Group #37571579 — subgroups

Subgroups — #37571579

Order 24448.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27133039	C₂	1	1	yes	maximal, characteristic
2	#27132863	C₂	1	2	yes	maximal, complemented
3	#27132407	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₁₉₁⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#559×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690274	—	2	2	no	—
7	C₂×#15690272	—	2	2	no	—
8	#20259319	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
10	#20259242	—	2	2	no	—
11	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
12	#127×C₁₉₁	C₂²	1	1	yes	characteristic
13	#124×C₁₉₁	—	2	2	no	—
14	#123×C₁₉₁	—	2	2	no	—
15	C₄×Q₈×C₁₉₁	C₂²	1	2	yes	—
16	C₄×D₄×C₁₉₁	—	2	2	no	—
17	C₂×D₇₆₄	—	2	1	no	—
18	C₂×D₇₆₄	—	2	2	no	—
19	#15690274	—	4	4	no	—
20	#15690272	—	4	4	no	—
21	C₂×C₁₉₁⋊C₈	—	2	1	no	—
22	C₂×C₁₉₁⋊C₈	—	2	2	no	—
23	C₂×Q₈×C₁₉₁	D₄	1	2	yes	—
24	C₂×D₄×C₁₉₁	—	2	2	no	—
25	C₂×D₄×C₁₉₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₉₁	—	2	2	no	—
27	C₄⋊C₄×C₁₉₁	—	2	2	no	—
28	C₄⋊C₄×C₁₉₁	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₉₁	—	2	4	no	—
30	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
31	C₄²×C₁₉₁	—	2	2	no	—
32	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₂²×D₁₉₁	—	4	1	no	—
34	D₇₆₄	—	4	2	no	—
35	D₇₆₄	—	4	2	no	—
36	D₇₆₄	—	4	2	no	—
37	C₁₉₁⋊C₈	—	2	2	no	—
38	C₁₉₁⋊C₈	—	4	2	no	—
39	C₂³×C₁₉₁	—	2	2	no	—
40	Q₈×C₁₉₁	—	2	2	no	—
41	Q₈×C₁₉₁	—	2	2	no	—
42	D₄×C₁₉₁	—	4	2	no	—
43	D₄×C₁₉₁	—	4	2	no	—
44	D₄×C₁₉₁	—	4	2	no	—
45	C₂×C₄×C₁₉₁	—	2	2	no	—
46	C₂×C₄×C₁₉₁	—	2	2	no	—
47	C₂×C₄×C₁₉₁	—	8	2	no	—
48	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
50	C₂²×C₁₉₁	—	4	4	no	—
51	C₂²×C₁₉₁	—	4	2	no	—
52	C₂²×C₁₉₁	#120	1	1	yes	characteristic
53	C₂×D₁₉₁	—	8	2	no	—
54	C₂×D₁₉₁	—	8	1	no	—
55	C₄×C₁₉₁	—	2	1	no	—
56	C₄×C₁₉₁	—	4	2	no	—
57	C₄×C₁₉₁	—	4	2	no	—
58	C₄×C₁₉₁	#135	1	2	yes	—
59	C₄×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₁₉₁	—	8	2	no	—
61	C₂×C₁₉₁	#449	1	2	yes	—
62	C₂×C₁₉₁	#453	1	1	yes	characteristic
63	D₁₉₁	—	16	1	no	—
64	C₁₉₁	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	191	1	no	maximal, Sylow
66	#559	D₁₉₁	1	1	yes	characteristic, complemented
67	#493	—	191	1	no	—
68	#460	—	191	2	no	—
69	#331	—	191	1	no	—
70	Q₈⋊C₈	—	191	2	no	—
71	C₂×SD₁₆	—	382	2	no	—
72	C₂×D₈	—	382	2	no	—
73	#127	—	191	1	no	—
74	#127	C₂×D₁₉₁	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₁₉₁	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	191	2	no	—
80	#102	—	382	2	no	—
81	C₄×C₈	—	191	1	no	—
82	C₂×Q₈	C₂²⋊D₁₉₁	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	382	2	no	—
85	C₂×D₄	—	382	1	no	—
86	C₂×D₄	C₂²⋊D₁₉₁	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	764	4	no	—
89	D₈	—	764	4	no	—
90	C₂×C₈	—	382	1	no	—
91	C₂×C₈	—	382	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₁₉₁	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	764	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	764	2	no	—
106	D₄	—	764	2	no	—
107	D₄	—	764	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
113	C₈	—	382	2	no	—
114	C₈	—	764	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	1528	2	no	—
118	C₂²	—	1528	1	no	—
119	C₂²	#20259291	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#20259342	1	2	yes	—
124	C₄	#20259347	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	3056	1	no	—
127	C₂	#27132808	1	2	yes	—
128	C₂	#27132831	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133039

C₂

yes

maximal, characteristic

#27132863

C₂

yes

maximal, complemented

#27132407

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₁₉₁⋊C₈)

C₂

yes

maximal, complemented

#559×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690274

—

C₂×#15690272

—

#20259319

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

#20259242

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

#127×C₁₉₁

C₂²

yes

characteristic

#124×C₁₉₁

—

#123×C₁₉₁

—

C₄×Q₈×C₁₉₁

C₂²

yes

—

C₄×D₄×C₁₉₁

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

#15690274

—

#15690272

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×Q₈×C₁₉₁

D₄

yes

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₂²⋊C₄×C₁₉₁

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#120

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#135

yes

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#449

yes

—

C₂×C₁₉₁

#453

yes

characteristic

D₁₉₁

—

C₁₉₁

#1620

yes

characteristic, complemented, Sylow

#1620

—

191

maximal, Sylow

#559

D₁₉₁

yes

characteristic, complemented

#493

—

191

—

#460

—

191

—

#331

—

191

—

Q₈⋊C₈

—

191

—

C₂×SD₁₆

—

382

—

C₂×D₈

—

382

—

#127

—

191

—

#127

C₂×D₁₉₁

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₉₁

yes

—

C₄×D₄

—

C₄⋊C₈

—

191

—

#102

—

382

—

C₄×C₈

—

191

—

C₂×Q₈

C₂²⋊D₁₉₁

yes

—

C₂×D₄

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

—

SD₁₆

—

764

—

D₈

—

764

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₉₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

—

C₂³

—

764

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

764

—

106

D₄

—

764

—

107

D₄

—

764

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

113

C₈

—

382

—

114

C₈

—

764

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

1528

—

118

C₂²

—

1528

—

119

C₂²

#20259291

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#20259342

yes

—

124

C₄

#20259347

yes

—

125

C₂

—

126

C₂

—

3056

—

127

C₂

#27132808

yes

—

128

C₂

#27132831

yes

characteristic

	subgroup	G/H
center	C₂²	#20259291
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#559×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#120
prime radical	#27133039	C₂
hypercenter	#559	D₁₉₁
hypocenter	C₁₉₁	#1620
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#20259291

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#559×C₁₉₁

C₂

C₂²×C₁₉₁

C₂

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #27132863	C₂	semidirect
3. #27132407	C₂	semidirect
4. Q₈⋊(C₁₉₁⋊C₈)	C₂	semidirect
5. #559×C₁₉₁	C₂	semidirect
64. C₁₉₁	#1620	semidirect
66. #559	D₁₉₁	semidirect

H ≅ G/N

type

2. #27132863

C₂

semidirect

3. #27132407

C₂

semidirect

4. Q₈⋊(C₁₉₁⋊C₈)

C₂

semidirect

5. #559×C₁₉₁

C₂

semidirect

64. C₁₉₁

#1620

semidirect

66. #559

D₁₉₁

semidirect

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

subgroup	index
#27133039	2
#27132863	2
#27132407	2
Q₈⋊(C₁₉₁⋊C₈)	2
#559×C₁₉₁	2
#1620	191

subgroup

index

#27133039

#27132863

#27132407

Q₈⋊(C₁₉₁⋊C₈)

#559×C₁₉₁

#1620

191

Subgroups — #37571579

Conjugacy classes of subgroups

Complements

Maximal subgroups