Group #37510471 — subgroups

Subgroups — #37510471

Order 22912.

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

normal subgroups only

This group has 230 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	C₂×#19153046	C₂	1	2	yes	maximal
2	#26038309	C₂	1	1	yes	maximal, characteristic
3	#26038201	C₂	1	2	yes	maximal
4	#26038194	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₇₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₇₉	—	2	1	no	—
8	C₂×C₄⋊Dic₁₇₉	—	2	1	no	—
9	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
10	#19153046	—	2	4	no	—
11	C₂×C₄⋊C₄×C₁₇₉	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
15	#95×C₁₇₉	—	2	1	no	—
16	C₂²×Dic₁₇₉	—	2	2	no	—
17	C₂×C₁₇₉⋊C₈	—	2	2	no	—
18	C₂×C₁₇₉⋊C₈	—	2	2	no	—
19	C₂×C₁₇₉⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	C₄⋊Dic₁₇₉	—	2	2	no	—
22	C₄⋊Dic₁₇₉	—	4	2	no	—
23	C₄⋊Dic₁₇₉	—	4	2	no	—
24	C₂²×C₄×C₁₇₉	—	2	1	no	—
25	C₂²×C₄×C₁₇₉	—	2	1	no	—
26	C₂²×C₄×C₁₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₇₉	—	2	1	no	—
30	C₄⋊C₄×C₁₇₉	—	2	2	no	—
31	C₄⋊C₄×C₁₇₉	—	4	2	no	—
32	C₄⋊C₄×C₁₇₉	D₄	1	2	yes	—
33	C₄²×C₁₇₉	—	2	2	no	—
34	C₂×Dic₁₇₉	—	4	2	no	—
35	C₂×Dic₁₇₉	—	4	2	no	—
36	C₂×Dic₁₇₉	—	4	2	no	—
37	C₁₇₉⋊C₈	—	2	4	no	—
38	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇₉	—	2	1	no	—
40	C₂×C₄×C₁₇₉	—	2	1	no	—
41	C₂×C₄×C₁₇₉	—	2	1	no	—
42	C₂×C₄×C₁₇₉	—	2	1	no	—
43	C₂×C₄×C₁₇₉	—	2	2	no	—
44	C₂×C₄×C₁₇₉	—	4	1	no	—
45	C₂×C₄×C₁₇₉	—	4	1	no	—
46	C₂×C₄×C₁₇₉	—	4	1	no	—
47	C₂×C₄×C₁₇₉	—	4	1	no	—
48	C₂×C₄×C₁₇₉	—	4	1	no	—
49	C₂×C₄×C₁₇₉	D₈	1	2	yes	—
50	C₂×C₄×C₁₇₉	Q₁₆	1	2	yes	—
51	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₇₉	#41	1	2	yes	—
54	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
55	C₂²×C₁₇₉	#121	1	1	yes	characteristic
56	C₂²×C₁₇₉	#123	1	1	yes	characteristic
57	C₂²×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₁₇₉	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₁₇₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₁₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₁₇₉	#137	1	1	yes	characteristic
62	C₄×C₁₇₉	—	2	2	no	—
63	C₄×C₁₇₉	—	2	2	no	—
64	C₄×C₁₇₉	—	4	2	no	—
65	C₄×C₁₇₉	—	8	1	no	—
66	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₇₉	#125	1	2	yes	—
68	Dic₁₇₉	—	8	2	no	—
69	C₂×C₁₇₉	#398	1	1	yes	characteristic
70	C₂×C₁₇₉	#459	1	1	yes	characteristic
71	C₂×C₁₇₉	#462	1	1	yes	characteristic
72	C₂×C₁₇₉	#480	1	1	yes	characteristic
73	C₂×C₁₇₉	#484	1	1	yes	characteristic
74	C₂×C₁₇₉	#500	1	1	yes	characteristic
75	C₂×C₁₇₉	#501	1	1	yes	characteristic
76	C₁₇₉	#2083	1	1	yes	characteristic, complemented, Sylow
77	#2083	—	179	1	no	maximal, Sylow
78	C₂×#107	—	179	2	no	—
79	#389	D₁₇₉	1	1	yes	characteristic
80	#384	—	179	1	no	—
81	#340	—	179	2	no	—
82	#336	—	179	1	no	—
83	C₂²×C₈	—	179	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	358	1	no	—
86	C₂×C₄⋊C₄	—	358	1	no	—
87	C₂×C₄⋊C₄	—	179	2	no	—
88	C₂×C₄⋊C₄	C₂×D₁₇₉	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₁₇₉	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
91	#107	—	358	4	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	358	2	no	—
96	C₂²×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
99	C₂×C₈	—	358	2	no	—
100	C₂×C₈	—	358	2	no	—
101	C₂×C₈	—	179	4	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	358	2	no	—
106	C₄⋊C₄	—	358	2	no	—
107	C₄⋊C₄	—	716	2	no	—
108	C₄⋊C₄	—	716	2	no	—
109	C₄⋊C₄	D₇₁₆	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	716	2	no	—
123	C₂×C₄	—	716	2	no	—
124	C₂×C₄	—	716	2	no	—
125	C₂×C₄	#15668244	1	2	yes	—
126	C₂×C₄	#15668249	1	2	yes	—
127	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₇₉	1	2	yes	—
129	C₂×C₄	#15668258	1	2	yes	—
130	C₂×C₄	#15668259	1	1	yes	characteristic
131	C₈	—	358	4	no	—
132	C₂²	#19153087	1	1	yes	characteristic
133	C₂²	#19153094	1	1	yes	characteristic
134	C₂²	#19153111	1	1	yes	characteristic
135	C₂²	C₂×#15668244	1	1	yes	characteristic
136	C₂²	C₂×#15668249	1	1	yes	characteristic
137	C₂²	#19153138	1	1	yes	characteristic
138	C₂²	#19153143	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	1	no	—
143	C₄	—	1432	2	no	—
144	C₄	#19153093	1	2	yes	—
145	C₄	#19153103	1	2	yes	—
146	C₂	#26038369	1	1	yes	characteristic
147	C₂	#26038634	1	1	yes	characteristic
148	C₂	#26038649	1	1	yes	characteristic
149	C₂	#26038750	1	1	yes	characteristic
150	C₂	#26038772	1	1	yes	characteristic
151	C₂	#26038842	1	1	yes	characteristic
152	C₂	#26038849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153046

C₂

yes

maximal

#26038309

C₂

yes

maximal, characteristic

#26038201

C₂

yes

maximal

#26038194

C₂

yes

maximal, characteristic

#389×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

—

C₂×C₄⋊Dic₁₇₉

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

#19153046

—

C₂×C₄⋊C₄×C₁₇₉

—

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

characteristic

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

#95×C₁₇₉

—

C₂²×Dic₁₇₉

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

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

D₄

yes

—

C₄²×C₁₇₉

—

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

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

D₈

yes

—

C₂×C₄×C₁₇₉

Q₁₆

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇₉

C₂×D₈

yes

characteristic

C₂²×C₁₇₉

C₂×Q₁₆

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

#125

yes

—

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

#2083

—

179

maximal, Sylow

C₂×#107

—

179

—

#389

D₁₇₉

yes

characteristic

#384

—

179

—

#340

—

179

—

#336

—

179

—

C₂²×C₈

—

179

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

characteristic

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#107

—

358

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇₉

yes

characteristic

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

358

—

100

C₂×C₈

—

358

—

101

C₂×C₈

—

179

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

358

—

106

C₄⋊C₄

—

358

—

107

C₄⋊C₄

—

716

—

108

C₄⋊C₄

—

716

—

109

C₄⋊C₄

D₇₁₆

yes

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

716

—

123

C₂×C₄

—

716

—

124

C₂×C₄

—

716

—

125

C₂×C₄

#15668244

yes

—

126

C₂×C₄

#15668249

yes

—

127

C₂×C₄

C₂×D₇₁₆

yes

characteristic

128

C₂×C₄

Q₈×D₁₇₉

yes

—

129

C₂×C₄

#15668258

yes

—

130

C₂×C₄

#15668259

yes

characteristic

131

C₈

—

358

—

132

C₂²

#19153087

yes

characteristic

133

C₂²

#19153094

yes

characteristic

134

C₂²

#19153111

yes

characteristic

135

C₂²

C₂×#15668244

yes

characteristic

136

C₂²

C₂×#15668249

yes

characteristic

137

C₂²

#19153138

yes

characteristic

138

C₂²

#19153143

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

1432

—

144

C₄

#19153093

yes

—

145

C₄

#19153103

yes

—

146

C₂

#26038369

yes

characteristic

147

C₂

#26038634

yes

characteristic

148

C₂

#26038649

yes

characteristic

149

C₂

#26038750

yes

characteristic

150

C₂

#26038772

yes

characteristic

151

C₂

#26038842

yes

characteristic

152

C₂

#26038849

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₉
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#389×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
hypercenter	#389	D₁₇₉
hypocenter	C₁₇₉	#2083
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₉

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#389×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
76. C₁₇₉	#2083	semidirect

H ≅ G/N

type

76. C₁₇₉

#2083

semidirect

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

subgroup	index
C₂×#19153046	2
#26038309	2
#26038201	2
#26038194	2
#389×C₁₇₉	2
#2083	179

subgroup

index

#389×C₁₇₉

179

Subgroups — #37510471

Conjugacy classes of subgroups

Complements

Maximal subgroups