Group #37508081 — subgroups

Subgroups — #37508081

Order 22912.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#26039155	C₂	1	1	yes	maximal, characteristic, complemented
2	#26038631	C₂	1	2	yes	maximal, complemented
3	#26038188	C₂	1	1	yes	maximal, characteristic, complemented
4	#26038168	C₂	1	2	yes	maximal, complemented
5	#492×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668244	—	2	2	no	—
7	#19153102	—	2	2	no	—
8	#19153088	—	2	2	no	—
9	C₄×D₇₁₆	C₂²	1	2	yes	—
10	#19153071	—	2	2	no	—
11	C₄⋊(C₁₇₉⋊C₈)	C₂²	1	2	yes	—
12	#19153034	—	2	2	no	—
13	C₂×SD₁₆×C₁₇₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
15	#127×C₁₇₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
17	C₂×D₇₁₆	—	2	2	no	—
18	C₂×D₇₁₆	D₄	1	2	yes	—
19	C₂×C₄×D₁₇₉	—	2	2	no	—
20	#15668244	—	4	4	no	—
21	C₂×C₁₇₉⋊C₈	—	2	2	no	—
22	C₄⋊Dic₁₇₉	D₄	1	2	yes	—
23	(C₂×D₁₇₉)⋊C₄	—	2	4	no	—
24	(C₂×D₁₇₉)⋊C₄	—	2	2	no	—
25	C₄×Dic₁₇₉	—	2	2	no	—
26	C₂×Q₈×C₁₇₉	D₄	1	2	yes	—
27	C₂×D₄×C₁₇₉	—	2	1	no	—
28	C₂×D₄×C₁₇₉	—	2	2	no	—
29	SD₁₆×C₁₇₉	—	4	4	no	—
30	C₂×C₈×C₁₇₉	—	2	1	no	—
31	C₄⋊C₄×C₁₇₉	—	2	2	no	—
32	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
33	C₂²×D₁₇₉	—	2	2	no	—
34	D₇₁₆	—	2	2	no	—
35	D₇₁₆	—	2	2	no	—
36	D₇₁₆	—	8	2	no	—
37	C₂×Dic₁₇₉	—	2	2	no	—
38	C₄×D₁₇₉	—	4	2	no	—
39	C₄×D₁₇₉	—	4	2	no	—
40	C₁₇₉⋊C₈	—	4	2	no	—
41	C₂³×C₁₇₉	—	4	1	no	—
42	Q₈×C₁₇₉	—	4	2	no	—
43	D₄×C₁₇₉	—	4	2	no	—
44	D₄×C₁₇₉	—	4	2	no	—
45	D₄×C₁₇₉	—	4	2	no	—
46	C₂×C₄×C₁₇₉	—	2	2	no	—
47	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₇₉	—	2	2	no	—
50	C₂²×C₁₇₉	—	8	2	no	—
51	C₂²×C₁₇₉	—	8	1	no	—
52	C₂²×C₁₇₉	#120	1	1	yes	characteristic
53	C₂×D₁₇₉	—	4	2	no	—
54	C₂×D₁₇₉	—	4	2	no	—
55	C₂×D₁₇₉	—	4	2	no	—
56	C₄×C₁₇₉	—	2	1	no	—
57	C₄×C₁₇₉	—	4	2	no	—
58	C₄×C₁₇₉	#135	1	2	yes	—
59	C₄×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₁₇₉	—	4	2	no	—
61	C₂×C₁₇₉	—	16	1	no	—
62	C₂×C₁₇₉	#449	1	2	yes	—
63	C₂×C₁₇₉	#453	1	1	yes	characteristic
64	D₁₇₉	—	8	2	no	—
65	C₁₇₉	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	179	1	no	maximal, Sylow
67	#555	—	179	1	no	—
68	#492	D₁₇₉	1	1	yes	characteristic, complemented
69	#459	—	179	2	no	—
70	#333	—	179	1	no	—
71	D₄⋊C₈	—	179	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	358	2	no	—
74	C₄⋊Q₈	C₂×D₁₇₉	1	1	yes	characteristic
75	#127	C₂×D₁₇₉	1	1	yes	characteristic
76	#124	—	358	2	no	—
77	#121	—	358	2	no	—
78	C₄×D₄	—	179	2	no	—
79	#117	—	358	2	no	—
80	C₄⋊C₈	—	179	2	no	—
81	#102	—	358	2	no	—
82	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₁₇₉	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	358	2	no	—
87	C₂×D₄	—	179	2	no	—
88	C₂²×C₄	—	358	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	716	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	358	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	179	2	no	—
95	C₂²⋊C₄	—	358	4	no	—
96	C₂²⋊C₄	—	358	2	no	—
97	C₄²	—	358	2	no	—
98	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	358	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	358	2	no	—
106	D₄	—	358	2	no	—
107	D₄	—	1432	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	358	2	no	—
110	C₂×C₄	—	716	2	no	—
111	C₂×C₄	—	716	2	no	—
112	C₂×C₄	D₄×D₁₇₉	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	716	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	716	2	no	—
119	C₂²	—	716	2	no	—
120	C₂²	—	716	2	no	—
121	C₂²	#19153081	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	716	2	no	—
125	C₄	#19153134	1	2	yes	—
126	C₄	#19153140	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	1432	2	no	—
129	C₂	#26038583	1	2	yes	—
130	C₂	#26038606	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26039155

C₂

yes

maximal, characteristic, complemented

#26038631

C₂

yes

maximal, complemented

#26038188

C₂

yes

maximal, characteristic, complemented

#26038168

C₂

yes

maximal, complemented

#492×C₁₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#15668244

—

#19153102

—

#19153088

—

C₄×D₇₁₆

C₂²

yes

—

#19153071

—

C₄⋊(C₁₇₉⋊C₈)

C₂²

yes

—

#19153034

—

C₂×SD₁₆×C₁₇₉

—

C₄⋊Q₈×C₁₇₉

C₂²

yes

characteristic

#127×C₁₇₉

C₂²

yes

characteristic

C₄×C₈×C₁₇₉

C₂²

yes

characteristic

C₂×D₇₁₆

—

C₂×D₇₁₆

D₄

yes

—

C₂×C₄×D₁₇₉

—

#15668244

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

D₄

yes

—

(C₂×D₁₇₉)⋊C₄

—

(C₂×D₁₇₉)⋊C₄

—

C₄×Dic₁₇₉

—

C₂×Q₈×C₁₇₉

D₄

yes

—

C₂×D₄×C₁₇₉

—

C₂×D₄×C₁₇₉

—

SD₁₆×C₁₇₉

—

C₂×C₈×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄²×C₁₇₉

C₂³

yes

characteristic

C₂²×D₁₇₉

—

D₇₁₆

—

D₇₁₆

—

D₇₁₆

—

C₂×Dic₁₇₉

—

C₄×D₁₇₉

—

C₄×D₁₇₉

—

C₁₇₉⋊C₈

—

C₂³×C₁₇₉

—

Q₈×C₁₇₉

—

D₄×C₁₇₉

—

D₄×C₁₇₉

—

D₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₈×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

—

C₂²×C₁₇₉

#120

yes

characteristic

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₂×D₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

#135

yes

—

C₄×C₁₇₉

AGL(1,ℤ/8)

yes

—

Dic₁₇₉

—

C₂×C₁₇₉

—

C₂×C₁₇₉

#449

yes

—

C₂×C₁₇₉

#453

yes

characteristic

D₁₇₉

—

C₁₇₉

#1624

yes

characteristic, complemented, Sylow

#1624

—

179

maximal, Sylow

#555

—

179

—

#492

D₁₇₉

yes

characteristic, complemented

#459

—

179

—

#333

—

179

—

D₄⋊C₈

—

179

—

C₂×SD₁₆

—

C₂×D₈

—

358

—

C₄⋊Q₈

C₂×D₁₇₉

yes

characteristic

#127

C₂×D₁₇₉

yes

characteristic

#124

—

358

—

#121

—

358

—

C₄×D₄

—

179

—

#117

—

358

—

C₄⋊C₈

—

179

—

#102

—

358

—

C₄×C₈

C₂×D₁₇₉

yes

characteristic

C₂×Q₈

C₂²⋊D₁₇₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

358

—

C₂×D₄

—

179

—

C₂²×C₄

—

358

—

SD₁₆

—

D₈

—

716

—

C₂×C₈

—

C₂×C₈

—

358

—

C₄⋊C₄

—

C₄⋊C₄

—

179

—

C₂²⋊C₄

—

358

—

C₂²⋊C₄

—

358

—

C₄²

—

358

—

C₄²

C₂²×D₁₇₉

yes

characteristic

C₂³

—

100

C₂³

—

358

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

358

—

106

D₄

—

358

—

107

D₄

—

1432

—

108

C₂×C₄

—

109

C₂×C₄

—

358

—

110

C₂×C₄

—

716

—

111

C₂×C₄

—

716

—

112

C₂×C₄

D₄×D₁₇₉

yes

—

113

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

114

C₈

—

115

C₈

—

716

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

716

—

119

C₂²

—

716

—

120

C₂²

—

716

—

121

C₂²

#19153081

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

716

—

125

C₄

#19153134

yes

—

126

C₄

#19153140

yes

—

127

C₂

—

128

C₂

—

1432

—

129

C₂

#26038583

yes

—

130

C₂

#26038606

yes

characteristic

	subgroup	G/H
center	C₂²	#19153081
commutator, cosocle	C₄²×C₁₇₉	C₂³
Frattini	C₄²	C₂²×D₁₇₉
Fitting	#492×C₁₇₉	C₂
socle	C₂²×C₁₇₉	#120
hypercenter	#492	D₁₇₉
hypocenter	C₁₇₉	#1624
LCS	G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#19153081

commutator, cosocle

C₄²×C₁₇₉

C₂³

Frattini

C₄²

C₂²×D₁₇₉

Fitting

#492×C₁₇₉

C₂

C₂²×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #26039155	C₂	semidirect
2. #26038631	C₂	semidirect
3. #26038188	C₂	semidirect
4. #26038168	C₂	semidirect
5. #492×C₁₇₉	C₂	semidirect
65. C₁₇₉	#1624	semidirect
68. #492	D₁₇₉	semidirect

H ≅ G/N

type

1. #26039155

C₂

semidirect

2. #26038631

C₂

semidirect

3. #26038188

C₂

semidirect

4. #26038168

C₂

semidirect

5. #492×C₁₇₉

C₂

semidirect

65. C₁₇₉

#1624

semidirect

68. #492

D₁₇₉

semidirect

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

subgroup	index
#26039155	2
#26038631	2
#26038188	2
#26038168	2
#492×C₁₇₉	2
#1624	179

subgroup

index

#492×C₁₇₉

179

Subgroups — #37508081

Conjugacy classes of subgroups

Complements

Maximal subgroups