Group #37509988 — subgroups

Subgroups — #37509988

Order 22912.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19153113	C₂	1	1	yes	maximal, characteristic
2	C₄×#15668243	C₂	1	1	yes	maximal, characteristic
3	#26038207	C₂	1	4	yes	maximal
4	#423×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₈	C₄	1	2	yes	—
6	C₂×#15668243	C₂²	1	2	yes	—
7	#19153113	—	2	1	no	—
8	#19153113	—	2	1	no	—
9	#19153113	—	2	1	no	—
10	#19153113	—	2	1	no	—
11	C₂×C₄⋊Dic₁₇₉	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
13	#19153024	—	2	1	no	—
14	C₄×C₁₇₉⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₁₇₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₇₉	—	2	1	no	—
18	C₄⋊C₈×C₁₇₉	—	2	1	no	—
19	C₂×Dic₃₅₈	—	2	2	no	—
20	C₂×Dic₃₅₈	—	2	2	no	—
21	C₂×Dic₃₅₈	—	4	4	no	—
22	C₂²×Dic₁₇₉	C₂×C₄	1	4	yes	—
23	#15668243	—	2	2	no	—
24	#15668243	Q₈	1	4	yes	complemented
25	C₂×C₁₇₉⋊C₈	—	2	2	no	—
26	C₄⋊Dic₁₇₉	—	2	2	no	—
27	C₄⋊Dic₁₇₉	—	2	2	no	—
28	C₄⋊Dic₁₇₉	—	2	2	no	—
29	C₄⋊Dic₁₇₉	—	2	2	no	—
30	C₂²×C₄×C₁₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₇₉	—	2	4	no	—
34	C₂×C₈×C₁₇₉	—	2	2	no	—
35	C₄²×C₁₇₉	D₄	1	1	yes	characteristic
36	C₄²×C₁₇₉	D₄	1	1	yes	characteristic
37	C₄²×C₁₇₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₇₉	D₄	1	1	yes	characteristic
39	Dic₃₅₈	—	4	4	no	—
40	Dic₃₅₈	—	4	4	no	—
41	C₂×Dic₁₇₉	—	2	4	no	—
42	C₂×Dic₁₇₉	—	4	4	no	—
43	C₁₇₉⋊C₈	—	2	4	no	—
44	C₂³×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	—	2	2	no	—
46	C₂×C₄×C₁₇₉	—	2	2	no	—
47	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₇₉	#41	1	2	yes	—
55	C₈×C₁₇₉	—	4	2	no	—
56	C₂²×C₁₇₉	—	2	1	no	—
57	C₂²×C₁₇₉	—	2	1	no	—
58	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₇₉	C₄×Q₈	1	2	yes	—
60	C₄×C₁₇₉	—	2	1	no	—
61	C₄×C₁₇₉	—	2	1	no	—
62	C₄×C₁₇₉	#100	1	2	yes	—
63	C₄×C₁₇₉	#101	1	2	yes	—
64	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₇₉	#124	1	1	yes	characteristic
66	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₇₉	—	4	4	no	—
68	C₂×C₁₇₉	—	2	2	no	—
69	C₂×C₁₇₉	#391	1	1	yes	characteristic
70	C₂×C₁₇₉	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₇₉	C₂×#101	1	1	yes	characteristic
72	C₁₇₉	#1997	1	1	yes	characteristic, complemented, Sylow
73	#1997	—	179	1	no	maximal, Sylow
74	C₂×C₄⋊Q₈	—	179	1	no	—
75	#423	D₁₇₉	1	1	yes	characteristic
76	C₄×M₁₆	—	179	1	no	—
77	#343	—	179	4	no	—
78	C₂²×Q₈	—	179	2	no	—
79	C₂×M₁₆	—	179	2	no	—
80	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
81	C₄⋊Q₈	—	358	1	no	—
82	C₄⋊Q₈	—	358	1	no	—
83	C₄⋊Q₈	—	358	1	no	—
84	C₄⋊Q₈	—	358	1	no	—
85	C₂×C₄⋊C₄	—	179	2	no	—
86	C₂×C₄⋊C₄	—	179	2	no	—
87	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	358	1	no	—
91	C₄×C₈	—	358	1	no	—
92	C₂×Q₈	—	358	2	no	—
93	C₂×Q₈	—	358	2	no	—
94	C₂×Q₈	—	716	4	no	—
95	C₂²×C₄	—	179	4	no	—
96	C₂²×C₄	C₄×D₁₇₉	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₁₇₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	358	2	no	—
101	M₁₆	—	179	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	358	2	no	—
104	C₄⋊C₄	—	358	2	no	—
105	C₄⋊C₄	—	358	2	no	—
106	C₄⋊C₄	—	358	2	no	—
107	C₄⋊C₄	—	358	2	no	—
108	C₄²	C₂²⋊D₁₇₉	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₇₉	1	1	yes	characteristic
110	C₄²	D₇₁₆	1	1	yes	characteristic
111	C₄²	D₇₁₆	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₁₇₉	1	1	yes	characteristic
113	Q₈	—	716	4	no	—
114	Q₈	—	716	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	358	4	no	—
118	C₂×C₄	—	716	4	no	—
119	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
124	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₇₉	1	2	yes	—
126	C₂×C₄	#15668258	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₈	—	358	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
132	C₂²	#19153078	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	716	4	no	—
136	C₄	#19153029	1	2	yes	—
137	C₄	#19153031	1	2	yes	—
138	C₄	#19153093	1	2	yes	—
139	C₄	#19153100	1	1	yes	characteristic
140	C₄	#19153111	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#26038344	1	1	yes	characteristic
143	C₂	C₂×#19153029	1	1	yes	characteristic
144	C₂	C₂×#19153031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153113

C₂

yes

maximal, characteristic

C₄×#15668243

C₂

yes

maximal, characteristic

#26038207

C₂

yes

maximal

#423×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₈

C₄

yes

—

C₂×#15668243

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₁₇₉

C₄

yes

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

#19153024

—

C₄×C₁₇₉⋊C₈

—

C₂×M₁₆×C₁₇₉

C₂²

yes

—

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₉

—

C₄⋊C₈×C₁₇₉

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂×Dic₃₅₈

—

C₂²×Dic₁₇₉

C₂×C₄

yes

—

#15668243

—

#15668243

Q₈

yes

complemented

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₄²×C₁₇₉

D₄

yes

characteristic

C₄²×C₁₇₉

D₄

yes

characteristic

C₄²×C₁₇₉

D₄

yes

characteristic

C₄²×C₁₇₉

D₄

yes

characteristic

Dic₃₅₈

—

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

characteristic

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×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₉

C₄×Q₈

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#100

yes

—

C₄×C₁₇₉

#101

yes

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

#124

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1997

—

179

maximal, Sylow

C₂×C₄⋊Q₈

—

179

—

#423

D₁₇₉

yes

characteristic

C₄×M₁₆

—

179

—

#343

—

179

—

C₂²×Q₈

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

C₄⋊Q₈

—

358

—

C₄⋊Q₈

—

358

—

C₄⋊Q₈

—

358

—

C₄⋊Q₈

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

358

—

C₄×C₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

716

—

C₂²×C₄

—

179

—

C₂²×C₄

C₄×D₁₇₉

yes

characteristic

C₂²×C₄

C₄×D₁₇₉

yes

characteristic

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

100

M₁₆

—

358

—

101

M₁₆

—

179

—

102

C₂×C₈

—

103

C₂×C₈

—

358

—

104

C₄⋊C₄

—

358

—

105

C₄⋊C₄

—

358

—

106

C₄⋊C₄

—

358

—

107

C₄⋊C₄

—

358

—

108

C₄²

C₂²⋊D₁₇₉

yes

characteristic

109

C₄²

C₂²⋊D₁₇₉

yes

characteristic

110

C₄²

D₇₁₆

yes

characteristic

111

C₄²

D₇₁₆

yes

characteristic

112

C₂³

C₂×C₄×D₁₇₉

yes

characteristic

113

Q₈

—

716

—

114

Q₈

—

716

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

358

—

118

C₂×C₄

—

716

—

119

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₁₇₉)⋊C₄

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

124

C₂×C₄

C₂×D₇₁₆

yes

characteristic

125

C₂×C₄

Q₈×D₁₇₉

yes

—

126

C₂×C₄

#15668258

yes

—

127

C₈

—

128

C₈

—

358

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×(C₂×D₁₇₉)⋊C₄

yes

characteristic

132

C₂²

#19153078

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

716

—

136

C₄

#19153029

yes

—

137

C₄

#19153031

yes

—

138

C₄

#19153093

yes

—

139

C₄

#19153100

yes

characteristic

140

C₄

#19153111

yes

characteristic

141

C₂

—

142

C₂

#26038344

yes

characteristic

143

C₂

C₂×#19153029

yes

characteristic

144

C₂

C₂×#19153031

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₇₉)⋊C₄
commutator, prime radical	C₂³×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#423×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#423	D₁₇₉
hypocenter	C₁₇₉	#1997
LCS	G ▷ C₂³×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₇₉)⋊C₄

commutator, prime radical

C₂³×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#423×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂³×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #15668243	Q₈	semidirect
72. C₁₇₉	#1997	semidirect

H ≅ G/N

type

24. #15668243

Q₈

semidirect

72. C₁₇₉

#1997

semidirect

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

subgroup	index
C₂×#19153113	2
C₄×#15668243	2
#26038207	2
#423×C₁₇₉	2
#1997	179

subgroup

index

#423×C₁₇₉

179

Subgroups — #37509988

Conjugacy classes of subgroups

Complements

Maximal subgroups