Group #37509788 — subgroups

Subgroups — #37509788

Order 22912.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#26038487	C₂	1	2	yes	maximal
2	C₂×#19153049	C₂	1	1	yes	maximal, characteristic
3	#26038394	C₂	1	1	yes	maximal, characteristic
4	#26038210	C₂	1	2	yes	maximal
5	C₂×#108×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668243	C₂²	1	2	yes	—
7	C₂×#15668243	C₂²	1	1	yes	characteristic
8	#19153072	C₂²	1	1	yes	characteristic
9	#19153049	C₄	1	4	yes	complemented
10	#19153048	—	2	4	no	—
11	#19153023	—	2	1	no	—
12	C₈×Dic₁₇₉	—	2	1	no	—
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	2	no	—
19	#15668243	C₂×C₄	1	4	yes	—
20	#15668243	D₄	1	2	yes	—
21	C₂×C₁₇₉⋊C₈	—	2	1	no	—
22	C₂×C₁₇₉⋊C₈	—	2	2	no	—
23	C₄⋊Dic₁₇₉	—	2	1	no	—
24	C₂²⋊Dic₁₇₉	Q₈	1	2	yes	—
25	C₄×Dic₁₇₉	—	2	1	no	—
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	2	yes	—
31	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇₉	Q₈	1	2	yes	—
34	C₂×Dic₁₇₉	—	2	2	no	—
35	C₁₇₉⋊C₈	—	2	4	no	—
36	C₁₇₉⋊C₈	—	2	2	no	—
37	C₂³×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₇₉	#41	1	1	yes	characteristic
44	C₈×C₁₇₉	—	4	2	no	—
45	C₈×C₁₇₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₇₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₇₉	—	4	1	no	—
48	C₂²×C₁₇₉	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₁₇₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₁₇₉	C₄×D₄	1	1	yes	characteristic
53	C₄×C₁₇₉	#121	1	1	yes	characteristic
54	C₄×C₁₇₉	#125	1	1	yes	characteristic
55	Dic₁₇₉	—	4	2	no	—
56	C₂×C₁₇₉	—	2	1	no	—
57	C₂×C₁₇₉	—	2	1	no	—
58	C₂×C₁₇₉	—	2	1	no	—
59	C₂×C₁₇₉	#389	1	1	yes	characteristic
60	C₁₇₉	#1953	1	1	yes	characteristic, complemented, Sylow
61	#1953	—	179	1	no	maximal, Sylow
62	#430	—	179	2	no	—
63	C₂×#108	—	179	1	no	—
64	C₂×#108	D₁₇₉	1	1	yes	characteristic
65	#405	—	179	1	no	—
66	#344	—	179	2	no	—
67	C₂×M₁₆	—	179	2	no	—
68	C₂×M₁₆	—	179	1	no	—
69	C₂×M₁₆	C₂×D₁₇₉	1	2	yes	—
70	C₂²×C₈	C₂×D₁₇₉	1	1	yes	characteristic
71	#117	—	179	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	358	4	no	—
75	#108	—	179	4	no	—
76	#97	—	358	1	no	—
77	C₄×C₈	—	358	1	no	—
78	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	358	1	no	—
82	M₁₆	—	358	2	no	—
83	M₁₆	—	179	4	no	—
84	M₁₆	—	179	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	358	2	no	—
87	C₂×C₈	—	358	1	no	—
88	C₂×C₈	C₄×D₁₇₉	1	2	yes	—
89	C₂×C₈	C₂²⋊D₁₇₉	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₁₇₉	1	1	yes	characteristic
91	C₂×C₈	Dic₃₅₈	1	2	yes	—
92	C₄⋊C₄	—	358	1	no	—
93	C₂²⋊C₄	—	179	2	no	—
94	C₄²	—	358	1	no	—
95	C₂³	Q₈×D₁₇₉	1	1	yes	characteristic
96	C₂×C₄	—	358	2	no	—
97	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₃₅₈	1	1	yes	characteristic
101	C₂×C₄	#15668258	1	1	yes	characteristic
102	C₂×C₄	#15668260	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	358	4	no	—
105	C₈	—	358	2	no	—
106	C₈	Dic₁₇₉⋊C₄	1	2	yes	—
107	C₈	Dic₁₇₉⋊C₄	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#19153078	1	1	yes	characteristic
110	C₂²	#19153089	1	1	yes	characteristic
111	C₂²	#19153112	1	1	yes	characteristic
112	C₄	—	716	2	no	—
113	C₄	C₂×Dic₁₇₉⋊C₄	1	1	yes	characteristic
114	C₄	#19153075	1	1	yes	characteristic
115	C₄	#19153084	1	1	yes	characteristic
116	C₄	#19153104	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#26038335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26038487

C₂

yes

maximal

C₂×#19153049

C₂

yes

maximal, characteristic

#26038394

C₂

yes

maximal, characteristic

#26038210

C₂

yes

maximal

C₂×#108×C₁₇₉

C₂

yes

maximal, characteristic

C₂×#15668243

C₂²

yes

—

C₂×#15668243

C₂²

yes

characteristic

#19153072

C₂²

yes

characteristic

#19153049

C₄

yes

complemented

#19153048

—

#19153023

—

C₈×Dic₁₇₉

—

C₂×M₁₆×C₁₇₉

C₂²

yes

—

C₂²×C₈×C₁₇₉

C₂²

yes

characteristic

#108×C₁₇₉

—

#108×C₁₇₉

—

#15668243

—

#15668243

—

#15668243

C₂×C₄

yes

—

#15668243

D₄

yes

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₂²⋊Dic₁₇₉

Q₈

yes

—

C₄×Dic₁₇₉

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

M₁₆×C₁₇₉

—

M₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₂×C₈×C₁₇₉

C₂×C₄

yes

—

C₂×C₈×C₁₇₉

D₄

yes

characteristic

C₂×C₈×C₁₇₉

D₄

yes

characteristic

C₂×C₈×C₁₇₉

Q₈

yes

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

C₂×Q₈

yes

characteristic

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

C₂×C₄×C₁₇₉

yes

C₂×C₄×C₁₇₉

yes

C₈×C₁₇₉

—

C₈×C₁₇₉

C₄⋊C₄

yes

—

C₈×C₁₇₉

C₄⋊C₄

yes

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

C₄×Q₈

yes

characteristic

C₂²×C₁₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇₉

C₄×D₄

yes

C₄×C₁₇₉

yes

C₄×C₁₇₉

yes

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#389

yes

characteristic

C₁₇₉

#1953

yes

characteristic, complemented, Sylow

#1953

—

179

maximal, Sylow

#430

—

179

—

C₂×#108

—

179

—

C₂×#108

D₁₇₉

yes

characteristic

#405

—

179

—

#344

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

—

179

—

C₂×M₁₆

C₂×D₁₇₉

yes

—

C₂²×C₈

C₂×D₁₇₉

yes

characteristic

#117

—

179

—

#108

—

#108

—

#108

—

358

—

#108

—

179

—

#97

—

358

—

C₄×C₈

—

358

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

358

—

M₁₆

—

358

—

M₁₆

—

179

—

M₁₆

—

179

—

C₂×C₈

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

C₄×D₁₇₉

yes

—

C₂×C₈

C₂²⋊D₁₇₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₇₉

yes

characteristic

C₂×C₈

Dic₃₅₈

yes

—

C₄⋊C₄

—

358

—

C₂²⋊C₄

—

179

—

C₄²

—

358

—

C₂³

Q₈×D₁₇₉

yes

characteristic

C₂×C₄

—

358

—

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

C₂×C₄

D₄×D₁₇₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

100

C₂×C₄

C₂×Dic₃₅₈

yes

characteristic

101

C₂×C₄

#15668258

yes

characteristic

102

C₂×C₄

#15668260

yes

characteristic

103

C₈

—

104

C₈

—

358

—

105

C₈

—

358

—

106

C₈

Dic₁₇₉⋊C₄

yes

—

107

C₈

Dic₁₇₉⋊C₄

yes

—

108

C₂²

—

109

C₂²

#19153078

yes

characteristic

110

C₂²

#19153089

yes

characteristic

111

C₂²

#19153112

yes

characteristic

112

C₄

—

716

—

113

C₄

C₂×Dic₁₇₉⋊C₄

yes

characteristic

114

C₄

#19153075

yes

characteristic

115

C₄

#19153084

yes

characteristic

116

C₄

#19153104

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#26038335

yes

characteristic

	subgroup	G/H
center	C₄	#19153084
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	C₂×#108×C₁₇₉	C₂
socle	C₂×C₁₇₉	#389
prime radical	C₂³×C₁₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	C₂×#108	D₁₇₉
hypocenter	C₁₇₉	#1953
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#19153084

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

C₂×#108×C₁₇₉

C₂

socle

C₂×C₁₇₉

#389

prime radical

C₂³×C₁₇₉

C₂×Q₈

cosocle

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
9. #19153049	C₄	semidirect
60. C₁₇₉	#1953	semidirect

H ≅ G/N

type

9. #19153049

C₄

semidirect

60. C₁₇₉

#1953

semidirect

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

subgroup	index
#26038487	2
C₂×#19153049	2
#26038394	2
#26038210	2
C₂×#108×C₁₇₉	2
#1953	179

subgroup

index

C₂×#108×C₁₇₉

179

Subgroups — #37509788

Conjugacy classes of subgroups

Complements

Maximal subgroups