Group #38753034 — subgroups

Subgroups — #38753034

Order 25216.

400 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 141 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, complemented
2	#27161433	C₂	1	4	yes	maximal, complemented
3	#27161429	C₂	1	2	yes	maximal, complemented
4	#27161425	C₂	1	2	yes	maximal, complemented
5	#27161075	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×#13096802	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13096802	C₂²	1	2	yes	—
9	#20263326	C₂²	1	2	yes	—
10	#20263318	C₂²	1	2	yes	—
11	#20263305	—	2	2	no	—
12	C₄×Dic₃₉₄	—	2	2	no	—
13	#20263285	C₂²	1	2	yes	—
14	#20263250	C₂²	1	8	yes	—
15	#20263247	C₂²	1	8	yes	—
16	C₄×#13096802	C₂²	1	2	yes	—
17	C₄×#13096802	C₂²	1	2	yes	—
18	C₂×#41×C₁₉₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
20	#127×C₁₉₇	C₂²	1	1	yes	characteristic
21	#124×C₁₉₇	C₂²	1	2	yes	—
22	#121×C₁₉₇	—	2	2	no	—
23	C₄×D₄×C₁₉₇	—	2	2	no	—
24	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
25	C₂×Dic₃₉₄	—	2	2	no	—
26	C₂×#13096802	—	2	4	no	—
27	C₂×#13096802	C₂³	1	4	yes	—
28	C₄⋊Dic₁₉₇	—	2	2	no	—
29	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
30	#15691337	—	2	4	no	—
31	C₂²⋊Dic₁₉₇	—	2	2	no	—
32	C₄×Dic₁₉₇	—	2	2	no	—
33	#41×C₁₉₇	—	2	8	no	—
34	C₂×Q₈×C₁₉₇	C₂³	1	2	yes	—
35	C₂×D₄×C₁₉₇	—	2	2	no	—
36	C₂×D₄×C₁₉₇	—	2	2	no	—
37	C₂×D₄×C₁₉₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₉₇	—	2	2	no	—
39	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
40	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₉₇	—	2	2	no	—
42	C₂²⋊C₄×C₁₉₇	—	2	4	no	—
43	C₄²×C₁₉₇	D₄	1	2	yes	—
44	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
45	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
46	Dic₃₉₄	—	4	4	no	—
47	C₂×Dic₁₉₇	—	2	4	no	—
48	#13096802	—	2	4	no	—
49	C₂³×C₁₉₇	—	2	2	no	—
50	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₉₇	—	2	4	no	—
52	D₄×C₁₉₇	—	2	4	no	—
53	D₄×C₁₉₇	—	4	4	no	—
54	D₄×C₁₉₇	—	4	4	no	—
55	C₂×C₄×C₁₉₇	—	2	4	no	—
56	C₂×C₄×C₁₉₇	—	2	2	no	—
57	C₂×C₄×C₁₉₇	—	4	4	no	—
58	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉₇	#41	1	4	yes	—
62	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
63	C₂²×C₁₉₇	—	2	2	no	—
64	C₂²×C₁₉₇	—	4	2	no	—
65	C₂²×C₁₉₇	—	4	4	no	—
66	C₂²×C₁₉₇	#124	1	2	yes	—
67	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₉₇	—	2	2	no	—
69	C₄×C₁₉₇	—	4	2	no	—
70	C₄×C₁₉₇	#124	1	2	yes	—
71	C₄×C₁₉₇	#135	1	4	yes	—
72	C₄×C₁₉₇	C₂×#41	1	2	yes	—
73	Dic₁₉₇	—	4	4	no	—
74	C₂×C₁₉₇	—	2	2	no	—
75	C₂×C₁₉₇	—	8	2	no	—
76	C₂×C₁₉₇	C₂×#124	1	1	yes	characteristic
77	C₂×C₁₉₇	C₂×#135	1	2	yes	—
78	C₁₉₇	#3131	1	1	yes	characteristic, complemented, Sylow
79	#3131	—	197	1	no	maximal, Sylow
80	#533	—	197	1	no	—
81	#532	D₁₉₇	1	1	yes	characteristic
82	#488	—	197	4	no	—
83	#487	—	197	2	no	—
84	#486	—	197	2	no	—
85	#416	—	197	4	no	—
86	C₂×C₄×C₈	—	197	1	no	—
87	C₂×#41	C₂×D₁₉₇	1	2	yes	—
88	C₂²×C₈	—	197	2	no	—
89	C₄⋊Q₈	—	197	2	no	—
90	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
91	#127	C₂×D₁₉₇	1	1	yes	characteristic
92	#125	—	197	2	no	—
93	#124	C₂×D₁₉₇	1	2	yes	—
94	C₂²⋊Q₈	—	394	2	no	—
95	#121	—	2	2	no	—
96	C₄×Q₈	—	394	2	no	—
97	C₄×D₄	—	2	2	no	—
98	#117	—	197	2	no	—
99	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
100	#103	—	197	8	no	—
101	#102	—	197	8	no	—
102	C₄×C₈	—	197	2	no	—
103	C₄×C₈	—	197	2	no	—
104	#41	—	2	8	no	—
105	C₂×Q₈	—	394	2	no	—
106	C₂×Q₈	C₂²×D₁₉₇	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²×D₁₉₇	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
113	C₂×C₈	—	394	4	no	—
114	C₂×C₈	—	197	4	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	394	4	no	—
117	C₄⋊C₄	—	394	2	no	—
118	C₄⋊C₄	—	197	4	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	394	2	no	—
121	C₄²	—	394	2	no	—
122	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
123	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
124	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
127	Q₈	—	2	4	no	—
128	Q₈	—	788	4	no	—
129	D₄	—	2	4	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	394	4	no	—
136	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
138	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
139	C₂×C₄	#15691357	1	4	yes	—
140	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
141	C₈	—	394	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	4	no	—
144	C₂²	—	4	2	no	—
145	C₂²	#20263313	1	2	yes	—
146	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	788	4	no	—
150	C₄	#20263313	1	2	yes	—
151	C₄	#20263346	1	4	yes	—
152	C₄	C₂×#15691357	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	8	2	no	—
155	C₂	C₂×#20263313	1	1	yes	characteristic
156	C₂	C₂×#20263346	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161636

C₂

yes

maximal, characteristic, complemented

#27161433

C₂

yes

maximal, complemented

#27161429

C₂

yes

maximal, complemented

#27161425

C₂

yes

maximal, complemented

#27161075

C₂

yes

maximal, complemented

C₂×C₄×#13096802

C₂

yes

maximal, characteristic, complemented

#532×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

#20263326

C₂²

yes

—

#20263318

C₂²

yes

—

#20263305

—

C₄×Dic₃₉₄

—

#20263285

C₂²

yes

—

#20263250

C₂²

yes

—

#20263247

C₂²

yes

—

C₄×#13096802

C₂²

yes

—

C₄×#13096802

C₂²

yes

—

C₂×#41×C₁₉₇

C₂²

yes

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

#127×C₁₉₇

C₂²

yes

characteristic

#124×C₁₉₇

C₂²

yes

—

#121×C₁₉₇

—

C₄×D₄×C₁₉₇

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×#13096802

—

C₂×#13096802

C₂³

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂³

yes

—

#15691337

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

#41×C₁₉₇

—

C₂×Q₈×C₁₉₇

C₂³

yes

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

C₂³

yes

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

C₂³

yes

characteristic

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

Q₈×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#124

yes

—

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#124

yes

—

C₄×C₁₉₇

#135

yes

—

C₄×C₁₉₇

C₂×#41

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

C₂×#124

yes

characteristic

C₂×C₁₉₇

C₂×#135

yes

—

C₁₉₇

#3131

yes

characteristic, complemented, Sylow

#3131

—

197

maximal, Sylow

#533

—

197

—

#532

D₁₉₇

yes

characteristic

#488

—

197

—

#487

—

197

—

#486

—

197

—

#416

—

197

—

C₂×C₄×C₈

—

197

—

C₂×#41

C₂×D₁₉₇

yes

—

C₂²×C₈

—

197

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

#127

C₂×D₁₉₇

yes

characteristic

#125

—

197

—

#124

C₂×D₁₉₇

yes

—

C₂²⋊Q₈

—

394

—

#121

—

C₄×Q₈

—

394

—

C₄×D₄

—

#117

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

100

#103

—

197

—

101

#102

—

197

—

102

C₄×C₈

—

197

—

103

C₄×C₈

—

197

—

104

#41

—

105

C₂×Q₈

—

394

—

106

C₂×Q₈

C₂²×D₁₉₇

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₁₉₇

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

112

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

113

C₂×C₈

—

394

—

114

C₂×C₈

—

197

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

394

—

117

C₄⋊C₄

—

394

—

118

C₄⋊C₄

—

197

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

394

—

121

C₄²

—

394

—

122

C₄²

C₂²⋊D₁₉₇

yes

—

123

C₄²

C₂²×D₁₉₇

yes

characteristic

124

C₄²

C₂²×D₁₉₇

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

127

Q₈

—

128

Q₈

—

788

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

394

—

136

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

138

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

139

C₂×C₄

#15691357

yes

—

140

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

141

C₈

—

394

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#20263313

yes

—

146

C₂²

C₂²×C₂²⋊D₁₉₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

788

—

150

C₄

#20263313

yes

—

151

C₄

#20263346

yes

—

152

C₄

C₂×#15691357

yes

—

153

C₂

—

154

C₂

—

155

C₂

C₂×#20263313

yes

characteristic

156

C₂

C₂×#20263346

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting, prime radical	#532×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂²×D₄
hypercenter	#532	D₁₉₇
hypocenter	C₁₉₇	#3131
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₉₇

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting, prime radical

#532×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂²×D₄

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #27161636	C₂	semidirect
2. #27161433	C₂	semidirect
3. #27161429	C₂	semidirect
4. #27161425	C₂	semidirect
5. #27161075	C₂	semidirect
6. C₂×C₄×#13096802	C₂	semidirect
78. C₁₉₇	#3131	semidirect

H ≅ G/N

type

1. #27161636

C₂

semidirect

2. #27161433

C₂

semidirect

3. #27161429

C₂

semidirect

4. #27161425

C₂

semidirect

5. #27161075

C₂

semidirect

6. C₂×C₄×#13096802

C₂

semidirect

78. C₁₉₇

#3131

semidirect

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

subgroup	index
#27161636	2
#27161433	2
#27161429	2
#27161425	2
#27161075	2
C₂×C₄×#13096802	2
#532×C₁₉₇	2
#3131	197

subgroup

index

#532×C₁₉₇

197

Subgroups — #38753034

Conjugacy classes of subgroups

Complements

Maximal subgroups