Group #38744274 — subgroups

Subgroups — #38744274

Order 25216.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#27160977	C₂	1	1	yes	maximal, characteristic
2	#27160841	C₂	1	2	yes	maximal
3	#27160841	C₂	1	2	yes	maximal
4	#27160834	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13096802	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
9	C₂×#15691337	—	2	1	no	—
10	#20263233	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₇	—	2	1	no	—
16	C₂²×Dic₁₉₇	—	2	1	no	—
17	C₂×#13096802	—	2	2	no	—
18	C₂×#13096802	—	2	2	no	—
19	C₂×#13096802	—	2	2	no	—
20	C₄⋊Dic₁₉₇	—	2	1	no	—
21	C₄⋊Dic₁₉₇	—	2	2	no	—
22	C₄⋊Dic₁₉₇	Q₈	1	2	yes	—
23	#15691337	—	4	2	no	—
24	C₂²×C₄×C₁₉₇	—	2	2	no	—
25	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₉₇	—	2	4	no	—
29	C₄⋊C₄×C₁₉₇	—	4	2	no	—
30	C₄⋊C₄×C₁₉₇	—	4	2	no	—
31	C₄²×C₁₉₇	—	2	2	no	—
32	C₂×Dic₁₉₇	—	2	2	no	—
33	C₂×Dic₁₉₇	—	4	1	no	—
34	C₂×Dic₁₉₇	—	4	1	no	—
35	C₂×Dic₁₉₇	—	4	1	no	—
36	C₂×Dic₁₉₇	—	4	1	no	—
37	C₂×Dic₁₉₇	—	4	1	no	—
38	#13096802	—	4	2	no	—
39	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₇	—	2	1	no	—
41	C₂×C₄×C₁₉₇	—	2	1	no	—
42	C₂×C₄×C₁₉₇	—	2	1	no	—
43	C₂×C₄×C₁₉₇	—	2	1	no	—
44	C₂×C₄×C₁₉₇	—	4	2	no	—
45	C₂×C₄×C₁₉₇	—	4	2	no	—
46	C₂×C₄×C₁₉₇	—	4	2	no	—
47	C₂×C₄×C₁₉₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₉₇	Q₁₆	1	2	yes	—
49	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₉₇	#41	1	2	yes	—
51	C₂×C₄×C₁₉₇	#41	1	2	yes	—
52	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
53	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₉₇	#123	1	1	yes	characteristic
55	C₂²×C₁₉₇	#124	1	1	yes	characteristic
56	C₂²×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₁₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₁₉₇	—	2	2	no	—
61	C₄×C₁₉₇	—	2	2	no	—
62	C₄×C₁₉₇	—	8	2	no	—
63	C₄×C₁₉₇	#125	1	2	yes	—
64	C₄×C₁₉₇	#126	1	2	yes	—
65	Dic₁₉₇	—	4	2	no	—
66	Dic₁₉₇	—	8	1	no	—
67	C₂×C₁₉₇	#400	1	1	yes	characteristic
68	C₂×C₁₉₇	#476	1	1	yes	characteristic
69	C₂×C₁₉₇	#477	1	1	yes	characteristic
70	C₂×C₁₉₇	#481	1	1	yes	characteristic
71	C₂×C₁₉₇	#484	1	1	yes	characteristic
72	C₂×C₁₉₇	#487	1	1	yes	characteristic
73	C₂×C₁₉₇	#490	1	1	yes	characteristic
74	C₁₉₇	#2100	1	1	yes	characteristic, complemented, Sylow
75	#2100	—	197	1	no	maximal, Sylow
76	#389	—	197	1	no	—
77	#384	D₁₉₇	1	1	yes	characteristic
78	#340	—	197	2	no	—
79	#340	—	197	2	no	—
80	#336	—	197	1	no	—
81	C₂²×C₈	—	197	2	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	394	1	no	—
85	C₂×C₄⋊C₄	—	197	1	no	—
86	C₂×C₄⋊C₄	—	197	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
88	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
89	#95	—	394	1	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	394	1	no	—
92	C₂²×C₄	—	394	1	no	—
93	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
96	C₂×C₈	—	394	2	no	—
97	C₂×C₈	—	394	2	no	—
98	C₂×C₈	—	394	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	788	2	no	—
103	C₄⋊C₄	—	394	2	no	—
104	C₄⋊C₄	—	394	1	no	—
105	C₄⋊C₄	—	197	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	788	1	no	—
116	C₂×C₄	—	788	1	no	—
117	C₂×C₄	—	788	1	no	—
118	C₂×C₄	—	788	1	no	—
119	C₂×C₄	—	788	1	no	—
120	C₂×C₄	—	394	2	no	—
121	C₂×C₄	#15691345	1	2	yes	—
122	C₂×C₄	#15691348	1	2	yes	—
123	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
124	C₂×C₄	#15691357	1	2	yes	—
125	C₂×C₄	#15691358	1	2	yes	—
126	C₂×C₄	#15691359	1	1	yes	characteristic
127	C₈	—	788	2	no	—
128	C₂²	#20263302	1	1	yes	characteristic
129	C₂²	#20263309	1	1	yes	characteristic
130	C₂²	#20263313	1	1	yes	characteristic
131	C₂²	C₂×#15691345	1	1	yes	characteristic
132	C₂²	C₂×#15691348	1	1	yes	characteristic
133	C₂²	#20263352	1	1	yes	characteristic
134	C₂²	#20263352	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	788	2	no	—
139	C₄	—	1576	1	no	—
140	C₄	#20263316	1	2	yes	—
141	C₄	#20263321	1	2	yes	—
142	C₂	#27161017	1	1	yes	characteristic
143	C₂	#27161366	1	1	yes	characteristic
144	C₂	#27161371	1	1	yes	characteristic
145	C₂	#27161397	1	1	yes	characteristic
146	C₂	#27161414	1	1	yes	characteristic
147	C₂	#27161429	1	1	yes	characteristic
148	C₂	#27161443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27160977

C₂

yes

maximal, characteristic

#27160841

C₂

yes

maximal

#27160841

C₂

yes

maximal

#27160834

C₂

yes

maximal, characteristic

#384×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₇

C₂²

yes

characteristic

C₂×#15691337

—

#20263233

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

—

C₂²×Dic₁₉₇

—

C₂×#13096802

—

C₂×#13096802

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

Q₈

yes

—

#15691337

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

SD₁₆

yes

—

C₂×C₄×C₁₉₇

Q₁₆

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

characteristic

C₂²×C₁₉₇

C₂²⋊Q₈

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₇

C₂×Q₁₆

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#125

yes

—

C₄×C₁₉₇

#126

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#2100

—

197

maximal, Sylow

#389

—

197

—

#384

D₁₉₇

yes

characteristic

#340

—

197

—

#340

—

197

—

#336

—

197

—

C₂²×C₈

—

197

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#95

—

394

—

C₂²×C₄

—

C₂²×C₄

—

394

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

788

—

103

C₄⋊C₄

—

394

—

104

C₄⋊C₄

—

394

—

105

C₄⋊C₄

—

197

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

788

—

116

C₂×C₄

—

788

—

117

C₂×C₄

—

788

—

118

C₂×C₄

—

788

—

119

C₂×C₄

—

788

—

120

C₂×C₄

—

394

—

121

C₂×C₄

#15691345

yes

—

122

C₂×C₄

#15691348

yes

—

123

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

124

C₂×C₄

#15691357

yes

—

125

C₂×C₄

#15691358

yes

—

126

C₂×C₄

#15691359

yes

characteristic

127

C₈

—

788

—

128

C₂²

#20263302

yes

characteristic

129

C₂²

#20263309

yes

characteristic

130

C₂²

#20263313

yes

characteristic

131

C₂²

C₂×#15691345

yes

characteristic

132

C₂²

C₂×#15691348

yes

characteristic

133

C₂²

#20263352

yes

characteristic

134

C₂²

#20263352

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

788

—

139

C₄

—

1576

—

140

C₄

#20263316

yes

—

141

C₄

#20263321

yes

—

142

C₂

#27161017

yes

characteristic

143

C₂

#27161366

yes

characteristic

144

C₂

#27161371

yes

characteristic

145

C₂

#27161397

yes

characteristic

146

C₂

#27161414

yes

characteristic

147

C₂

#27161429

yes

characteristic

148

C₂

#27161443

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂²×C₄×C₁₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#384×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂×D₄
hypercenter	#384	D₁₉₇
hypocenter	C₁₉₇	#2100
LCS	G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₇

commutator, cosocle

C₂²×C₄×C₁₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#384×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂×D₄

D₁₉₇

C₁₉₇

G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
74. C₁₉₇	#2100	semidirect

H ≅ G/N

type

74. C₁₉₇

#2100

semidirect

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

subgroup	index
#27160977	2
#27160841	2
#27160834	2
#384×C₁₉₇	2
#2100	197

subgroup

index

#384×C₁₉₇

197

Subgroups — #38744274

Conjugacy classes of subgroups

Complements

Maximal subgroups