Group #38744254 — subgroups

Subgroups — #38744254

Order 25216.

254 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 95 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×#20263258	C₂	1	2	yes	maximal
2	C₂×#20263251	C₂	1	2	yes	maximal
3	#27160985	C₂	1	1	yes	maximal, characteristic
4	#27160949	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₉₇	—	2	2	no	—
8	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
10	#20263258	—	2	4	no	—
11	#20263251	—	2	4	no	—
12	#20263233	—	2	2	no	—
13	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₉₄	—	2	2	no	—
16	C₂×Dic₃₉₄	—	2	1	no	—
17	C₂×Dic₃₉₄	—	2	2	no	—
18	C₂×Dic₃₉₄	Q₈	1	2	yes	—
19	C₂²×Dic₁₉₇	—	2	1	no	—
20	C₂²×Dic₁₉₇	—	2	1	no	—
21	C₂²×Dic₁₉₇	—	2	2	no	—
22	C₄⋊Dic₁₉₇	—	2	2	no	—
23	C₄⋊Dic₁₉₇	—	2	2	no	—
24	C₄⋊Dic₁₉₇	—	2	2	no	—
25	C₄⋊Dic₁₉₇	—	4	4	no	—
26	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₉₇	—	2	2	no	—
29	C₂×C₈×C₁₉₇	—	2	2	no	—
30	C₂×C₈×C₁₉₇	D₄	1	4	yes	—
31	C₄²×C₁₉₇	—	2	2	no	—
32	Dic₃₉₄	—	2	4	no	—
33	Dic₃₉₄	—	4	2	no	—
34	C₂×Dic₁₉₇	—	2	2	no	—
35	C₂×Dic₁₉₇	—	4	1	no	—
36	C₂×Dic₁₉₇	—	4	1	no	—
37	C₂×Dic₁₉₇	—	4	2	no	—
38	C₂×Dic₁₉₇	—	4	1	no	—
39	C₂×Dic₁₉₇	—	4	4	no	—
40	C₂×Dic₁₉₇	—	4	2	no	—
41	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	—	2	2	no	—
43	C₂×C₄×C₁₉₇	—	2	2	no	—
44	C₂×C₄×C₁₉₇	SD₁₆	1	4	yes	—
45	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₉₇	#41	1	2	yes	—
48	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
49	C₈×C₁₉₇	—	2	4	no	—
50	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₉₇	#123	1	1	yes	characteristic
52	C₂²×C₁₉₇	#127	1	1	yes	characteristic
53	C₂²×C₁₉₇	C₂×SD₁₆	1	2	yes	—
54	C₂²×C₁₉₇	#137	1	2	yes	—
55	C₄×C₁₉₇	—	2	4	no	—
56	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₉₇	#123	1	2	yes	—
58	Dic₁₉₇	—	4	2	no	—
59	Dic₁₉₇	—	8	2	no	—
60	Dic₁₉₇	—	8	1	no	—
61	C₂×C₁₉₇	#399	1	1	yes	characteristic
62	C₂×C₁₉₇	#475	1	2	yes	—
63	C₂×C₁₉₇	#483	1	2	yes	—
64	C₂×C₁₉₇	#492	1	1	yes	characteristic
65	C₂×C₁₉₇	#497	1	1	yes	characteristic
66	C₁₉₇	#2096	1	1	yes	characteristic, complemented, Sylow
67	#2096	—	197	1	no	maximal, Sylow
68	C₂×#106	—	197	2	no	—
69	C₂×#103	—	197	2	no	—
70	#391	—	197	1	no	—
71	#384	—	197	1	no	—
72	#336	D₁₉₇	1	1	yes	characteristic
73	C₂²×Q₈	—	197	1	no	—
74	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
75	C₂×C₄⋊C₄	—	394	2	no	—
76	C₂×C₄⋊C₄	—	197	2	no	—
77	C₂×C₄⋊C₄	—	197	1	no	—
78	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
79	#106	—	394	4	no	—
80	#103	—	394	4	no	—
81	#95	—	394	2	no	—
82	C₂×Q₈	—	394	2	no	—
83	C₂×Q₈	—	394	2	no	—
84	C₂×Q₈	—	394	1	no	—
85	C₂×Q₈	—	197	2	no	—
86	C₂²×C₄	—	394	1	no	—
87	C₂²×C₄	—	394	2	no	—
88	C₂²×C₄	—	394	1	no	—
89	C₂²×C₄	D₇₈₈	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₇₈₈	1	4	yes	—
94	C₄⋊C₄	—	788	4	no	—
95	C₄⋊C₄	—	394	2	no	—
96	C₄⋊C₄	—	394	2	no	—
97	C₄⋊C₄	—	394	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
100	Q₈	—	788	2	no	—
101	Q₈	—	394	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	788	1	no	—
105	C₂×C₄	—	788	1	no	—
106	C₂×C₄	—	788	2	no	—
107	C₂×C₄	—	788	1	no	—
108	C₂×C₄	—	788	4	no	—
109	C₂×C₄	—	788	2	no	—
110	C₂×C₄	—	394	2	no	—
111	C₂×C₄	#15691347	1	4	yes	—
112	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
113	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
114	C₂×C₄	#15691357	1	2	yes	—
115	C₂×C₄	#15691357	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₂²	#20263306	1	1	yes	characteristic
118	C₂²	#20263310	1	1	yes	characteristic
119	C₂²	#20263323	1	1	yes	characteristic
120	C₂²	C₂×#15691347	1	2	yes	—
121	C₂²	#20263359	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	788	2	no	—
124	C₄	—	1576	2	no	—
125	C₄	—	1576	1	no	—
126	C₄	#20263306	1	2	yes	—
127	C₄	#20263310	1	2	yes	—
128	C₂	#27161016	1	1	yes	characteristic
129	C₂	#27161365	1	2	yes	—
130	C₂	#27161410	1	2	yes	—
131	C₂	#27161452	1	1	yes	characteristic
132	C₂	#27161473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263258

C₂

yes

maximal

C₂×#20263251

C₂

yes

maximal

#27160985

C₂

yes

maximal, characteristic

#27160949

C₂

yes

maximal, characteristic

#336×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₇

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

#20263258

—

#20263251

—

#20263233

—

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

Q₈

yes

—

C₂²×Dic₁₉₇

—

C₂²×Dic₁₉₇

—

C₂²×Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

—

Dic₃₉₄

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

SD₁₆

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

C₂²⋊Q₈

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

C₂×SD₁₆

yes

—

C₂²×C₁₉₇

#137

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

C₄×C₁₉₇

#123

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

#399

yes

characteristic

C₂×C₁₉₇

#475

yes

—

C₂×C₁₉₇

#483

yes

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#2096

—

197

maximal, Sylow

C₂×#106

—

197

—

C₂×#103

—

197

—

#391

—

197

—

#384

—

197

—

#336

D₁₉₇

yes

characteristic

C₂²×Q₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

—

C₂×C₄⋊C₄

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#106

—

394

—

#103

—

394

—

#95

—

394

—

C₂×Q₈

—

394

—

C₂×Q₈

—

394

—

C₂×Q₈

—

394

—

C₂×Q₈

—

197

—

C₂²×C₄

—

394

—

C₂²×C₄

—

394

—

C₂²×C₄

—

394

—

C₂²×C₄

D₇₈₈

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₇₈₈

yes

—

C₄⋊C₄

—

788

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄²

—

C₂³

C₂×D₇₈₈

yes

characteristic

100

Q₈

—

788

—

101

Q₈

—

394

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

788

—

105

C₂×C₄

—

788

—

106

C₂×C₄

—

788

—

107

C₂×C₄

—

788

—

108

C₂×C₄

—

788

—

109

C₂×C₄

—

788

—

110

C₂×C₄

—

394

—

111

C₂×C₄

#15691347

yes

—

112

C₂×C₄

C₂×D₇₈₈

yes

—

113

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

114

C₂×C₄

#15691357

yes

—

115

C₂×C₄

#15691357

yes

characteristic

116

C₈

—

117

C₂²

#20263306

yes

characteristic

118

C₂²

#20263310

yes

characteristic

119

C₂²

#20263323

yes

characteristic

120

C₂²

C₂×#15691347

yes

—

121

C₂²

#20263359

yes

—

122

C₄

—

123

C₄

—

788

—

124

C₄

—

1576

—

125

C₄

—

1576

—

126

C₄

#20263306

yes

—

127

C₄

#20263310

yes

—

128

C₂

#27161016

yes

characteristic

129

C₂

#27161365

yes

—

130

C₂

#27161410

yes

—

131

C₂

#27161452

yes

characteristic

132

C₂

#27161473

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₈₈
commutator, cosocle	C₂²×C₄×C₁₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#336×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂×D₄
hypercenter	#336	D₁₉₇
hypocenter	C₁₉₇	#2096
LCS	G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₇₈₈

commutator, cosocle

C₂²×C₄×C₁₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#336×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂×D₄

D₁₉₇

C₁₉₇

G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
66. C₁₉₇	#2096	semidirect

H ≅ G/N

type

66. C₁₉₇

#2096

semidirect

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

subgroup	index
C₂×#20263258	2
C₂×#20263251	2
#27160985	2
#27160949	2
#336×C₁₉₇	2
#2096	197

subgroup

index

#336×C₁₉₇

197

Subgroups — #38744254

Conjugacy classes of subgroups

Complements

Maximal subgroups