Group #38751646 — subgroups

Subgroups — #38751646

Order 25216.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#27161636	C₂	1	1	yes	maximal, characteristic
2	#27161209	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₃₉₄	C₂	1	4	yes	maximal, complemented
4	#27161105	C₂	1	2	yes	maximal
5	#27161103	C₂	1	1	yes	maximal, characteristic
6	#27161034	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#15691342	C₂²	1	2	yes	—
9	#20263326	C₄	1	2	yes	—
10	#20263318	C₄	1	2	yes	—
11	#20263305	C₄	1	4	yes	—
12	C₄×Dic₃₉₄	C₂²	1	4	yes	—
13	#20263285	C₂²	1	2	yes	—
14	C₄⋊#13096802	C₂²	1	4	yes	—
15	#20263255	C₂²	1	8	yes	—
16	#20263236	C₂²	1	4	yes	—
17	C₈×Dic₁₉₇	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₉₇	C₂²	1	2	yes	—
22	C₄×C₈×C₁₉₇	C₂²	1	2	yes	—
23	C₂×Dic₃₉₄	C₂×C₄	1	4	yes	—
24	#15691342	—	2	4	no	—
25	C₂×#13096802	C₂³	1	4	yes	—
26	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
28	#15691337	C₂×C₄	1	8	yes	—
29	C₂²⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
30	C₄×Dic₁₉₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₉₇	—	2	2	no	—
34	C₂×C₈×C₁₉₇	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
36	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
38	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₉₇	C₂³	1	2	yes	—
41	Dic₃₉₄	—	2	8	no	—
42	C₂×Dic₁₉₇	C₂²×C₄	1	8	yes	—
43	#13096802	—	2	4	no	—
44	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₇	—	2	4	no	—
46	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₉₇	#41	1	2	yes	—
51	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
52	C₈×C₁₉₇	—	2	2	no	—
53	C₈×C₁₉₇	C₂×Q₈	1	2	yes	—
54	C₈×C₁₉₇	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₉₇	—	2	2	no	—
56	C₂²×C₁₉₇	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₉₇	—	2	2	no	—
59	C₄×C₁₉₇	C₄×Q₈	1	2	yes	—
60	C₄×C₁₉₇	#131	1	4	yes	—
61	C₄×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
63	Dic₁₉₇	—	2	8	no	—
64	C₂×C₁₉₇	—	2	2	no	—
65	C₂×C₁₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₉₇	C₂×#131	1	2	yes	—
67	C₁₉₇	#2960	1	1	yes	characteristic, complemented, Sylow
68	#2960	—	197	1	no	maximal, Sylow
69	#533	—	197	1	no	—
70	#446	—	197	4	no	—
71	C₈×Q₈	—	197	4	no	—
72	#424	—	197	2	no	—
73	#423	—	197	1	no	—
74	#405	—	197	2	no	—
75	C₂×C₄×C₈	D₁₉₇	1	1	yes	characteristic
76	C₂×M₁₆	—	197	2	no	—
77	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
79	C₄⋊Q₈	—	197	2	no	—
80	#125	—	197	2	no	—
81	C₂²⋊Q₈	—	197	4	no	—
82	C₄×Q₈	—	197	4	no	—
83	#117	—	197	2	no	—
84	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
85	C₄⋊C₈	—	197	8	no	—
86	C₄⋊C₈	—	197	4	no	—
87	#97	—	197	4	no	—
88	C₄×C₈	—	197	4	no	—
89	C₄×C₈	C₂×D₁₉₇	1	2	yes	—
90	C₄×C₈	C₂×D₁₉₇	1	2	yes	—
91	C₂×Q₈	—	197	4	no	—
92	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
94	M₁₆	—	394	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	197	4	no	—
97	C₂×C₈	Dic₃₉₄	1	4	yes	complemented
98	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
99	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
101	C₄⋊C₄	—	197	8	no	—
102	C₄⋊C₄	—	197	4	no	—
103	C₄⋊C₄	—	197	4	no	—
104	C₂²⋊C₄	—	197	4	no	—
105	C₄²	—	197	4	no	—
106	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
107	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₉₇	1	2	yes	—
109	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
110	Q₈	—	394	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	197	8	no	—
113	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
116	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
117	C₂×C₄	#15691358	1	2	yes	—
118	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	394	4	no	—
121	C₈	C₂×Dic₃₉₄	1	2	yes	—
122	C₈	C₂×Dic₃₉₄	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₃₉₄	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	394	8	no	—
128	C₄	C₄×Dic₃₉₄	1	2	yes	—
129	C₄	#20263332	1	4	yes	—
130	C₄	C₂²×Dic₃₉₄	1	1	yes	characteristic
131	C₄	C₂×#15691358	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₃₉₄	1	1	yes	characteristic
134	C₂	C₂×#20263332	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161636

C₂

yes

maximal, characteristic

#27161209

C₂

yes

maximal, complemented

C₈×Dic₃₉₄

C₂

yes

maximal, complemented

#27161105

C₂

yes

maximal

#27161103

C₂

yes

maximal, characteristic

#27161034

C₂

yes

maximal

C₂×C₄×C₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

#20263326

C₄

yes

—

#20263318

C₄

yes

—

#20263305

C₄

yes

—

C₄×Dic₃₉₄

C₂²

yes

—

#20263285

C₂²

yes

—

C₄⋊#13096802

C₂²

yes

—

#20263255

C₂²

yes

—

#20263236

C₂²

yes

—

C₈×Dic₁₉₇

C₂²

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

characteristic

C₂²×C₈×C₁₉₇

C₂²

yes

characteristic

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄×C₈×C₁₉₇

C₂²

yes

—

C₄×C₈×C₁₉₇

C₂²

yes

—

C₂×Dic₃₉₄

C₂×C₄

yes

—

#15691342

—

C₂×#13096802

C₂³

yes

—

C₄⋊Dic₁₉₇

C₂×C₄

yes

—

C₄⋊Dic₁₉₇

C₂×C₄

yes

—

#15691337

C₂×C₄

yes

—

C₂²⋊Dic₁₉₇

C₂×C₄

yes

—

C₄×Dic₁₉₇

C₂³

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

Q₈

yes

complemented

C₂×C₈×C₁₉₇

C₂³

yes

—

C₂×C₈×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

C₂³

yes

characteristic

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

C₂³

yes

—

Dic₃₉₄

—

C₂×Dic₁₉₇

C₂²×C₄

yes

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₈×C₁₉₇

—

C₈×C₁₉₇

C₂×Q₈

yes

—

C₈×C₁₉₇

C₂×Q₈

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₄×Q₈

yes

—

C₂²×C₁₉₇

C₂³×C₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄×Q₈

yes

—

C₄×C₁₉₇

#131

yes

—

C₄×C₁₉₇

C₂²×Q₈

yes

characteristic

C₄×C₁₉₇

C₂×#41

yes

characteristic

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₉₇

C₂×#131

yes

—

C₁₉₇

#2960

yes

characteristic, complemented, Sylow

#2960

—

197

maximal, Sylow

#533

—

197

—

#446

—

197

—

C₈×Q₈

—

197

—

#424

—

197

—

#423

—

197

—

#405

—

197

—

C₂×C₄×C₈

D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

characteristic

C₂²×C₈

C₂×D₁₉₇

yes

characteristic

C₄⋊Q₈

—

197

—

#125

—

197

—

C₂²⋊Q₈

—

197

—

C₄×Q₈

—

197

—

#117

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

197

—

C₄⋊C₈

—

197

—

#97

—

197

—

C₄×C₈

—

197

—

C₄×C₈

C₂×D₁₉₇

yes

—

C₄×C₈

C₂×D₁₉₇

yes

—

C₂×Q₈

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

394

—

C₂×C₈

—

C₂×C₈

—

197

—

C₂×C₈

Dic₃₉₄

yes

complemented

C₂×C₈

C₂²×D₁₉₇

yes

—

C₂×C₈

C₂²×D₁₉₇

yes

characteristic

100

C₂×C₈

C₂²×D₁₉₇

yes

characteristic

101

C₄⋊C₄

—

197

—

102

C₄⋊C₄

—

197

—

103

C₄⋊C₄

—

197

—

104

C₂²⋊C₄

—

197

—

105

C₄²

—

197

—

106

C₄²

C₄×D₁₉₇

yes

characteristic

107

C₄²

C₄×D₁₉₇

yes

characteristic

108

C₄²

C₂²×D₁₉₇

yes

—

109

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

110

Q₈

—

394

—

111

C₂×C₄

—

112

C₂×C₄

—

197

—

113

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

114

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

115

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

116

C₂×C₄

C₂×Dic₃₉₄

yes

—

117

C₂×C₄

#15691358

yes

—

118

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

119

C₈

—

120

C₈

—

394

—

121

C₈

C₂×Dic₃₉₄

yes

—

122

C₈

C₂×Dic₃₉₄

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₃₉₄

yes

—

125

C₂²

C₂²×C₄×D₁₉₇

yes

characteristic

126

C₄

—

127

C₄

—

394

—

128

C₄

C₄×Dic₃₉₄

yes

—

129

C₄

#20263332

yes

—

130

C₄

C₂²×Dic₃₉₄

yes

characteristic

131

C₄

C₂×#15691358

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₃₉₄

yes

characteristic

134

C₂

C₂×#20263332

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂×C₄×C₈×C₁₉₇	C₂
prime radical	C₂³×C₁₉₇	C₂²×C₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₉₇
hypocenter	C₁₉₇	#2960
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₁₉₇

commutator, socle

C₂²×C₁₉₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting

C₂×C₄×C₈×C₁₉₇

C₂

prime radical

C₂³×C₁₉₇

C₂²×C₄

cosocle

C₂×C₄×C₁₉₇

C₂⁴

hypercenter

C₂×C₄×C₈

D₁₉₇

C₁₉₇

G ▷ C₂²×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #27161209	C₂	semidirect
3. C₈×Dic₃₉₄	C₂	semidirect
34. C₂×C₈×C₁₉₇	Q₈	semidirect
67. C₁₉₇	#2960	semidirect
97. C₂×C₈	Dic₃₉₄	semidirect

H ≅ G/N

type

2. #27161209

C₂

semidirect

3. C₈×Dic₃₉₄

C₂

semidirect

34. C₂×C₈×C₁₉₇

Q₈

semidirect

67. C₁₉₇

#2960

semidirect

97. C₂×C₈

Dic₃₉₄

semidirect

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

subgroup	index
#27161636	2
#27161209	2
C₈×Dic₃₉₄	2
#27161105	2
#27161103	2
#27161034	2
C₂×C₄×C₈×C₁₉₇	2
#2960	197

subgroup

index

C₈×Dic₃₉₄

C₂×C₄×C₈×C₁₉₇

197

Subgroups — #38751646

Conjugacy classes of subgroups

Complements

Maximal subgroups