Group #38751642 — subgroups

Subgroups — #38751642

Order 25216.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#27161206	C₂	1	4	yes	maximal, complemented
3	#27161203	C₂	1	4	yes	maximal, complemented
4	#27161107	C₂	1	2	yes	maximal
5	#27161037	C₂	1	2	yes	maximal
6	C₂×C₄×#13096802	C₂	1	1	yes	maximal, characteristic
7	#423×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13096802	C₂²	1	1	yes	characteristic
9	C₂²×#13096802	C₂²	1	1	yes	characteristic
10	#20263326	C₄	1	2	yes	—
11	#20263318	C₄	1	2	yes	—
12	#20263305	C₄	1	4	yes	—
13	C₄×Dic₃₉₄	C₂²	1	4	yes	—
14	#20263285	C₂²	1	2	yes	—
15	#20263255	C₂²	1	8	yes	—
16	#20263236	C₂²	1	4	yes	—
17	C₄×#13096802	C₂²	1	2	yes	—
18	C₄×#13096802	C₂²	1	2	yes	—
19	C₈×Dic₁₉₇	C₂²	1	4	yes	—
20	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₉₇	C₂²	1	4	yes	—
23	C₂×Dic₃₉₄	C₂×C₄	1	4	yes	—
24	C₂×#13096802	—	2	2	no	—
25	C₂×#13096802	Q₈	1	4	yes	complemented
26	C₂×#13096802	C₂³	1	2	yes	—
27	C₂×#13096802	C₂³	1	1	yes	characteristic
28	C₂×#13096802	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
31	#15691337	C₂×C₄	1	8	yes	—
32	C₂²⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₉₇	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
36	M₁₆×C₁₉₇	—	2	4	no	—
37	C₂×C₈×C₁₉₇	C₂³	1	4	yes	—
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	2	no	—
44	#13096802	C₂×Q₈	1	2	yes	—
45	#13096802	C₂×Q₈	1	2	yes	—
46	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₇	—	2	4	no	—
48	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₁₉₇	#41	1	2	yes	—
53	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₉₇	—	2	4	no	—
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	D₁₉₇	1	1	yes	characteristic
74	#405	—	197	2	no	—
75	C₂×C₄×C₈	—	197	1	no	—
76	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
77	C₂²×C₈	—	197	1	no	—
78	C₂²×C₈	—	197	1	no	—
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	C₂×D₁₉₇	1	4	yes	—
87	#97	—	197	4	no	—
88	C₄×C₈	—	197	4	no	—
89	C₄×C₈	—	197	2	no	—
90	C₄×C₈	—	197	2	no	—
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	—	2	4	no	—
95	C₂×C₈	—	394	2	no	—
96	C₂×C₈	—	197	2	no	—
97	C₂×C₈	—	197	4	no	—
98	C₂×C₈	—	197	1	no	—
99	C₂×C₈	—	197	1	no	—
100	C₂×C₈	C₂²×D₁₉₇	1	4	yes	—
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	Q₈×D₁₉₇	1	2	yes	—
117	C₂×C₄	#15691357	1	2	yes	—
118	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	394	2	no	—
121	C₈	—	197	2	no	—
122	C₈	—	197	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#20263291	1	2	yes	—
125	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	394	8	no	—
128	C₄	#20263291	1	2	yes	—
129	C₄	#20263334	1	4	yes	—
130	C₄	C₂×Q₈×D₁₉₇	1	1	yes	characteristic
131	C₄	C₂×#15691357	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#20263291	1	1	yes	characteristic
134	C₂	C₂×#20263334	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161636

C₂

yes

maximal, characteristic

#27161206

C₂

yes

maximal, complemented

#27161203

C₂

yes

maximal, complemented

#27161107

C₂

yes

maximal

#27161037

C₂

yes

maximal

C₂×C₄×#13096802

C₂

yes

maximal, characteristic

#423×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

characteristic

C₂²×#13096802

C₂²

yes

characteristic

#20263326

C₄

yes

—

#20263318

C₄

yes

—

#20263305

C₄

yes

—

C₄×Dic₃₉₄

C₂²

yes

—

#20263285

C₂²

yes

—

#20263255

C₂²

yes

—

#20263236

C₂²

yes

—

C₄×#13096802

C₂²

yes

—

C₄×#13096802

C₂²

yes

—

C₈×Dic₁₉₇

C₂²

yes

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

C₂²

yes

—

C₂×Dic₃₉₄

C₂×C₄

yes

—

C₂×#13096802

—

C₂×#13096802

Q₈

yes

complemented

C₂×#13096802

C₂³

yes

—

C₂×#13096802

C₂³

yes

characteristic

C₂×#13096802

C₂³

yes

characteristic

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

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂³

yes

—

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

C₂³

yes

—

Dic₃₉₄

—

C₂×Dic₁₉₇

C₂²×C₄

yes

—

#13096802

—

#13096802

C₂×Q₈

yes

—

#13096802

C₂×Q₈

yes

—

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

—

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

D₁₉₇

yes

characteristic

#405

—

197

—

C₂×C₄×C₈

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₂²×C₈

—

197

—

C₂²×C₈

—

197

—

C₄⋊Q₈

—

197

—

#125

—

197

—

C₂²⋊Q₈

—

197

—

C₄×Q₈

—

197

—

#117

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

197

—

C₄⋊C₈

C₂×D₁₉₇

yes

—

#97

—

197

—

C₄×C₈

—

197

—

C₄×C₈

—

197

—

C₄×C₈

—

197

—

C₂×Q₈

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

100

C₂×C₈

C₂²×D₁₉₇

yes

—

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

Q₈×D₁₉₇

yes

—

117

C₂×C₄

#15691357

yes

—

118

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

119

C₈

—

120

C₈

—

394

—

121

C₈

—

197

—

122

C₈

—

197

—

123

C₂²

—

124

C₂²

#20263291

yes

—

125

C₂²

C₂²×C₄×D₁₉₇

yes

characteristic

126

C₄

—

127

C₄

—

394

—

128

C₄

#20263291

yes

—

129

C₄

#20263334

yes

—

130

C₄

C₂×Q₈×D₁₉₇

yes

characteristic

131

C₄

C₂×#15691357

yes

characteristic

132

C₂

—

133

C₂

C₂×#20263291

yes

characteristic

134

C₂

C₂×#20263334

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Fitting	#423×C₁₉₇	C₂
prime radical	C₂³×C₁₉₇	C₂²×C₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#423	D₁₉₇
hypocenter	C₁₉₇	#2960
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉₇

commutator, socle

C₂²×C₁₉₇

C₂³×C₄

Fitting

#423×C₁₉₇

C₂

prime radical

C₂³×C₁₉₇

C₂²×C₄

cosocle

C₂×C₄×C₁₉₇

C₂⁴

D₁₉₇

C₁₉₇

G ▷ C₂²×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #27161206	C₂	semidirect
3. #27161203	C₂	semidirect
25. C₂×#13096802	Q₈	semidirect
67. C₁₉₇	#2960	semidirect

H ≅ G/N

type

2. #27161206

C₂

semidirect

3. #27161203

C₂

semidirect

25. C₂×#13096802

Q₈

semidirect

67. C₁₉₇

#2960

semidirect

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

subgroup	index
#27161636	2
#27161206	2
#27161203	2
#27161107	2
#27161037	2
C₂×C₄×#13096802	2
#423×C₁₉₇	2
#2960	197

subgroup

index

#423×C₁₉₇

197

Subgroups — #38751642

Conjugacy classes of subgroups

Complements

Maximal subgroups