Group #38743234 — subgroups

Subgroups — #38743234

Order 25216.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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	#27161650	C₂	1	1	yes	maximal, characteristic
2	#27161089	C₂	1	2	yes	maximal, complemented
3	#27161053	C₂	1	1	yes	maximal, characteristic
4	#27160836	C₂	1	2	yes	maximal, complemented
5	#568×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₄	C₂²	1	1	yes	characteristic
7	#20263324	C₄	1	2	yes	complemented
8	#20263313	C₄	1	2	yes	complemented
9	#20263307	—	2	2	no	—
10	#20263305	—	2	2	no	—
11	#20263285	C₂²	1	2	yes	—
12	#20263285	C₂²	1	1	yes	characteristic
13	#20263251	—	2	4	no	—
14	#20263241	—	2	2	no	—
15	C₂×#41×C₁₉₇	C₂²	1	1	yes	characteristic
16	#131×C₁₉₇	—	2	4	no	—
17	C₂×M₁₆×C₁₉₇	—	2	2	no	—
18	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
19	C₂×Dic₃₉₄	—	2	1	no	—
20	C₂×Dic₃₉₄	—	2	2	no	—
21	C₂×Dic₃₉₄	—	2	2	no	—
22	C₂×Dic₃₉₄	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₉₇	—	2	1	no	—
24	C₄⋊Dic₁₉₇	—	2	1	no	—
25	C₄⋊Dic₁₉₇	—	2	2	no	—
26	#15691337	—	2	4	no	—
27	C₂²⋊Dic₁₉₇	—	2	4	no	—
28	C₂²⋊Dic₁₉₇	—	2	1	no	—
29	C₂²⋊Dic₁₉₇	—	2	2	no	—
30	C₄×Dic₁₉₇	—	2	2	no	—
31	C₄×Dic₁₉₇	C₂×C₄	1	2	yes	—
32	#41×C₁₉₇	—	2	2	no	—
33	#41×C₁₉₇	—	2	2	no	—
34	C₂×Q₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₁₉₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
38	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
39	M₁₆×C₁₉₇	—	2	4	no	—
40	M₁₆×C₁₉₇	—	2	2	no	—
41	M₁₆×C₁₉₇	—	2	2	no	—
42	C₂×C₈×C₁₉₇	—	2	4	no	—
43	C₂×C₈×C₁₉₇	D₄	1	4	yes	—
44	Dic₃₉₄	—	2	4	no	—
45	Dic₃₉₄	—	4	1	no	—
46	Dic₃₉₄	—	4	1	no	—
47	C₂×Dic₁₉₇	—	2	2	no	—
48	C₂×Dic₁₉₇	—	2	2	no	—
49	C₂×Dic₁₉₇	—	4	2	no	—
50	C₂×Dic₁₉₇	—	4	2	no	—
51	C₂³×C₁₉₇	C₂²⋊C₄	1	2	yes	—
52	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
53	Q₈×C₁₉₇	—	2	2	no	—
54	D₄×C₁₉₇	—	2	4	no	—
55	D₄×C₁₉₇	—	2	2	no	—
56	C₂×C₄×C₁₉₇	—	2	2	no	—
57	C₂×C₄×C₁₉₇	—	2	2	no	—
58	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₇	#41	1	2	yes	—
63	C₈×C₁₉₇	—	2	4	no	—
64	C₂²×C₁₉₇	—	2	2	no	—
65	C₂²×C₁₉₇	—	4	1	no	—
66	C₂²×C₁₉₇	—	4	2	no	—
67	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₁₉₇	#121	1	2	yes	—
69	C₄×C₁₉₇	—	2	2	no	—
70	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
71	C₄×C₁₉₇	#120	1	1	yes	characteristic
72	C₄×C₁₉₇	#123	1	1	yes	characteristic
73	Dic₁₉₇	—	4	2	no	—
74	Dic₁₉₇	—	4	2	no	—
75	Dic₁₉₇	—	8	2	no	—
76	C₂×C₁₉₇	—	2	1	no	—
77	C₂×C₁₉₇	—	2	2	no	—
78	C₂×C₁₉₇	—	4	2	no	—
79	C₂×C₁₉₇	#386	1	1	yes	characteristic
80	C₁₉₇	#1909	1	1	yes	characteristic, complemented, Sylow
81	#1909	—	197	1	no	maximal, Sylow
82	#568	D₁₉₇	1	1	yes	characteristic
83	#536	—	197	1	no	—
84	#419	—	197	2	no	—
85	#410	—	197	1	no	—
86	H(3,ℤ/4)	—	197	2	no	—
87	C₂×#41	C₂×D₁₉₇	1	1	yes	characteristic
88	C₂²×Q₈	—	197	1	no	—
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
92	C₄⋊Q₈	—	197	2	no	—
93	#124	—	197	2	no	—
94	#123	—	394	2	no	—
95	C₂²⋊Q₈	—	394	2	no	—
96	#117	—	197	2	no	—
97	#117	—	197	1	no	—
98	#103	—	394	4	no	—
99	#99	—	394	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	394	2	no	—
103	C₂×Q₈	—	394	2	no	—
104	C₂×Q₈	—	394	1	no	—
105	C₂×Q₈	—	197	2	no	—
106	C₂×Q₈	C₄×D₁₉₇	1	1	yes	characteristic
107	C₂×D₄	C₄×D₁₉₇	1	1	yes	characteristic
108	C₂×D₄	D₇₈₈	1	2	yes	—
109	C₂²×C₄	—	394	1	no	—
110	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	C₂²⋊D₁₉₇	1	4	yes	—
117	C₄⋊C₄	—	394	2	no	—
118	C₄⋊C₄	—	394	4	no	—
119	C₄⋊C₄	—	394	1	no	—
120	C₂²⋊C₄	—	394	2	no	—
121	C₂²⋊C₄	—	394	4	no	—
122	C₂²⋊C₄	—	394	1	no	—
123	C₄²	—	394	2	no	—
124	C₄²	—	197	2	no	—
125	C₂³	#15691335	1	2	yes	—
126	C₂³	C₂×D₇₈₈	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	788	1	no	—
129	Q₈	—	788	1	no	—
130	Q₈	—	394	4	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	788	2	no	—
136	C₂×C₄	—	788	2	no	—
137	C₂×C₄	—	394	2	no	—
138	C₂×C₄	—	394	2	no	—
139	C₂×C₄	#15691335	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
143	C₂×C₄	#15691358	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×#15691335	1	1	yes	characteristic
149	C₂²	#20263300	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	788	2	no	—
152	C₄	—	788	2	no	—
153	C₄	—	1576	2	no	—
154	C₄	C₄×C₂²⋊D₁₉₇	1	2	yes	—
155	C₄	#20263295	1	1	yes	characteristic
156	C₄	#20263309	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#27160958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161650

C₂

yes

maximal, characteristic

#27161089

C₂

yes

maximal, complemented

#27161053

C₂

yes

maximal, characteristic

#27160836

C₂

yes

maximal, complemented

#568×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₄

C₂²

yes

characteristic

#20263324

C₄

yes

complemented

#20263313

C₄

yes

—

—

C₂²

yes

—

#20263285

C₂²

yes

characteristic

#20263251

—

#20263241

—

C₂×#41×C₁₉₇

C₂²

yes

characteristic

#131×C₁₉₇

—

C₂×M₁₆×C₁₉₇

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

C₂×C₄

yes

—

C₂²×Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

#15691337

—

C₂²⋊Dic₁₉₇

—

C₂²⋊Dic₁₉₇

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

C₂×C₄

yes

—

#41×C₁₉₇

—

#41×C₁₉₇

—

C₂×Q₈×C₁₉₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

Dic₃₉₄

—

Dic₃₉₄

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂³×C₁₉₇

C₂²⋊C₄

yes

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

Q₈×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

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

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

#121

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄×D₄

yes

—

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#386

yes

characteristic

C₁₉₇

#1909

yes

characteristic, complemented, Sylow

#1909

—

197

maximal, Sylow

#568

D₁₉₇

yes

characteristic

#536

—

197

—

#419

—

197

—

#410

—

197

—

H(3,ℤ/4)

—

197

—

C₂×#41

C₂×D₁₉₇

yes

characteristic

C₂²×Q₈

—

197

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₄⋊Q₈

—

197

—

#124

—

197

—

#123

—

394

—

C₂²⋊Q₈

—

394

—

#117

—

197

—

#117

—

197

—

#103

—

394

—

#99

—

394

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

394

—

103

C₂×Q₈

—

394

—

104

C₂×Q₈

—

394

—

105

C₂×Q₈

—

197

—

106

C₂×Q₈

C₄×D₁₉₇

yes

characteristic

107

C₂×D₄

C₄×D₁₉₇

yes

characteristic

108

C₂×D₄

D₇₈₈

yes

—

109

C₂²×C₄

—

394

—

110

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

111

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊D₁₉₇

yes

—

117

C₄⋊C₄

—

394

—

118

C₄⋊C₄

—

394

—

119

C₄⋊C₄

—

394

—

120

C₂²⋊C₄

—

394

—

121

C₂²⋊C₄

—

394

—

122

C₂²⋊C₄

—

394

—

123

C₄²

—

394

—

124

C₄²

—

197

—

125

C₂³

#15691335

yes

—

126

C₂³

C₂×D₇₈₈

yes

characteristic

127

Q₈

—

128

Q₈

—

788

—

129

Q₈

—

788

—

130

Q₈

—

394

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

788

—

136

C₂×C₄

—

788

—

137

C₂×C₄

—

394

—

138

C₂×C₄

—

394

—

139

C₂×C₄

#15691335

yes

—

140

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

143

C₂×C₄

#15691358

yes

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×#15691335

yes

characteristic

149

C₂²

#20263300

yes

—

150

C₄

—

151

C₄

—

788

—

152

C₄

—

788

—

153

C₄

—

1576

—

154

C₄

C₄×C₂²⋊D₁₉₇

yes

—

155

C₄

#20263295

yes

characteristic

156

C₄

#20263309

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#27160958

yes

characteristic

	subgroup	G/H
center	C₂	#27160958
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#568×C₁₉₇	C₂
socle	C₂×C₁₉₇	#386
prime radical	C₂×#41×C₁₉₇	C₂²
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#568	D₁₉₇
hypocenter	C₁₉₇	#1909
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#27160958

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#568×C₁₉₇

C₂

socle

C₂×C₁₉₇

#386

prime radical

C₂×#41×C₁₉₇

C₂²

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
2. #27161089	C₂	semidirect
4. #27160836	C₂	semidirect
7. #20263324	C₄	semidirect
8. #20263313	C₄	semidirect
80. C₁₉₇	#1909	semidirect

H ≅ G/N

type

2. #27161089

C₂

semidirect

4. #27160836

C₂

semidirect

7. #20263324

C₄

semidirect

8. #20263313

C₄

semidirect

80. C₁₉₇

#1909

semidirect

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

subgroup	index
#27161650	2
#27161089	2
#27161053	2
#27160836	2
#568×C₁₉₇	2
#1909	197

subgroup

index

#568×C₁₉₇

197

Subgroups — #38743234

Conjugacy classes of subgroups

Complements

Maximal subgroups