Group #37571254 — subgroups

Subgroups — #37571254

Order 24448.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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	#27133214	C₂	1	1	yes	maximal, characteristic
2	#27132685	C₂	1	2	yes	maximal, complemented
3	#27132412	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132406	C₂	1	2	yes	maximal, complemented
5	#333×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690288	C₄	1	2	yes	—
7	C₂×#15690270	—	2	2	no	—
8	#20259319	C₂²	1	1	yes	characteristic
9	#20259311	C₄	1	2	yes	—
10	#20259297	—	2	2	no	—
11	#20259284	—	2	2	no	—
12	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
13	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
14	Dic₁₉₁⋊C₈	—	2	2	no	—
15	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
17	#15690288	—	4	4	no	—
18	C₂×D₇₆₄	—	2	2	no	—
19	C₂×D₇₆₄	—	2	2	no	—
20	C₂×D₇₆₄	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₁₉₁	—	2	2	no	—
22	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₉₁	D₄	1	2	yes	—
24	#15690270	—	4	4	no	—
25	C₂×C₁₉₁⋊C₈	—	2	2	no	—
26	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
27	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₉₁	D₄	1	2	yes	—
29	C₂×Q₈×C₁₉₁	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₉₁	—	2	2	no	—
31	C₄⋊C₄×C₁₉₁	—	2	2	no	—
32	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₂²×D₁₉₁	—	2	2	no	—
34	C₂²×D₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
35	D₇₆₄	—	4	2	no	—
36	D₇₆₄	—	4	4	no	—
37	D₇₆₄	—	4	2	no	—
38	D₇₆₄	—	4	2	no	—
39	C₂×Dic₁₉₁	C₂²⋊C₄	1	2	yes	—
40	C₂×Dic₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×D₁₉₁	—	2	2	no	—
42	C₄×D₁₉₁	—	2	2	no	—
43	C₄×D₁₉₁	—	2	2	no	—
44	C₄×D₁₉₁	—	4	4	no	—
45	C₁₉₁⋊C₈	—	4	2	no	—
46	Q₈×C₁₉₁	—	4	2	no	—
47	C₂×C₄×C₁₉₁	—	2	2	no	—
48	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
50	C₈×C₁₉₁	—	4	2	no	—
51	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₁₉₁	—	2	1	no	—
53	C₂×D₁₉₁	—	2	2	no	—
54	C₂×D₁₉₁	—	4	2	no	—
55	C₂×D₁₉₁	—	4	4	no	—
56	C₄×C₁₉₁	—	2	1	no	—
57	C₄×C₁₉₁	—	4	2	no	—
58	C₄×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
59	C₄×C₁₉₁	#137	1	2	yes	—
60	Dic₁₉₁	—	2	2	no	—
61	Dic₁₉₁	#100	1	2	yes	—
62	C₂×C₁₉₁	C₂×#100	1	1	yes	characteristic
63	C₂×C₁₉₁	#417	1	2	yes	—
64	D₁₉₁	—	4	1	no	—
65	D₁₉₁	—	8	2	no	—
66	C₁₉₁	#1560	1	1	yes	characteristic, complemented, Sylow
67	#1560	—	191	1	no	maximal, Sylow
68	#532	—	191	1	no	—
69	#423	—	191	2	no	—
70	#333	—	191	1	no	—
71	#333	D₁₉₁	1	1	yes	characteristic, complemented
72	#331	—	191	2	no	—
73	C₂×#41	—	191	2	no	—
74	C₂×M₁₆	—	382	2	no	—
75	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
76	#127	—	191	1	no	—
77	#124	—	191	2	no	—
78	#121	—	382	2	no	—
79	C₄×D₄	—	382	2	no	—
80	C₂×C₄²	—	191	1	no	—
81	C₄⋊C₈	—	382	2	no	—
82	C₄⋊C₈	—	191	2	no	—
83	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
84	#41	—	764	4	no	—
85	C₂×Q₈	C₄×D₁₉₁	1	2	yes	—
86	C₂×D₄	—	382	2	no	—
87	C₂×D₄	—	382	2	no	—
88	C₂×D₄	—	191	2	no	—
89	C₂²×C₄	—	382	2	no	—
90	C₂²×C₄	—	191	1	no	—
91	C₂²×C₄	—	191	2	no	—
92	M₁₆	—	764	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	382	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	382	4	no	—
97	C₄²	—	191	1	no	—
98	C₄²	—	191	2	no	—
99	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
100	C₂³	—	382	2	no	—
101	C₂³	—	191	1	no	—
102	Q₈	—	4	2	no	—
103	D₄	—	764	2	no	—
104	D₄	—	764	4	no	—
105	D₄	—	764	2	no	—
106	D₄	—	764	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	382	2	no	—
109	C₂×C₄	—	382	2	no	—
110	C₂×C₄	—	382	2	no	—
111	C₂×C₄	—	191	2	no	—
112	C₂×C₄	—	191	1	no	—
113	C₂×C₄	—	764	4	no	—
114	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
115	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	764	2	no	—
118	C₂²	—	382	1	no	—
119	C₂²	—	382	2	no	—
120	C₂²	—	764	2	no	—
121	C₂²	—	764	4	no	—
122	C₂²	C₂²⋊C₄×D₁₉₁	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	4	2	no	—
125	C₄	—	382	2	no	—
126	C₄	—	191	2	no	—
127	C₄	#20259349	1	2	yes	—
128	C₄	#20259354	1	2	yes	—
129	C₂	—	1528	2	no	—
130	C₂	—	764	1	no	—
131	C₂	#100×D₁₉₁	1	1	yes	characteristic
132	C₂	#27132663	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133214

C₂

yes

maximal, characteristic

#27132685

C₂

yes

maximal, complemented

#27132412

C₂

yes

maximal, characteristic, complemented

#27132406

C₂

yes

maximal, complemented

#333×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690288

C₄

yes

—

C₂×#15690270

—

#20259319

C₂²

yes

characteristic

#20259311

C₄

yes

—

#20259297

—

#20259284

—

C₄²×D₁₉₁

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

Dic₁₉₁⋊C₈

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

—

#15690288

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

C₂×C₄

yes

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₉₁

D₄

yes

—

#15690270

—

C₂×C₁₉₁⋊C₈

—

(C₂×D₁₉₁)⋊C₄

—

C₄×Dic₁₉₁

C₂×C₄

yes

characteristic

C₄×Dic₁₉₁

D₄

yes

—

C₂×Q₈×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

—

C₂²×D₁₉₁

C₂²⋊C₄

yes

characteristic

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

characteristic

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₁

#137

yes

—

Dic₁₉₁

—

Dic₁₉₁

#100

yes

—

C₂×C₁₉₁

C₂×#100

yes

characteristic

C₂×C₁₉₁

#417

yes

—

D₁₉₁

—

D₁₉₁

—

C₁₉₁

#1560

yes

characteristic, complemented, Sylow

#1560

—

191

maximal, Sylow

#532

—

191

—

#423

—

191

—

#333

—

191

—

#333

D₁₉₁

yes

characteristic, complemented

#331

—

191

—

C₂×#41

—

191

—

C₂×M₁₆

—

382

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

#127

—

191

—

#124

—

191

—

#121

—

382

—

C₄×D₄

—

382

—

C₂×C₄²

—

191

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

191

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

#41

—

764

—

C₂×Q₈

C₄×D₁₉₁

yes

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

191

—

C₂²×C₄

—

382

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

M₁₆

—

764

—

C₂×C₈

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₂²⋊C₄

—

382

—

C₄²

—

191

—

C₄²

—

191

—

C₄²

C₂²×D₁₉₁

yes

characteristic

100

C₂³

—

382

—

101

C₂³

—

191

—

102

Q₈

—

103

D₄

—

764

—

104

D₄

—

764

—

105

D₄

—

764

—

106

D₄

—

764

—

107

C₂×C₄

—

108

C₂×C₄

—

382

—

109

C₂×C₄

—

382

—

110

C₂×C₄

—

382

—

111

C₂×C₄

—

191

—

112

C₂×C₄

—

191

—

113

C₂×C₄

—

764

—

114

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

115

C₂×C₄

D₄×D₁₉₁

yes

—

116

C₈

—

117

C₈

—

764

—

118

C₂²

—

382

—

119

C₂²

—

382

—

120

C₂²

—

764

—

121

C₂²

—

764

—

122

C₂²

C₂²⋊C₄×D₁₉₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

382

—

126

C₄

—

191

—

127

C₄

#20259349

yes

—

128

C₄

#20259354

yes

—

129

C₂

—

1528

—

130

C₂

—

764

—

131

C₂

#100×D₁₉₁

yes

characteristic

132

C₂

#27132663

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#333×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
prime radical	#27133214	C₂
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	#333	D₁₉₁
hypocenter	C₁₉₁	#1560
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#333×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

prime radical

#27133214

C₂

cosocle

C₄²×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #27132685	C₂	semidirect
3. #27132412	C₂	semidirect
4. #27132406	C₂	semidirect
5. #333×C₁₉₁	C₂	semidirect
66. C₁₉₁	#1560	semidirect
71. #333	D₁₉₁	semidirect

H ≅ G/N

type

2. #27132685

C₂

semidirect

3. #27132412

C₂

semidirect

4. #27132406

C₂

semidirect

5. #333×C₁₉₁

C₂

semidirect

66. C₁₉₁

#1560

semidirect

71. #333

D₁₉₁

semidirect

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

subgroup	index
#27133214	2
#27132685	2
#27132412	2
#27132406	2
#333×C₁₉₁	2
#1560	191

subgroup

index

#333×C₁₉₁

191

Subgroups — #37571254

Conjugacy classes of subgroups

Complements

Maximal subgroups