Group #37508994 — subgroups

Subgroups — #37508994

Order 22912.

272 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 95 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×Q₈×Dic₁₇₉	C₂	1	1	yes	maximal, characteristic
2	#26038462	C₂	1	2	yes	maximal
3	C₂×#19153032	C₂	1	1	yes	maximal, characteristic
4	#26038207	C₂	1	2	yes	maximal
5	C₂×#101×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₂²	1	2	yes	—
7	Q₈×Dic₁₇₉	—	2	2	no	—
8	Q₈×Dic₁₇₉	—	2	2	no	—
9	C₂×C₄⋊Dic₁₇₉	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₁₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇₉	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
13	Dic₁₇₉⋊C₈	—	2	2	no	—
14	Dic₁₇₉⋊C₈	—	2	2	no	—
15	#19153032	—	2	2	no	—
16	C₂²×Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇₉	C₂²	1	2	yes	—
18	#101×C₁₇₉	—	2	2	no	—
19	C₂²×Dic₁₇₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₇₉	C₂×C₄	1	2	yes	—
21	C₂²×Dic₁₇₉	C₂×C₄	1	1	yes	characteristic
22	#15668243	—	2	2	no	—
23	#15668243	—	2	2	no	—
24	C₂×C₁₇₉⋊C₈	—	2	2	no	—
25	C₄⋊Dic₁₇₉	—	2	4	no	—
26	C₄⋊Dic₁₇₉	—	2	2	no	—
27	C₄×Dic₁₇₉	—	2	2	no	—
28	C₄×Dic₁₇₉	—	2	2	no	—
29	C₄×Dic₁₇₉	D₄	1	2	yes	—
30	C₄×Dic₁₇₉	D₄	1	2	yes	—
31	C₂×Q₈×C₁₇₉	—	2	2	no	—
32	C₂×Q₈×C₁₇₉	—	2	2	no	—
33	C₂×Q₈×C₁₇₉	D₄	1	2	yes	—
34	C₂×Q₈×C₁₇₉	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₇₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₇₉	—	2	2	no	—
38	M₁₆×C₁₇₉	—	2	2	no	—
39	C₂×C₈×C₁₇₉	—	2	2	no	—
40	C₂×Dic₁₇₉	—	2	2	no	—
41	C₂×Dic₁₇₉	—	2	2	no	—
42	C₂×Dic₁₇₉	—	4	1	no	—
43	C₂×Dic₁₇₉	—	4	2	no	—
44	C₂×Dic₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×Dic₁₇₉	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₇₉⋊C₈	—	4	2	no	—
48	C₂³×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₇₉	—	2	4	no	—
50	Q₈×C₁₇₉	—	2	4	no	—
51	C₂×C₄×C₁₇₉	—	2	2	no	—
52	C₂×C₄×C₁₇₉	—	2	2	no	—
53	C₂×C₄×C₁₇₉	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇₉	#41	1	2	yes	—
58	C₈×C₁₇₉	—	4	2	no	—
59	C₂²×C₁₇₉	—	2	1	no	—
60	C₂²×C₁₇₉	—	2	1	no	—
61	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₇₉	#117	1	1	yes	characteristic
64	C₄×C₁₇₉	—	2	4	no	—
65	C₄×C₁₇₉	#121	1	2	yes	—
66	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
67	Dic₁₇₉	—	2	1	no	—
68	Dic₁₇₉	—	2	2	no	—
69	Dic₁₇₉	—	4	2	no	—
70	Dic₁₇₉	#101	1	2	yes	complemented
71	C₂×C₁₇₉	—	2	1	no	—
72	C₂×C₁₇₉	—	2	1	no	—
73	C₂×C₁₇₉	#380	1	1	yes	characteristic
74	C₂×C₁₇₉	C₂×#101	1	1	yes	characteristic
75	C₂×C₁₇₉	#413	1	1	yes	characteristic
76	C₁₇₉	#1801	1	1	yes	characteristic, complemented, Sylow
77	#1801	—	179	1	no	maximal, Sylow
78	C₂×C₄×Q₈	—	179	1	no	—
79	#423	—	179	2	no	—
80	C₂×#101	—	179	1	no	—
81	C₂×#101	D₁₇₉	1	1	yes	characteristic
82	#343	—	179	2	no	—
83	C₂²×Q₈	C₂×D₁₇₉	1	1	yes	characteristic
84	C₂×M₁₆	—	179	2	no	—
85	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
86	C₄×Q₈	—	358	2	no	—
87	C₄×Q₈	—	358	2	no	—
88	C₂×C₄⋊C₄	—	179	2	no	—
89	C₂×C₄⋊C₄	—	179	1	no	—
90	C₂×C₄²	—	179	2	no	—
91	C₂×C₄²	—	179	1	no	—
92	C₄⋊C₈	—	358	2	no	—
93	C₄⋊C₈	—	358	2	no	—
94	#101	—	2	2	no	—
95	#101	—	358	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂²⋊D₁₇₉	1	2	yes	—
99	C₂×Q₈	Dic₃₅₈	1	2	yes	—
100	C₂²×C₄	—	179	1	no	—
101	C₂²×C₄	—	179	2	no	—
102	C₂²×C₄	—	179	1	no	—
103	C₂²×C₄	C₄×D₁₇₉	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	358	2	no	—
108	M₁₆	—	358	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	358	2	no	—
111	C₄⋊C₄	—	358	4	no	—
112	C₄⋊C₄	—	358	2	no	—
113	C₄²	—	358	2	no	—
114	C₄²	—	358	2	no	—
115	C₄²	—	179	2	no	—
116	C₄²	—	179	2	no	—
117	C₂³	C₂×C₄×D₁₇₉	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	358	2	no	—
123	C₂×C₄	—	358	2	no	—
124	C₂×C₄	—	179	1	no	—
125	C₂×C₄	—	179	2	no	—
126	C₂×C₄	—	179	1	no	—
127	C₂×C₄	—	716	1	no	—
128	C₂×C₄	—	716	2	no	—
129	C₂×C₄	Dic₁₇₉⋊C₄	1	4	yes	—
130	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
132	C₂×C₄	C₂×Dic₃₅₈	1	1	yes	characteristic
133	C₂×C₄	#15668258	1	2	yes	—
134	C₈	—	4	2	no	—
135	C₈	—	716	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	C₂²⋊C₄×D₁₇₉	1	1	yes	characteristic
139	C₂²	C₂×Dic₁₇₉⋊C₄	1	1	yes	characteristic
140	C₂²	#19153069	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	358	1	no	—
143	C₄	—	358	2	no	—
144	C₄	—	179	2	no	—
145	C₄	—	716	2	no	—
146	C₄	#19153085	1	2	yes	—
147	C₄	#19153090	1	2	yes	—
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	#26038297	1	1	yes	characteristic
151	C₂	#101×D₁₇₉	1	1	yes	characteristic
152	C₂	#26038422	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₇₉

C₂

yes

maximal, characteristic

#26038462

C₂

yes

maximal

C₂×#19153032

C₂

yes

maximal, characteristic

#26038207

C₂

yes

maximal

C₂×#101×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

Q₈×Dic₁₇₉

—

Q₈×Dic₁₇₉

—

C₂×C₄⋊Dic₁₇₉

C₄

yes

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₉

C₄

yes

—

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

Dic₁₇₉⋊C₈

—

Dic₁₇₉⋊C₈

—

#19153032

—

C₂²×Q₈×C₁₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₇₉

C₂²

yes

—

#101×C₁₇₉

—

C₂²×Dic₁₇₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₇₉

C₂×C₄

yes

—

C₂²×Dic₁₇₉

C₂×C₄

yes

characteristic

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

D₄

yes

—

C₄×Dic₁₇₉

D₄

yes

—

C₂×Q₈×C₁₇₉

—

C₂×Q₈×C₁₇₉

—

C₂×Q₈×C₁₇₉

D₄

yes

—

C₂×Q₈×C₁₇₉

Q₈

yes

—

C₂²×C₄×C₁₇₉

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇₉

C₂²⋊C₄

yes

—

C₂×Dic₁₇₉

C₂²⋊C₄

yes

characteristic

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂²×C₄

yes

characteristic

Q₈×C₁₇₉

—

Q₈×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇₉

#117

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

#121

yes

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

#101

yes

complemented

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1801

—

179

maximal, Sylow

C₂×C₄×Q₈

—

179

—

#423

—

179

—

C₂×#101

—

179

—

C₂×#101

D₁₇₉

yes

characteristic

#343

—

179

—

C₂²×Q₈

C₂×D₁₇₉

yes

characteristic

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

C₄×Q₈

—

358

—

C₄×Q₈

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄²

—

179

—

C₂×C₄²

—

179

—

C₄⋊C₈

—

358

—

C₄⋊C₈

—

358

—

#101

—

#101

—

358

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₇₉

yes

—

C₂×Q₈

Dic₃₅₈

yes

—

100

C₂²×C₄

—

179

—

101

C₂²×C₄

—

179

—

102

C₂²×C₄

—

179

—

103

C₂²×C₄

C₄×D₁₇₉

yes

—

104

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

358

—

108

M₁₆

—

358

—

109

C₂×C₈

—

110

C₂×C₈

—

358

—

111

C₄⋊C₄

—

358

—

112

C₄⋊C₄

—

358

—

113

C₄²

—

358

—

114

C₄²

—

358

—

115

C₄²

—

179

—

116

C₄²

—

179

—

117

C₂³

C₂×C₄×D₁₇₉

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

358

—

123

C₂×C₄

—

358

—

124

C₂×C₄

—

179

—

125

C₂×C₄

—

179

—

126

C₂×C₄

—

179

—

127

C₂×C₄

—

716

—

128

C₂×C₄

—

716

—

129

C₂×C₄

Dic₁₇₉⋊C₄

yes

—

130

C₂×C₄

D₄×D₁₇₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

132

C₂×C₄

C₂×Dic₃₅₈

yes

characteristic

133

C₂×C₄

#15668258

yes

—

134

C₈

—

135

C₈

—

716

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂²⋊C₄×D₁₇₉

yes

characteristic

139

C₂²

C₂×Dic₁₇₉⋊C₄

yes

characteristic

140

C₂²

#19153069

yes

characteristic

141

C₄

—

142

C₄

—

358

—

143

C₄

—

358

—

144

C₄

—

179

—

145

C₄

—

716

—

146

C₄

#19153085

yes

—

147

C₄

#19153090

yes

—

148

C₂

—

149

C₂

—

150

C₂

#26038297

yes

characteristic

151

C₂

#101×D₁₇₉

yes

characteristic

152

C₂

#26038422

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₇₉
commutator, prime radical	C₂³×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂×#101×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₂×#101	D₁₇₉
hypocenter	C₁₇₉	#1801
LCS	G ▷ C₂³×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₇₉

commutator, prime radical

C₂³×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂×#101×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂³×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
70. Dic₁₇₉	#101	semidirect
76. C₁₇₉	#1801	semidirect

H ≅ G/N

type

70. Dic₁₇₉

#101

semidirect

76. C₁₇₉

#1801

semidirect

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

subgroup	index
C₂×Q₈×Dic₁₇₉	2
#26038462	2
C₂×#19153032	2
#26038207	2
C₂×#101×C₁₇₉	2
#1801	179

subgroup

index

C₂×Q₈×Dic₁₇₉

C₂×#101×C₁₇₉

179

Subgroups — #37508994

Conjugacy classes of subgroups

Complements

Maximal subgroups