Group #37571275 — subgroups

Subgroups — #37571275

Order 24448.

184 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 97 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	#27132687	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#27132413	C₂	1	4	yes	maximal, complemented
4	#423×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15690271	—	2	1	no	—
6	C₂²×C₁₉₁⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₁₉₁⋊C₈)	—	2	1	no	—
8	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₁₉₁⋊C₈	C₄	1	1	yes	characteristic
10	C₄×C₁₉₁⋊C₈	C₄	1	1	yes	characteristic
11	C₄×C₁₉₁⋊C₈	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₉₁	—	2	1	no	—
13	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₉₁	—	2	1	no	—
15	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
16	#15690271	—	4	2	no	—
17	C₂×C₁₉₁⋊C₈	—	2	4	no	—
18	C₂×C₁₉₁⋊C₈	—	2	2	no	—
19	C₂×C₁₉₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₉₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉₁	Q₈	1	1	yes	characteristic
24	M₁₆×C₁₉₁	—	4	2	no	—
25	C₂×C₈×C₁₉₁	—	2	2	no	—
26	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
28	C₄²×C₁₉₁	Q₈	1	1	yes	characteristic
29	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
30	C₁₉₁⋊C₈	—	2	2	no	—
31	C₁₉₁⋊C₈	—	4	2	no	—
32	C₁₉₁⋊C₈	M₁₆	1	4	yes	complemented
33	C₂³×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₁	—	2	2	no	—
35	C₂×C₄×C₁₉₁	—	2	2	no	—
36	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
40	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
41	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₁₉₁	—	4	2	no	—
45	C₂²×C₁₉₁	—	2	1	no	—
46	C₂²×C₁₉₁	—	2	1	no	—
47	C₂²×C₁₉₁	#106	1	2	yes	—
48	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₁₉₁	—	2	1	no	—
50	C₄×C₁₉₁	—	2	1	no	—
51	C₄×C₁₉₁	#106	1	2	yes	—
52	C₄×C₁₉₁	#108	1	2	yes	—
53	C₄×C₁₉₁	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₁₉₁	—	2	2	no	—
57	C₂×C₁₉₁	#423	1	1	yes	characteristic
58	C₂×C₁₉₁	C₂×#106	1	1	yes	characteristic
59	C₂×C₁₉₁	C₂×#108	1	1	yes	characteristic
60	C₁₉₁	#1566	1	1	yes	characteristic, complemented, Sylow
61	#1566	—	191	1	no	maximal, Sylow
62	#423	—	191	1	no	—
63	#423	D₁₉₁	1	1	yes	characteristic
64	C₂×C₄×C₈	—	191	1	no	—
65	#334	—	191	4	no	—
66	C₂×M₁₆	—	2	1	no	—
67	C₂×M₁₆	—	382	1	no	—
68	C₂²×C₈	—	191	2	no	—
69	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	382	1	no	—
72	C₄⋊C₈	—	191	2	no	—
73	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
74	C₄×C₈	—	191	1	no	—
75	C₄×C₈	—	191	1	no	—
76	C₄×C₈	—	191	2	no	—
77	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
79	C₂²×C₄	Dic₃₈₂	1	1	yes	characteristic
80	M₁₆	—	4	2	no	—
81	M₁₆	—	764	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	382	4	no	—
84	C₂×C₈	—	382	2	no	—
85	C₂×C₈	—	191	2	no	—
86	C₂×C₈	—	191	2	no	—
87	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
88	C₄²	C₂²⋊D₁₉₁	1	1	yes	characteristic
89	C₄²	Dic₃₈₂	1	1	yes	characteristic
90	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
91	C₂³	Dic₁₉₁⋊C₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	Dic₁₉₁⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₁₉₁⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₁₉₁⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#15690274	1	2	yes	—
98	C₂×C₄	#15690275	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	382	2	no	—
104	C₈	—	191	4	no	—
105	C₈	—	764	2	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#20259253	1	2	yes	—
109	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#20259253	1	2	yes	—
113	C₄	#20259257	1	2	yes	—
114	C₄	M₁₆×D₁₉₁	1	2	yes	—
115	C₄	C₂×#15690274	1	1	yes	characteristic
116	C₄	C₂×#15690275	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#27132686	1	1	yes	characteristic
119	C₂	C₂×#20259253	1	1	yes	characteristic
120	C₂	C₂×#20259257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132687

C₂

yes

maximal, characteristic

C₂×C₄×C₁₉₁⋊C₈

C₂

yes

maximal, characteristic

#27132413

C₂

yes

maximal, complemented

#423×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

—

C₂²×C₁₉₁⋊C₈

C₄

yes

—

C₄⋊(C₁₉₁⋊C₈)

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₄

yes

characteristic

C₄×C₁₉₁⋊C₈

C₄

yes

characteristic

C₄×C₁₉₁⋊C₈

C₂²

yes

—

C₂×M₁₆×C₁₉₁

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

—

C₄⋊C₈×C₁₉₁

C₂²

yes

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

Q₈

yes

characteristic

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

Q₈

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

M₁₆

yes

complemented

C₂³×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#106

yes

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#106

yes

—

C₄×C₁₉₁

#108

yes

—

C₄×C₁₉₁

C₂×M₁₆

yes

—

C₄×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1566

—

191

maximal, Sylow

#423

—

191

—

#423

D₁₉₁

yes

characteristic

C₂×C₄×C₈

—

191

—

#334

—

191

—

C₂×M₁₆

—

C₂×M₁₆

—

382

—

C₂²×C₈

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

191

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

C₂²×C₄

Dic₃₈₂

yes

characteristic

M₁₆

—

M₁₆

—

764

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₄²

C₄×D₁₉₁

yes

characteristic

C₄²

C₂²⋊D₁₉₁

yes

characteristic

C₄²

Dic₃₈₂

yes

characteristic

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

Dic₁₉₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₁₉₁⋊C₄

yes

characteristic

C₂×C₄

Dic₁₉₁⋊C₄

yes

characteristic

C₂×C₄

Dic₁₉₁⋊C₄

yes

characteristic

C₂×C₄

#15690274

yes

—

C₂×C₄

#15690275

yes

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

101

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

102

C₈

—

103

C₈

—

382

—

104

C₈

—

191

—

105

C₈

—

764

—

106

C₂²

—

107

C₂²

—

108

C₂²

#20259253

yes

—

109

C₂²

C₂×Dic₁₉₁⋊C₄

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#20259253

yes

—

113

C₄

#20259257

yes

—

114

C₄

M₁₆×D₁₉₁

yes

—

115

C₄

C₂×#15690274

yes

characteristic

116

C₄

C₂×#15690275

yes

characteristic

117

C₂

—

118

C₂

#27132686

yes

characteristic

119

C₂

C₂×#20259253

yes

characteristic

120

C₂

C₂×#20259257

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#423×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₁	C₄⋊C₄
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	#423	D₁₉₁
hypocenter	C₁₉₁	#1566
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#423×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₁

C₄⋊C₄

cosocle

C₄²×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #27132413	C₂	semidirect
32. C₁₉₁⋊C₈	M₁₆	semidirect
60. C₁₉₁	#1566	semidirect

H ≅ G/N

type

3. #27132413

C₂

semidirect

32. C₁₉₁⋊C₈

M₁₆

semidirect

60. C₁₉₁

#1566

semidirect

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

subgroup	index
#27132687	2
C₂×C₄×C₁₉₁⋊C₈	2
#27132413	2
#423×C₁₉₁	2
#1566	191

subgroup

index

#27132687

C₂×C₄×C₁₉₁⋊C₈

#27132413

#423×C₁₉₁

#1566

191

Subgroups — #37571275

Conjugacy classes of subgroups

Complements

Maximal subgroups