Group #37510495 — subgroups

Subgroups — #37510495

Order 22912.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#19153047	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#26038338	C₂	1	2	yes	maximal
4	#26038201	C₂	1	2	yes	maximal
5	#340×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₇₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
10	#19153047	—	2	2	no	—
11	Dic₁₇₉⋊C₈	—	2	2	no	—
12	C₁₇₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
15	C₂²×Dic₁₇₉	—	2	2	no	—
16	C₂²×Dic₁₇₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₁₇₉	D₄	1	1	yes	characteristic
18	C₂×C₁₇₉⋊C₈	—	2	1	no	—
19	C₂×C₁₇₉⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	C₄⋊Dic₁₇₉	—	2	2	no	—
22	C₄⋊Dic₁₇₉	—	4	4	no	—
23	C₄×Dic₁₇₉	—	2	1	no	—
24	C₄×Dic₁₇₉	—	2	1	no	—
25	C₂²×C₄×C₁₇₉	—	2	2	no	—
26	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₇₉	—	2	2	no	—
28	C₂×C₈×C₁₇₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₇₉	—	2	2	no	—
30	C₄⋊C₄×C₁₇₉	Q₈	1	4	yes	—
31	C₂×Dic₁₇₉	—	2	1	no	—
32	C₂×Dic₁₇₉	—	2	1	no	—
33	C₂×Dic₁₇₉	—	2	1	no	—
34	C₂×Dic₁₇₉	—	2	2	no	—
35	C₂×Dic₁₇₉	—	4	2	no	—
36	C₂×Dic₁₇₉	—	4	2	no	—
37	C₂×Dic₁₇₉	—	4	2	no	—
38	C₂×Dic₁₇₉	Q₁₆	1	2	yes	—
39	C₁₇₉⋊C₈	—	4	1	no	—
40	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	—	2	4	no	—
42	C₂×C₄×C₁₇₉	—	4	2	no	—
43	C₂×C₄×C₁₇₉	—	4	2	no	—
44	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₇₉	#41	1	2	yes	—
47	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
48	C₈×C₁₇₉	—	2	2	no	—
49	C₂²×C₁₇₉	#121	1	1	yes	characteristic
50	C₂²×C₁₇₉	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₁₇₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₁₇₉	#135	1	1	yes	characteristic
53	C₂²×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₁₇₉	—	4	4	no	—
55	C₄×C₁₇₉	#123	1	1	yes	characteristic
56	C₄×C₁₇₉	#125	1	2	yes	—
57	C₄×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₁₇₉	—	2	2	no	—
59	Dic₁₇₉	—	4	1	no	—
60	Dic₁₇₉	—	8	2	no	—
61	C₂×C₁₇₉	#398	1	1	yes	characteristic
62	C₂×C₁₇₉	#467	1	1	yes	characteristic
63	C₂×C₁₇₉	#469	1	1	yes	characteristic
64	C₂×C₁₇₉	#477	1	2	yes	—
65	C₂×C₁₇₉	#478	1	2	yes	—
66	C₁₇₉	#2087	1	1	yes	characteristic, complemented, Sylow
67	#2087	—	179	1	no	maximal, Sylow
68	C₂×#107	—	179	1	no	—
69	C₂×C₄⋊C₈	—	179	1	no	—
70	#389	—	179	2	no	—
71	#340	—	179	2	no	—
72	#340	D₁₇₉	1	1	yes	characteristic
73	C₂²×C₈	—	179	1	no	—
74	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	358	2	no	—
76	C₂×C₄⋊C₄	—	179	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
78	C₂×C₄²	—	179	1	no	—
79	#107	—	358	2	no	—
80	C₄⋊C₈	—	358	2	no	—
81	#95	—	358	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	358	2	no	—
84	C₂²×C₄	—	179	1	no	—
85	C₂²×C₄	—	179	1	no	—
86	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	358	1	no	—
89	C₂×C₈	—	358	2	no	—
90	C₂×C₈	D₇₁₆	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	358	2	no	—
93	C₄⋊C₄	—	358	2	no	—
94	C₄⋊C₄	—	716	4	no	—
95	C₄⋊C₄	Dic₃₅₈	1	4	yes	—
96	C₄²	—	358	1	no	—
97	C₄²	—	358	1	no	—
98	C₂³	D₄×D₁₇₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	358	1	no	—
103	C₂×C₄	—	358	1	no	—
104	C₂×C₄	—	358	1	no	—
105	C₂×C₄	—	358	2	no	—
106	C₂×C₄	—	179	2	no	—
107	C₂×C₄	—	716	2	no	—
108	C₂×C₄	—	716	2	no	—
109	C₂×C₄	—	716	2	no	—
110	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₃₅₈	1	2	yes	—
112	C₂×C₄	#15668258	1	2	yes	—
113	C₂×C₄	#15668260	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	716	1	no	—
116	C₂²	#19153088	1	1	yes	characteristic
117	C₂²	#19153090	1	2	yes	—
118	C₂²	Q₁₆×D₁₇₉	1	1	yes	characteristic
119	C₂²	#19153132	1	1	yes	characteristic
120	C₂²	#19153140	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	358	2	no	—
123	C₄	—	1432	2	no	—
124	C₄	—	716	1	no	—
125	C₄	#19153097	1	1	yes	characteristic
126	C₄	#19153104	1	2	yes	—
127	C₄	#19153113	1	1	yes	characteristic
128	C₂	#26038371	1	1	yes	characteristic
129	C₂	#26038681	1	1	yes	characteristic
130	C₂	#26038693	1	1	yes	characteristic
131	C₂	#26038732	1	2	yes	—
132	C₂	#26038737	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153047

C₂

yes

maximal, characteristic

C₂×Dic₁₇₉⋊C₈

C₂

yes

maximal, characteristic

#26038338

C₂

yes

maximal

#26038201

C₂

yes

maximal

#340×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇₉

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

C₂×C₄×Dic₁₇₉

C₂²

yes

characteristic

#19153047

—

Dic₁₇₉⋊C₈

—

C₁₇₉⋊#95

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

—

C₂²×Dic₁₇₉

—

C₂²×Dic₁₇₉

D₄

yes

characteristic

C₂²×Dic₁₇₉

D₄

yes

characteristic

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

Q₈

yes

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

Q₁₆

yes

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₂×C₄×C₁₇₉

#41

yes

characteristic

C₈×C₁₇₉

—

C₂²×C₁₇₉

#121

yes

characteristic

C₂²×C₁₇₉

C₂²⋊Q₈

yes

—

C₂²×C₁₇₉

C₂×Q₁₆

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#123

yes

characteristic

C₄×C₁₇₉

#125

yes

—

C₄×C₁₇₉

C₄⋊Q₈

yes

characteristic

Dic₁₇₉

—

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

—

C₂×C₁₇₉

#478

yes

—

C₁₇₉

#2087

yes

characteristic, complemented, Sylow

#2087

—

179

maximal, Sylow

C₂×#107

—

179

—

C₂×C₄⋊C₈

—

179

—

#389

—

179

—

#340

—

179

—

#340

D₁₇₉

yes

characteristic

C₂²×C₈

—

179

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂×C₄²

—

179

—

#107

—

358

—

C₄⋊C₈

—

358

—

#95

—

358

—

C₂²×C₄

—

C₂²×C₄

—

358

—

C₂²×C₄

—

179

—

C₂²×C₄

—

179

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

D₇₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

716

—

C₄⋊C₄

Dic₃₅₈

yes

—

C₄²

—

358

—

C₄²

—

358

—

C₂³

D₄×D₁₇₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

358

—

103

C₂×C₄

—

358

—

104

C₂×C₄

—

358

—

105

C₂×C₄

—

358

—

106

C₂×C₄

—

179

—

107

C₂×C₄

—

716

—

108

C₂×C₄

—

716

—

109

C₂×C₄

—

716

—

110

C₂×C₄

C₂×D₇₁₆

yes

characteristic

111

C₂×C₄

C₂×Dic₃₅₈

yes

—

112

C₂×C₄

#15668258

yes

—

113

C₂×C₄

#15668260

yes

characteristic

114

C₈

—

115

C₈

—

716

—

116

C₂²

#19153088

yes

characteristic

117

C₂²

#19153090

yes

—

118

C₂²

Q₁₆×D₁₇₉

yes

characteristic

119

C₂²

#19153132

yes

characteristic

120

C₂²

#19153140

yes

—

121

C₄

—

122

C₄

—

358

—

123

C₄

—

1432

—

124

C₄

—

716

—

125

C₄

#19153097

yes

characteristic

126

C₄

#19153104

yes

—

127

C₄

#19153113

yes

characteristic

128

C₂

#26038371

yes

characteristic

129

C₂

#26038681

yes

characteristic

130

C₂

#26038693

yes

characteristic

131

C₂

#26038732

yes

—

132

C₂

#26038737

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₇₉
commutator, cosocle	C₂²×C₄×C₁₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#340×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
hypercenter	#340	D₁₇₉
hypocenter	C₁₇₉	#2087
LCS	G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₇₉

commutator, cosocle

C₂²×C₄×C₁₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#340×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

D₁₇₉

C₁₇₉

G ▷ C₂²×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
66. C₁₇₉	#2087	semidirect

H ≅ G/N

type

66. C₁₇₉

#2087

semidirect

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

subgroup	index
C₂×#19153047	2
C₂×Dic₁₇₉⋊C₈	2
#26038338	2
#26038201	2
#340×C₁₇₉	2
#2087	179

subgroup

index

C₂×Dic₁₇₉⋊C₈

#340×C₁₇₉

179

Subgroups — #37510495

Conjugacy classes of subgroups

Complements

Maximal subgroups