Group #37573248 — subgroups

Subgroups — #37573248

Order 24448.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#20259254	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259254	C₂	1	1	yes	maximal, characteristic
3	#27132561	C₂	1	2	yes	maximal
4	#27132426	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
8	C₂×Dic₁₉₁⋊C₄	—	2	2	no	—
9	#20259254	—	2	2	no	—
10	#20259254	C₄	1	4	yes	complemented
11	C₁₉₁⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₉₁	—	2	2	no	—
16	C₂²×Dic₁₉₁	—	2	2	no	—
17	C₂²×Dic₁₉₁	—	2	2	no	—
18	C₄⋊Dic₁₉₁	—	2	2	no	—
19	C₄⋊Dic₁₉₁	—	2	2	no	—
20	C₄⋊Dic₁₉₁	—	2	2	no	—
21	C₄⋊Dic₁₉₁	C₂×C₄	1	4	yes	—
22	Dic₁₉₁⋊C₄	—	4	4	no	—
23	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
24	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₉₁	—	2	1	no	—
26	C₂×C₈×C₁₉₁	—	2	1	no	—
27	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
29	C₂×C₈×C₁₉₁	Q₈	1	2	yes	—
30	C₂×C₈×C₁₉₁	Q₈	1	2	yes	—
31	C₄²×C₁₉₁	—	2	1	no	—
32	C₄²×C₁₉₁	—	2	1	no	—
33	C₂×Dic₁₉₁	—	2	4	no	—
34	C₂×Dic₁₉₁	—	4	2	no	—
35	C₂×Dic₁₉₁	—	4	2	no	—
36	C₂×Dic₁₉₁	—	4	2	no	—
37	C₂×Dic₁₉₁	—	4	2	no	—
38	C₂×Dic₁₉₁	—	4	2	no	—
39	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₁	—	2	2	no	—
41	C₂×C₄×C₁₉₁	—	2	2	no	—
42	C₂×C₄×C₁₉₁	SD₁₆	1	4	yes	—
43	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
49	C₈×C₁₉₁	—	2	2	no	—
50	C₈×C₁₉₁	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₁₉₁	#121	1	1	yes	characteristic
53	C₂²×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₉₁	C₂×SD₁₆	1	2	yes	—
55	C₂²×C₁₉₁	#135	1	2	yes	—
56	C₄×C₁₉₁	—	2	4	no	—
57	C₄×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₁₉₁	C₄×Q₈	1	1	yes	characteristic
59	C₄×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₁₉₁	#125	1	1	yes	characteristic
61	Dic₁₉₁	—	4	4	no	—
62	Dic₁₉₁	—	8	2	no	—
63	C₂×C₁₉₁	#389	1	1	yes	characteristic
64	C₂×C₁₉₁	C₄×SD₁₆	1	2	yes	—
65	C₂×C₁₉₁	#465	1	2	yes	—
66	C₂×C₁₉₁	#498	1	1	yes	characteristic
67	C₂×C₁₉₁	#499	1	1	yes	characteristic
68	C₁₉₁	#1941	1	1	yes	characteristic, complemented, Sylow
69	#1941	—	191	1	no	maximal, Sylow
70	C₂×#106	—	191	1	no	—
71	C₂×#106	—	191	1	no	—
72	C₂×C₄×C₈	D₁₉₁	1	1	yes	characteristic
73	#389	—	191	2	no	—
74	#340	—	191	2	no	—
75	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	382	2	no	—
78	C₂×C₄⋊C₄	—	191	2	no	—
79	C₂×C₄⋊C₄	—	191	2	no	—
80	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
81	#106	—	382	2	no	—
82	#106	—	191	4	no	—
83	C₄×C₈	—	2	2	no	—
84	#95	—	382	2	no	—
85	C₂²×C₄	—	382	2	no	—
86	C₂²×C₄	—	382	2	no	—
87	C₂²×C₄	D₇₆₄	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₄×D₁₉₁	1	2	yes	—
92	C₂×C₈	C₂²⋊D₁₉₁	1	2	yes	—
93	C₂×C₈	Dic₃₈₂	1	2	yes	—
94	C₂×C₈	Dic₃₈₂	1	2	yes	—
95	C₄⋊C₄	—	382	2	no	—
96	C₄⋊C₄	—	382	2	no	—
97	C₄⋊C₄	—	382	2	no	—
98	C₄⋊C₄	—	191	4	no	—
99	C₄⋊C₄	—	764	4	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	382	4	no	—
106	C₂×C₄	—	764	2	no	—
107	C₂×C₄	—	764	2	no	—
108	C₂×C₄	—	764	2	no	—
109	C₂×C₄	—	764	2	no	—
110	C₂×C₄	—	764	2	no	—
111	C₂×C₄	#15690276	1	4	yes	—
112	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
114	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
116	C₂×C₄	#15690287	1	1	yes	characteristic
117	C₂×C₄	#15690287	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	Dic₁₉₁⋊C₄	1	4	yes	complemented
120	C₂²	C₄×D₇₆₄	1	1	yes	characteristic
121	C₂²	#20259296	1	1	yes	characteristic
122	C₂²	#20259322	1	1	yes	characteristic
123	C₂²	C₂×#15690276	1	2	yes	—
124	C₂²	#20259344	1	2	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	1528	2	no	—
127	C₄	—	764	4	no	—
128	C₄	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
129	C₄	C₄×Dic₃₈₂	1	1	yes	characteristic
130	C₄	#20259301	1	1	yes	characteristic
131	C₄	#20259314	1	1	yes	characteristic
132	C₂	#27132561	1	1	yes	characteristic
133	C₂	C₄×#15690276	1	2	yes	—
134	C₂	#27132892	1	2	yes	—
135	C₂	#27133061	1	1	yes	characteristic
136	C₂	#27133065	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259254

C₂

yes

maximal, characteristic

C₂×#20259254

C₂

yes

maximal, characteristic

#27132561

C₂

yes

maximal

#27132426

C₂

yes

maximal

C₂×C₄×C₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×Dic₁₉₁⋊C₄

—

#20259254

—

#20259254

C₄

yes

complemented

C₁₉₁⋊#95

—

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

Q₈

yes

—

C₂×C₈×C₁₉₁

Q₈

yes

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

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

C₂×C₄×C₁₉₁

C₂×Q₈

yes

C₂×C₄×C₁₉₁

yes

C₂×C₄×C₁₉₁

yes

C₈×C₁₉₁

—

C₈×C₁₉₁

C₄⋊C₄

yes

complemented

C₂²×C₁₉₁

C₄×D₄

yes

characteristic

C₂²×C₁₉₁

#121

yes

characteristic

C₂²×C₁₉₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉₁

C₂×SD₁₆

yes

—

C₂²×C₁₉₁

#135

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₁

C₄×Q₈

yes

characteristic

C₄×C₁₉₁

C₂²⋊Q₈

yes

characteristic

C₄×C₁₉₁

#125

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#389

yes

characteristic

C₂×C₁₉₁

C₄×SD₁₆

yes

—

C₂×C₁₉₁

#465

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1941

—

191

maximal, Sylow

C₂×#106

—

191

—

C₂×#106

—

191

—

C₂×C₄×C₈

D₁₉₁

yes

characteristic

#389

—

191

—

#340

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#106

—

382

—

#106

—

191

—

C₄×C₈

—

#95

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

D₇₆₄

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₉₁

yes

—

C₂×C₈

C₂²⋊D₁₉₁

yes

—

C₂×C₈

Dic₃₈₂

yes

—

C₂×C₈

Dic₃₈₂

yes

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

764

—

100

C₄²

—

101

C₄²

—

102

C₂³

C₂×D₇₆₄

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

382

—

106

C₂×C₄

—

764

—

107

C₂×C₄

—

764

—

108

C₂×C₄

—

764

—

109

C₂×C₄

—

764

—

110

C₂×C₄

—

764

—

111

C₂×C₄

#15690276

yes

—

112

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

114

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

115

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

116

C₂×C₄

#15690287

yes

characteristic

117

C₂×C₄

#15690287

yes

characteristic

118

C₈

—

119

C₈

Dic₁₉₁⋊C₄

yes

complemented

120

C₂²

C₄×D₇₆₄

yes

characteristic

121

C₂²

#20259296

yes

characteristic

122

C₂²

#20259322

yes

characteristic

123

C₂²

C₂×#15690276

yes

—

124

C₂²

#20259344

yes

—

125

C₄

—

126

C₄

—

1528

—

127

C₄

—

764

—

128

C₄

C₂×Dic₁₉₁⋊C₄

yes

characteristic

129

C₄

C₄×Dic₃₈₂

yes

characteristic

130

C₄

#20259301

yes

characteristic

131

C₄

#20259314

yes

characteristic

132

C₂

#27132561

yes

characteristic

133

C₂

C₄×#15690276

yes

—

134

C₂

#27132892

yes

—

135

C₂

#27133061

yes

characteristic

136

C₂

#27133065

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₆₄
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×C₄×C₈×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×C₄×C₈	D₁₉₁
hypocenter	C₁₉₁	#1941
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₇₆₄

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×C₄×C₈×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×C₄×C₈

D₁₉₁

hypocenter

C₁₉₁

#1941

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
10. #20259254	C₄	semidirect
50. C₈×C₁₉₁	C₄⋊C₄	semidirect
68. C₁₉₁	#1941	semidirect
119. C₈	Dic₁₉₁⋊C₄	semidirect

H ≅ G/N

type

10. #20259254

C₄

semidirect

50. C₈×C₁₉₁

C₄⋊C₄

semidirect

68. C₁₉₁

#1941

semidirect

119. C₈

Dic₁₉₁⋊C₄

semidirect

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

subgroup	index
C₂×#20259254	2
#27132561	2
#27132426	2
C₂×C₄×C₈×C₁₉₁	2
#1941	191

subgroup

index

C₂×C₄×C₈×C₁₉₁

191

Subgroups — #37573248

Conjugacy classes of subgroups

Complements

Maximal subgroups