Group #37583265 — subgroups

Subgroups — #37583265

Order 24448.

358 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 133 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	#27133220	C₂	1	2	yes	maximal
2	#27133061	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133043	C₂	1	1	yes	maximal, characteristic, complemented
4	#27133014	C₂	1	2	yes	maximal, complemented
5	#27132921	C₂	1	4	yes	maximal
6	#27132771	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₆₄	C₂²	1	4	yes	—
9	#20259322	C₂²	1	2	yes	—
10	#20259322	C₂²	1	2	yes	—
11	#20259314	—	2	2	no	—
12	#20259301	—	2	2	no	—
13	#20259301	C₂²	1	4	yes	—
14	C₄×Dic₃₈₂	—	2	2	no	—
15	C₄×Dic₃₈₂	C₂²	1	4	yes	—
16	#20259281	—	2	2	no	—
17	#20259254	C₂²	1	4	yes	—
18	#20259247	C₂²	1	8	yes	—
19	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
21	#97×C₁₉₁	C₂²	1	2	yes	—
22	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
24	C₂×Dic₃₈₂	—	2	2	no	—
25	C₂×Dic₃₈₂	C₂³	1	4	yes	—
26	Dic₇₆₄	—	2	8	no	—
27	C₄⋊Dic₁₉₁	—	2	2	no	—
28	C₄⋊Dic₁₉₁	—	2	2	no	—
29	C₄⋊Dic₁₉₁	—	2	2	no	—
30	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
31	Dic₁₉₁⋊C₄	—	2	4	no	—
32	Dic₁₉₁⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₁₉₁	—	2	4	no	—
34	C₄×Dic₁₉₁	—	2	4	no	—
35	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
36	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₉₁	—	2	4	no	—
38	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
39	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
40	C₄²×C₁₉₁	D₄	1	2	yes	—
41	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
42	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
43	Dic₃₈₂	—	2	8	no	—
44	Dic₃₈₂	—	4	4	no	—
45	C₂×Dic₁₉₁	—	2	4	no	—
46	C₂×Dic₁₉₁	—	2	4	no	—
47	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₁	—	2	4	no	—
49	C₂×C₄×C₁₉₁	—	2	2	no	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₉₁	—	2	2	no	—
55	C₈×C₁₉₁	#41	1	4	yes	—
56	C₂²×C₁₉₁	—	2	1	no	—
57	C₂²×C₁₉₁	—	2	2	no	—
58	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₉₁	—	2	1	no	—
60	C₄×C₁₉₁	—	2	2	no	—
61	C₄×C₁₉₁	#137	1	4	yes	—
62	C₄×C₁₉₁	C₂×#41	1	2	yes	—
63	Dic₁₉₁	—	4	4	no	—
64	Dic₁₉₁	—	4	4	no	—
65	C₂×C₁₉₁	—	4	1	no	—
66	C₂×C₁₉₁	#532	1	1	yes	characteristic
67	C₂×C₁₉₁	C₂×#137	1	2	yes	—
68	C₁₉₁	#3185	1	1	yes	characteristic, complemented, Sylow
69	#3185	—	191	1	no	maximal, Sylow
70	#533	—	191	2	no	—
71	#498	—	191	1	no	—
72	#494	—	191	1	no	—
73	#487	—	191	2	no	—
74	#470	—	191	4	no	—
75	#441	—	191	4	no	—
76	C₄×M₁₆	D₁₉₁	1	1	yes	characteristic
77	C₂×Q₁₆	—	191	4	no	—
78	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
79	C₄⋊Q₈	—	191	2	no	—
80	C₄⋊Q₈	—	191	2	no	—
81	#125	—	382	2	no	—
82	C₂²⋊Q₈	—	382	2	no	—
83	C₂²⋊Q₈	—	191	4	no	—
84	C₄×Q₈	—	382	2	no	—
85	C₄×Q₈	—	191	4	no	—
86	#117	—	382	2	no	—
87	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
88	#106	—	191	4	no	—
89	#103	—	191	8	no	—
90	#97	C₂×D₁₉₁	1	2	yes	—
91	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
92	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
93	C₂×Q₈	—	382	2	no	—
94	C₂×Q₈	—	191	4	no	—
95	C₂²×C₄	D₇₆₄	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
97	Q₁₆	—	382	8	no	—
98	M₁₆	—	2	4	no	—
99	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
100	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
101	C₄⋊C₄	—	382	4	no	—
102	C₄⋊C₄	—	382	4	no	—
103	C₄⋊C₄	—	382	2	no	—
104	C₄⋊C₄	—	382	2	no	—
105	C₄⋊C₄	—	382	2	no	—
106	C₄⋊C₄	—	191	4	no	—
107	C₂²⋊C₄	—	382	4	no	—
108	C₄²	—	382	4	no	—
109	C₄²	D₇₆₄	1	2	yes	—
110	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
111	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
112	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
113	Q₈	—	382	8	no	—
114	Q₈	—	764	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	382	4	no	—
118	C₂×C₄	—	382	4	no	—
119	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
120	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
121	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
122	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#15690287	1	4	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	764	4	no	—
131	C₄	—	764	4	no	—
132	C₄	#20259355	1	4	yes	—
133	C₄	C₂×#15690287	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#27133213	1	1	yes	characteristic
136	C₂	C₂×#20259355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133220

C₂

yes

maximal

#27133061

C₂

yes

maximal, characteristic, complemented

#27133043

C₂

yes

maximal, characteristic, complemented

#27133014

C₂

yes

maximal, complemented

#27132921

C₂

yes

maximal

#27132771

C₂

yes

maximal, complemented

C₄×M₁₆×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

C₂²

yes

—

#20259322

C₂²

yes

—

#20259322

C₂²

yes

—

#20259314

—

#20259301

—

#20259301

C₂²

yes

—

C₄×Dic₃₈₂

—

C₄×Dic₃₈₂

C₂²

yes

—

#20259281

—

#20259254

C₂²

yes

—

#20259247

C₂²

yes

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#97×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

C₂³

yes

—

Dic₇₆₄

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#137

yes

—

C₄×C₁₉₁

C₂×#41

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#532

yes

characteristic

C₂×C₁₉₁

C₂×#137

yes

—

C₁₉₁

#3185

yes

characteristic, complemented, Sylow

#3185

—

191

maximal, Sylow

#533

—

191

—

#498

—

191

—

#494

—

191

—

#487

—

191

—

#470

—

191

—

#441

—

191

—

C₄×M₁₆

D₁₉₁

yes

characteristic

C₂×Q₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

—

191

—

#125

—

382

—

C₂²⋊Q₈

—

382

—

C₂²⋊Q₈

—

191

—

C₄×Q₈

—

382

—

C₄×Q₈

—

191

—

#117

—

382

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#106

—

191

—

#103

—

191

—

#97

C₂×D₁₉₁

yes

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂²×C₄

D₇₆₄

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

Q₁₆

—

382

—

M₁₆

—

C₂×C₈

C₂²×D₁₉₁

yes

—

100

C₂×C₈

C₂²×D₁₉₁

yes

—

101

C₄⋊C₄

—

382

—

102

C₄⋊C₄

—

382

—

103

C₄⋊C₄

—

382

—

104

C₄⋊C₄

—

382

—

105

C₄⋊C₄

—

382

—

106

C₄⋊C₄

—

191

—

107

C₂²⋊C₄

—

382

—

108

C₄²

—

382

—

109

C₄²

D₇₆₄

yes

—

110

C₄²

C₂²×D₁₉₁

yes

characteristic

111

C₄²

C₂²×D₁₉₁

yes

characteristic

112

C₂³

C₂×D₇₆₄

yes

characteristic

113

Q₈

—

382

—

114

Q₈

—

764

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

382

—

118

C₂×C₄

—

382

—

119

C₂×C₄

C₂×D₇₆₄

yes

—

120

C₂×C₄

C₂×D₇₆₄

yes

characteristic

121

C₂×C₄

C₂×D₇₆₄

yes

—

122

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

123

C₈

—

124

C₈

#15690287

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₇₆₄

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

764

—

131

C₄

—

764

—

132

C₄

#20259355

yes

—

133

C₄

C₂×#15690287

yes

—

134

C₂

—

135

C₂

#27133213

yes

characteristic

136

C₂

C₂×#20259355

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₇₆₄
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	C₄×M₁₆×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	C₄×M₁₆	D₁₉₁
hypocenter	C₁₉₁	#3185
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₇₆₄

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

C₄×M₁₆×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

prime radical

C₂³×C₁₉₁

C₂×D₄

hypercenter

C₄×M₁₆

D₁₉₁

hypocenter

C₁₉₁

#3185

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #27133061	C₂	semidirect
3. #27133043	C₂	semidirect
4. #27133014	C₂	semidirect
6. #27132771	C₂	semidirect
68. C₁₉₁	#3185	semidirect

H ≅ G/N

type

2. #27133061

C₂

semidirect

3. #27133043

C₂

semidirect

4. #27133014

C₂

semidirect

6. #27132771

C₂

semidirect

68. C₁₉₁

#3185

semidirect

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

subgroup	index
#27133220	2
#27133061	2
#27133043	2
#27133014	2
#27132921	2
#27132771	2
C₄×M₁₆×C₁₉₁	2
#3185	191

subgroup

index

C₄×M₁₆×C₁₉₁

191

Subgroups — #37583265

Conjugacy classes of subgroups

Complements

Maximal subgroups