Group #37510414 — subgroups

Subgroups — #37510414

Order 22912.

258 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 79 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×#19153033	C₂	1	2	yes	maximal
2	#26038341	C₂	1	1	yes	maximal, characteristic
3	#26038201	C₂	1	2	yes	maximal, complemented
4	#26038194	C₂	1	1	yes	maximal, characteristic, complemented
5	#382×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₇₉⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₇₉	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₁₇₉)⋊C₄	—	2	2	no	—
10	#19153033	—	2	4	no	—
11	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
13	#95×C₁₇₉	—	2	2	no	—
14	C₂³×D₁₇₉	—	2	1	no	—
15	C₂×D₇₁₆	—	2	2	no	—
16	C₂×D₇₁₆	—	2	2	no	—
17	C₂×D₇₁₆	—	2	1	no	—
18	C₂×D₇₁₆	D₄	1	2	yes	—
19	C₂²×Dic₁₇₉	—	2	1	no	—
20	C₂×C₁₇₉⋊C₈	—	2	2	no	—
21	C₂×C₁₇₉⋊C₈	—	2	2	no	—
22	C₂×C₁₇₉⋊C₈	—	2	2	no	—
23	C₄⋊Dic₁₇₉	—	2	2	no	—
24	(C₂×D₁₇₉)⋊C₄	—	4	4	no	—
25	C₂²×C₄×C₁₇₉	—	2	2	no	—
26	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₇₉	—	2	2	no	—
29	C₄⋊C₄×C₁₇₉	—	2	2	no	—
30	C₄²×C₁₇₉	—	2	2	no	—
31	C₂²×D₁₇₉	—	2	2	no	—
32	C₂²×D₁₇₉	—	4	1	no	—
33	C₂²×D₁₇₉	—	4	1	no	—
34	C₂²×D₁₇₉	—	4	2	no	—
35	C₂²×D₁₇₉	—	4	2	no	—
36	D₇₁₆	—	2	4	no	—
37	D₇₁₆	—	4	2	no	—
38	C₂×Dic₁₇₉	—	4	1	no	—
39	C₂×Dic₁₇₉	—	4	2	no	—
40	C₁₇₉⋊C₈	—	4	2	no	—
41	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	—	2	2	no	—
43	C₂×C₄×C₁₇₉	—	2	2	no	—
44	C₂×C₄×C₁₇₉	—	2	2	no	—
45	C₂×C₄×C₁₇₉	—	4	2	no	—
46	C₂×C₄×C₁₇₉	—	4	2	no	—
47	C₂×C₄×C₁₇₉	—	4	2	no	—
48	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₁₇₉	#41	1	2	yes	—
51	C₂×C₄×C₁₇₉	#41	1	2	yes	—
52	C₂²×C₁₇₉	#121	1	1	yes	characteristic
53	C₂²×C₁₇₉	#123	1	1	yes	characteristic
54	C₂²×C₁₇₉	#124	1	1	yes	characteristic
55	C₂²×C₁₇₉	#135	1	2	yes	—
56	C₂²×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₂×D₁₇₉	—	4	2	no	—
58	C₂×D₁₇₉	—	4	4	no	—
59	C₂×D₁₇₉	—	8	2	no	—
60	C₂×D₁₇₉	—	8	1	no	—
61	C₂×D₁₇₉	—	8	1	no	—
62	C₄×C₁₇₉	—	4	2	no	—
63	C₄×C₁₇₉	—	8	2	no	—
64	C₄×C₁₇₉	#123	1	2	yes	—
65	C₄×C₁₇₉	#126	1	2	yes	—
66	Dic₁₇₉	—	8	1	no	—
67	C₂×C₁₇₉	#397	1	1	yes	characteristic
68	C₂×C₁₇₉	#463	1	2	yes	—
69	C₂×C₁₇₉	#482	1	2	yes	—
70	C₂×C₁₇₉	#488	1	1	yes	characteristic
71	C₂×C₁₇₉	#490	1	1	yes	characteristic
72	D₁₇₉	—	8	2	no	—
73	C₁₇₉	#2073	1	1	yes	characteristic, complemented, Sylow
74	#2073	—	179	1	no	maximal, Sylow
75	C₂×#102	—	179	2	no	—
76	#390	—	179	1	no	—
77	#382	D₁₇₉	1	1	yes	characteristic, complemented
78	#340	—	179	2	no	—
79	#336	—	179	1	no	—
80	C₂²×D₄	—	179	1	no	—
81	C₂²×C₈	—	179	2	no	—
82	C₂×C₄⋊C₄	—	179	1	no	—
83	C₂×C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
84	C₂×C₂²⋊C₄	—	358	2	no	—
85	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
86	#102	—	358	4	no	—
87	#95	—	2	2	no	—
88	C₂⁴	—	358	1	no	—
89	C₂×D₄	—	358	1	no	—
90	C₂×D₄	—	358	2	no	—
91	C₂×D₄	—	358	2	no	—
92	C₂×D₄	—	179	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	358	1	no	—
95	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
97	C₂×C₈	—	358	2	no	—
98	C₂×C₈	—	358	2	no	—
99	C₂×C₈	—	358	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	358	2	no	—
103	C₂²⋊C₄	—	716	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	358	2	no	—
106	C₂³	—	716	1	no	—
107	C₂³	—	716	1	no	—
108	C₂³	—	716	2	no	—
109	C₂³	—	716	2	no	—
110	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
111	D₄	—	358	4	no	—
112	D₄	—	716	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	716	1	no	—
120	C₂×C₄	—	716	2	no	—
121	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
122	C₂×C₄	#15668258	1	1	yes	characteristic
123	C₂×C₄	#15668259	1	2	yes	—
124	C₂×C₄	#15668260	1	2	yes	—
125	C₈	—	716	2	no	—
126	C₂²	—	1432	1	no	—
127	C₂²	—	1432	1	no	—
128	C₂²	—	1432	2	no	—
129	C₂²	—	716	2	no	—
130	C₂²	—	716	4	no	—
131	C₂²	#19153084	1	1	yes	characteristic
132	C₂²	#19153096	1	1	yes	characteristic
133	C₂²	#19153102	1	1	yes	characteristic
134	C₂²	#19153133	1	2	yes	—
135	C₂²	#19153138	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	1432	1	no	—
139	C₄	#19153098	1	2	yes	—
140	C₄	#19153108	1	2	yes	—
141	C₂	—	1432	2	no	—
142	C₂	#26038365	1	1	yes	characteristic
143	C₂	#26038651	1	2	yes	—
144	C₂	#26038763	1	2	yes	—
145	C₂	#26038796	1	1	yes	characteristic
146	C₂	#26038805	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153033

C₂

yes

maximal

#26038341

C₂

yes

maximal, characteristic

#26038201

C₂

yes

maximal, complemented

#26038194

C₂

yes

maximal, characteristic, complemented

#382×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₁₆

C₂²

yes

characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

characteristic

C₂×(C₂×D₁₇₉)⋊C₄

—

#19153033

—

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

#95×C₁₇₉

—

C₂³×D₁₇₉

—

C₂×D₇₁₆

—

C₂×D₇₁₆

—

C₂×D₇₁₆

—

C₂×D₇₁₆

D₄

yes

—

C₂²×Dic₁₇₉

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

(C₂×D₁₇₉)⋊C₄

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₂²×D₁₇₉

—

C₂²×D₁₇₉

—

C₂²×D₁₇₉

—

C₂²×D₁₇₉

—

C₂²×D₁₇₉

—

D₇₁₆

—

D₇₁₆

—

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×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

yes

—

C₂²×C₁₇₉

AGL(1,ℤ/8)

yes

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#123

yes

—

C₄×C₁₇₉

#126

yes

—

Dic₁₇₉

—

C₂×C₁₇₉

#397

yes

characteristic

C₂×C₁₇₉

#463

yes

—

C₂×C₁₇₉

#482

yes

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

D₁₇₉

—

C₁₇₉

#2073

yes

characteristic, complemented, Sylow

#2073

—

179

maximal, Sylow

C₂×#102

—

179

—

#390

—

179

—

#382

D₁₇₉

yes

characteristic, complemented

#340

—

179

—

#336

—

179

—

C₂²×D₄

—

179

—

C₂²×C₈

—

179

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂×C₂²⋊C₄

—

358

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#102

—

358

—

#95

—

C₂⁴

—

358

—

C₂×D₄

—

358

—

C₂×D₄

—

358

—

C₂×D₄

—

358

—

C₂×D₄

—

179

—

C₂²×C₄

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

358

—

103

C₂²⋊C₄

—

716

—

104

C₄²

—

105

C₂³

—

358

—

106

C₂³

—

716

—

107

C₂³

—

716

—

108

C₂³

—

716

—

109

C₂³

—

716

—

110

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

111

D₄

—

358

—

112

D₄

—

716

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

716

—

120

C₂×C₄

—

716

—

121

C₂×C₄

D₄×D₁₇₉

yes

characteristic

122

C₂×C₄

#15668258

yes

characteristic

123

C₂×C₄

#15668259

yes

—

124

C₂×C₄

#15668260

yes

—

125

C₈

—

716

—

126

C₂²

—

1432

—

127

C₂²

—

1432

—

128

C₂²

—

1432

—

129

C₂²

—

716

—

130

C₂²

—

716

—

131

C₂²

#19153084

yes

characteristic

132

C₂²

#19153096

yes

characteristic

133

C₂²

#19153102

yes

characteristic

134

C₂²

#19153133

yes

—

135

C₂²

#19153138

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

1432

—

139

C₄

#19153098

yes

—

140

C₄

#19153108

yes

—

141

C₂

—

1432

—

142

C₂

#26038365

yes

characteristic

143

C₂

#26038651

yes

—

144

C₂

#26038763

yes

—

145

C₂

#26038796

yes

characteristic

146

C₂

#26038805

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₉
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#382×C₁₇₉	C₂
socle	C₂³×C₁₇₉	C₂×D₄
prime radical	C₂²×D₇₁₆	C₂²
hypercenter	#382	D₁₇₉
hypocenter	C₁₇₉	#2073
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₉

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#382×C₁₇₉

C₂

socle

C₂³×C₁₇₉

C₂×D₄

prime radical

C₂²×D₇₁₆

C₂²

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
3. #26038201	C₂	semidirect
4. #26038194	C₂	semidirect
5. #382×C₁₇₉	C₂	semidirect
73. C₁₇₉	#2073	semidirect
77. #382	D₁₇₉	semidirect

H ≅ G/N

type

3. #26038201

C₂

semidirect

4. #26038194

C₂

semidirect

5. #382×C₁₇₉

C₂

semidirect

73. C₁₇₉

#2073

semidirect

77. #382

D₁₇₉

semidirect

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

subgroup	index
C₂×#19153033	2
#26038341	2
#26038201	2
#26038194	2
#382×C₁₇₉	2
#2073	179

subgroup

index

#382×C₁₇₉

179

Subgroups — #37510414

Conjugacy classes of subgroups

Complements

Maximal subgroups