Group #37509752 — subgroups

Subgroups — #37509752

Order 22912.

200 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 83 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×#19153048	C₂	1	1	yes	maximal, characteristic
2	#26038475	C₂	1	1	yes	maximal, characteristic
3	#26038467	C₂	1	1	yes	maximal, characteristic
4	#26038466	C₂	1	1	yes	maximal, characteristic
5	C₂×#19153024	C₂	1	1	yes	maximal, characteristic
6	#26038199	C₂	1	1	yes	maximal, characteristic
7	#339×C₁₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668243	C₂²	1	1	yes	characteristic
9	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	#19153072	C₂²	1	1	yes	characteristic
12	#19153048	—	2	2	no	—
13	#19153046	C₄	1	2	yes	complemented
14	#19153044	C₄	1	2	yes	complemented
15	C₄⋊(C₁₇₉⋊C₈)	—	2	2	no	—
16	Dic₁₇₉⋊C₈	—	2	2	no	—
17	#19153024	—	2	2	no	—
18	C₂×M₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
19	#117×C₁₇₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
21	#15668243	—	2	1	no	—
22	#15668243	—	2	1	no	—
23	C₂×C₁₇₉⋊C₈	—	2	1	no	—
24	C₂×C₁₇₉⋊C₈	—	2	1	no	—
25	C₂×C₁₇₉⋊C₈	—	2	1	no	—
26	C₂×C₁₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
29	C₂×C₁₇₉⋊C₈	D₄	1	2	yes	—
30	C₂×C₁₇₉⋊C₈	Q₈	1	2	yes	—
31	C₄⋊Dic₁₇₉	C₂×C₄	1	2	yes	—
32	C₂²⋊Dic₁₇₉	—	2	1	no	—
33	C₄×Dic₁₇₉	—	2	1	no	—
34	C₂²×C₄×C₁₇₉	—	2	1	no	—
35	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₇₉	—	2	1	no	—
37	M₁₆×C₁₇₉	—	2	1	no	—
38	C₂×C₈×C₁₇₉	—	2	1	no	—
39	C₄⋊C₄×C₁₇₉	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₁₇₉	—	2	1	no	—
41	C₄²×C₁₇₉	—	2	1	no	—
42	C₄²×C₁₇₉	—	2	1	no	—
43	C₄²×C₁₇₉	Q₈	1	2	yes	—
44	C₂×Dic₁₇₉	—	2	2	no	—
45	C₁₇₉⋊C₈	—	2	2	no	—
46	C₁₇₉⋊C₈	—	4	1	no	—
47	C₁₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₁₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂³×C₁₇₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₁₇₉	—	2	2	no	—
51	C₂×C₄×C₁₇₉	—	2	2	no	—
52	C₂×C₄×C₁₇₉	—	4	2	no	—
53	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
59	C₈×C₁₇₉	—	4	1	no	—
60	C₂²×C₁₇₉	—	2	2	no	—
61	C₂²×C₁₇₉	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₁₇₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₇₉	#125	1	1	yes	characteristic
64	C₄×C₁₇₉	—	4	2	no	—
65	C₄×C₁₇₉	—	4	2	no	—
66	C₄×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₁₇₉	C₄×Q₈	1	1	yes	characteristic
68	C₄×C₁₇₉	#121	1	1	yes	characteristic
69	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
70	Dic₁₇₉	—	4	2	no	—
71	C₂×C₁₇₉	—	2	1	no	—
72	C₂×C₁₇₉	—	2	1	no	—
73	C₂×C₁₇₉	#389	1	1	yes	characteristic
74	C₂×C₁₇₉	#444	1	2	yes	—
75	C₁₇₉	#1946	1	1	yes	characteristic, complemented, Sylow
76	#1946	—	179	1	no	maximal, Sylow
77	C₂×#108	—	179	1	no	—
78	#427	—	179	1	no	—
79	#424	—	179	1	no	—
80	#424	—	179	1	no	—
81	C₂×#97	—	179	1	no	—
82	#339	—	179	1	no	—
83	#339	D₁₇₉	1	1	yes	characteristic
84	C₂×M₁₆	—	179	1	no	—
85	C₂×M₁₆	C₂×D₁₇₉	1	1	yes	characteristic
86	C₂²×C₈	—	179	1	no	—
87	C₂²×C₈	—	179	1	no	—
88	#117	—	179	1	no	—
89	#117	C₂×D₁₇₉	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
91	#108	—	358	2	no	—
92	#107	—	179	2	no	—
93	#106	—	179	2	no	—
94	C₄⋊C₈	—	358	2	no	—
95	C₄⋊C₈	—	358	2	no	—
96	#97	—	358	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	358	1	no	—
102	M₁₆	—	358	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	358	1	no	—
105	C₂×C₈	—	358	1	no	—
106	C₂×C₈	—	358	1	no	—
107	C₂×C₈	—	179	2	no	—
108	C₂×C₈	—	179	1	no	—
109	C₂×C₈	—	179	1	no	—
110	C₂×C₈	—	179	2	no	—
111	C₂×C₈	—	179	2	no	—
112	C₄⋊C₄	—	179	2	no	—
113	C₄⋊C₄	C₄×D₁₇₉	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	358	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	358	1	no	—
119	C₄²	Dic₃₅₈	1	2	yes	—
120	C₂³	#15668258	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	358	2	no	—
125	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₇₉	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₃₅₈	1	1	yes	characteristic
130	C₂×C₄	#15668259	1	1	yes	characteristic
131	C₈	—	4	1	no	—
132	C₈	—	358	2	no	—
133	C₈	—	179	2	no	—
134	C₈	—	179	2	no	—
135	C₈	—	716	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	(C₂²⋊D₁₇₉)⋊C₄	1	1	yes	characteristic
138	C₂²	#19153090	1	1	yes	characteristic
139	C₂²	#19153103	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	716	2	no	—
143	C₄	C₄⋊C₄×D₁₇₉	1	1	yes	characteristic
144	C₄	C₄×Dic₃₅₈	1	1	yes	characteristic
145	C₄	#19153086	1	1	yes	characteristic
146	C₄	#19153112	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#26038336	1	1	yes	characteristic
150	C₂	#26038558	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153048

C₂

yes

maximal, characteristic

#26038475

C₂

yes

maximal, characteristic

#26038467

C₂

yes

maximal, characteristic

#26038466

C₂

yes

maximal, characteristic

C₂×#19153024

C₂

yes

maximal, characteristic

#26038199

C₂

yes

maximal, characteristic

#339×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

#19153072

C₂²

yes

characteristic

#19153048

—

#19153046

C₄

yes

complemented

#19153044

C₄

yes

complemented

C₄⋊(C₁₇₉⋊C₈)

—

Dic₁₇₉⋊C₈

—

#19153024

—

C₂×M₁₆×C₁₇₉

C₂²

yes

characteristic

#117×C₁₇₉

C₂²

yes

characteristic

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×C₁₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₁₇₉

C₂×C₄

yes

—

C₂²⋊Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₄⋊C₄×C₁₇₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

—

C₄²×C₁₇₉

—

C₄²×C₁₇₉

Q₈

yes

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

C₄⋊C₄

yes

—

C₁₇₉⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

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

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₄×D₄

yes

characteristic

C₂²×C₁₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇₉

#125

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇₉

C₄×Q₈

yes

characteristic

C₄×C₁₇₉

#121

yes

characteristic

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#389

yes

characteristic

C₂×C₁₇₉

#444

yes

—

C₁₇₉

#1946

yes

characteristic, complemented, Sylow

#1946

—

179

maximal, Sylow

C₂×#108

—

179

—

#427

—

179

—

#424

—

179

—

#424

—

179

—

C₂×#97

—

179

—

#339

—

179

—

#339

D₁₇₉

yes

characteristic

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

characteristic

C₂²×C₈

—

179

—

C₂²×C₈

—

179

—

#117

—

179

—

#117

C₂×D₁₇₉

yes

characteristic

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#108

—

358

—

#107

—

179

—

#106

—

179

—

C₄⋊C₈

—

358

—

C₄⋊C₈

—

358

—

#97

—

358

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

358

—

102

M₁₆

—

358

—

103

C₂×C₈

—

104

C₂×C₈

—

358

—

105

C₂×C₈

—

358

—

106

C₂×C₈

—

358

—

107

C₂×C₈

—

179

—

108

C₂×C₈

—

179

—

109

C₂×C₈

—

179

—

110

C₂×C₈

—

179

—

111

C₂×C₈

—

179

—

112

C₄⋊C₄

—

179

—

113

C₄⋊C₄

C₄×D₁₇₉

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

358

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

358

—

119

C₄²

Dic₃₅₈

yes

—

120

C₂³

#15668258

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

358

—

125

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

126

C₂×C₄

D₄×D₁₇₉

yes

characteristic

127

C₂×C₄

D₄×D₁₇₉

yes

characteristic

128

C₂×C₄

Q₈×D₁₇₉

yes

characteristic

129

C₂×C₄

C₂×Dic₃₅₈

yes

characteristic

130

C₂×C₄

#15668259

yes

characteristic

131

C₈

—

132

C₈

—

358

—

133

C₈

—

179

—

134

C₈

—

179

—

135

C₈

—

716

—

136

C₂²

—

137

C₂²

(C₂²⋊D₁₇₉)⋊C₄

yes

characteristic

138

C₂²

#19153090

yes

characteristic

139

C₂²

#19153103

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

716

—

143

C₄

C₄⋊C₄×D₁₇₉

yes

characteristic

144

C₄

C₄×Dic₃₅₈

yes

characteristic

145

C₄

#19153086

yes

characteristic

146

C₄

#19153112

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#26038336

yes

characteristic

150

C₂

#26038558

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#339×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₄×D₄
prime radical	C₂³×C₁₇₉	#41
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#339	D₁₇₉
hypocenter	C₁₇₉	#1946
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

subgroup

G/H

center

C₂×C₄

D₄×D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#339×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₄×D₄

prime radical

C₂³×C₁₇₉

#41

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

N	H ≅ G/N	type
13. #19153046	C₄	semidirect
14. #19153044	C₄	semidirect
75. C₁₇₉	#1946	semidirect

H ≅ G/N

type

13. #19153046

C₄

semidirect

14. #19153044

C₄

semidirect

75. C₁₇₉

#1946

semidirect

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

subgroup	index
C₂×#19153048	2
#26038475	2
#26038467	2
#26038466	2
C₂×#19153024	2
#26038199	2
#339×C₁₇₉	2
#1946	179

subgroup

index

#339×C₁₇₉

179

Subgroups — #37509752

Conjugacy classes of subgroups

Complements

Maximal subgroups