Group #37510409 — subgroups

Subgroups — #37510409

Order 22912.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#19153036	C₂	1	2	yes	maximal
2	#26038345	C₂	1	1	yes	maximal, characteristic
3	#26038201	C₂	1	2	yes	maximal
4	#26038194	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₅₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₇₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₇₉	C₂²	1	1	yes	characteristic
9	#19153036	—	2	4	no	—
10	C₁₇₉⋊#95	—	2	1	no	—
11	C₁₇₉⋊#95	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₇₉	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₇₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₈	—	2	1	no	—
17	C₂×Dic₃₅₈	—	2	2	no	—
18	C₂×Dic₃₅₈	—	2	2	no	—
19	C₂×Dic₃₅₈	D₄	1	2	yes	—
20	C₂²×Dic₁₇₉	—	2	1	no	—
21	C₂²×Dic₁₇₉	—	2	1	no	—
22	C₂×C₁₇₉⋊C₈	—	2	2	no	—
23	C₂×C₁₇₉⋊C₈	—	2	2	no	—
24	C₂×C₁₇₉⋊C₈	—	2	2	no	—
25	C₄⋊Dic₁₇₉	—	2	2	no	—
26	C₂²×C₄×C₁₇₉	—	2	2	no	—
27	C₂²×C₄×C₁₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₇₉	—	2	2	no	—
31	C₄⋊C₄×C₁₇₉	—	2	2	no	—
32	C₄⋊C₄×C₁₇₉	—	4	2	no	—
33	C₄⋊C₄×C₁₇₉	—	4	2	no	—
34	C₄²×C₁₇₉	—	2	2	no	—
35	Dic₃₅₈	—	2	4	no	—
36	Dic₃₅₈	—	4	2	no	—
37	C₂×Dic₁₇₉	—	2	2	no	—
38	C₂×Dic₁₇₉	—	4	1	no	—
39	C₂×Dic₁₇₉	—	4	1	no	—
40	C₂×Dic₁₇₉	—	4	1	no	—
41	C₂×Dic₁₇₉	—	4	1	no	—
42	C₂×Dic₁₇₉	—	4	1	no	—
43	C₁₇₉⋊C₈	—	4	2	no	—
44	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	—	2	1	no	—
46	C₂×C₄×C₁₇₉	—	2	1	no	—
47	C₂×C₄×C₁₇₉	—	2	1	no	—
48	C₂×C₄×C₁₇₉	—	2	1	no	—
49	C₂×C₄×C₁₇₉	—	4	2	no	—
50	C₂×C₄×C₁₇₉	—	4	2	no	—
51	C₂×C₄×C₁₇₉	—	4	2	no	—
52	C₂×C₄×C₁₇₉	SD₁₆	1	2	yes	—
53	C₂×C₄×C₁₇₉	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₇₉	#41	1	2	yes	—
57	C₂×C₄×C₁₇₉	#41	1	2	yes	—
58	C₂²×C₁₇₉	#121	1	1	yes	characteristic
59	C₂²×C₁₇₉	#123	1	1	yes	characteristic
60	C₂²×C₁₇₉	#124	1	1	yes	characteristic
61	C₂²×C₁₇₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₁₇₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₁₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₁₇₉	#137	1	1	yes	characteristic
65	C₄×C₁₇₉	—	2	2	no	—
66	C₄×C₁₇₉	—	2	2	no	—
67	C₄×C₁₇₉	—	8	2	no	—
68	C₄×C₁₇₉	#123	1	2	yes	—
69	C₄×C₁₇₉	#126	1	2	yes	—
70	Dic₁₇₉	—	4	2	no	—
71	Dic₁₇₉	—	8	1	no	—
72	C₂×C₁₇₉	#397	1	1	yes	characteristic
73	C₂×C₁₇₉	#460	1	1	yes	characteristic
74	C₂×C₁₇₉	#462	1	1	yes	characteristic
75	C₂×C₁₇₉	#483	1	1	yes	characteristic
76	C₂×C₁₇₉	#484	1	1	yes	characteristic
77	C₂×C₁₇₉	#487	1	1	yes	characteristic
78	C₂×C₁₇₉	#489	1	1	yes	characteristic
79	C₁₇₉	#2072	1	1	yes	characteristic, complemented, Sylow
80	#2072	—	179	1	no	maximal, Sylow
81	C₂×#103	—	179	2	no	—
82	#391	—	179	1	no	—
83	#384	D₁₇₉	1	1	yes	characteristic
84	#340	—	179	2	no	—
85	#336	—	179	1	no	—
86	C₂²×Q₈	—	179	1	no	—
87	C₂²×C₈	—	179	2	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	179	1	no	—
91	C₂×C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
92	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
93	#103	—	358	4	no	—
94	#95	—	358	1	no	—
95	#95	—	358	1	no	—
96	C₂×Q₈	—	358	1	no	—
97	C₂×Q₈	—	358	2	no	—
98	C₂×Q₈	—	358	2	no	—
99	C₂×Q₈	—	179	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	358	1	no	—
102	C₂²×C₄	—	358	1	no	—
103	C₂²×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
106	C₂×C₈	—	358	2	no	—
107	C₂×C₈	—	358	2	no	—
108	C₂×C₈	—	358	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	358	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
116	Q₈	—	358	4	no	—
117	Q₈	—	716	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	358	2	no	—
126	C₂×C₄	—	716	1	no	—
127	C₂×C₄	—	716	1	no	—
128	C₂×C₄	—	716	1	no	—
129	C₂×C₄	—	716	1	no	—
130	C₂×C₄	—	716	1	no	—
131	C₂×C₄	#15668247	1	2	yes	—
132	C₂×C₄	#15668249	1	2	yes	—
133	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
134	C₂×C₄	#15668258	1	1	yes	characteristic
135	C₂×C₄	#15668259	1	2	yes	—
136	C₂×C₄	#15668260	1	2	yes	—
137	C₈	—	716	2	no	—
138	C₂²	#19153084	1	1	yes	characteristic
139	C₂²	#19153096	1	1	yes	characteristic
140	C₂²	#19153102	1	1	yes	characteristic
141	C₂²	C₂×#15668247	1	1	yes	characteristic
142	C₂²	C₂×#15668249	1	1	yes	characteristic
143	C₂²	#19153139	1	1	yes	characteristic
144	C₂²	#19153144	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	8	2	no	—
148	C₄	—	1432	1	no	—
149	C₄	—	716	2	no	—
150	C₄	#19153098	1	2	yes	—
151	C₄	#19153108	1	2	yes	—
152	C₂	#26038365	1	1	yes	characteristic
153	C₂	#26038636	1	1	yes	characteristic
154	C₂	#26038646	1	1	yes	characteristic
155	C₂	#26038769	1	1	yes	characteristic
156	C₂	#26038774	1	1	yes	characteristic
157	C₂	#26038791	1	1	yes	characteristic
158	C₂	#26038801	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153036

C₂

yes

maximal

#26038345

C₂

yes

maximal, characteristic

#26038201

C₂

yes

maximal

#26038194

C₂

yes

maximal, characteristic

#384×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₈

C₂²

yes

characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

characteristic

#19153036

—

C₁₇₉⋊#95

—

C₁₇₉⋊#95

—

C₂×C₄⋊C₄×C₁₇₉

—

C₂×C₄⋊C₄×C₁₇₉

—

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

D₄

yes

—

C₂²×Dic₁₇₉

—

C₂²×Dic₁₇₉

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

characteristic

C₂²×C₄×C₁₇₉

D₄

yes

characteristic

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

SD₁₆

yes

—

C₂×C₄×C₁₇₉

Q₁₆

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

C₂×C₄×C₁₇₉

yes

C₂×C₄×C₁₇₉

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇₉

C₂×Q₁₆

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#123

yes

—

C₄×C₁₇₉

#126

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#2072

—

179

maximal, Sylow

C₂×#103

—

179

—

#391

—

179

—

#384

D₁₇₉

yes

characteristic

#340

—

179

—

#336

—

179

—

C₂²×Q₈

—

179

—

C₂²×C₈

—

179

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#103

—

358

—

#95

—

358

—

#95

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

100

C₂²×C₄

—

101

C₂²×C₄

—

358

—

102

C₂²×C₄

—

358

—

103

C₂²×C₄

C₂²⋊D₁₇₉

yes

characteristic

104

C₂²×C₄

C₂²⋊D₁₇₉

yes

characteristic

105

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

106

C₂×C₈

—

358

—

107

C₂×C₈

—

358

—

108

C₂×C₈

—

358

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

358

—

114

C₄²

—

115

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

116

Q₈

—

358

—

117

Q₈

—

716

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

358

—

126

C₂×C₄

—

716

—

127

C₂×C₄

—

716

—

128

C₂×C₄

—

716

—

129

C₂×C₄

—

716

—

130

C₂×C₄

—

716

—

131

C₂×C₄

#15668247

yes

—

132

C₂×C₄

#15668249

yes

—

133

C₂×C₄

D₄×D₁₇₉

yes

characteristic

134

C₂×C₄

#15668258

yes

characteristic

135

C₂×C₄

#15668259

yes

—

136

C₂×C₄

#15668260

yes

—

137

C₈

—

716

—

138

C₂²

#19153084

yes

characteristic

139

C₂²

#19153096

yes

characteristic

140

C₂²

#19153102

yes

characteristic

141

C₂²

C₂×#15668247

yes

characteristic

142

C₂²

C₂×#15668249

yes

characteristic

143

C₂²

#19153139

yes

characteristic

144

C₂²

#19153144

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

1432

—

149

C₄

—

716

—

150

C₄

#19153098

yes

—

151

C₄

#19153108

yes

—

152

C₂

#26038365

yes

characteristic

153

C₂

#26038636

yes

characteristic

154

C₂

#26038646

yes

characteristic

155

C₂

#26038769

yes

characteristic

156

C₂

#26038774

yes

characteristic

157

C₂

#26038791

yes

characteristic

158

C₂

#26038801

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₉
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#384×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
hypercenter	#384	D₁₇₉
hypocenter	C₁₇₉	#2072
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₉

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#384×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
79. C₁₇₉	#2072	semidirect

H ≅ G/N

type

79. C₁₇₉

#2072

semidirect

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

subgroup	index
C₂×#19153036	2
#26038345	2
#26038201	2
#26038194	2
#384×C₁₇₉	2
#2072	179

subgroup

index

#384×C₁₇₉

179

Subgroups — #37510409

Conjugacy classes of subgroups

Complements

Maximal subgroups