Group #37509169 — subgroups

Subgroups — #37509169

Order 22912.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#19153046	C₂	1	2	yes	maximal
2	C₂×#19153044	C₂	1	2	yes	maximal
3	C₂×#19153024	C₂	1	1	yes	maximal, characteristic
4	#26038309	C₂	1	1	yes	maximal, characteristic
5	#382×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₇₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₇₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
9	#19153046	—	2	4	no	—
10	#19153044	—	2	4	no	—
11	#19153024	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
15	#95×C₁₇₉	—	2	2	no	—
16	C₂²×Dic₁₇₉	—	2	2	no	—
17	C₂×C₁₇₉⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₇₉⋊C₈	D₄	1	4	yes	—
19	C₂×C₁₇₉⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	C₄⋊Dic₁₇₉	—	2	2	no	—
22	C₄⋊Dic₁₇₉	—	4	4	no	—
23	C₂²×C₄×C₁₇₉	—	2	1	no	—
24	C₂²×C₄×C₁₇₉	—	2	1	no	—
25	C₂²×C₄×C₁₇₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₇₉	—	2	2	no	—
28	C₄⋊C₄×C₁₇₉	—	2	2	no	—
29	C₄²×C₁₇₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₇₉	Q₈	1	2	yes	—
31	C₂×Dic₁₇₉	—	4	2	no	—
32	C₂×Dic₁₇₉	—	4	4	no	—
33	C₁₇₉⋊C₈	C₄⋊C₄	1	8	yes	complemented
34	C₂³×C₁₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇₉	—	2	2	no	—
36	C₂×C₄×C₁₇₉	—	2	2	no	—
37	C₂×C₄×C₁₇₉	—	4	1	no	—
38	C₂×C₄×C₁₇₉	—	4	1	no	—
39	C₂×C₄×C₁₇₉	—	4	2	no	—
40	C₂×C₄×C₁₇₉	—	4	2	no	—
41	C₂×C₄×C₁₇₉	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₇₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₁₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₇₉	#127	1	1	yes	characteristic
48	C₂²×C₁₇₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₇₉	#137	1	2	yes	—
51	C₄×C₁₇₉	—	2	4	no	—
52	C₄×C₁₇₉	—	8	1	no	—
53	C₄×C₁₇₉	—	8	1	no	—
54	C₄×C₁₇₉	C₂×C₄⋊C₄	1	2	yes	—
55	C₄×C₁₇₉	C₄⋊Q₈	1	2	yes	—
56	Dic₁₇₉	—	8	2	no	—
57	C₂×C₁₇₉	#384	1	1	yes	characteristic
58	C₂×C₁₇₉	#428	1	2	yes	—
59	C₂×C₁₇₉	#496	1	1	yes	characteristic
60	C₂×C₁₇₉	#497	1	1	yes	characteristic
61	C₂×C₁₇₉	#501	1	2	yes	—
62	C₁₇₉	#1847	1	1	yes	characteristic, complemented, Sylow
63	#1847	—	179	1	no	maximal, Sylow
64	C₂×#107	—	179	2	no	—
65	C₂×#106	—	179	2	no	—
66	C₂×#97	—	179	1	no	—
67	#384	—	179	1	no	—
68	#382	D₁₇₉	1	1	yes	characteristic
69	C₂²×C₈	—	179	2	no	—
70	C₂×C₄⋊C₄	—	358	2	no	—
71	C₂×C₄⋊C₄	—	179	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₇₉	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×D₁₇₉	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₁₇₉	1	1	yes	characteristic
75	#107	—	358	4	no	—
76	#106	—	358	4	no	—
77	#97	—	179	4	no	—
78	#95	—	2	2	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	358	2	no	—
82	C₂²×C₄	C₂²⋊D₁₇₉	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
84	C₂×C₈	—	179	4	no	—
85	C₂×C₈	—	179	4	no	—
86	C₂×C₈	—	179	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	358	2	no	—
90	C₄⋊C₄	—	358	2	no	—
91	C₄⋊C₄	—	716	4	no	—
92	C₄²	C₄×D₁₇₉	1	2	yes	—
93	C₄²	Dic₃₅₈	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	716	2	no	—
102	C₂×C₄	—	716	4	no	—
103	C₂×C₄	Dic₁₇₉⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
106	C₂×C₄	Q₈×D₁₇₉	1	2	yes	—
107	C₂×C₄	C₂×Dic₃₅₈	1	1	yes	characteristic
108	C₈	—	179	8	no	—
109	C₂²	C₂×Dic₁₇₉⋊C₄	1	1	yes	characteristic
110	C₂²	#19153109	1	1	yes	characteristic
111	C₂²	#19153111	1	1	yes	characteristic
112	C₂²	#19153138	1	2	yes	—
113	C₂²	#19153143	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	1432	2	no	—
118	C₄	C₄⋊C₄×D₁₇₉	1	2	yes	—
119	C₄	#19153112	1	2	yes	—
120	C₂	#26038308	1	1	yes	characteristic
121	C₂	#26038482	1	2	yes	—
122	C₂	#26038829	1	1	yes	characteristic
123	C₂	#26038834	1	1	yes	characteristic
124	C₂	#26038849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153046

C₂

yes

maximal

C₂×#19153044

C₂

yes

maximal

C₂×#19153024

C₂

yes

maximal, characteristic

#26038309

C₂

yes

maximal, characteristic

#382×C₁₇₉

C₂

yes

maximal, characteristic

C₂²×C₁₇₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₇₉

—

C₂×C₄⋊Dic₁₇₉

C₂²

yes

—

#19153046

—

#19153044

—

#19153024

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₉

C₂²

yes

characteristic

C₂×C₄²×C₁₇₉

C₂²

yes

characteristic

#95×C₁₇₉

—

C₂²×Dic₁₇₉

—

C₂×C₁₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₇₉⋊C₈

D₄

yes

—

C₂×C₁₇₉⋊C₈

Q₈

yes

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₄⋊Dic₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

—

C₂²×C₄×C₁₇₉

D₄

yes

—

C₂²×C₄×C₁₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂×C₄

yes

—

C₄²×C₁₇₉

Q₈

yes

—

C₂×Dic₁₇₉

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

C₄⋊C₄

yes

complemented

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

C₄⋊C₄

yes

—

C₂×C₄×C₁₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×Q₈

yes

—

C₂×C₄×C₁₇₉

C₂×Q₈

yes

characteristic

C₂²×C₁₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇₉

#127

yes

characteristic

C₂²×C₁₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₇₉

#137

yes

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₇₉

C₄⋊Q₈

yes

—

Dic₁₇₉

—

C₂×C₁₇₉

#384

yes

characteristic

C₂×C₁₇₉

#428

yes

—

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

C₂×C₁₇₉

yes

—

C₁₇₉

#1847

yes

characteristic, complemented, Sylow

#1847

—

179

maximal, Sylow

C₂×#107

—

179

—

C₂×#106

—

179

—

C₂×#97

—

179

—

#384

—

179

—

#382

D₁₇₉

yes

characteristic

C₂²×C₈

—

179

—

C₂×C₄⋊C₄

—

358

—

C₂×C₄⋊C₄

—

179

—

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₇₉

yes

characteristic

C₂×C₄²

C₂×D₁₇₉

yes

characteristic

#107

—

358

—

#106

—

358

—

#97

—

179

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

358

—

C₂²×C₄

C₂²⋊D₁₇₉

yes

—

C₂²×C₄

C₂²×D₁₇₉

yes

characteristic

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₂×C₈

—

179

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

358

—

C₄⋊C₄

—

716

—

C₄²

C₄×D₁₇₉

yes

—

C₄²

Dic₃₅₈

yes

—

C₂³

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

716

—

102

C₂×C₄

—

716

—

103

C₂×C₄

Dic₁₇₉⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₁₇₉

yes

characteristic

105

C₂×C₄

D₄×D₁₇₉

yes

—

106

C₂×C₄

Q₈×D₁₇₉

yes

—

107

C₂×C₄

C₂×Dic₃₅₈

yes

characteristic

108

C₈

—

179

—

109

C₂²

C₂×Dic₁₇₉⋊C₄

yes

characteristic

110

C₂²

#19153109

yes

characteristic

111

C₂²

#19153111

yes

characteristic

112

C₂²

#19153138

yes

—

113

C₂²

#19153143

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

1432

—

118

C₄

C₄⋊C₄×D₁₇₉

yes

—

119

C₄

#19153112

yes

—

120

C₂

#26038308

yes

characteristic

121

C₂

#26038482

yes

—

122

C₂

#26038829

yes

characteristic

123

C₂

#26038834

yes

characteristic

124

C₂

#26038849

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇₉
commutator	C₂×C₄×C₁₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₉
Fitting	#382×C₁₇₉	C₂
socle, prime radical	C₂³×C₁₇₉	C₂×D₄
cosocle	C₂²×C₄×C₁₇₉	C₂³
hypercenter	#382	D₁₇₉
hypocenter	C₁₇₉	#1847
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇₉

commutator

C₂×C₄×C₁₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₉

Fitting

#382×C₁₇₉

C₂

socle, prime radical

C₂³×C₁₇₉

C₂×D₄

cosocle

C₂²×C₄×C₁₇₉

C₂³

D₁₇₉

C₁₇₉

G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
11. #19153024	C₄	semidirect
33. C₁₇₉⋊C₈	C₄⋊C₄	semidirect
62. C₁₇₉	#1847	semidirect

H ≅ G/N

type

11. #19153024

C₄

semidirect

33. C₁₇₉⋊C₈

C₄⋊C₄

semidirect

62. C₁₇₉

#1847

semidirect

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

subgroup	index
C₂×#19153046	2
C₂×#19153044	2
C₂×#19153024	2
#26038309	2
#382×C₁₇₉	2
#1847	179

subgroup

index

#382×C₁₇₉

179

Subgroups — #37509169

Conjugacy classes of subgroups

Complements

Maximal subgroups