Group #38742926 — subgroups

Subgroups — #38742926

Order 25216.

272 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 89 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	#27161557	C₂	1	1	yes	maximal, characteristic
2	#27161119	C₂	1	2	yes	maximal
3	C₂×#20263245	C₂	1	1	yes	maximal, characteristic
4	#27160839	C₂	1	2	yes	maximal
5	#391×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	Q₈×Dic₁₉₇	—	2	2	no	—
8	Q₈×Dic₁₉₇	—	2	2	no	—
9	#20263285	—	2	2	no	—
10	#20263285	C₂²	1	2	yes	—
11	#20263259	—	2	2	no	—
12	#20263257	—	2	2	no	—
13	#20263245	C₄	1	4	yes	complemented
14	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
17	#95×C₁₉₇	—	2	2	no	—
18	#15691342	—	2	2	no	—
19	#15691342	C₂×C₄	1	4	yes	—
20	C₂×#13096802	—	2	2	no	—
21	C₄⋊Dic₁₉₇	—	2	2	no	—
22	C₄⋊Dic₁₉₇	—	2	2	no	—
23	C₄⋊Dic₁₉₇	D₄	1	2	yes	—
24	C₄⋊Dic₁₉₇	D₄	1	2	yes	—
25	C₂²⋊Dic₁₉₇	—	2	2	no	—
26	C₄×Dic₁₉₇	—	2	4	no	—
27	C₄×Dic₁₉₇	—	2	2	no	—
28	C₂×Q₈×C₁₉₇	—	2	2	no	—
29	C₂×Q₈×C₁₉₇	—	2	2	no	—
30	C₂×Q₈×C₁₉₇	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₁₉₇	Q₈	1	2	yes	—
32	C₂²×C₄×C₁₉₇	—	2	1	no	—
33	C₂²×C₄×C₁₉₇	—	2	1	no	—
34	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
35	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₉₇	—	2	2	no	—
37	C₄²×C₁₉₇	—	2	2	no	—
38	C₂×Dic₁₉₇	—	2	2	no	—
39	C₂×Dic₁₉₇	—	2	2	no	—
40	#13096802	—	2	4	no	—
41	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₁₉₇	—	2	4	no	—
43	Q₈×C₁₉₇	—	2	4	no	—
44	C₂×C₄×C₁₉₇	—	2	2	no	—
45	C₂×C₄×C₁₉₇	—	2	2	no	—
46	C₂×C₄×C₁₉₇	—	2	2	no	—
47	C₂×C₄×C₁₉₇	—	4	1	no	—
48	C₂×C₄×C₁₉₇	—	4	1	no	—
49	C₂×C₄×C₁₉₇	—	4	2	no	—
50	C₂×C₄×C₁₉₇	—	4	1	no	—
51	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
52	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₉₇	#41	1	2	yes	—
56	C₂²×C₁₉₇	—	2	1	no	—
57	C₂²×C₁₉₇	—	2	1	no	—
58	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₉₇	#120	1	1	yes	characteristic
60	C₂²×C₁₉₇	#123	1	1	yes	characteristic
61	C₄×C₁₉₇	—	2	4	no	—
62	C₄×C₁₉₇	—	4	2	no	—
63	C₄×C₁₉₇	—	8	1	no	—
64	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
65	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
66	Dic₁₉₇	—	4	2	no	—
67	Dic₁₉₇	—	4	2	no	—
68	C₂×C₁₉₇	—	2	1	no	—
69	C₂×C₁₉₇	—	2	1	no	—
70	C₂×C₁₉₇	#385	1	1	yes	characteristic
71	C₂×C₁₉₇	#454	1	1	yes	characteristic
72	C₂×C₁₉₇	#456	1	1	yes	characteristic
73	C₁₉₇	#1857	1	1	yes	characteristic, complemented, Sylow
74	#1857	—	197	1	no	maximal, Sylow
75	#519	—	197	1	no	—
76	#428	—	197	2	no	—
77	C₂×#101	—	197	1	no	—
78	#391	D₁₉₇	1	1	yes	characteristic
79	#339	—	197	2	no	—
80	C₂²×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
81	C₂×M₁₆	—	197	2	no	—
82	C₄×Q₈	—	394	2	no	—
83	C₄×Q₈	—	394	2	no	—
84	#117	—	394	2	no	—
85	#117	—	197	2	no	—
86	C₂×C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
88	#107	—	394	2	no	—
89	#106	—	394	2	no	—
90	#101	—	197	4	no	—
91	#95	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	C₄×D₁₉₇	1	2	yes	—
95	C₂×Q₈	Dic₃₉₄	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
100	M₁₆	—	394	2	no	—
101	M₁₆	—	197	4	no	—
102	C₂×C₈	—	394	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	394	2	no	—
105	C₄⋊C₄	—	394	2	no	—
106	C₄⋊C₄	—	197	2	no	—
107	C₄⋊C₄	—	197	2	no	—
108	C₂²⋊C₄	—	394	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	394	4	no	—
111	C₄²	—	394	2	no	—
112	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	394	2	no	—
123	C₂×C₄	—	394	2	no	—
124	C₂×C₄	#15691337	1	4	yes	—
125	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
127	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
128	C₂×C₄	#15691357	1	2	yes	—
129	C₈	—	394	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×#15691337	1	1	yes	characteristic
133	C₂²	#20263294	1	1	yes	characteristic
134	C₂²	#20263307	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	788	2	no	—
139	C₄	—	788	2	no	—
140	C₄	#20263286	1	2	yes	—
141	C₄	#20263303	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#27160951	1	1	yes	characteristic
145	C₂	#27161251	1	1	yes	characteristic
146	C₂	#27161261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161557

C₂

yes

maximal, characteristic

#27161119

C₂

yes

maximal

C₂×#20263245

C₂

yes

maximal, characteristic

#27160839

C₂

yes

maximal

#391×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

Q₈×Dic₁₉₇

—

Q₈×Dic₁₉₇

—

#20263285

—

#20263285

C₂²

yes

—

#20263259

—

#20263257

—

#20263245

C₄

yes

complemented

C₂²×Q₈×C₁₉₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

characteristic

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

#95×C₁₉₇

—

#15691342

—

#15691342

C₂×C₄

yes

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

D₄

yes

—

C₄⋊Dic₁₉₇

D₄

yes

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

C₂×C₄

yes

—

C₂×Q₈×C₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄×D₄

yes

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1857

—

197

maximal, Sylow

#519

—

197

—

#428

—

197

—

C₂×#101

—

197

—

#391

D₁₉₇

yes

characteristic

#339

—

197

—

C₂²×Q₈

C₂×D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₄×Q₈

—

394

—

C₄×Q₈

—

394

—

#117

—

394

—

#117

—

197

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

characteristic

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#107

—

394

—

#106

—

394

—

#101

—

197

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₁₉₇

yes

—

C₂×Q₈

Dic₃₉₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

100

M₁₆

—

394

—

101

M₁₆

—

197

—

102

C₂×C₈

—

394

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

394

—

105

C₄⋊C₄

—

394

—

106

C₄⋊C₄

—

197

—

107

C₄⋊C₄

—

197

—

108

C₂²⋊C₄

—

394

—

109

C₄²

—

110

C₄²

—

394

—

111

C₄²

—

394

—

112

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

394

—

123

C₂×C₄

—

394

—

124

C₂×C₄

#15691337

yes

—

125

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

126

C₂×C₄

D₄×D₁₉₇

yes

—

127

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

128

C₂×C₄

#15691357

yes

—

129

C₈

—

394

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#15691337

yes

characteristic

133

C₂²

#20263294

yes

characteristic

134

C₂²

#20263307

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

788

—

139

C₄

—

788

—

140

C₄

#20263286

yes

—

141

C₄

#20263303

yes

—

142

C₂

—

143

C₂

—

144

C₂

#27160951

yes

characteristic

145

C₂

#27161251

yes

characteristic

146

C₂

#27161261

yes

characteristic

	subgroup	G/H
center	C₂²	#20263294
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#391×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#120
prime radical	C₂³×C₁₉₇	C₂×D₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#391	D₁₉₇
hypocenter	C₁₉₇	#1857
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#20263294

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#391×C₁₉₇

C₂

socle

C₂²×C₁₉₇

#120

prime radical

C₂³×C₁₉₇

C₂×D₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

N	H ≅ G/N	type
13. #20263245	C₄	semidirect
73. C₁₉₇	#1857	semidirect

H ≅ G/N

type

13. #20263245

C₄

semidirect

73. C₁₉₇

#1857

semidirect

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

subgroup	index
#27161557	2
#27161119	2
C₂×#20263245	2
#27160839	2
#391×C₁₉₇	2
#1857	197

subgroup

index

#391×C₁₉₇

197

Subgroups — #38742926

Conjugacy classes of subgroups

Complements

Maximal subgroups