Group #37509125 — subgroups

Subgroups — #37509125

Order 22912.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#26038477	C₂	1	1	yes	maximal, characteristic
2	#26038394	C₂	1	1	yes	maximal, characteristic
3	#26038211	C₂	1	2	yes	maximal
4	#26038196	C₂	1	2	yes	maximal
5	C₂×#108×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₂²	1	1	yes	characteristic
7	#19153072	C₂²	1	2	yes	—
8	#19153072	C₂²	1	1	yes	characteristic
9	#19153047	—	2	1	no	—
10	#19153045	—	2	1	no	—
11	#19153023	C₄	1	2	yes	complemented
12	C₈×Dic₁₇₉	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₇₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
15	#108×C₁₇₉	—	2	1	no	—
16	#108×C₁₇₉	—	2	1	no	—
17	#15668243	—	2	1	no	—
18	#15668243	—	2	1	no	—
19	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	C₄⋊Dic₁₇₉	—	2	1	no	—
22	C₂²⋊Dic₁₇₉	—	2	1	no	—
23	C₂²⋊Dic₁₇₉	—	2	2	no	—
24	C₄×Dic₁₇₉	—	2	2	no	—
25	C₄×Dic₁₇₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₇₉	—	2	2	no	—
28	M₁₆×C₁₇₉	—	2	2	no	—
29	C₂×C₈×C₁₇₉	—	2	2	no	—
30	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇₉	Q₈	1	1	yes	characteristic
35	C₂×C₈×C₁₇₉	Q₈	1	1	yes	characteristic
36	C₂×Dic₁₇₉	—	2	2	no	—
37	C₂×Dic₁₇₉	—	4	2	no	—
38	C₁₇₉⋊C₈	—	2	2	no	—
39	C₂³×C₁₇₉	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
44	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
46	C₈×C₁₇₉	—	4	2	no	—
47	C₈×C₁₇₉	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₇₉	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₁₇₉	C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₇₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₇₉	—	4	1	no	—
52	C₂²×C₁₇₉	#117	1	1	yes	characteristic
53	C₂²×C₁₇₉	#124	1	1	yes	characteristic
54	C₂²×C₁₇₉	#125	1	1	yes	characteristic
55	C₄×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₁₇₉	#117	1	1	yes	characteristic
57	C₄×C₁₇₉	#124	1	1	yes	characteristic
58	C₄×C₁₇₉	#125	1	1	yes	characteristic
59	Dic₁₇₉	—	4	2	no	—
60	Dic₁₇₉	—	8	2	no	—
61	C₂×C₁₇₉	—	2	1	no	—
62	C₂×C₁₇₉	—	2	1	no	—
63	C₂×C₁₇₉	—	2	1	no	—
64	C₂×C₁₇₉	#382	1	1	yes	characteristic
65	C₁₇₉	#1833	1	1	yes	characteristic, complemented, Sylow
66	#1833	—	179	1	no	maximal, Sylow
67	C₂×#108	D₁₇₉	1	1	yes	characteristic
68	#427	—	179	1	no	—
69	#405	—	179	1	no	—
70	#344	—	179	2	no	—
71	H(3,ℤ/4)	—	179	2	no	—
72	C₂×M₁₆	—	179	1	no	—
73	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
74	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
75	#117	—	179	2	no	—
76	#117	—	179	1	no	—
77	#108	—	2	1	no	—
78	#108	—	2	1	no	—
79	#107	—	358	1	no	—
80	#106	—	358	1	no	—
81	#97	—	179	2	no	—
82	C₄×C₈	—	179	2	no	—
83	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	358	1	no	—
87	M₁₆	—	358	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	179	2	no	—
90	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
91	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₇₉	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₇₉	1	1	yes	characteristic
94	C₂×C₈	Dic₃₅₈	1	1	yes	characteristic
95	C₂×C₈	Dic₃₅₈	1	1	yes	characteristic
96	C₄⋊C₄	—	358	1	no	—
97	C₄⋊C₄	—	358	2	no	—
98	C₂²⋊C₄	—	358	1	no	—
99	C₂²⋊C₄	—	358	2	no	—
100	C₄²	—	358	2	no	—
101	C₄²	—	179	2	no	—
102	C₂³	#15668260	1	1	yes	characteristic
103	C₂×C₄	—	358	2	no	—
104	C₂×C₄	—	716	2	no	—
105	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₃₅₈	1	1	yes	characteristic
108	C₂×C₄	#15668258	1	1	yes	characteristic
109	C₂×C₄	#15668258	1	1	yes	characteristic
110	C₂×C₄	#15668260	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	358	2	no	—
113	C₈	Dic₁₇₉⋊C₄	1	1	yes	characteristic
114	C₈	Dic₁₇₉⋊C₄	1	1	yes	characteristic
115	C₈	Dic₁₇₉⋊C₄	1	1	yes	characteristic
116	C₈	Dic₁₇₉⋊C₄	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#19153071	1	1	yes	characteristic
119	C₂²	#19153102	1	1	yes	characteristic
120	C₂²	#19153104	1	1	yes	characteristic
121	C₄	—	1432	2	no	—
122	C₄	—	716	2	no	—
123	C₄	C₂×Dic₁₇₉⋊C₄	1	1	yes	characteristic
124	C₄	#19153071	1	1	yes	characteristic
125	C₄	#19153100	1	1	yes	characteristic
126	C₄	#19153104	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#26038304	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038477

C₂

yes

maximal, characteristic

#26038394

C₂

yes

maximal, characteristic

#26038211

C₂

yes

maximal

#26038196

C₂

yes

maximal

C₂×#108×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

characteristic

#19153072

C₂²

yes

—

#19153072

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₁₇₉

C₄

yes

complemented

C₂×M₁₆×C₁₇₉

C₂²

yes

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

#108×C₁₇₉

—

#108×C₁₇₉

—

#15668243

—

#15668243

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²⋊Dic₁₇₉

—

C₂²⋊Dic₁₇₉

—

C₄×Dic₁₇₉

—

C₄×Dic₁₇₉

C₂×C₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇₉

D₄

yes

characteristic

C₂×C₈×C₁₇₉

D₄

yes

characteristic

C₂×C₈×C₁₇₉

Q₈

yes

characteristic

C₂×C₈×C₁₇₉

Q₈

yes

characteristic

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

#41

yes

characteristic

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×Q₈

yes

C₂×C₄×C₁₇₉

yes

C₂×C₄×C₁₇₉

yes

C₂×C₄×C₁₇₉

yes

C₈×C₁₇₉

—

C₈×C₁₇₉

C₄⋊C₄

yes

characteristic

C₈×C₁₇₉

C₄⋊C₄

yes

characteristic

C₈×C₁₇₉

C₄⋊C₄

yes

characteristic

C₈×C₁₇₉

C₄⋊C₄

yes

characteristic

C₂²×C₁₇₉

—

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₂²×C₁₇₉

yes

C₄×C₁₇₉

C₂×C₄⋊C₄

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

Dic₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#382

yes

characteristic

C₁₇₉

#1833

yes

characteristic, complemented, Sylow

#1833

—

179

maximal, Sylow

C₂×#108

D₁₇₉

yes

characteristic

#427

—

179

—

#405

—

179

—

#344

—

179

—

H(3,ℤ/4)

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

#117

—

179

—

#117

—

179

—

#108

—

#108

—

#107

—

358

—

#106

—

358

—

#97

—

179

—

C₄×C₈

—

179

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

358

—

M₁₆

—

358

—

C₂×C₈

—

C₂×C₈

—

179

—

C₂×C₈

C₄×D₁₇₉

yes

characteristic

C₂×C₈

C₄×D₁₇₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₇₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₇₉

yes

characteristic

C₂×C₈

Dic₃₅₈

yes

characteristic

C₂×C₈

Dic₃₅₈

yes

characteristic

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₂²⋊C₄

—

358

—

C₂²⋊C₄

—

358

—

100

C₄²

—

358

—

101

C₄²

—

179

—

102

C₂³

#15668260

yes

characteristic

103

C₂×C₄

—

358

—

104

C₂×C₄

—

716

—

105

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

107

C₂×C₄

C₂×Dic₃₅₈

yes

characteristic

108

C₂×C₄

#15668258

yes

characteristic

109

C₂×C₄

#15668258

yes

characteristic

110

C₂×C₄

#15668260

yes

characteristic

111

C₈

—

112

C₈

—

358

—

113

C₈

Dic₁₇₉⋊C₄

yes

characteristic

114

C₈

Dic₁₇₉⋊C₄

yes

characteristic

115

C₈

Dic₁₇₉⋊C₄

yes

characteristic

116

C₈

Dic₁₇₉⋊C₄

yes

characteristic

117

C₂²

—

118

C₂²

#19153071

yes

characteristic

119

C₂²

#19153102

yes

characteristic

120

C₂²

#19153104

yes

characteristic

121

C₄

—

1432

—

122

C₄

—

716

—

123

C₄

C₂×Dic₁₇₉⋊C₄

yes

characteristic

124

C₄

#19153071

yes

characteristic

125

C₄

#19153100

yes

characteristic

126

C₄

#19153104

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#26038304

yes

characteristic

	subgroup	G/H
center	C₄	#19153071
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂×#108×C₁₇₉	C₂
socle	C₂×C₁₇₉	#382
prime radical	C₂³×C₁₇₉	#41
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₂×#108	D₁₇₉
hypocenter	C₁₇₉	#1833
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#19153071

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂×#108×C₁₇₉

C₂

C₂×C₁₇₉

C₂³×C₁₇₉

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. #19153023	C₄	semidirect
12. C₈×Dic₁₇₉	C₄	semidirect
65. C₁₇₉	#1833	semidirect

H ≅ G/N

type

11. #19153023

C₄

semidirect

12. C₈×Dic₁₇₉

C₄

semidirect

65. C₁₇₉

#1833

semidirect

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

subgroup	index
#26038477	2
#26038394	2
#26038211	2
#26038196	2
C₂×#108×C₁₇₉	2
#1833	179

subgroup

index

C₂×#108×C₁₇₉

179

Subgroups — #37509125

Conjugacy classes of subgroups

Complements

Maximal subgroups