Group #15682049 — subgroups

Subgroups — #15682049

Order 2944.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#67288	C₂	1	1	yes	maximal, characteristic
2	#13092476	C₂	1	4	yes	maximal, complemented
3	#13092472	C₂	1	4	yes	maximal, complemented
4	#13092445	C₂	1	2	yes	maximal, complemented
5	#13092385	C₂	1	2	yes	maximal
6	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#67288	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×C₂²⋊D₂₃	C₄	1	4	yes	—
12	#10625097	C₂²	1	1	yes	characteristic
13	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
14	C₄×D₉₂	C₄	1	2	yes	—
15	#10625043	C₂²	1	2	yes	—
16	C₄²×D₂₃	C₂²	1	2	yes	—
17	C₄×Dic₄₆	C₄	1	2	yes	—
18	C₂₃×#97	C₂²	1	2	yes	—
19	C₄×C₈×C₂₃	C₂²	1	2	yes	—
20	(C₂×D₂₃)⋊C₈	C₂²	1	8	yes	—
21	Dic₂₃⋊C₈	C₂²	1	8	yes	—
22	#10624985	C₂²	1	2	yes	—
23	#10624985	C₂²	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
27	#67288	—	2	4	no	—
28	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₂₃	C₂³	1	2	yes	—
31	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
32	M₁₆×C₂₃	—	2	4	no	—
33	C₂×C₈×C₂₃	C₂³	1	4	yes	—
34	C₄²×C₂₃	C₂×C₄	1	2	yes	—
35	C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₄²×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
38	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
40	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
41	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
42	C₄×Dic₂₃	C₂³	1	2	yes	—
43	C₂×C₂₃⋊C₈	—	2	4	no	—
44	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₃	—	4	2	no	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	4	yes	—
52	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₂₃	—	2	4	no	—
54	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
55	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
56	D₉₂	—	2	2	no	—
57	C₄×D₂₃	—	2	4	no	—
58	C₄×D₂₃	—	4	4	no	—
59	Dic₄₆	—	2	2	no	—
60	C₈×C₂₃	—	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	#117	1	2	yes	—
65	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
66	C₂×D₂₃	—	2	2	no	—
67	C₂×D₂₃	—	2	4	no	—
68	C₄×C₂₃	—	2	2	no	—
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	#117	1	2	yes	—
71	C₄×C₂₃	C₂×#41	1	2	yes	—
72	Dic₂₃	—	2	2	no	—
73	Dic₂₃	—	2	4	no	—
74	C₄×#41	—	23	1	no	—
75	#433	—	23	4	no	—
76	#432	—	23	4	no	—
77	#424	—	23	2	no	—
78	#408	—	23	2	no	—
79	C₄×M₁₆	D₂₃	1	1	yes	characteristic, complemented
80	C₂×#97	—	23	1	no	—
81	C₂×C₂₃	—	2	2	no	—
82	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
83	C₂×C₂₃	#568	1	2	yes	—
84	D₂₃	—	4	2	no	—
85	C₂×#41	—	23	1	no	—
86	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
87	C₂²×C₈	—	23	2	no	—
88	C₄×Q₈	—	23	2	no	—
89	C₄×D₄	—	23	4	no	—
90	C₄×D₄	—	23	2	no	—
91	#117	—	23	1	no	—
92	#117	—	23	2	no	—
93	C₂×C₄²	—	23	2	no	—
94	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
95	C₄⋊C₈	—	23	8	no	—
96	C₂²⋊C₈	—	23	8	no	—
97	#97	—	23	2	no	—
98	#97	—	23	2	no	—
99	#97	C₂×D₂₃	1	2	yes	—
100	C₄×C₈	C₂×D₂₃	1	2	yes	—
101	C₂₃	#2923	1	1	yes	characteristic, complemented, Sylow
102	#41	—	46	4	no	—
103	C₂×Q₈	—	23	1	no	—
104	C₂×D₄	—	23	2	no	—
105	C₂×D₄	—	23	1	no	—
106	C₂²×C₄	—	23	2	no	—
107	C₂²×C₄	—	23	4	no	—
108	C₂²×C₄	C₄×D₂₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	23	4	no	—
112	C₂×C₈	—	46	4	no	—
113	C₂×C₈	C₂²×D₂₃	1	4	yes	—
114	C₄⋊C₄	—	23	4	no	—
115	C₄⋊C₄	—	23	2	no	—
116	C₂²⋊C₄	—	23	2	no	—
117	C₂²⋊C₄	—	23	4	no	—
118	C₄²	—	23	4	no	—
119	C₄²	—	23	2	no	—
120	C₄²	C₄×D₂₃	1	2	yes	—
121	C₄²	C₂²×D₂₃	1	1	yes	characteristic
122	C₄²	C₂²×D₂₃	1	1	yes	characteristic
123	C₂³	—	23	2	no	—
124	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
125	Q₈	—	46	2	no	—
126	D₄	—	46	4	no	—
127	D₄	—	46	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	23	4	no	—
130	C₂×C₄	—	23	2	no	—
131	C₂×C₄	—	46	4	no	—
132	C₂×C₄	—	92	4	no	—
133	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
136	C₂×C₄	#67288	1	4	yes	—
137	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	46	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	46	2	no	—
142	C₂²	—	46	4	no	—
143	C₂²	#10625043	1	2	yes	—
144	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	46	2	no	—
148	C₄	—	46	4	no	—
149	C₄	#10625043	1	2	yes	—
150	C₄	C₂×#67288	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	92	2	no	—
153	C₂	C₂×#10625043	1	1	yes	characteristic
154	C₂	#13093183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#67288

C₂

yes

maximal, characteristic

#13092476

C₂

yes

maximal, complemented

#13092472

C₂

yes

maximal, complemented

#13092445

C₂

yes

maximal, complemented

#13092385

C₂

yes

maximal

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₄×C₂²⋊D₂₃

C₄

yes

—

#10625097

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₄×D₉₂

C₄

yes

—

#10625043

C₂²

yes

—

C₄²×D₂₃

C₂²

yes

—

C₄×Dic₄₆

C₄

yes

—

C₂₃×#97

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₈

C₂²

yes

—

Dic₂₃⋊C₈

C₂²

yes

—

#10624985

C₂²

yes

—

#10624985

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

C₂×C₄

yes

—

#67288

—

C₂×D₉₂

C₂×C₄

yes

characteristic

C₂×C₄×D₂₃

C₂×C₄

yes

—

C₂×C₄×D₂₃

C₂³

yes

—

C₂×Dic₄₆

C₂×C₄

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

C₂×C₄

yes

—

(C₂×D₂₃)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

C₂×C₄

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂²×D₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

—

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

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂²×C₄

yes

—

C₂×Dic₂₃

C₂²×C₄

yes

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#2923

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

C₂×#41

yes

—

Dic₂₃

—

Dic₂₃

—

C₄×#41

—

—

—

—

—

C₄×M₁₆

D₂₃

yes

characteristic, complemented

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#117

yes

characteristic

C₂×C₂₃

#568

yes

—

D₂₃

—

C₂×#41

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

C₂×D₂₃

yes

—

100

C₄×C₈

C₂×D₂₃

yes

—

101

C₂₃

#2923

yes

characteristic, complemented, Sylow

102

#41

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₂₃

yes

—

109

C₂²×C₄

C₂²×D₂₃

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₂₃

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₂₃

yes

—

121

C₄²

C₂²×D₂₃

yes

characteristic

122

C₄²

C₂²×D₂₃

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₄×D₂₃

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₄×D₂₃

yes

—

134

C₂×C₄

C₂×C₄×D₂₃

yes

—

135

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

136

C₂×C₄

#67288

yes

—

137

C₂×C₄

C₂³×D₂₃

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#10625043

yes

—

144

C₂²

C₂²×C₄×D₂₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#10625043

yes

—

150

C₄

C₂×#67288

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#10625043

yes

characteristic

154

C₂

#13093183

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₄×M₁₆×C₂₃	C₂
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×M₁₆	D₂₃
hypocenter	C₂₃	#2923
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₃

commutator, socle

C₂²×C₂₃

C₂³×C₄

Fitting

C₄×M₁₆×C₂₃

C₂

prime radical

C₂×#67288

C₂²

cosocle

C₂×C₄×C₂₃

C₂⁴

hypercenter

C₄×M₁₆

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #13092476	C₂	semidirect
3. #13092472	C₂	semidirect
4. #13092445	C₂	semidirect
6. C₂×#10624985	C₂	semidirect
7. C₄×M₁₆×C₂₃	C₂	semidirect
79. C₄×M₁₆	D₂₃	semidirect
101. C₂₃	#2923	semidirect

H ≅ G/N

type

2. #13092476

C₂

semidirect

3. #13092472

C₂

semidirect

4. #13092445

C₂

semidirect

6. C₂×#10624985

C₂

semidirect

7. C₄×M₁₆×C₂₃

C₂

semidirect

79. C₄×M₁₆

D₂₃

semidirect

101. C₂₃

#2923

semidirect

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

subgroup	index
C₄×#67288	2
#13092476	2
#13092472	2
#13092445	2
#13092385	2
C₂×#10624985	2
C₄×M₁₆×C₂₃	2
#2923	23

subgroup

index

C₄×M₁₆×C₂₃

Subgroups — #15682049

Conjugacy classes of subgroups

Complements

Maximal subgroups