Group #38754093 — subgroups

Subgroups — #38754093

Order 25216.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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	#27161629	C₂	1	2	yes	maximal, complemented
2	#27161370	C₂	1	4	yes	maximal, complemented
3	#27161332	C₂	1	4	yes	maximal, complemented
4	#27161275	C₂	1	4	yes	maximal, complemented
5	#423×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691358	—	2	2	no	—
7	C₂×D₁₅₇₆	C₂²	1	4	yes	—
8	#20263326	C₂²	1	2	yes	—
9	#20263323	C₂²	1	2	yes	—
10	#20263314	—	2	2	no	—
11	#20263300	—	2	2	no	—
12	#20263300	C₂²	1	4	yes	—
13	C₄×D₇₈₈	—	2	2	no	—
14	C₄×D₇₈₈	C₂²	1	4	yes	—
15	#20263258	C₂²	1	4	yes	—
16	#20263248	C₂²	1	8	yes	—
17	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₉₇	C₂²	1	4	yes	—
20	#15691358	—	4	8	no	—
21	C₂×Dic₃₉₄	—	2	2	no	—
22	C₂×D₇₈₈	—	2	2	no	—
23	C₂×D₇₈₈	—	2	2	no	—
24	C₂×D₇₈₈	C₂³	1	4	yes	—
25	C₂×C₂²⋊D₁₉₇	—	2	4	no	—
26	C₂×C₄×D₁₉₇	—	2	4	no	—
27	D₁₅₇₆	—	4	4	no	—
28	C₄⋊Dic₁₉₇	—	2	2	no	—
29	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
30	#15691335	—	2	4	no	—
31	#15691335	—	2	4	no	—
32	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
33	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₉₇	—	4	2	no	—
35	C₂×C₈×C₁₉₇	C₂³	1	4	yes	—
36	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₉₇	C₂³	1	2	yes	—
39	C₂²×D₁₉₇	—	2	4	no	—
40	Dic₃₉₄	—	4	4	no	—
41	D₇₈₈	—	2	4	no	—
42	D₇₈₈	—	2	4	no	—
43	D₇₈₈	—	4	4	no	—
44	D₇₈₈	—	4	4	no	—
45	C₂²⋊D₁₉₇	—	4	8	no	—
46	C₂×Dic₁₉₇	—	2	4	no	—
47	C₄×D₁₉₇	—	4	8	no	—
48	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₇	—	2	4	no	—
50	C₂×C₄×C₁₉₇	—	2	1	no	—
51	C₂×C₄×C₁₉₇	—	2	1	no	—
52	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
56	C₈×C₁₉₇	—	2	4	no	—
57	C₂²×C₁₉₇	—	2	1	no	—
58	C₂²×C₁₉₇	—	2	2	no	—
59	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
60	C₂×D₁₉₇	—	4	4	no	—
61	C₂×D₁₉₇	—	4	4	no	—
62	C₂×D₁₉₇	—	4	4	no	—
63	C₄×C₁₉₇	—	2	1	no	—
64	C₄×C₁₉₇	—	2	2	no	—
65	C₄×C₁₉₇	AGL(1,ℤ/8)	1	4	yes	—
66	C₄×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
67	C₄×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₁₉₇	—	4	4	no	—
69	C₂×C₁₉₇	—	4	1	no	—
70	C₂×C₁₉₇	#537	1	1	yes	characteristic
71	C₂×C₁₉₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
72	D₁₉₇	—	8	4	no	—
73	C₁₉₇	#3253	1	1	yes	characteristic, complemented, Sylow
74	#3253	—	197	1	no	maximal, Sylow
75	#532	—	197	2	no	—
76	#476	—	197	4	no	—
77	#469	—	197	4	no	—
78	#459	—	197	4	no	—
79	#423	D₁₉₇	1	1	yes	characteristic, complemented
80	C₂×#41	—	394	2	no	—
81	C₂×D₈	—	197	4	no	—
82	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
83	C₄⋊Q₈	—	197	2	no	—
84	#127	—	197	2	no	—
85	#124	—	394	2	no	—
86	#121	—	394	2	no	—
87	#121	—	197	4	no	—
88	C₄×D₄	—	394	2	no	—
89	C₄×D₄	—	197	4	no	—
90	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
91	#106	—	197	4	no	—
92	C₄⋊C₈	C₂×D₁₉₇	1	4	yes	—
93	#102	—	197	8	no	—
94	#41	—	788	8	no	—
95	C₂×Q₈	—	394	2	no	—
96	C₂×D₄	—	394	4	no	—
97	C₂×D₄	—	394	2	no	—
98	C₂×D₄	—	394	2	no	—
99	C₂×D₄	—	197	4	no	—
100	C₂²×C₄	—	394	4	no	—
101	C₂²×C₄	D₇₈₈	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
103	D₈	—	788	4	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	C₂²×D₁₉₇	1	4	yes	—
106	C₄⋊C₄	—	394	2	no	—
107	C₄⋊C₄	—	197	4	no	—
108	C₂²⋊C₄	—	394	4	no	—
109	C₂²⋊C₄	—	394	4	no	—
110	C₄²	D₇₈₈	1	1	yes	characteristic
111	C₄²	D₇₈₈	1	1	yes	characteristic
112	C₄²	C₂²×D₁₉₇	1	2	yes	—
113	C₂³	—	394	4	no	—
114	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
115	Q₈	—	788	4	no	—
116	D₄	—	788	4	no	—
117	D₄	—	788	8	no	—
118	D₄	—	788	4	no	—
119	D₄	—	394	4	no	—
120	D₄	—	394	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	788	8	no	—
125	C₂×C₄	—	394	4	no	—
126	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
127	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
128	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
129	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	788	4	no	—
134	C₂²	—	788	4	no	—
135	C₂²	—	788	4	no	—
136	C₂²	C₂²×D₇₈₈	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	788	4	no	—
140	C₄	#20263354	1	4	yes	—
141	C₄	#20263371	1	1	yes	characteristic
142	C₄	#20263374	1	1	yes	characteristic
143	C₂	—	4	1	no	—
144	C₂	—	1576	4	no	—
145	C₂	#27161660	1	1	yes	characteristic
146	C₂	C₂×#20263354	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161629

C₂

yes

maximal, complemented

#27161370

C₂

yes

maximal, complemented

#27161332

C₂

yes

maximal, complemented

#27161275

C₂

yes

maximal, complemented

#423×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#15691358

—

C₂×D₁₅₇₆

C₂²

yes

—

#20263326

C₂²

yes

—

#20263323

C₂²

yes

—

#20263314

—

#20263300

—

#20263300

C₂²

yes

—

C₄×D₇₈₈

—

C₄×D₇₈₈

C₂²

yes

—

#20263258

C₂²

yes

—

#20263248

C₂²

yes

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

C₂²

yes

—

#15691358

—

C₂×Dic₃₉₄

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

C₂×D₇₈₈

C₂³

yes

—

C₂×C₂²⋊D₁₉₇

—

C₂×C₄×D₁₉₇

—

D₁₅₇₆

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂³

yes

—

#15691335

—

#15691335

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂³

yes

—

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

C₂³

yes

—

C₂²×D₁₉₇

—

Dic₃₉₄

—

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

C₂²⋊D₁₉₇

—

C₂×Dic₁₉₇

—

C₄×D₁₉₇

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₉₇

2₋¹⁺⁴

yes

characteristic

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#537

yes

characteristic

C₂×C₁₉₇

C₂×AGL(1,ℤ/8)

yes

—

D₁₉₇

—

C₁₉₇

#3253

yes

characteristic, complemented, Sylow

#3253

—

197

maximal, Sylow

#532

—

197

—

#476

—

197

—

#469

—

197

—

#459

—

197

—

#423

D₁₉₇

yes

characteristic, complemented

C₂×#41

—

394

—

C₂×D₈

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₄⋊Q₈

—

197

—

#127

—

197

—

#124

—

394

—

#121

—

394

—

#121

—

197

—

C₄×D₄

—

394

—

C₄×D₄

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#106

—

197

—

C₄⋊C₈

C₂×D₁₉₇

yes

—

#102

—

197

—

#41

—

788

—

C₂×Q₈

—

394

—

C₂×D₄

—

394

—

C₂×D₄

—

394

—

C₂×D₄

—

394

—

C₂×D₄

—

197

—

100

C₂²×C₄

—

394

—

101

C₂²×C₄

D₇₈₈

yes

—

102

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

103

D₈

—

788

—

104

M₁₆

—

105

C₂×C₈

C₂²×D₁₉₇

yes

—

106

C₄⋊C₄

—

394

—

107

C₄⋊C₄

—

197

—

108

C₂²⋊C₄

—

394

—

109

C₂²⋊C₄

—

394

—

110

C₄²

D₇₈₈

yes

characteristic

111

C₄²

D₇₈₈

yes

characteristic

112

C₄²

C₂²×D₁₉₇

yes

—

113

C₂³

—

394

—

114

C₂³

C₂×D₇₈₈

yes

characteristic

115

Q₈

—

788

—

116

D₄

—

788

—

117

D₄

—

788

—

118

D₄

—

788

—

119

D₄

—

394

—

120

D₄

—

394

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

788

—

125

C₂×C₄

—

394

—

126

C₂×C₄

C₂×D₇₈₈

yes

characteristic

127

C₂×C₄

C₂×D₇₈₈

yes

—

128

C₂×C₄

C₂×D₇₈₈

yes

—

129

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

788

—

134

C₂²

—

788

—

135

C₂²

—

788

—

136

C₂²

C₂²×D₇₈₈

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

788

—

140

C₄

#20263354

yes

—

141

C₄

#20263371

yes

characteristic

142

C₄

#20263374

yes

characteristic

143

C₂

—

144

C₂

—

1576

—

145

C₂

#27161660

yes

characteristic

146

C₂

C₂×#20263354

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₇₈₈
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#423×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
hypercenter	#423	D₁₉₇
hypocenter	C₁₉₇	#3253
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₇₈₈

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting

#423×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂²×D₄

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #27161629	C₂	semidirect
2. #27161370	C₂	semidirect
3. #27161332	C₂	semidirect
4. #27161275	C₂	semidirect
5. #423×C₁₉₇	C₂	semidirect
73. C₁₉₇	#3253	semidirect
79. #423	D₁₉₇	semidirect

H ≅ G/N

type

1. #27161629

C₂

semidirect

2. #27161370

C₂

semidirect

3. #27161332

C₂

semidirect

4. #27161275

C₂

semidirect

5. #423×C₁₉₇

C₂

semidirect

73. C₁₉₇

#3253

semidirect

79. #423

D₁₉₇

semidirect

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

subgroup	index
#27161629	2
#27161370	2
#27161332	2
#27161275	2
#423×C₁₉₇	2
#3253	197

subgroup

index

#423×C₁₉₇

197

Subgroups — #38754093

Conjugacy classes of subgroups

Complements

Maximal subgroups