Group #38742761 — subgroups

Subgroups — #38742761

Order 25216.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#20263334	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263261	C₂	1	2	yes	maximal, complemented
3	#27161061	C₂	1	2	yes	maximal
4	#27160837	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691358	C₂²	1	1	yes	characteristic
7	#20263334	—	2	4	no	—
8	C₂×#15691342	C₂²	1	2	yes	—
9	C₂×#15691341	C₄	1	2	yes	—
10	C₂²×#13096802	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₉₇	C₄	1	2	yes	—
12	#20263261	—	2	4	no	—
13	#20263244	—	2	2	no	—
14	#20263242	—	2	2	no	—
15	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
17	#15691358	—	2	2	no	—
18	#15691358	—	2	2	no	—
19	C₂×Dic₃₉₄	D₄	1	1	yes	characteristic
20	C₂×D₇₈₈	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₉₇	Q₈	1	2	yes	—
22	C₂×C₄×D₁₉₇	C₂×C₄	1	2	yes	—
23	#15691342	—	2	2	no	—
24	#15691342	—	2	2	no	—
25	#15691341	—	2	4	no	—
26	C₂×#13096802	—	2	2	no	—
27	C₂×#13096802	C₂×C₄	1	1	yes	characteristic
28	C₂×#13096802	C₂×C₄	1	1	yes	characteristic
29	C₂×#13096802	D₄	1	4	yes	—
30	C₈×D₁₉₇	—	2	4	no	—
31	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₉₇	—	2	2	no	—
33	M₁₆×C₁₉₇	—	2	2	no	—
34	M₁₆×C₁₉₇	—	2	2	no	—
35	C₂×C₈×C₁₉₇	—	2	2	no	—
36	C₂×C₈×C₁₉₇	C₂×C₄	1	2	yes	—
37	C₂²×D₁₉₇	C₄⋊C₄	1	2	yes	—
38	Dic₃₉₄	—	2	2	no	—
39	D₇₈₈	—	2	2	no	—
40	C₂²⋊D₁₉₇	—	2	4	no	—
41	C₂×Dic₁₉₇	C₄⋊C₄	1	2	yes	—
42	C₄×D₁₉₇	—	2	2	no	—
43	C₄×D₁₉₇	—	2	2	no	—
44	#13096802	—	4	2	no	—
45	#13096802	C₂²⋊C₄	1	2	yes	—
46	#13096802	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₇	#41	1	2	yes	—
52	C₈×C₁₉₇	—	2	2	no	—
53	C₈×C₁₉₇	—	4	2	no	—
54	C₂²×C₁₉₇	—	4	1	no	—
55	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₁₉₇	—	2	2	no	—
58	C₂×D₁₉₇	—	4	2	no	—
59	C₄×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₉₇	#117	1	1	yes	characteristic
61	C₄×C₁₉₇	#121	1	2	yes	—
62	Dic₁₉₇	—	2	2	no	—
63	C₂×C₁₉₇	—	2	1	no	—
64	C₂×C₁₉₇	—	2	2	no	—
65	C₂×C₁₉₇	#380	1	1	yes	characteristic
66	D₁₉₇	—	4	2	no	—
67	C₁₉₇	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	197	1	no	maximal, Sylow
69	C₂×#131	—	197	1	no	—
70	C₂×#108	—	197	2	no	—
71	#413	—	197	2	no	—
72	#338	—	197	1	no	—
73	#338	D₁₉₇	1	1	yes	characteristic, complemented
74	C₂×#41	—	197	1	no	—
75	#131	—	394	4	no	—
76	C₂×M₁₆	—	197	2	no	—
77	C₂×M₁₆	—	197	2	no	—
78	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
79	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
80	C₂²×C₈	—	197	2	no	—
81	C₂²×C₈	—	197	1	no	—
82	#108	—	394	4	no	—
83	#101	—	394	2	no	—
84	#100	—	394	2	no	—
85	#41	—	394	2	no	—
86	#41	—	394	2	no	—
87	C₂×Q₈	—	197	1	no	—
88	C₂×D₄	—	197	2	no	—
89	C₂×D₄	—	197	1	no	—
90	C₂²×C₄	—	197	2	no	—
91	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	394	4	no	—
96	M₁₆	—	394	2	no	—
97	M₁₆	—	394	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	394	4	no	—
100	C₂×C₈	—	394	2	no	—
101	C₂×C₈	—	197	1	no	—
102	C₂×C₈	—	197	1	no	—
103	C₂×C₈	—	197	4	no	—
104	C₂×C₈	C₄×D₁₉₇	1	2	yes	—
105	C₂³	—	197	2	no	—
106	C₂³	Q₈×D₁₉₇	1	1	yes	characteristic
107	Q₈	—	394	2	no	—
108	D₄	—	394	4	no	—
109	D₄	—	394	2	no	—
110	C₂×C₄	—	394	2	no	—
111	C₂×C₄	—	394	2	no	—
112	C₂×C₄	—	197	2	no	—
113	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
115	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
116	C₂×C₄	#15691358	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	788	2	no	—
120	C₈	—	197	2	no	—
121	C₈	—	197	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	788	2	no	—
124	C₂²	—	394	2	no	—
125	C₂²	C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
126	C₂²	#20263304	1	2	yes	—
127	C₄	—	394	2	no	—
128	C₄	C₂²⋊C₄×D₁₉₇	1	1	yes	characteristic
129	C₄	#20263283	1	1	yes	characteristic
130	C₄	#20263299	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	788	2	no	—
134	C₂	#27160938	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263334

C₂

yes

maximal, characteristic

C₂×#20263261

C₂

yes

maximal, complemented

#27161061

C₂

yes

maximal

#27160837

C₂

yes

maximal, characteristic, complemented

#338×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#15691358

C₂²

yes

characteristic

#20263334

—

C₂×#15691342

C₂²

yes

—

C₂×#15691341

C₄

yes

—

C₂²×#13096802

C₂²

yes

characteristic

C₂×C₈×D₁₉₇

C₄

yes

—

#20263261

—

#20263244

—

#20263242

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×M₁₆×C₁₉₇

C₂²

yes

characteristic

#15691358

—

#15691358

—

C₂×Dic₃₉₄

D₄

yes

characteristic

C₂×D₇₈₈

D₄

yes

characteristic

C₂×C₂²⋊D₁₉₇

Q₈

yes

—

C₂×C₄×D₁₉₇

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#13096802

C₂×C₄

yes

characteristic

C₂×#13096802

D₄

yes

—

C₈×D₁₉₇

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂×C₄

yes

—

C₂²×D₁₉₇

C₄⋊C₄

yes

—

Dic₃₉₄

—

D₇₈₈

—

C₂²⋊D₁₉₇

—

C₂×Dic₁₉₇

C₄⋊C₄

yes

—

C₄×D₁₉₇

—

C₄×D₁₉₇

—

#13096802

—

#13096802

C₂²⋊C₄

yes

—

#13096802

C₂²⋊C₄

yes

—

C₂³×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₂²⋊Q₈

yes

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

C₂×C₂²⋊C₄

yes

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#380

yes

characteristic

D₁₉₇

—

C₁₉₇

#1813

yes

characteristic, complemented, Sylow

#1813

—

197

maximal, Sylow

C₂×#131

—

197

—

C₂×#108

—

197

—

#413

—

197

—

#338

—

197

—

#338

D₁₉₇

yes

characteristic, complemented

C₂×#41

—

197

—

#131

—

394

—

C₂×M₁₆

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₂×M₁₆

C₂×D₁₉₇

yes

characteristic

C₂²×C₈

—

197

—

C₂²×C₈

—

197

—

#108

—

394

—

#101

—

394

—

#100

—

394

—

#41

—

394

—

#41

—

394

—

C₂×Q₈

—

197

—

C₂×D₄

—

197

—

C₂×D₄

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

394

—

M₁₆

—

394

—

M₁₆

—

394

—

C₂×C₈

—

C₂×C₈

—

394

—

100

C₂×C₈

—

394

—

101

C₂×C₈

—

197

—

102

C₂×C₈

—

197

—

103

C₂×C₈

—

197

—

104

C₂×C₈

C₄×D₁₉₇

yes

—

105

C₂³

—

197

—

106

C₂³

Q₈×D₁₉₇

yes

characteristic

107

Q₈

—

394

—

108

D₄

—

394

—

109

D₄

—

394

—

110

C₂×C₄

—

394

—

111

C₂×C₄

—

394

—

112

C₂×C₄

—

197

—

113

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

114

C₂×C₄

D₄×D₁₉₇

yes

—

115

C₂×C₄

D₄×D₁₉₇

yes

characteristic

116

C₂×C₄

#15691358

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

788

—

120

C₈

—

197

—

121

C₈

—

197

—

122

C₂²

—

123

C₂²

—

788

—

124

C₂²

—

394

—

125

C₂²

C₄⋊C₄×D₁₉₇

yes

characteristic

126

C₂²

#20263304

yes

—

127

C₄

—

394

—

128

C₄

C₂²⋊C₄×D₁₉₇

yes

characteristic

129

C₄

#20263283

yes

characteristic

130

C₄

#20263299

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

788

—

134

C₂

#27160938

yes

characteristic

	subgroup	G/H
center	C₄	#20263283
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#338×C₁₉₇	C₂
socle	C₂×C₁₉₇	#380
prime radical	C₂×#15691358	C₂²
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#338	D₁₉₇
hypocenter	C₁₉₇	#1813
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#20263283

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#338×C₁₉₇

C₂

C₂×C₁₉₇

C₂²

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#20263261	C₂	semidirect
4. #27160837	C₂	semidirect
5. #338×C₁₉₇	C₂	semidirect
67. C₁₉₇	#1813	semidirect
73. #338	D₁₉₇	semidirect

H ≅ G/N

type

2. C₂×#20263261

C₂

semidirect

4. #27160837

C₂

semidirect

5. #338×C₁₉₇

C₂

semidirect

67. C₁₉₇

#1813

semidirect

73. #338

D₁₉₇

semidirect

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

subgroup	index
C₂×#20263334	2
C₂×#20263261	2
#27161061	2
#27160837	2
#338×C₁₉₇	2
#1813	197

subgroup

index

#338×C₁₉₇

197

Subgroups — #38742761

Conjugacy classes of subgroups

Complements

Maximal subgroups