Group #38753235 — subgroups

Subgroups — #38753235

Order 25216.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#27161632	C₂	1	2	yes	maximal
2	#27161487	C₂	1	4	yes	maximal, complemented
3	#27161486	C₂	1	4	yes	maximal, complemented
4	#27161117	C₂	1	1	yes	maximal, characteristic
5	#27161115	C₂	1	2	yes	maximal
6	C₂×#20263236	C₂	1	1	yes	maximal, characteristic
7	#427×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13096802	C₂²	1	1	yes	characteristic
9	#20263325	C₂²	1	4	yes	—
10	#20263317	C₂²	1	4	yes	—
11	#20263303	—	2	4	no	—
12	C₄×Dic₃₉₄	—	2	4	no	—
13	#20263285	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
15	#20263260	C₂²	1	2	yes	—
16	#20263259	C₂²	1	4	yes	—
17	#20263258	C₂²	1	2	yes	—
18	#20263257	C₂²	1	4	yes	—
19	#20263236	C₂²	1	2	yes	—
20	#20263236	C₂²	1	2	yes	—
21	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
22	#117×C₁₉₇	C₂²	1	2	yes	—
23	#107×C₁₉₇	C₂²	1	2	yes	—
24	#106×C₁₉₇	C₂²	1	2	yes	—
25	C₂×Dic₃₉₄	—	2	4	no	—
26	C₂²×Dic₁₉₇	D₄	1	2	yes	—
27	C₂×#13096802	Q₈	1	4	yes	complemented
28	C₂×#13096802	C₂³	1	2	yes	—
29	C₄⋊Dic₁₉₇	—	2	4	no	—
30	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
31	#15691337	—	2	4	no	—
32	#15691337	—	2	4	no	—
33	C₂²⋊Dic₁₉₇	—	2	2	no	—
34	C₄×Dic₁₉₇	—	2	2	no	—
35	C₄×Dic₁₉₇	D₄	1	2	yes	—
36	C₄×Dic₁₉₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₉₇	Q₈	1	4	yes	complemented
39	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
43	C₄²×C₁₉₇	—	2	2	no	—
44	Dic₃₉₄	—	4	8	no	—
45	C₂×Dic₁₉₇	—	2	4	no	—
46	C₂×Dic₁₉₇	—	4	2	no	—
47	C₂×Dic₁₉₇	C₂×D₄	1	4	yes	—
48	#13096802	C₂×Q₈	1	4	yes	—
49	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₇	—	2	4	no	—
51	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
55	C₈×C₁₉₇	C₂×Q₈	1	2	yes	—
56	C₈×C₁₉₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₁₉₇	—	2	2	no	—
58	C₂²×C₁₉₇	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₁₉₇	—	4	4	no	—
61	C₄×C₁₉₇	C₄⋊Q₈	1	2	yes	—
62	C₄×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₉₇	C₂²×Q₈	1	1	yes	characteristic
64	Dic₁₉₇	—	2	4	no	—
65	Dic₁₉₇	—	4	4	no	—
66	C₂×C₁₉₇	—	2	2	no	—
67	C₂×C₁₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₁₉₇	#575	1	2	yes	—
69	C₁₉₇	#3162	1	1	yes	characteristic, complemented, Sylow
70	#3162	—	197	1	no	maximal, Sylow
71	#533	—	197	2	no	—
72	#501	—	197	4	no	—
73	#501	—	197	4	no	—
74	#427	—	197	1	no	—
75	#427	—	197	2	no	—
76	#427	D₁₉₇	1	1	yes	characteristic
77	C₂×#97	—	197	1	no	—
78	C₂²×C₈	—	197	1	no	—
79	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
80	C₄⋊Q₈	—	197	4	no	—
81	#125	—	197	4	no	—
82	C₂²⋊Q₈	—	394	4	no	—
83	C₄×Q₈	—	394	4	no	—
84	#117	—	197	2	no	—
85	#117	C₂×D₁₉₇	1	2	yes	—
86	C₂×C₄²	—	197	1	no	—
87	#107	—	197	2	no	—
88	#107	—	197	4	no	—
89	#107	C₂×D₁₉₇	1	2	yes	—
90	#106	—	197	2	no	—
91	#106	—	197	4	no	—
92	#106	C₂×D₁₉₇	1	2	yes	—
93	#97	—	197	2	no	—
94	#97	—	197	2	no	—
95	C₂×Q₈	—	394	4	no	—
96	C₂²×C₄	—	197	2	no	—
97	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
98	C₂×C₈	—	197	4	no	—
99	C₂×C₈	—	197	2	no	—
100	C₂×C₈	Dic₃₉₄	1	4	yes	complemented
101	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
103	C₄⋊C₄	—	394	4	no	—
104	C₄⋊C₄	—	394	4	no	—
105	C₄⋊C₄	—	394	4	no	—
106	C₄⋊C₄	—	197	4	no	—
107	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	394	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	394	2	no	—
112	C₄²	—	197	2	no	—
113	C₄²	—	197	2	no	—
114	C₂³	D₄×D₁₉₇	1	1	yes	characteristic
115	Q₈	—	788	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	788	2	no	—
118	C₂×C₄	—	394	4	no	—
119	C₂×C₄	—	197	4	no	—
120	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₁₉₇	1	2	yes	—
122	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
123	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
124	C₈	—	197	4	no	—
125	C₈	C₂×Dic₃₉₄	1	2	yes	—
126	C₈	C₂×Dic₃₉₄	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#20263325	1	2	yes	—
129	C₂²	C₂×D₄×D₁₉₇	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	788	4	no	—
132	C₄	—	394	4	no	—
133	C₄	#20263325	1	2	yes	—
134	C₄	C₂×Q₈×D₁₉₇	1	1	yes	characteristic
135	C₄	C₂²×Dic₃₉₄	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#20263325	1	1	yes	characteristic
138	C₂	#27161876	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161632

C₂

yes

maximal

#27161487

C₂

yes

maximal, complemented

#27161486

C₂

yes

maximal, complemented

#27161117

C₂

yes

maximal, characteristic

#27161115

C₂

yes

maximal

C₂×#20263236

C₂

yes

maximal, characteristic

#427×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

characteristic

#20263325

C₂²

yes

—

#20263317

C₂²

yes

—

#20263303

—

C₄×Dic₃₉₄

—

#20263285

C₂²

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

#20263260

C₂²

yes

—

#20263259

C₂²

yes

—

#20263258

C₂²

yes

—

#20263257

C₂²

yes

—

#20263236

C₂²

yes

—

#20263236

C₂²

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

characteristic

#117×C₁₉₇

C₂²

yes

—

#107×C₁₉₇

C₂²

yes

—

#106×C₁₉₇

C₂²

yes

—

C₂×Dic₃₉₄

—

C₂²×Dic₁₉₇

D₄

yes

—

C₂×#13096802

Q₈

yes

complemented

C₂×#13096802

C₂³

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂³

yes

—

#15691337

—

#15691337

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

D₄

yes

—

C₄×Dic₁₉₇

C₂³

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

Q₈

yes

complemented

C₂×C₈×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

C₂³

yes

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₂×D₄

yes

—

#13096802

C₂×Q₈

yes

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₈×C₁₉₇

C₂×Q₈

yes

—

C₈×C₁₉₇

C₂×Q₈

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₄⋊Q₈

yes

—

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄⋊Q₈

yes

—

C₄×C₁₉₇

C₂²×Q₈

yes

characteristic

C₄×C₁₉₇

C₂²×Q₈

yes

characteristic

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₉₇

#575

yes

—

C₁₉₇

#3162

yes

characteristic, complemented, Sylow

#3162

—

197

maximal, Sylow

#533

—

197

—

#501

—

197

—

#501

—

197

—

#427

—

197

—

#427

—

197

—

#427

D₁₉₇

yes

characteristic

C₂×#97

—

197

—

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

characteristic

C₄⋊Q₈

—

197

—

#125

—

197

—

C₂²⋊Q₈

—

394

—

C₄×Q₈

—

394

—

#117

—

197

—

#117

C₂×D₁₉₇

yes

—

C₂×C₄²

—

197

—

#107

—

197

—

#107

—

197

—

#107

C₂×D₁₉₇

yes

—

#106

—

197

—

#106

—

197

—

#106

C₂×D₁₉₇

yes

—

#97

—

197

—

#97

—

197

—

C₂×Q₈

—

394

—

C₂²×C₄

—

197

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

197

—

C₂×C₈

—

197

—

100

C₂×C₈

Dic₃₉₄

yes

complemented

101

C₂×C₈

C₂²×D₁₉₇

yes

characteristic

102

C₂×C₈

C₂²×D₁₉₇

yes

characteristic

103

C₄⋊C₄

—

394

—

104

C₄⋊C₄

—

394

—

105

C₄⋊C₄

—

394

—

106

C₄⋊C₄

—

197

—

107

C₄⋊C₄

C₂²×D₁₉₇

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

394

—

110

C₄²

—

111

C₄²

—

394

—

112

C₄²

—

197

—

113

C₄²

—

197

—

114

C₂³

D₄×D₁₉₇

yes

characteristic

115

Q₈

—

788

—

116

C₂×C₄

—

117

C₂×C₄

—

788

—

118

C₂×C₄

—

394

—

119

C₂×C₄

—

197

—

120

C₂×C₄

D₄×D₁₉₇

yes

characteristic

121

C₂×C₄

Q₈×D₁₉₇

yes

—

122

C₂×C₄

C₂×Dic₃₉₄

yes

—

123

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

124

C₈

—

197

—

125

C₈

C₂×Dic₃₉₄

yes

—

126

C₈

C₂×Dic₃₉₄

yes

—

127

C₂²

—

128

C₂²

#20263325

yes

—

129

C₂²

C₂×D₄×D₁₉₇

yes

characteristic

130

C₄

—

131

C₄

—

788

—

132

C₄

—

394

—

133

C₄

#20263325

yes

—

134

C₄

C₂×Q₈×D₁₉₇

yes

characteristic

135

C₄

C₂²×Dic₃₉₄

yes

characteristic

136

C₂

—

137

C₂

C₂×#20263325

yes

characteristic

138

C₂

#27161876

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	#427×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
prime radical	C₂³×C₁₉₇	C₂×D₄
hypercenter	#427	D₁₉₇
hypocenter	C₁₉₇	#3162
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₉₇

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting

#427×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂²×D₄

prime radical

C₂³×C₁₉₇

C₂×D₄

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #27161487	C₂	semidirect
3. #27161486	C₂	semidirect
27. C₂×#13096802	Q₈	semidirect
38. C₂×C₈×C₁₉₇	Q₈	semidirect
69. C₁₉₇	#3162	semidirect
100. C₂×C₈	Dic₃₉₄	semidirect

H ≅ G/N

type

2. #27161487

C₂

semidirect

3. #27161486

C₂

semidirect

27. C₂×#13096802

Q₈

semidirect

38. C₂×C₈×C₁₉₇

Q₈

semidirect

69. C₁₉₇

#3162

semidirect

100. C₂×C₈

Dic₃₉₄

semidirect

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

subgroup	index
#27161632	2
#27161487	2
#27161486	2
#27161117	2
#27161115	2
C₂×#20263236	2
#427×C₁₉₇	2
#3162	197

subgroup

index

#427×C₁₉₇

197

Subgroups — #38753235

Conjugacy classes of subgroups

Complements

Maximal subgroups