Group #15674121 — subgroups

Subgroups — #15674121

Order 2944.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
3	C₂×#10624985	C₂	1	1	yes	maximal, characteristic
4	#13092172	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂₃×#95	C₄	1	1	yes	characteristic
13	C₂₃×#95	C₄	1	1	yes	characteristic
14	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
15	C₄⋊(C₂₃⋊C₈)	—	2	4	no	—
16	#10624985	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₃	—	2	4	no	—
23	C₄²×C₂₃	—	2	2	no	—
24	C₄²×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	D₄	1	2	yes	—
26	C₄²×C₂₃	Q₈	1	2	yes	—
27	C₂×C₂₃⋊C₈	—	2	1	no	—
28	C₂×C₂₃⋊C₈	—	2	1	no	—
29	C₂×C₂₃⋊C₈	—	2	1	no	—
30	C₂×C₂₃⋊C₈	—	2	1	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
35	C₂×C₂₃⋊C₈	Q₈	1	2	yes	—
36	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
47	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
53	C₂₃⋊C₈	—	2	2	no	—
54	C₂₃⋊C₈	—	2	2	no	—
55	C₂₃⋊C₈	—	4	2	no	—
56	#1940	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
59	C₂²×C₂₃	C₄×Q₈	1	1	yes	characteristic
60	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
61	C₂²×C₂₃	#131	1	2	yes	—
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	2	2	no	—
64	C₄×C₂₃	—	2	4	no	—
65	C₄×C₂₃	—	4	2	no	—
66	C₄×C₂₃	#121	1	1	yes	characteristic
67	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₄×C₂₃	#125	1	1	yes	characteristic
69	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄⋊C₈	—	23	2	no	—
71	C₂×C₄⋊C₈	—	23	1	no	—
72	C₂×#97	—	23	1	no	—
73	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic
74	#336	—	23	2	no	—
75	C₂×C₂₃	#389	1	1	yes	characteristic
76	C₂×C₂₃	#423	1	1	yes	characteristic
77	C₂×C₂₃	#424	1	1	yes	characteristic
78	C₂×C₂₃	#435	1	2	yes	—
79	C₂×C₂₃	#446	1	2	yes	—
80	C₂²×C₈	—	23	1	no	—
81	C₂²×C₈	—	23	1	no	—
82	C₂²×C₈	—	23	2	no	—
83	C₂×C₄⋊C₄	Dic₂₃	1	2	yes	—
84	C₂×C₄²	C₂×D₂₃	1	2	yes	—
85	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
86	C₄⋊C₈	—	46	4	no	—
87	C₄⋊C₈	—	46	2	no	—
88	#97	—	46	2	no	—
89	#95	Dic₂₃	1	1	yes	characteristic
90	#95	Dic₂₃	1	1	yes	characteristic
91	C₂₃	#1940	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
93	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
94	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
97	C₂×C₈	—	23	2	no	—
98	C₂×C₈	—	23	2	no	—
99	C₂×C₈	—	46	1	no	—
100	C₂×C₈	—	46	1	no	—
101	C₂×C₈	—	46	1	no	—
102	C₂×C₈	—	46	1	no	—
103	C₂×C₈	—	46	2	no	—
104	C₂×C₈	—	46	2	no	—
105	C₂×C₈	—	46	2	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	Dic₄₆	1	2	yes	—
110	C₄²	D₉₂	1	2	yes	—
111	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	#67267	1	4	yes	—
120	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
121	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
122	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
123	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
124	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
125	C₂×C₄	#67290	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
127	C₂×C₄	#67292	1	1	yes	characteristic
128	C₈	—	46	2	no	—
129	C₈	—	46	2	no	—
130	C₈	—	92	2	no	—
131	C₂²	C₂×#67267	1	2	yes	—
132	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
133	C₂²	D₄×Dic₂₃	1	1	yes	characteristic
134	C₂²	Q₈×Dic₂₃	1	1	yes	characteristic
135	C₂²	#10625119	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	#10625059	1	1	yes	characteristic
141	C₄	#10625061	1	1	yes	characteristic
142	C₄	#10625066	1	1	yes	characteristic
143	C₄	#10625068	1	1	yes	characteristic
144	C₂	#13092316	1	1	yes	characteristic
145	C₂	#13092441	1	1	yes	characteristic
146	C₂	#13092444	1	1	yes	characteristic
147	C₂	#13092489	1	2	yes	—
148	C₂	#13092548	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal

C₂×#10624985

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₃

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#95

C₄

yes

characteristic

C₂₃×#95

C₄

yes

characteristic

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

—

#10624985

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

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

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1940

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₂²×C₂₃

C₄×Q₈

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₄×C₂₃

#125

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄×C₄⋊C₄

D₂₃

yes

characteristic

#336

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#446

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

Dic₂₃

yes

characteristic

#95

Dic₂₃

yes

characteristic

C₂₃

#1940

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₄₆

yes

—

110

C₄²

D₉₂

yes

—

111

C₂³

C₂²×Dic₂₃

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#67267

yes

—

120

C₂×C₄

C₄⋊Dic₂₃

yes

—

121

C₂×C₄

C₄⋊Dic₂₃

yes

—

122

C₂×C₄

C₂×Dic₄₆

yes

characteristic

123

C₂×C₄

C₂×D₉₂

yes

characteristic

124

C₂×C₄

D₄×D₂₃

yes

characteristic

125

C₂×C₄

#67290

yes

characteristic

126

C₂×C₄

Q₈×D₂₃

yes

characteristic

127

C₂×C₄

#67292

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₂²

C₂×#67267

yes

—

132

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

133

C₂²

D₄×Dic₂₃

yes

characteristic

134

C₂²

Q₈×Dic₂₃

yes

characteristic

135

C₂²

#10625119

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#10625059

yes

characteristic

141

C₄

#10625061

yes

characteristic

142

C₄

#10625066

yes

characteristic

143

C₄

#10625068

yes

characteristic

144

C₂

#13092316

yes

characteristic

145

C₂

#13092441

yes

characteristic

146

C₂

#13092444

yes

characteristic

147

C₂

#13092489

yes

—

148

C₂

#13092548

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle, prime radical	C₂³×C₂₃	C₂²×C₄
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#1940
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle, prime radical

C₂³×C₂₃

C₂²×C₄

Fitting

C₄×C₄⋊C₄×C₂₃

C₂

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
91. C₂₃	#1940	semidirect

H ≅ G/N

type

91. C₂₃

#1940

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×#10624985	2
#13092172	2
C₄×C₄⋊C₄×C₂₃	2
#1940	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×#10624985

#13092172

C₄×C₄⋊C₄×C₂₃

#1940

Subgroups — #15674121

Conjugacy classes of subgroups

Complements

Maximal subgroups