Group #15674312 — subgroups

Subgroups — #15674312

Order 2944.

276 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×Dic₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625001	C₂	1	1	yes	maximal, characteristic
3	#13092425	C₂	1	2	yes	maximal
4	C₂×#10625006	C₂	1	2	yes	maximal
5	C₂₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₆	C₂²	1	2	yes	—
7	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#117	C₂²	1	1	yes	characteristic
10	C₂×#67267	C₂²	1	2	yes	—
11	C₂×Dic₉₂	—	2	2	no	—
12	C₂×Dic₉₂	—	2	1	no	—
13	C₂×Dic₉₂	—	2	1	no	—
14	C₂×Dic₉₂	C₄	1	4	yes	complemented
15	C₂₃×#97	—	2	1	no	—
16	C₄×C₈×C₂₃	—	2	1	no	—
17	#10625006	—	2	4	no	—
18	#10625001	—	2	2	no	—
19	#10624992	—	2	4	no	—
20	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₂₃	—	2	2	no	—
22	C₂×Dic₄₆	—	2	2	no	—
23	C₂×Dic₄₆	—	2	2	no	—
24	C₂×Dic₄₆	—	2	2	no	—
25	C₂×Dic₄₆	—	2	2	no	—
26	C₂×Dic₄₆	—	2	2	no	—
27	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
28	M₁₆×C₂₃	—	2	1	no	—
29	M₁₆×C₂₃	D₄	1	2	yes	—
30	C₂×C₈×C₂₃	—	2	1	no	—
31	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₂₃	D₄	1	2	yes	—
33	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
36	C₂²⋊C₄×C₂₃	—	2	1	no	—
37	C₄²×C₂₃	—	2	1	no	—
38	#67267	—	2	2	no	—
39	#67267	—	2	2	no	—
40	C₂×C₂₃⋊C₈	—	2	2	no	—
41	Dic₉₂	—	2	4	no	—
42	Dic₉₂	—	2	4	no	—
43	C₂³×C₂₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
51	C₂×Dic₂₃	—	2	4	no	—
52	C₂×Dic₂₃	—	4	2	no	—
53	C₂×Dic₂₃	—	4	2	no	—
54	Dic₄₆	—	2	4	no	—
55	Dic₄₆	—	2	4	no	—
56	Dic₄₆	—	4	2	no	—
57	Dic₄₆	—	4	2	no	—
58	C₈×C₂₃	—	2	2	no	—
59	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
60	C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
61	C₂₃⋊C₈	—	4	2	no	—
62	#1979	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
65	C₂²×C₂₃	#121	1	1	yes	characteristic
66	C₂²×C₂₃	#124	1	1	yes	characteristic
67	C₄×C₂₃	—	4	2	no	—
68	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
70	C₄×C₂₃	#121	1	1	yes	characteristic
71	C₄×C₂₃	#127	1	1	yes	characteristic
72	Dic₂₃	—	4	4	no	—
73	C₂²×Q₁₆	—	23	1	no	—
74	C₂×#108	—	23	1	no	—
75	#419	—	23	2	no	—
76	C₂×#101	—	23	2	no	—
77	#405	D₂₃	1	1	yes	characteristic
78	C₂×C₂₃	—	2	1	no	—
79	C₂×C₂₃	—	2	1	no	—
80	C₂×C₂₃	#390	1	1	yes	characteristic
81	C₂×C₂₃	#473	1	2	yes	—
82	C₂²×Q₈	—	23	2	no	—
83	C₂×Q₁₆	—	23	4	no	—
84	C₂×Q₁₆	—	46	2	no	—
85	C₂×Q₁₆	—	46	1	no	—
86	C₂×Q₁₆	—	46	1	no	—
87	C₂×M₁₆	—	23	2	no	—
88	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
90	#117	C₂×D₂₃	1	1	yes	characteristic
91	#108	—	46	2	no	—
92	#103	—	46	4	no	—
93	#101	—	46	4	no	—
94	#97	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₂₃	#1979	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	23	4	no	—
98	C₂×Q₈	—	46	2	no	—
99	C₂×Q₈	—	46	2	no	—
100	C₂×Q₈	—	46	2	no	—
101	C₂×Q₈	—	46	2	no	—
102	C₂×Q₈	—	46	2	no	—
103	C₂²×C₄	—	46	2	no	—
104	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
105	Q₁₆	—	46	4	no	—
106	Q₁₆	—	46	4	no	—
107	M₁₆	—	2	1	no	—
108	M₁₆	—	46	2	no	—
109	M₁₆	—	46	2	no	—
110	M₁₆	D₉₂	1	2	yes	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	46	2	no	—
113	C₂×C₈	C₄×D₂₃	1	2	yes	—
114	C₂×C₈	D₉₂	1	2	yes	—
115	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
117	C₄⋊C₄	D₉₂	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₂³	#67288	1	1	yes	characteristic
121	Q₈	—	46	4	no	—
122	Q₈	—	46	4	no	—
123	Q₈	—	92	2	no	—
124	Q₈	—	92	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	46	4	no	—
127	C₂×C₄	—	92	2	no	—
128	C₂×C₄	—	92	2	no	—
129	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
130	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
131	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
132	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
133	C₂×C₄	#67288	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	92	2	no	—
137	C₈	(C₂×D₂₃)⋊C₄	1	2	yes	—
138	C₈	(C₂×D₂₃)⋊C₄	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₄×D₉₂	1	1	yes	characteristic
141	C₂²	#10625046	1	1	yes	characteristic
142	C₂²	#10625101	1	1	yes	characteristic
143	C₄	—	4	2	no	—
144	C₄	—	92	4	no	—
145	C₄	C₄×D₉₂	1	1	yes	characteristic
146	C₄	#10625045	1	1	yes	characteristic
147	C₄	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
148	C₄	#10625101	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#13092319	1	1	yes	characteristic
152	C₂	#13092692	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₉₂

C₂

yes

maximal, characteristic

C₂×#10625001

C₂

yes

maximal, characteristic

#13092425

C₂

yes

maximal

C₂×#10625006

C₂

yes

maximal

C₂₃×#405

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×Dic₉₂

—

C₂×Dic₉₂

—

C₂×Dic₉₂

—

C₂×Dic₉₂

C₄

yes

complemented

C₂₃×#97

—

C₄×C₈×C₂₃

—

#10625006

—

#10625001

—

#10624992

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

C₂×C₄

yes

—

M₁₆×C₂₃

—

M₁₆×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

Dic₉₂

—

Dic₉₂

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₈×C₂₃

C₂²⋊C₄

yes

—

C₂₃⋊C₈

—

#1979

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×D₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

C₂²×Q₁₆

—

—

—

—

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#473

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

C₂×D₂₃

yes

—

—

—

—

C₄×C₈

—

C₂₃

#1979

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₂₃

yes

characteristic

105

Q₁₆

—

106

Q₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

D₉₂

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₂₃

yes

—

114

C₂×C₈

D₉₂

yes

—

115

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

116

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

117

C₄⋊C₄

D₉₂

yes

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

#67288

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

130

C₂×C₄

C₂×D₉₂

yes

characteristic

131

C₂×C₄

C₂×D₉₂

yes

characteristic

132

C₂×C₄

C₂×D₉₂

yes

characteristic

133

C₂×C₄

#67288

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

(C₂×D₂₃)⋊C₄

yes

—

138

C₈

(C₂×D₂₃)⋊C₄

yes

—

139

C₂²

—

140

C₂²

C₄×D₉₂

yes

characteristic

141

C₂²

#10625046

yes

characteristic

142

C₂²

#10625101

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

C₄×D₉₂

yes

characteristic

146

C₄

#10625045

yes

characteristic

147

C₄

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

148

C₄

#10625101

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#13092319

yes

characteristic

152

C₂

#13092692

yes

—

	subgroup	G/H
center	C₂²	#10625101
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#405	C₂
socle	C₂²×C₂₃	#121
prime radical	C₂³×C₂₃	#41
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#405	D₂₃
hypocenter	C₂₃	#1979
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#10625101

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#405

C₂

C₂²×C₂₃

C₂³×C₂₃

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
14. C₂×Dic₉₂	C₄	semidirect
96. C₂₃	#1979	semidirect

H ≅ G/N

type

14. C₂×Dic₉₂

C₄

semidirect

96. C₂₃

#1979

semidirect

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

subgroup	index
C₂²×Dic₉₂	2
C₂×#10625001	2
#13092425	2
C₂×#10625006	2
C₂₃×#405	2
#1979	23

subgroup

index

C₂²×Dic₉₂

C₂₃×#405

Subgroups — #15674312

Conjugacy classes of subgroups

Complements

Maximal subgroups