Group #37572568 — subgroups

Subgroups — #37572568

Order 24448.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20259329	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259257	C₂	1	2	yes	maximal, complemented
3	#27132647	C₂	1	1	yes	maximal, characteristic
4	C₁₉₁⋊#338	C₂	1	2	yes	maximal, complemented
5	#413×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	#20259329	—	2	4	no	—
7	C₂×#15690271	C₄	1	2	yes	—
8	C₂×#15690271	C₂²	1	2	yes	—
9	C₂×#15690271	C₂²	1	1	yes	characteristic
10	C₂²×C₁₉₁⋊C₈	C₄	1	2	yes	—
11	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
12	#20259257	—	2	4	no	—
13	#20259241	—	2	1	no	—
14	#20259239	—	2	1	no	—
15	C₂×#41×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
17	#101×C₁₉₁	—	2	1	no	—
18	#100×C₁₉₁	—	2	1	no	—
19	#15690271	—	2	4	no	—
20	#15690271	—	2	2	no	—
21	#15690271	—	2	2	no	—
22	#15690271	—	2	2	no	—
23	C₂×C₁₉₁⋊C₈	—	2	4	no	—
24	C₂×C₁₉₁⋊C₈	—	2	2	no	—
25	C₂×C₁₉₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₉₁⋊C₈	D₄	1	4	yes	—
29	#41×C₁₉₁	—	2	2	no	—
30	#41×C₁₉₁	—	2	2	no	—
31	C₂×Q₈×C₁₉₁	D₄	1	1	yes	characteristic
32	C₂×D₄×C₁₉₁	D₄	1	1	yes	characteristic
33	C₂×D₄×C₁₉₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₉₁	—	2	2	no	—
37	M₁₆×C₁₉₁	—	2	2	no	—
38	C₂×C₈×C₁₉₁	—	2	2	no	—
39	C₁₉₁⋊C₈	—	2	2	no	—
40	C₁₉₁⋊C₈	—	4	2	no	—
41	C₁₉₁⋊C₈	C₂²⋊C₄	1	2	yes	—
42	C₁₉₁⋊C₈	C₂²⋊C₄	1	2	yes	—
43	C₂³×C₁₉₁	C₄⋊C₄	1	2	yes	—
44	C₂³×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
45	Q₈×C₁₉₁	—	2	2	no	—
46	D₄×C₁₉₁	—	2	4	no	—
47	D₄×C₁₉₁	—	2	2	no	—
48	C₂×C₄×C₁₉₁	—	2	2	no	—
49	C₂×C₄×C₁₉₁	—	2	2	no	—
50	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₁	#41	1	2	yes	—
55	C₈×C₁₉₁	—	4	2	no	—
56	C₂²×C₁₉₁	—	2	2	no	—
57	C₂²×C₁₉₁	—	4	1	no	—
58	C₂²×C₁₉₁	—	4	2	no	—
59	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₁₉₁	—	2	2	no	—
62	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₁₉₁	#117	1	1	yes	characteristic
64	C₄×C₁₉₁	#121	1	2	yes	—
65	C₂×C₁₉₁	—	2	1	no	—
66	C₂×C₁₉₁	—	2	2	no	—
67	C₂×C₁₉₁	—	4	2	no	—
68	C₂×C₁₉₁	#380	1	1	yes	characteristic
69	C₁₉₁	#1813	1	1	yes	characteristic, complemented, Sylow
70	#1813	—	191	1	no	maximal, Sylow
71	C₂×#131	—	191	1	no	—
72	C₂×#108	—	191	2	no	—
73	#413	—	191	1	no	—
74	#413	D₁₉₁	1	1	yes	characteristic
75	#338	—	191	2	no	—
76	C₂×#41	C₂×D₁₉₁	1	1	yes	characteristic
77	#131	—	382	4	no	—
78	C₂×M₁₆	—	191	2	no	—
79	C₂×M₁₆	—	191	2	no	—
80	C₂×M₁₆	—	191	1	no	—
81	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
82	C₂²×C₈	—	191	2	no	—
83	C₂²×C₈	—	191	1	no	—
84	#108	—	382	4	no	—
85	#101	—	2	1	no	—
86	#101	—	382	1	no	—
87	#100	—	2	1	no	—
88	#100	—	382	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
92	C₂×D₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
93	C₂×D₄	Dic₃₈₂	1	2	yes	—
94	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	382	2	no	—
99	M₁₆	—	382	4	no	—
100	M₁₆	—	382	2	no	—
101	M₁₆	—	382	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	382	4	no	—
104	C₂×C₈	—	382	2	no	—
105	C₂×C₈	—	191	2	no	—
106	C₂×C₈	—	191	1	no	—
107	C₂×C₈	—	191	1	no	—
108	C₂×C₈	—	191	4	no	—
109	C₂³	Dic₁₉₁⋊C₄	1	2	yes	—
110	C₂³	C₂×Dic₃₈₂	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	Dic₁₉₁⋊C₄	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
118	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
120	C₂×C₄	#15690286	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₈	—	382	2	no	—
123	C₈	—	191	2	no	—
124	C₈	—	191	2	no	—
125	C₈	—	764	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
130	C₂²	#20259299	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₁₉₁	1	1	yes	characteristic
133	C₄	#20259278	1	1	yes	characteristic
134	C₄	#20259294	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#27132521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259329

C₂

yes

maximal, characteristic

C₂×#20259257

C₂

yes

maximal, complemented

#27132647

C₂

yes

maximal, characteristic

C₁₉₁⋊#338

C₂

yes

maximal, complemented

#413×C₁₉₁

C₂

yes

maximal, characteristic

#20259329

—

C₂×#15690271

C₄

yes

—

C₂×#15690271

C₂²

yes

—

C₂×#15690271

C₂²

yes

characteristic

C₂²×C₁₉₁⋊C₈

C₄

yes

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

—

—

C₂×#41×C₁₉₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₁

C₂²

yes

—

#101×C₁₉₁

—

#100×C₁₉₁

—

—

—

—

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₉₁⋊C₈

D₄

yes

—

#41×C₁₉₁

—

#41×C₁₉₁

—

C₂×Q₈×C₁₉₁

D₄

yes

characteristic

C₂×D₄×C₁₉₁

D₄

yes

characteristic

C₂×D₄×C₁₉₁

Q₈

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

C₂²⋊C₄

yes

—

C₁₉₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₉₁

C₄⋊C₄

yes

—

C₂³×C₁₉₁

C₂×Q₈

yes

characteristic

Q₈×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₂²⋊Q₈

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×C₂²⋊C₄

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#380

yes

characteristic

C₁₉₁

#1813

yes

characteristic, complemented, Sylow

#1813

—

191

maximal, Sylow

C₂×#131

—

191

—

C₂×#108

—

191

—

#413

—

191

—

#413

D₁₉₁

yes

characteristic

#338

—

191

—

C₂×#41

C₂×D₁₉₁

yes

characteristic

#131

—

382

—

C₂×M₁₆

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₂²×C₈

—

191

—

C₂²×C₈

—

191

—

#108

—

382

—

#101

—

#101

—

382

—

#100

—

#100

—

382

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₁₉₁

yes

characteristic

C₂×D₄

C₂²⋊D₁₉₁

yes

characteristic

C₂×D₄

Dic₃₈₂

yes

—

C₂²×C₄

C₄×D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

382

—

M₁₆

—

382

—

100

M₁₆

—

382

—

101

M₁₆

—

382

—

102

C₂×C₈

—

103

C₂×C₈

—

382

—

104

C₂×C₈

—

382

—

105

C₂×C₈

—

191

—

106

C₂×C₈

—

191

—

107

C₂×C₈

—

191

—

108

C₂×C₈

—

191

—

109

C₂³

Dic₁₉₁⋊C₄

yes

—

110

C₂³

C₂×Dic₃₈₂

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

117

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

118

C₂×C₄

D₄×D₁₉₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

120

C₂×C₄

#15690286

yes

—

121

C₈

—

122

C₈

—

382

—

123

C₈

—

191

—

124

C₈

—

191

—

125

C₈

—

764

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×Dic₁₉₁⋊C₄

yes

characteristic

130

C₂²

#20259299

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₁₉₁

yes

characteristic

133

C₄

#20259278

yes

characteristic

134

C₄

#20259294

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#27132521

yes

characteristic

	subgroup	G/H
center	C₄	#20259278
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#413×C₁₉₁	C₂
socle	C₂×C₁₉₁	#380
prime radical	C₂×#41×C₁₉₁	C₂²
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#413	D₁₉₁
hypocenter	C₁₉₁	#1813
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#20259278

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#413×C₁₉₁

C₂

socle

C₂×C₁₉₁

#380

prime radical

C₂×#41×C₁₉₁

C₂²

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. C₂×#20259257	C₂	semidirect
4. C₁₉₁⋊#338	C₂	semidirect
69. C₁₉₁	#1813	semidirect

H ≅ G/N

type

2. C₂×#20259257

C₂

semidirect

4. C₁₉₁⋊#338

C₂

semidirect

69. C₁₉₁

#1813

semidirect

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

subgroup	index
C₂×#20259329	2
C₂×#20259257	2
#27132647	2
C₁₉₁⋊#338	2
#413×C₁₉₁	2
#1813	191

subgroup

index

C₂×#20259329

C₂×#20259257

#27132647

C₁₉₁⋊#338

#413×C₁₉₁

#1813

191

Subgroups — #37572568

Conjugacy classes of subgroups

Complements

Maximal subgroups