Group #37508856 — subgroups

Subgroups — #37508856

Order 22912.

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

normal subgroups only

This group has 260 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×Dic₁₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#19153023	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₁₇₉	C₂	1	1	yes	maximal, characteristic
5	#26038285	C₂	1	1	yes	maximal, characteristic
6	#26038194	C₂	1	1	yes	maximal, characteristic
7	#336×C₁₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
12	Dic₁₇₉⋊C₈	C₄	1	4	yes	complemented
13	Dic₁₇₉⋊C₈	C₄	1	4	yes	complemented
14	#19153023	—	2	2	no	—
15	C₈×Dic₁₇₉	—	2	2	no	—
16	C₁₇₉⋊#95	C₄	1	2	yes	—
17	C₁₇₉⋊#95	C₄	1	2	yes	—
18	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₁₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₇₉	C₂×C₄	1	2	yes	—
24	C₂×C₁₇₉⋊C₈	—	2	1	no	—
25	C₂×C₁₇₉⋊C₈	—	2	1	no	—
26	C₂×C₁₇₉⋊C₈	—	2	1	no	—
27	C₂×C₁₇₉⋊C₈	—	2	1	no	—
28	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×Dic₁₇₉	C₂×C₄	1	2	yes	—
31	C₄×Dic₁₇₉	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₇₉	D₄	1	2	yes	—
33	C₄×Dic₁₇₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₇₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₇₉	—	2	1	no	—
37	C₂×C₈×C₁₇₉	—	2	1	no	—
38	C₂×C₈×C₁₇₉	—	2	1	no	—
39	C₂×C₈×C₁₇₉	—	2	1	no	—
40	C₂×C₈×C₁₇₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₁₇₉	C₂×C₄	1	2	yes	—
42	C₄²×C₁₇₉	—	2	1	no	—
43	C₄²×C₁₇₉	—	2	1	no	—
44	C₂×Dic₁₇₉	—	2	2	no	—
45	C₂×Dic₁₇₉	—	2	2	no	—
46	C₂×Dic₁₇₉	—	2	2	no	—
47	C₂×Dic₁₇₉	—	2	2	no	—
48	C₂×Dic₁₇₉	C₄²	1	2	yes	—
49	C₂×Dic₁₇₉	C₄²	1	2	yes	—
50	C₂×Dic₁₇₉	C₄⋊C₄	1	4	yes	—
51	C₁₇₉⋊C₈	—	2	2	no	—
52	C₁₇₉⋊C₈	—	2	2	no	—
53	C₂³×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇₉	—	2	2	no	—
55	C₂×C₄×C₁₇₉	—	2	2	no	—
56	C₂×C₄×C₁₇₉	—	2	2	no	—
57	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
62	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
63	C₈×C₁₇₉	—	2	2	no	—
64	C₈×C₁₇₉	—	2	2	no	—
65	C₂²×C₁₇₉	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₁₇₉	#117	1	1	yes	characteristic
68	C₂²×C₁₇₉	#131	1	2	yes	—
69	C₂²×C₁₇₉	#131	1	2	yes	—
70	C₄×C₁₇₉	—	4	2	no	—
71	C₄×C₁₇₉	C₄×D₄	1	1	yes	characteristic
72	C₄×C₁₇₉	C₄×D₄	1	1	yes	characteristic
73	C₄×C₁₇₉	C₄×Q₈	1	1	yes	characteristic
74	C₄×C₁₇₉	C₄×Q₈	1	1	yes	characteristic
75	Dic₁₇₉	—	2	2	no	—
76	Dic₁₇₉	—	2	2	no	—
77	Dic₁₇₉	—	2	4	no	—
78	C₂×C₁₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₁₇₉	#405	1	1	yes	characteristic
80	C₂×C₁₇₉	#405	1	1	yes	characteristic
81	C₂×C₁₇₉	#424	1	2	yes	—
82	C₂×C₁₇₉	#433	1	2	yes	—
83	C₁₇₉	#1768	1	1	yes	characteristic, complemented, Sylow
84	#1768	—	179	1	no	maximal, Sylow
85	C₂×C₄⋊C₈	—	179	1	no	—
86	C₂×C₄⋊C₈	—	179	1	no	—
87	C₂×#97	—	179	1	no	—
88	C₂×C₄×C₈	—	179	1	no	—
89	#376	—	179	1	no	—
90	#336	—	179	1	no	—
91	#336	D₁₇₉	1	1	yes	characteristic
92	C₂²×C₈	—	179	1	no	—
93	C₂²×C₈	—	179	1	no	—
94	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
96	C₂×C₄²	—	179	1	no	—
97	C₂×C₄²	—	179	1	no	—
98	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
99	C₄⋊C₈	—	179	4	no	—
100	C₄⋊C₈	—	179	4	no	—
101	#97	—	358	2	no	—
102	C₄×C₈	—	358	2	no	—
103	#95	—	179	2	no	—
104	#95	—	179	2	no	—
105	C₂²×C₄	—	179	1	no	—
106	C₂²×C₄	—	179	2	no	—
107	C₂²×C₄	—	179	1	no	—
108	C₂²×C₄	C₄×D₁₇₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	358	1	no	—
115	C₂×C₈	—	358	1	no	—
116	C₂×C₈	—	358	1	no	—
117	C₂×C₈	—	358	1	no	—
118	C₂×C₈	—	179	2	no	—
119	C₂×C₈	—	179	2	no	—
120	C₂×C₈	C₄×D₁₇₉	1	2	yes	—
121	C₂×C₈	C₄×D₁₇₉	1	2	yes	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	179	2	no	—
125	C₄²	—	179	2	no	—
126	C₄²	—	179	2	no	—
127	C₄²	—	179	2	no	—
128	C₂³	C₂×C₄×D₁₇₉	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	358	2	no	—
133	C₂×C₄	—	358	2	no	—
134	C₂×C₄	—	358	2	no	—
135	C₂×C₄	—	358	2	no	—
136	C₂×C₄	—	179	2	no	—
137	C₂×C₄	—	179	2	no	—
138	C₂×C₄	—	179	4	no	—
139	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
141	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₁₇₉	1	1	yes	characteristic
143	C₂×C₄	#15668258	1	1	yes	characteristic
144	C₂×C₄	#15668258	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	358	2	no	—
148	C₈	—	358	2	no	—
149	C₂²	C₄²×D₁₇₉	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₁₇₉	1	1	yes	characteristic
151	C₂²	#19153069	1	1	yes	characteristic
152	C₂²	#19153119	1	2	yes	—
153	C₂²	#19153121	1	2	yes	—
154	C₄	—	4	2	no	—
155	C₄	—	358	2	no	—
156	C₄	—	358	2	no	—
157	C₄	—	358	4	no	—
158	C₄	(C₂²⋊D₁₇₉)⋊C₄	1	1	yes	characteristic
159	C₄	(C₂²⋊D₁₇₉)⋊C₄	1	1	yes	characteristic
160	C₄	#19153078	1	1	yes	characteristic
161	C₄	#19153078	1	1	yes	characteristic
162	C₂	#26038290	1	1	yes	characteristic
163	C₂	#26038393	1	1	yes	characteristic
164	C₂	#26038396	1	1	yes	characteristic
165	C₂	#26038464	1	2	yes	—
166	C₂	#26038497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₇₉⋊C₈

C₂

yes

maximal, characteristic

C₂×Dic₁₇₉⋊C₈

C₂

yes

maximal, characteristic

C₂×#19153023

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₇₉

C₂

yes

maximal, characteristic

#26038285

C₂

yes

maximal, characteristic

#26038194

C₂

yes

maximal, characteristic

#336×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

Dic₁₇₉⋊C₈

C₄

yes

complemented

Dic₁₇₉⋊C₈

C₄

yes

complemented

#19153023

—

C₈×Dic₁₇₉

—

C₁₇₉⋊#95

C₄

yes

—

C₁₇₉⋊#95

C₄

yes

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₂²×Dic₁₇₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₇₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₇₉

C₂×C₄

yes

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₇₉

C₂×C₄

yes

—

C₄×Dic₁₇₉

C₂×C₄

yes

—

C₄×Dic₁₇₉

D₄

yes

—

C₄×Dic₁₇₉

Q₈

yes

—

C₂²×C₄×C₁₇₉

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

—

C₄²×C₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

C₄²

yes

—

C₂×Dic₁₇₉

C₄²

yes

—

C₂×Dic₁₇₉

C₄⋊C₄

yes

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

C₂×C₄×C₁₇₉

yes

C₂×C₄×C₁₇₉

yes

C₈×C₁₇₉

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₄²

yes

characteristic

C₂²×C₁₇₉

C₂×C₄⋊C₄

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

—

C₂²×C₁₇₉

#131

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₄×D₄

yes

characteristic

C₄×C₁₇₉

C₄×D₄

yes

characteristic

C₄×C₁₇₉

C₄×Q₈

yes

characteristic

C₄×C₁₇₉

C₄×Q₈

yes

characteristic

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

C₄×C₄⋊C₄

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

—

C₂×C₁₇₉

#433

yes

—

C₁₇₉

#1768

yes

characteristic, complemented, Sylow

#1768

—

179

maximal, Sylow

C₂×C₄⋊C₈

—

179

—

C₂×C₄⋊C₈

—

179

—

C₂×#97

—

179

—

C₂×C₄×C₈

—

179

—

#376

—

179

—

#336

—

179

—

#336

D₁₇₉

yes

characteristic

C₂²×C₈

—

179

—

C₂²×C₈

—

179

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

C₂×C₄²

—

179

—

C₂×C₄²

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

C₄⋊C₈

—

179

—

100

C₄⋊C₈

—

179

—

101

#97

—

358

—

102

C₄×C₈

—

358

—

103

#95

—

179

—

104

#95

—

179

—

105

C₂²×C₄

—

179

—

106

C₂²×C₄

—

179

—

107

C₂²×C₄

—

179

—

108

C₂²×C₄

C₄×D₁₇₉

yes

—

109

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

358

—

115

C₂×C₈

—

358

—

116

C₂×C₈

—

358

—

117

C₂×C₈

—

358

—

118

C₂×C₈

—

179

—

119

C₂×C₈

—

179

—

120

C₂×C₈

C₄×D₁₇₉

yes

—

121

C₂×C₈

C₄×D₁₇₉

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

179

—

125

C₄²

—

179

—

126

C₄²

—

179

—

127

C₄²

—

179

—

128

C₂³

C₂×C₄×D₁₇₉

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

358

—

133

C₂×C₄

—

358

—

134

C₂×C₄

—

358

—

135

C₂×C₄

—

358

—

136

C₂×C₄

—

179

—

137

C₂×C₄

—

179

—

138

C₂×C₄

—

179

—

139

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

140

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

141

C₂×C₄

D₄×D₁₇₉

yes

characteristic

142

C₂×C₄

Q₈×D₁₇₉

yes

characteristic

143

C₂×C₄

#15668258

yes

characteristic

144

C₂×C₄

#15668258

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

358

—

148

C₈

—

358

—

149

C₂²

C₄²×D₁₇₉

yes

characteristic

150

C₂²

C₄⋊C₄×D₁₇₉

yes

characteristic

151

C₂²

#19153069

yes

characteristic

152

C₂²

#19153119

yes

—

153

C₂²

#19153121

yes

—

154

C₄

—

155

C₄

—

358

—

156

C₄

—

358

—

157

C₄

—

358

—

158

C₄

(C₂²⋊D₁₇₉)⋊C₄

yes

characteristic

159

C₄

(C₂²⋊D₁₇₉)⋊C₄

yes

characteristic

160

C₄

#19153078

yes

characteristic

161

C₄

#19153078

yes

characteristic

162

C₂

#26038290

yes

characteristic

163

C₂

#26038393

yes

characteristic

164

C₂

#26038396

yes

characteristic

165

C₂

#26038464

yes

—

166

C₂

#26038497

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₇₉
commutator	C₂²×C₁₇₉	C₂×C₄²
Fitting	#336×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#336	D₁₇₉
hypocenter	C₁₇₉	#1768
LCS	G ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₇₉

commutator

C₂²×C₁₇₉

C₂×C₄²

Fitting

#336×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂²×C₁₇₉ ▷ C₁₇₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
12. Dic₁₇₉⋊C₈	C₄	semidirect
13. Dic₁₇₉⋊C₈	C₄	semidirect
83. C₁₇₉	#1768	semidirect

H ≅ G/N

type

12. Dic₁₇₉⋊C₈

C₄

semidirect

13. Dic₁₇₉⋊C₈

C₄

semidirect

83. C₁₇₉

#1768

semidirect

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

subgroup	index
C₂×Dic₁₇₉⋊C₈	2
C₂×#19153023	2
C₂×C₈×Dic₁₇₉	2
#26038285	2
#26038194	2
#336×C₁₇₉	2
#1768	179

subgroup

index

C₂×Dic₁₇₉⋊C₈

C₂×C₈×Dic₁₇₉

#336×C₁₇₉

179

Subgroups — #37508856

Conjugacy classes of subgroups

Complements

Maximal subgroups