Group #15682048 — subgroups

Subgroups — #15682048

Order 2944.

384 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 213 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	#13092478	C₂	1	4	yes	maximal, complemented
2	#13092474	C₂	1	4	yes	maximal, complemented
3	#13092444	C₂	1	2	yes	maximal, complemented
4	#13092384	C₂	1	2	yes	maximal
5	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₂₃	C₄	1	2	yes	—
10	C₄×D₄×C₂₃	C₄	1	4	yes	—
11	C₄×D₄×C₂₃	C₄	1	2	yes	—
12	C₂₃×#117	C₂²	1	1	yes	characteristic
13	C₂₃×#117	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
16	C₂×#67267	C₂²	1	2	yes	—
17	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
18	C₂²⋊(C₂₃⋊C₈)	C₂²	1	8	yes	—
19	C₄⋊(C₂₃⋊C₈)	C₂²	1	8	yes	—
20	#10624985	C₂²	1	2	yes	—
21	#10624985	C₂²	1	2	yes	—
22	#10624985	C₂²	1	2	yes	—
23	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
24	#41×C₂₃	—	2	4	no	—
25	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
33	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
35	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
36	C₄²×C₂₃	C₂×C₄	1	4	yes	—
37	C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₄²×C₂₃	C₂³	1	2	yes	—
41	#67267	—	2	4	no	—
42	C₂×C₂₃⋊C₈	—	2	4	no	—
43	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
44	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
46	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₂₃	—	2	2	no	—
48	D₄×C₂₃	—	2	4	no	—
49	D₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	4	2	no	—
52	C₂×C₄×C₂₃	—	4	4	no	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	#41	1	4	yes	—
59	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂₃⋊C₈	—	2	4	no	—
61	C₂₃⋊C₈	—	2	4	no	—
62	#2923	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	—	2	2	no	—
65	C₂²×C₂₃	—	2	4	no	—
66	C₂²×C₂₃	#117	1	2	yes	—
67	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
68	C₄×C₂₃	—	2	2	no	—
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	—	2	2	no	—
71	C₄×C₂₃	—	2	4	no	—
72	C₄×C₂₃	#117	1	2	yes	—
73	C₄×C₂₃	C₂×#41	1	2	yes	—
74	C₄×#41	D₂₃	1	1	yes	characteristic
75	#433	—	23	4	no	—
76	#432	—	23	4	no	—
77	#424	—	23	2	no	—
78	#408	—	23	2	no	—
79	C₄×M₁₆	—	23	1	no	—
80	C₂×#97	—	23	1	no	—
81	C₂×C₂₃	—	2	2	no	—
82	C₂×C₂₃	—	4	2	no	—
83	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
84	C₂×C₂₃	#568	1	2	yes	—
85	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
86	C₂×M₁₆	—	23	2	no	—
87	C₂²×C₈	—	23	2	no	—
88	C₄×Q₈	Dic₂₃	1	2	yes	—
89	C₄×D₄	Dic₂₃	1	4	yes	—
90	C₄×D₄	Dic₂₃	1	2	yes	—
91	#117	C₂×D₂₃	1	1	yes	characteristic
92	#117	C₂×D₂₃	1	2	yes	—
93	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
94	C₂×C₄²	C₂×D₂₃	1	2	yes	—
95	C₄⋊C₈	—	23	8	no	—
96	C₂²⋊C₈	—	23	8	no	—
97	#97	—	23	2	no	—
98	#97	—	23	2	no	—
99	#97	—	23	2	no	—
100	C₄×C₈	—	23	2	no	—
101	C₂₃	#2923	1	1	yes	characteristic, complemented, Sylow
102	#41	—	2	4	no	—
103	C₂×Q₈	C₂×Dic₂₃	1	1	yes	characteristic
104	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
105	C₂×D₄	C₂×Dic₂₃	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
107	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
108	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
110	M₁₆	—	46	4	no	—
111	C₂×C₈	—	23	4	no	—
112	C₂×C₈	—	23	4	no	—
113	C₂×C₈	—	46	4	no	—
114	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
115	C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
116	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
117	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
118	C₄²	C₂×Dic₂₃	1	4	yes	—
119	C₄²	C₂×Dic₂₃	1	2	yes	—
120	C₄²	C₂²×D₂₃	1	1	yes	characteristic
121	C₄²	C₂²×D₂₃	1	1	yes	characteristic
122	C₄²	C₂²×D₂₃	1	2	yes	—
123	C₂³	C₂²×Dic₂₃	1	2	yes	—
124	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	#67288	1	4	yes	—
132	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
133	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
134	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
135	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
136	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
137	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
138	C₈	—	46	4	no	—
139	C₈	—	46	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	#10625097	1	2	yes	—
144	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	4	no	—
149	C₄	#10625097	1	2	yes	—
150	C₄	C₂×#67288	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	2	no	—
153	C₂	C₂×#10625097	1	1	yes	characteristic
154	C₂	#13093184	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092478

C₂

yes

maximal, complemented

#13092474

C₂

yes

maximal, complemented

#13092444

C₂

yes

maximal, complemented

#13092384

C₂

yes

maximal

C₄×#67267

C₂

yes

maximal, characteristic, complemented

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

C₄×#41×C₂₃

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₄×Q₈×C₂₃

C₄

yes

—

C₄×D₄×C₂₃

C₄

yes

—

C₄×D₄×C₂₃

C₄

yes

—

C₂₃×#117

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

#10624985

C₂²

yes

—

#10624985

C₂²

yes

—

#10624985

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂²×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×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

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#2923

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

C₂×#41

yes

—

C₄×#41

D₂₃

yes

—

—

—

—

C₄×M₁₆

—

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#117

yes

characteristic

C₂×C₂₃

#568

yes

—

C₂×#41

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

Dic₂₃

yes

—

C₄×D₄

Dic₂₃

yes

—

C₄×D₄

Dic₂₃

yes

—

#117

C₂×D₂₃

yes

characteristic

#117

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

100

C₄×C₈

—

101

C₂₃

#2923

yes

characteristic, complemented, Sylow

102

#41

—

103

C₂×Q₈

C₂×Dic₂₃

yes

characteristic

104

C₂×D₄

C₂×Dic₂₃

yes

—

105

C₂×D₄

C₂×Dic₂₃

yes

characteristic

106

C₂²×C₄

C₂×Dic₂₃

yes

—

107

C₂²×C₄

C₂×Dic₂₃

yes

—

108

C₂²×C₄

C₂²×D₂₃

yes

—

109

C₂²×C₄

C₂²×D₂₃

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

C₂×Dic₂₃

yes

—

115

C₄⋊C₄

C₂×Dic₂₃

yes

—

116

C₂²⋊C₄

C₂×Dic₂₃

yes

—

117

C₂²⋊C₄

C₂×Dic₂₃

yes

—

118

C₄²

C₂×Dic₂₃

yes

—

119

C₄²

C₂×Dic₂₃

yes

—

120

C₄²

C₂²×D₂₃

yes

characteristic

121

C₄²

C₂²×D₂₃

yes

characteristic

122

C₄²

C₂²×D₂₃

yes

—

123

C₂³

C₂²×Dic₂₃

yes

—

124

C₂³

C₂²×Dic₂₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#67288

yes

—

132

C₂×C₄

C₂²×Dic₂₃

yes

—

133

C₂×C₄

C₂²×Dic₂₃

yes

—

134

C₂×C₄

C₂²×Dic₂₃

yes

—

135

C₂×C₄

C₂²×Dic₂₃

yes

—

136

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

137

C₂×C₄

C₂³×D₂₃

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#10625097

yes

—

144

C₂²

C₂³×Dic₂₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#10625097

yes

—

150

C₄

C₂×#67288

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#10625097

yes

characteristic

154

C₂

#13093184

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₄×#41×C₂₃	C₂
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×#41	D₂₃
hypocenter	C₂₃	#2923
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₃

commutator, socle

C₂²×C₂₃

C₂³×C₄

Fitting

C₄×#41×C₂₃

C₂

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂×C₄×C₂₃

C₂⁴

hypercenter

C₄×#41

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. #13092478	C₂	semidirect
2. #13092474	C₂	semidirect
3. #13092444	C₂	semidirect
5. C₄×#67267	C₂	semidirect
6. C₂×#10624985	C₂	semidirect
101. C₂₃	#2923	semidirect

H ≅ G/N

type

1. #13092478

C₂

semidirect

2. #13092474

C₂

semidirect

3. #13092444

C₂

semidirect

5. C₄×#67267

C₂

semidirect

6. C₂×#10624985

C₂

semidirect

101. C₂₃

#2923

semidirect

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

subgroup	index
#13092478	2
#13092474	2
#13092444	2
#13092384	2
C₄×#67267	2
C₂×#10624985	2
C₄×#41×C₂₃	2
#2923	23

subgroup

index

C₄×#41×C₂₃

Subgroups — #15682048

Conjugacy classes of subgroups

Complements

Maximal subgroups