	—	46	1	no	—
81	C₂×C₄⋊C₄	—	46	1	no	—
82	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
85	C₄⋊C₈	—	46	2	no	—
86	#103	—	46	8	no	—
87	#95	—	2	1	no	—
88	C₂₃	#1981	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	23	4	no	—
90	C₂×Q₈	—	46	2	no	—
91	C₂×Q₈	—	46	4	no	—
92	C₂×Q₈	—	46	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	46	2	no	—
95	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
98	C₂×C₈	—	46	4	no	—
99	C₂×C₈	—	46	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	46	4	no	—
102	C₄⋊C₄	—	46	4	no	—
103	C₄⋊C₄	D₉₂	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₂₃	1	2	yes	—
106	C₄²	D₉₂	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
108	Q₈	—	46	8	no	—
109	Q₈	—	92	2	no	—
110	Q₈	—	92	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	46	4	no	—
116	C₂×C₄	—	92	4	no	—
117	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
118	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
119	C₂×C₄	#67273	1	2	yes	—
120	C₂×C₄	#67275	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
122	C₂×C₄	C₂×D₉₂	1	2	yes	—
123	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
124	C₂×C₄	#67288	1	2	yes	—
125	C₈	—	92	2	no	—
126	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
127	C₂²	#10625101	1	2	yes	—
128	C₂²	C₂×#67273	1	1	yes	characteristic
129	C₂²	C₂×#67275	1	1	yes	characteristic
130	C₂²	#10625122	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	92	4	no	—
134	C₄	#10624992	1	4	yes	complemented
135	C₄	C₄×D₉₂	1	2	yes	—
136	C₄	#10625045	1	1	yes	characteristic
137	C₄	#10625046	1	1	yes	characteristic
138	C₂	#13092319	1	1	yes	characteristic
139	C₂	C₂×#10624992	1	1	yes	characteristic
140	C₂	#13092425	1	1	yes	characteristic
141	C₂	#13092622	1	2	yes	—
142	C₂	#13092627	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625040

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×#10624992

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₃

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625040

—

#10625040

C₄

yes

complemented

C₂₃×#95

—

#10624992

—

C₄⋊(C₂₃⋊C₈)

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₂₃⋊C₈

—

#1981

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×Q₁₆

yes

characteristic

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#103

yes

complemented

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#462

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

C₂₃

#1981

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

D₉₂

yes

—

104

C₄²

—

105

C₄²

C₄×D₂₃

yes

—

106

C₄²

D₉₂

yes

—

107

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

118

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

119

C₂×C₄

#67273

yes

—

120

C₂×C₄

#67275

yes

—

121

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

122

C₂×C₄

C₂×D₉₂

yes

—

123

C₂×C₄

C₂×D₉₂

yes

characteristic

124

C₂×C₄

#67288

yes

—

125

C₈

—

126

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

127

C₂²

#10625101

yes

—

128

C₂²

C₂×#67273

yes

characteristic

129

C₂²

C₂×#67275

yes

characteristic

130

C₂²

#10625122

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#10624992

yes

complemented

135

C₄

C₄×D₉₂

yes

—

136

C₄

#10625045

yes

characteristic

137

C₄

#10625046

yes

characteristic

138

C₂

#13092319

yes

characteristic

139

C₂

C₂×#10624992

yes

characteristic

140

C₂

#13092425

yes

characteristic

141

C₂

#13092622

yes

—

142

C₂

#13092627

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#1981
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₄×C₄⋊C₄×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₃

hypocenter

C₂₃

#1981

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
13. #10625040	C₄	semidirect
62. C₄×C₂₃	#103	semidirect
88. C₂₃	#1981	semidirect
134. C₄	#10624992	semidirect

H ≅ G/N

type

13. #10625040

C₄

semidirect

62. C₄×C₂₃

#103

semidirect

88. C₂₃

#1981

semidirect

134. C₄

#10624992

semidirect

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

subgroup	index
C₂×#10625040	2
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×#10624992	2
C₄×C₄⋊C₄×C₂₃	2
#1981	23

subgroup

index

C₂×#10625040

C₂×C₄⋊(C₂₃⋊C₈)

C₂×#10624992

C₄×C₄⋊C₄×C₂₃

#1981

Subgroups — #15674320

Conjugacy classes of subgroups

Complements

Maximal subgroups