Group #38753017 — subgroups

Subgroups — C₂×#27161425 #38753017

Order 25216.

528 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 181 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×#20263326	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161425	C₂	1	8	yes	maximal, direct product factor
3	C₂×#20263247	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×#13096802	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₄	—	2	1	no	—
7	C₂²×#13096802	C₂²	1	2	yes	—
8	#20263326	—	2	2	no	—
9	#20263326	C₂²	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₉₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
12	#20263247	C₂²	1	16	yes	complemented
13	C₄×#13096802	C₂²	1	4	yes	complemented
14	C₂²×D₄×C₁₉₇	—	2	1	no	—
15	C₂²×D₄×C₁₉₇	—	2	1	no	—
16	C₂²×D₄×C₁₉₇	C₂²	1	2	yes	—
17	#127×C₁₉₇	—	2	2	no	—
18	#127×C₁₉₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₉₄	—	2	4	no	—
21	C₂×Dic₃₉₄	—	4	4	no	—
22	C₂²×Dic₁₉₇	—	2	2	no	—
23	C₂×#13096802	—	2	4	no	—
24	C₂×#13096802	C₂³	1	4	yes	—
25	C₄⋊Dic₁₉₇	—	2	4	no	—
26	C₄⋊Dic₁₉₇	—	2	2	no	—
27	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
28	C₂⁴×C₁₉₇	—	2	2	no	—
29	C₂×D₄×C₁₉₇	—	2	2	no	—
30	C₂×D₄×C₁₉₇	—	2	4	no	—
31	C₂×D₄×C₁₉₇	—	2	4	no	—
32	C₂×D₄×C₁₉₇	—	2	8	no	—
33	C₂×D₄×C₁₉₇	—	4	4	no	—
34	C₂×D₄×C₁₉₇	—	4	4	no	—
35	C₂×D₄×C₁₉₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
39	C₄²×C₁₉₇	D₄	1	2	yes	—
40	C₄²×C₁₉₇	C₂³	1	2	yes	—
41	Dic₃₉₄	—	4	8	no	—
42	C₂×Dic₁₉₇	—	2	4	no	—
43	C₂×Dic₁₉₇	—	4	4	no	—
44	#13096802	—	2	4	no	—
45	C₂³×C₁₉₇	—	2	4	no	—
46	C₂³×C₁₉₇	—	4	4	no	—
47	C₂³×C₁₉₇	—	4	4	no	—
48	C₂³×C₁₉₇	—	4	4	no	—
49	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
50	D₄×C₁₉₇	—	2	8	no	—
51	D₄×C₁₉₇	—	4	4	no	—
52	D₄×C₁₉₇	—	4	8	no	—
53	D₄×C₁₉₇	—	4	8	no	—
54	C₂×C₄×C₁₉₇	D₈	1	4	yes	—
55	C₂×C₄×C₁₉₇	SD₁₆	1	4	yes	—
56	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉₇	#41	1	4	yes	—
60	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
61	C₂²×C₁₉₇	—	4	4	no	—
62	C₂²×C₁₉₇	—	4	4	no	—
63	C₂²×C₁₉₇	—	4	4	no	—
64	C₂²×C₁₉₇	—	8	8	no	—
65	C₂²×C₁₉₇	#124	1	2	yes	—
66	C₂²×C₁₉₇	C₂×D₈	1	2	yes	—
67	C₂²×C₁₉₇	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
69	C₄×C₁₉₇	#124	1	2	yes	—
70	C₄×C₁₉₇	C₂×D₈	1	4	yes	—
71	C₄×C₁₉₇	C₂×SD₁₆	1	4	yes	—
72	C₄×C₁₉₇	C₂×#41	1	2	yes	—
73	Dic₁₉₇	—	4	4	no	—
74	C₂×C₁₉₇	—	8	4	no	—
75	C₂×C₁₉₇	#486	1	4	yes	complemented
76	C₂×C₁₉₇	C₂×#124	1	1	yes	characteristic
77	C₂×C₁₉₇	C₂²×D₈	1	1	yes	characteristic
78	C₂×C₁₉₇	C₂²×SD₁₆	1	1	yes	characteristic
79	C₁₉₇	C₂×#486	1	1	yes	characteristic, complemented, Sylow
80	C₂×#486	—	197	1	no	maximal, Sylow
81	C₂×C₄⋊Q₈	—	197	1	no	—
82	C₂×#127	D₁₉₇	1	1	yes	characteristic
83	#486	—	197	8	no	—
84	C₂×#102	—	197	4	no	—
85	C₂×C₄×C₈	—	197	1	no	—
86	C₂²×Q₈	—	394	1	no	—
87	C₂²×D₄	—	2	1	no	—
88	C₂²×D₄	—	2	1	no	—
89	C₂²×D₄	C₂×D₁₉₇	1	2	yes	—
90	C₂²×C₈	—	197	2	no	—
91	C₄⋊Q₈	—	394	2	no	—
92	C₄⋊Q₈	—	197	4	no	—
93	#127	—	2	2	no	—
94	#127	C₂×D₁₉₇	1	4	yes	—
95	C₂×C₄⋊C₄	—	394	1	no	—
96	C₂×C₄⋊C₄	—	197	2	no	—
97	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
98	#102	—	197	16	no	—
99	C₄×C₈	—	197	4	no	—
100	C₂⁴	—	2	2	no	—
101	C₂×Q₈	—	788	4	no	—
102	C₂×Q₈	—	394	4	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	4	no	—
105	C₂×D₄	—	2	4	no	—
106	C₂×D₄	—	2	8	no	—
107	C₂×D₄	—	4	4	no	—
108	C₂×D₄	—	4	4	no	—
109	C₂×D₄	C₂²×D₁₉₇	1	4	yes	—
110	C₂²×C₄	—	394	2	no	—
111	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
114	C₂×C₈	—	394	4	no	—
115	C₂×C₈	—	197	4	no	—
116	C₄⋊C₄	—	394	4	no	—
117	C₄⋊C₄	—	394	2	no	—
118	C₄⋊C₄	—	197	4	no	—
119	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
120	C₄²	C₂²×D₁₉₇	1	2	yes	—
121	C₂³	—	2	4	no	—
122	C₂³	—	4	4	no	—
123	C₂³	—	4	4	no	—
124	C₂³	—	4	4	no	—
125	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
126	Q₈	—	788	8	no	—
127	D₄	—	2	8	no	—
128	D₄	—	4	4	no	—
129	D₄	—	4	8	no	—
130	D₄	—	4	8	no	—
131	C₂×C₄	—	788	4	no	—
132	C₂×C₄	—	394	4	no	—
133	C₂×C₄	#15691343	1	4	yes	—
134	C₂×C₄	#15691346	1	4	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
138	C₂×C₄	#15691357	1	4	yes	—
139	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
140	C₈	—	394	4	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	4	no	—
143	C₂²	—	4	4	no	—
144	C₂²	—	8	8	no	—
145	C₂²	#20263313	1	2	yes	—
146	C₂²	C₂×#15691343	1	2	yes	—
147	C₂²	C₂×#15691346	1	2	yes	—
148	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
149	C₄	—	788	4	no	—
150	C₄	#20263313	1	2	yes	—
151	C₄	C₂×#15691343	1	4	yes	—
152	C₄	C₂×#15691346	1	4	yes	—
153	C₄	C₂×#15691357	1	2	yes	—
154	C₂	—	8	4	no	—
155	C₂	#27161425	1	4	yes	direct product factor
156	C₂	C₂×#20263313	1	1	yes	characteristic
157	C₂	C₂²×#15691343	1	1	yes	characteristic
158	C₂	C₂²×#15691346	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263326

C₂

yes

maximal, characteristic, complemented

#27161425

C₂

yes

maximal, direct product factor

C₂×#20263247

C₂

yes

maximal, complemented

C₂×C₄×#13096802

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₄

—

C₂²×#13096802

C₂²

yes

—

#20263326

—

#20263326

C₂²

yes

complemented

C₂×C₄⋊Dic₁₉₇

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

#20263247

C₂²

yes

complemented

C₄×#13096802

C₂²

yes

complemented

C₂²×D₄×C₁₉₇

—

C₂²×D₄×C₁₉₇

—

C₂²×D₄×C₁₉₇

C₂²

yes

—

#127×C₁₉₇

—

#127×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂²×Dic₁₉₇

—

C₂×#13096802

—

C₂×#13096802

C₂³

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂³

yes

—

C₂⁴×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

C₂³

yes

—

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

C₂³

yes

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

D₄×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

D₈

yes

—

C₂×C₄×C₁₉₇

SD₁₆

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#124

yes

—

C₂²×C₁₉₇

C₂×D₈

yes

—

C₂²×C₁₉₇

C₂×SD₁₆

yes

—

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₄×C₁₉₇

#124

yes

—

C₄×C₁₉₇

C₂×D₈

yes

—

C₄×C₁₉₇

C₂×SD₁₆

yes

—

C₄×C₁₉₇

C₂×#41

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

C₂²×D₈

yes

characteristic

C₂×C₁₉₇

C₂²×SD₁₆

yes

characteristic

C₁₉₇

C₂×#486

yes

characteristic, complemented, Sylow

C₂×#486

—

197

maximal, Sylow

C₂×C₄⋊Q₈

—

197

—

C₂×#127

D₁₉₇

yes

characteristic

#486

—

197

—

C₂×#102

—

197

—

C₂×C₄×C₈

—

197

—

C₂²×Q₈

—

394

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₉₇

yes

—

C₂²×C₈

—

197

—

C₄⋊Q₈

—

394

—

C₄⋊Q₈

—

197

—

#127

—

#127

C₂×D₁₉₇

yes

—

C₂×C₄⋊C₄

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#102

—

197

—

C₄×C₈

—

197

—

100

C₂⁴

—

101

C₂×Q₈

—

788

—

102

C₂×Q₈

—

394

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²×D₁₉₇

yes

—

110

C₂²×C₄

—

394

—

111

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

112

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

113

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

114

C₂×C₈

—

394

—

115

C₂×C₈

—

197

—

116

C₄⋊C₄

—

394

—

117

C₄⋊C₄

—

394

—

118

C₄⋊C₄

—

197

—

119

C₄²

C₂²⋊D₁₉₇

yes

—

120

C₄²

C₂²×D₁₉₇

yes

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

126

Q₈

—

788

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

788

—

132

C₂×C₄

—

394

—

133

C₂×C₄

#15691343

yes

—

134

C₂×C₄

#15691346

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

137

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

138

C₂×C₄

#15691357

yes

—

139

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

140

C₈

—

394

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

#20263313

yes

—

146

C₂²

C₂×#15691343

yes

—

147

C₂²

C₂×#15691346

yes

—

148

C₂²

C₂²×C₂²⋊D₁₉₇

yes

characteristic

149

C₄

—

788

—

150

C₄

#20263313

yes

—

151

C₄

C₂×#15691343

yes

—

152

C₄

C₂×#15691346

yes

—

153

C₄

C₂×#15691357

yes

—

154

C₂

—

155

C₂

#27161425

yes

direct product factor

156

C₂

C₂×#20263313

yes

characteristic

157

C₂

C₂²×#15691343

yes

characteristic

158

C₂

C₂²×#15691346

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting, prime radical	C₂×#127×C₁₉₇	C₂
socle	C₂³×C₁₉₇	C₂×D₄
hypercenter	C₂×#127	D₁₉₇
hypocenter	C₁₉₇	C₂×#486
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₇

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting, prime radical

C₂×#127×C₁₉₇

C₂

socle

C₂³×C₁₉₇

C₂×D₄

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#20263326	C₂	semidirect
2. #27161425	C₂	direct (N × H)
3. C₂×#20263247	C₂	semidirect
4. C₂×C₄×#13096802	C₂	semidirect
9. #20263326	C₂²	semidirect
12. #20263247	C₂²	semidirect
13. C₄×#13096802	C₂²	semidirect
75. C₂×C₁₉₇	#486	semidirect
79. C₁₉₇	C₂×#486	semidirect
155. C₂	#27161425	direct (N × H)

H ≅ G/N

type

1. C₂×#20263326

C₂

semidirect

2. #27161425

C₂

direct (N × H)

3. C₂×#20263247

C₂

semidirect

4. C₂×C₄×#13096802

C₂

semidirect

9. #20263326

C₂²

semidirect

12. #20263247

C₂²

semidirect

13. C₄×#13096802

C₂²

semidirect

75. C₂×C₁₉₇

#486

semidirect

79. C₁₉₇

C₂×#486

semidirect

155. C₂

#27161425

direct (N × H)

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

subgroup	index
C₂×#20263326	2
#27161425	2
C₂×#20263247	2
C₂×C₄×#13096802	2
C₂×#127×C₁₉₇	2
C₂×#486	197

subgroup

index

C₂×#127×C₁₉₇

197

Subgroups — C₂×#27161425 #38753017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#27161425 #38753017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161425 #38753017