Group #37509080 — subgroups

Subgroups — #37509080

Order 22912.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#26038403	C₂	1	2	yes	maximal
2	#26038208	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄⋊C₄×C₁₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15668243	C₂²	1	4	yes	—
5	C₂²⋊(C₁₇₉⋊C₈)	—	2	2	no	—
6	C₂²⋊(C₁₇₉⋊C₈)	—	2	2	no	—
7	C₂³×C₄×C₁₇₉	C₄	1	2	yes	—
8	C₂³×C₄×C₁₇₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₇₉	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₇₉	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₇₉	C₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
13	#15668243	—	2	8	no	—
14	C₂×C₁₇₉⋊C₈	—	2	4	no	—
15	C₂⁴×C₁₇₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₇₉	—	2	4	no	—
17	C₂²×C₄×C₁₇₉	—	2	2	no	—
18	C₂²×C₄×C₁₇₉	—	2	2	no	—
19	C₂²×C₄×C₁₇₉	—	4	4	no	—
20	C₂²×C₄×C₁₇₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇₉	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
23	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
24	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₇₉	—	2	4	no	—
26	C₄⋊C₄×C₁₇₉	—	2	4	no	—
27	C₁₇₉⋊C₈	—	4	4	no	—
28	C₂³×C₁₇₉	—	2	2	no	—
29	C₂³×C₁₇₉	—	2	2	no	—
30	C₂³×C₁₇₉	—	2	2	no	—
31	C₂³×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂³×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂³×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇₉	—	2	2	no	—
35	C₂×C₄×C₁₇₉	—	2	4	no	—
36	C₂×C₄×C₁₇₉	—	2	4	no	—
37	C₂×C₄×C₁₇₉	—	2	4	no	—
38	C₂×C₄×C₁₇₉	—	4	4	no	—
39	C₂×C₄×C₁₇₉	—	4	4	no	—
40	C₂×C₄×C₁₇₉	—	4	4	no	—
41	C₂×C₄×C₁₇₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₇₉	#41	1	4	yes	—
44	C₂²×C₁₇₉	—	2	8	no	—
45	C₂²×C₁₇₉	—	2	1	no	—
46	C₂²×C₁₇₉	—	2	1	no	—
47	C₂²×C₁₇₉	—	2	1	no	—
48	C₂²×C₁₇₉	—	2	1	no	—
49	C₂²×C₁₇₉	—	2	1	no	—
50	C₂²×C₁₇₉	—	2	1	no	—
51	C₂²×C₁₇₉	#100	1	2	yes	—
52	C₂²×C₁₇₉	#101	1	2	yes	—
53	C₂²×C₁₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₇₉	#117	1	2	yes	—
55	C₄×C₁₇₉	—	2	2	no	—
56	C₄×C₁₇₉	—	4	4	no	—
57	C₄×C₁₇₉	#123	1	4	yes	—
58	C₂×C₁₇₉	—	2	2	no	—
59	C₂×C₁₇₉	—	2	2	no	—
60	C₂×C₁₇₉	—	2	2	no	—
61	C₂×C₁₇₉	#381	1	1	yes	characteristic
62	C₂×C₁₇₉	C₂×#100	1	1	yes	characteristic
63	C₂×C₁₇₉	C₂×#101	1	1	yes	characteristic
64	C₁₇₉	#1821	1	1	yes	characteristic, complemented, Sylow
65	#1821	—	179	1	no	maximal, Sylow
66	C₂²×C₄⋊C₄	D₁₇₉	1	1	yes	characteristic
67	#407	—	179	2	no	—
68	#343	—	179	4	no	—
69	C₂³×C₄	Dic₁₇₉	1	2	yes	—
70	C₂³×C₄	C₂×D₁₇₉	1	1	yes	characteristic
71	C₂×M₁₆	—	179	4	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	Dic₁₇₉	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
76	C₂²⋊C₈	—	358	2	no	—
77	C₂²⋊C₈	—	358	2	no	—
78	C₂⁴	C₂×Dic₁₇₉	1	1	yes	characteristic
79	C₂²×C₄	—	2	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	4	4	no	—
83	C₂²×C₄	C₂×Dic₁₇₉	1	1	yes	characteristic
84	C₂²×C₄	C₂×Dic₁₇₉	1	4	yes	—
85	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
86	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
88	M₁₆	—	358	8	no	—
89	C₂×C₈	—	358	4	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₂³	—	2	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	2	no	—
95	C₂³	C₂²⋊Dic₁₇₉	1	1	yes	characteristic
96	C₂³	C₂²⋊Dic₁₇₉	1	1	yes	characteristic
97	C₂³	C₂²×Dic₁₇₉	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₁₇₉	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	2	yes	—
107	C₂×C₄	#15668259	1	4	yes	—
108	C₈	—	716	4	no	—
109	C₂²	—	2	8	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#19153030	1	2	yes	—
117	C₂²	#19153032	1	2	yes	—
118	C₂²	C₂×C₂²⋊Dic₁₇₉	1	1	yes	characteristic
119	C₂²	#19153072	1	2	yes	—
120	C₄	—	2	2	no	—
121	C₄	—	4	4	no	—
122	C₄	#19153096	1	4	yes	—
123	C₂	—	2	2	no	—
124	C₂	—	2	2	no	—
125	C₂	—	2	2	no	—
126	C₂	#26038302	1	1	yes	characteristic
127	C₂	C₂×#19153030	1	1	yes	characteristic
128	C₂	C₂×#19153032	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038403

C₂

yes

maximal

#26038208

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

C₂²⋊(C₁₇₉⋊C₈)

—

C₂²⋊(C₁₇₉⋊C₈)

—

C₂³×C₄×C₁₇₉

C₄

yes

—

C₂³×C₄×C₁₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₉

—

C₂×C₄⋊C₄×C₁₇₉

—

C₂×C₄⋊C₄×C₁₇₉

C₄

yes

—

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

—

#15668243

—

C₂×C₁₇₉⋊C₈

—

C₂⁴×C₁₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₉

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

—

C₂³×C₁₇₉

—

C₂³×C₁₇₉

—

C₂³×C₁₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₁₇₉

C₂²⋊C₄

yes

characteristic

C₂³×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

#41

yes

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#100

yes

—

C₂²×C₁₇₉

#101

yes

—

C₂²×C₁₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₉

#117

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#123

yes

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₁₇₉

yes

characteristic, complemented, Sylow

#1821

—

179

maximal, Sylow

C₂²×C₄⋊C₄

D₁₇₉

yes

characteristic

#407

—

179

—

#343

—

179

—

C₂³×C₄

Dic₁₇₉

yes

—

C₂³×C₄

C₂×D₁₇₉

yes

characteristic

C₂×M₁₆

—

179

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₇₉

yes

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

—

C₂²⋊C₈

—

358

—

C₂²⋊C₈

—

358

—

C₂⁴

C₂×Dic₁₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₇₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₉

yes

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

358

—

C₂×C₈

—

358

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₁₇₉

yes

characteristic

C₂³

C₂²⋊Dic₁₇₉

yes

characteristic

C₂³

C₂²×Dic₁₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₁₇₉

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

—

107

C₂×C₄

#15668259

yes

—

108

C₈

—

716

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#19153030

yes

—

117

C₂²

#19153032

yes

—

118

C₂²

C₂×C₂²⋊Dic₁₇₉

yes

characteristic

119

C₂²

#19153072

yes

—

120

C₄

—

121

C₄

—

122

C₄

#19153096

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#26038302

yes

characteristic

127

C₂

C₂×#19153030

yes

characteristic

128

C₂

C₂×#19153032

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₇₉
commutator	C₂³×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂²×C₄⋊C₄×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₇₉	C₂×C₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₇₉
hypocenter	C₁₇₉	#1821
LCS	G ▷ C₂³×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₇₉

commutator

C₂³×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂²×C₄⋊C₄×C₁₇₉

C₂

socle

C₂²×C₁₇₉

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₇₉

C₂×C₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₁₇₉

hypocenter

C₁₇₉

#1821

LCS

G ▷ C₂³×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #26038208	C₂	semidirect
64. C₁₇₉	#1821	semidirect

H ≅ G/N

type

2. #26038208

C₂

semidirect

64. C₁₇₉

#1821

semidirect

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

subgroup	index
#26038403	2
#26038208	2
C₂²×C₄⋊C₄×C₁₇₉	2
#1821	179

subgroup

index

#26038403

#26038208

C₂²×C₄⋊C₄×C₁₇₉

#1821

179

Subgroups — #37509080

Conjugacy classes of subgroups

Complements

Maximal subgroups