Group #37510148 — subgroups

Subgroups — #37510148

Order 22912.

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

normal subgroups only

This group has 258 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	C₂×#125×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	2	no	—
11	#125×C₁₇₉	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₇₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₅₈	—	2	2	no	—
16	C₂×Dic₃₅₈	—	2	1	no	—
17	C₂×Dic₃₅₈	—	2	2	no	—
18	C₂×Dic₃₅₈	Q₈	1	2	yes	—
19	C₂²×Dic₁₇₉	—	2	1	no	—
20	C₂²×Dic₁₇₉	—	2	1	no	—
21	C₂×C₁₇₉⋊C₈	—	2	2	no	—
22	C₂×C₁₇₉⋊C₈	—	2	2	no	—
23	C₂×C₁₇₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₇₉	—	2	2	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	8	no	—
29	C₄⋊C₄×C₁₇₉	—	2	2	no	—
30	C₄⋊C₄×C₁₇₉	D₄	1	4	yes	—
31	C₄²×C₁₇₉	—	2	2	no	—
32	Dic₃₅₈	—	2	4	no	—
33	Dic₃₅₈	—	4	2	no	—
34	C₂×Dic₁₇₉	—	2	2	no	—
35	C₂×Dic₁₇₉	—	4	1	no	—
36	C₂×Dic₁₇₉	—	4	1	no	—
37	C₂×Dic₁₇₉	—	4	1	no	—
38	C₂×Dic₁₇₉	—	4	2	no	—
39	C₁₇₉⋊C₈	—	4	2	no	—
40	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	—	2	4	no	—
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	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇₉	#41	1	2	yes	—
50	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
51	C₂²×C₁₇₉	#120	1	1	yes	characteristic
52	C₂²×C₁₇₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₇₉	#124	1	1	yes	characteristic
54	C₂²×C₁₇₉	#135	1	2	yes	—
55	C₂²×C₁₇₉	#137	1	2	yes	—
56	C₄×C₁₇₉	—	4	2	no	—
57	C₄×C₁₇₉	—	4	4	no	—
58	C₄×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₇₉	#124	1	2	yes	—
60	Dic₁₇₉	—	4	2	no	—
61	Dic₁₇₉	—	8	1	no	—
62	C₂×C₁₇₉	#393	1	1	yes	characteristic
63	C₂×C₁₇₉	#452	1	2	yes	—
64	C₂×C₁₇₉	#479	1	2	yes	—
65	C₂×C₁₇₉	#488	1	1	yes	characteristic
66	C₂×C₁₇₉	#491	1	1	yes	characteristic
67	C₁₇₉	#2026	1	1	yes	characteristic, complemented, Sylow
68	#2026	—	179	1	no	maximal, Sylow
69	C₂×#125	D₁₇₉	1	1	yes	characteristic
70	C₂×#103	—	179	2	no	—
71	#391	—	179	1	no	—
72	#340	—	179	2	no	—
73	#336	—	179	1	no	—
74	C₂²×Q₈	—	179	1	no	—
75	C₂²×C₈	—	179	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	179	1	no	—
79	C₂×C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
80	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
81	#103	—	358	4	no	—
82	#95	—	358	2	no	—
83	C₂×Q₈	—	358	1	no	—
84	C₂×Q₈	—	358	2	no	—
85	C₂×Q₈	—	358	2	no	—
86	C₂×Q₈	—	179	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	358	1	no	—
89	C₂²×C₄	—	358	1	no	—
90	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
92	C₂×C₈	—	358	2	no	—
93	C₂×C₈	—	358	2	no	—
94	C₂×C₈	—	358	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	358	2	no	—
98	C₄⋊C₄	C₂²⋊D₁₇₉	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
101	Q₈	—	358	4	no	—
102	Q₈	—	716	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	358	2	no	—
110	C₂×C₄	—	716	1	no	—
111	C₂×C₄	—	716	2	no	—
112	C₂×C₄	—	716	1	no	—
113	C₂×C₄	—	716	1	no	—
114	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	2	yes	—
115	C₂×C₄	Q₈×D₁₇₉	1	1	yes	characteristic
116	C₂×C₄	#15668260	1	2	yes	—
117	C₂×C₄	#15668260	1	1	yes	characteristic
118	C₈	—	716	2	no	—
119	C₂²	#19153082	1	1	yes	characteristic
120	C₂²	#19153089	1	1	yes	characteristic
121	C₂²	#19153102	1	1	yes	characteristic
122	C₂²	#19153133	1	2	yes	—
123	C₂²	#19153143	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	1432	1	no	—
127	C₄	—	716	2	no	—
128	C₄	#19153089	1	2	yes	—
129	C₄	#19153102	1	2	yes	—
130	C₂	#26038350	1	1	yes	characteristic
131	C₂	#26038601	1	2	yes	—
132	C₂	#26038743	1	2	yes	—
133	C₂	#26038796	1	1	yes	characteristic
134	C₂	#26038809	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

C₂×#125×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₈

C₂²

yes

characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

characteristic

#19153036

—

C₁₇₉⋊#95

—

#125×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

Q₈

yes

—

C₂²×Dic₁₇₉

—

C₂²×Dic₁₇₉

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

D₄

yes

—

C₄²×C₁₇₉

—

Dic₃₅₈

—

Dic₃₅₈

—

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×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

C₂²⋊Q₈

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

—

C₂²×C₁₇₉

#137

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂²⋊Q₈

yes

—

C₄×C₁₇₉

#124

yes

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

#393

yes

characteristic

C₂×C₁₇₉

#452

yes

—

C₂×C₁₇₉

#479

yes

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#2026

—

179

maximal, Sylow

C₂×#125

D₁₇₉

yes

characteristic

C₂×#103

—

179

—

#391

—

179

—

#340

—

179

—

#336

—

179

—

C₂²×Q₈

—

179

—

C₂²×C₈

—

179

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#103

—

358

—

#95

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

358

—

C₂×Q₈

—

179

—

C₂²×C₄

—

C₂²×C₄

—

358

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

C₂²⋊D₁₇₉

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

101

Q₈

—

358

—

102

Q₈

—

716

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

358

—

110

C₂×C₄

—

716

—

111

C₂×C₄

—

716

—

112

C₂×C₄

—

716

—

113

C₂×C₄

—

716

—

114

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

—

115

C₂×C₄

Q₈×D₁₇₉

yes

characteristic

116

C₂×C₄

#15668260

yes

—

117

C₂×C₄

#15668260

yes

characteristic

118

C₈

—

716

—

119

C₂²

#19153082

yes

characteristic

120

C₂²

#19153089

yes

characteristic

121

C₂²

#19153102

yes

characteristic

122

C₂²

#19153133

yes

—

123

C₂²

#19153143

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

1432

—

127

C₄

—

716

—

128

C₄

#19153089

yes

—

129

C₄

#19153102

yes

—

130

C₂

#26038350

yes

characteristic

131

C₂

#26038601

yes

—

132

C₂

#26038743

yes

—

133

C₂

#26038796

yes

characteristic

134

C₂

#26038809

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₉
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂×#125×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
hypercenter	C₂×#125	D₁₇₉
hypocenter	C₁₇₉	#2026
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₉

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂×#125×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
67. C₁₇₉	#2026	semidirect

H ≅ G/N

type

67. C₁₇₉

#2026

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
C₂×#125×C₁₇₉	2
#2026	179

subgroup

index

C₂×#125×C₁₇₉

179

Subgroups — #37510148

Conjugacy classes of subgroups

Complements

Maximal subgroups