Group #36475439 — subgroups

Subgroups — #36475439

Order 22784.

252 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19151437	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₈₄₈	C₂	1	2	yes	maximal, complemented
3	#26022513	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022502	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
7	C₂×#15667957	—	2	1	no	—
8	C₂×D₁₄₂₄	—	2	1	no	—
9	C₂×D₁₄₂₄	C₂²	1	2	yes	—
10	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
11	#19151982	—	2	1	no	—
12	#19151975	—	2	1	no	—
13	#19151437	—	2	4	no	—
14	D₂₈₄₈	—	2	4	no	—
15	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	—	2	1	no	—
18	C₂×#13083563	—	4	1	no	—
19	C₂×D₇₁₂	—	2	2	no	—
20	C₂×D₇₁₂	—	2	1	no	—
21	C₂×D₇₁₂	—	4	1	no	—
22	#15668009	—	4	1	no	—
23	#15668000	—	4	1	no	—
24	Dic₇₁₂	—	2	2	no	—
25	#15667957	—	4	2	no	—
26	D₁₄₂₄	—	2	2	no	—
27	D₁₄₂₄	—	2	2	no	—
28	D₁₄₂₄	—	2	2	no	—
29	D₁₄₂₄	—	4	2	no	—
30	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₃₂×C₈₉	D₄	1	4	yes	complemented
34	C₂×Dic₁₇₈	—	4	1	no	—
35	C₂×D₃₅₆	—	4	1	no	—
36	C₂×D₃₅₆	—	4	2	no	—
37	C₂×D₃₅₆	—	8	2	no	—
38	Dic₃₅₆	—	4	1	no	—
39	Dic₃₅₆	—	4	1	no	—
40	#13083563	—	8	2	no	—
41	D₇₁₂	—	4	2	no	—
42	D₇₁₂	—	4	1	no	—
43	D₇₁₂	—	4	2	no	—
44	D₇₁₂	—	4	1	no	—
45	D₇₁₂	—	8	2	no	—
46	#13083551	—	8	2	no	—
47	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₉	D₈	1	1	yes	characteristic
50	C₁₆×C₈₉	D₈	1	2	yes	—
51	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
52	C₂²×D₈₉	—	8	1	no	—
53	C₂²×D₈₉	—	8	2	no	—
54	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₈₉	—	8	1	no	—
57	D₃₅₆	—	8	2	no	—
58	D₃₅₆	—	8	1	no	—
59	D₃₅₆	—	8	2	no	—
60	D₃₅₆	—	8	1	no	—
61	D₃₅₆	—	16	4	no	—
62	Dic₁₇₈	—	8	1	no	—
63	Dic₁₇₈	—	8	1	no	—
64	C₈×C₈₉	D₁₆	1	2	yes	—
65	C₈×C₈₉	#127	1	1	yes	characteristic
66	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
68	C₂×D₈₉	—	16	2	no	—
69	C₂×D₈₉	—	16	1	no	—
70	C₂×D₈₉	—	16	4	no	—
71	C₂×D₈₉	—	16	2	no	—
72	C₄×C₈₉	—	2	2	no	—
73	C₄×C₈₉	#493	1	1	yes	characteristic
74	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
75	Dic₈₉	—	16	1	no	—
76	#13729	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	#2246	1	1	yes	characteristic
78	C₂×C₈₉	#2263	1	2	yes	—
79	D₈₉	—	32	2	no	—
80	D₈₉	—	32	1	no	—
81	C₂×SD₆₄	—	89	2	no	—
82	C₂×D₃₂	—	89	2	no	—
83	#2249	—	89	1	no	—
84	#2246	—	89	1	no	—
85	#1397	D₈₉	1	1	yes	characteristic, complemented
86	C₈₉	#13729	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₃₂	—	89	1	no	—
88	C₂×SD₃₂	—	178	1	no	—
89	C₂×D₁₆	—	89	2	no	—
90	C₂×D₁₆	—	89	1	no	—
91	C₂×D₁₆	—	178	1	no	—
92	#495	—	178	1	no	—
93	#493	—	178	1	no	—
94	SD₆₄	—	178	4	no	—
95	D₃₂	—	178	4	no	—
96	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
97	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
98	C₂×Q₁₆	—	178	1	no	—
99	C₂×SD₁₆	—	356	1	no	—
100	C₂×D₈	—	356	1	no	—
101	C₂×D₈	—	178	2	no	—
102	C₂×D₈	—	178	1	no	—
103	#127	—	356	1	no	—
104	#124	—	356	1	no	—
105	Q₃₂	—	178	2	no	—
106	SD₃₂	—	356	2	no	—
107	D₁₆	—	356	2	no	—
108	D₁₆	—	178	2	no	—
109	D₁₆	—	178	2	no	—
110	D₁₆	—	178	2	no	—
111	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
112	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
113	C₄×C₈	D₃₅₆	1	1	yes	characteristic
114	C₃₂	D₃₅₆	1	4	yes	complemented
115	C₂×Q₈	—	356	1	no	—
116	C₂×D₄	—	356	1	no	—
117	C₂×D₄	—	356	2	no	—
118	C₂×D₄	—	712	2	no	—
119	Q₁₆	—	356	1	no	—
120	Q₁₆	—	356	1	no	—
121	SD₁₆	—	712	2	no	—
122	D₈	—	356	1	no	—
123	D₈	—	356	2	no	—
124	D₈	—	356	2	no	—
125	D₈	—	356	1	no	—
126	D₈	—	712	2	no	—
127	C₂×C₈	D₇₁₂	1	1	yes	characteristic
128	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
129	C₂²⋊C₄	—	712	2	no	—
130	C₄²	D₇₁₂	1	1	yes	characteristic
131	C₁₆	D₇₁₂	1	2	yes	—
132	C₁₆	C₂×D₃₅₆	1	2	yes	—
133	C₂³	—	712	1	no	—
134	C₂³	—	712	2	no	—
135	Q₈	—	712	1	no	—
136	Q₈	—	712	1	no	—
137	D₄	—	1424	4	no	—
138	D₄	—	712	2	no	—
139	D₄	—	712	1	no	—
140	D₄	—	712	2	no	—
141	D₄	—	712	1	no	—
142	C₂×C₄	—	712	1	no	—
143	C₂×C₄	D₁₄₂₄	1	2	yes	—
144	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
145	C₈	D₁₄₂₄	1	2	yes	—
146	C₈	#15668009	1	1	yes	characteristic
147	C₈	C₂×D₇₁₂	1	1	yes	characteristic
148	C₂²	—	1424	2	no	—
149	C₂²	—	1424	1	no	—
150	C₂²	—	1424	4	no	—
151	C₂²	—	1424	2	no	—
152	C₂²	C₂×D₁₄₂₄	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	1424	1	no	—
155	C₄	#19151975	1	1	yes	characteristic
156	C₄	C₂×D₁₄₂₄	1	1	yes	characteristic
157	C₂	—	2848	2	no	—
158	C₂	—	2848	1	no	—
159	C₂	#26022502	1	1	yes	characteristic
160	C₂	#26022567	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151437

C₂

yes

maximal, complemented

C₂×D₂₈₄₈

C₂

yes

maximal, complemented

#26022513

C₂

yes

maximal, characteristic, complemented

#26022502

C₂

yes

maximal, characteristic, complemented

C₈₉×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₁₂

C₂²

yes

characteristic

C₂×#15667957

—

C₂×D₁₄₂₄

—

C₂×D₁₄₂₄

C₂²

yes

—

C₂×D₁₄₂₄

C₂²

yes

—

—

—

D₂₈₄₈

—

C₂×C₃₂×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×#13083563

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

#15668000

—

Dic₇₁₂

—

#15667957

—

D₁₄₂₄

—

D₁₄₂₄

—

D₁₄₂₄

—

D₁₄₂₄

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₃₂×C₈₉

D₄

yes

complemented

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083563

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

#13083551

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

D₈

yes

—

C₁₆×C₈₉

C₂×D₄

yes

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

D₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

D₁₆

yes

—

C₈×C₈₉

#127

yes

characteristic

C₈×C₈₉

C₂×D₈

yes

characteristic

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#493

yes

characteristic

C₄×C₈₉

C₂×D₁₆

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

D₈₉

—

C₂×SD₆₄

—

C₂×D₃₂

—

#2249

—

#2246

—

#1397

D₈₉

yes

characteristic, complemented

C₈₉

#13729

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

178

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

178

—

#495

—

178

—

#493

—

178

—

SD₆₄

—

178

—

D₃₂

—

178

—

C₂×C₃₂

C₂×D₈₉

yes

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₂×SD₁₆

—

356

—

100

C₂×D₈

—

356

—

101

C₂×D₈

—

178

—

102

C₂×D₈

—

178

—

103

#127

—

356

—

104

#124

—

356

—

105

Q₃₂

—

178

—

106

SD₃₂

—

356

—

107

D₁₆

—

356

—

108

D₁₆

—

178

—

109

D₁₆

—

178

—

110

D₁₆

—

178

—

111

C₂×C₁₆

D₃₅₆

yes

characteristic

112

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

113

C₄×C₈

D₃₅₆

yes

characteristic

114

C₃₂

D₃₅₆

yes

complemented

115

C₂×Q₈

—

356

—

116

C₂×D₄

—

356

—

117

C₂×D₄

—

356

—

118

C₂×D₄

—

712

—

119

Q₁₆

—

356

—

120

Q₁₆

—

356

—

121

SD₁₆

—

712

—

122

D₈

—

356

—

123

D₈

—

356

—

124

D₈

—

356

—

125

D₈

—

356

—

126

D₈

—

712

—

127

C₂×C₈

D₇₁₂

yes

characteristic

128

C₂×C₈

C₂×D₃₅₆

yes

characteristic

129

C₂²⋊C₄

—

712

—

130

C₄²

D₇₁₂

yes

characteristic

131

C₁₆

D₇₁₂

yes

—

132

C₁₆

C₂×D₃₅₆

yes

—

133

C₂³

—

712

—

134

C₂³

—

712

—

135

Q₈

—

712

—

136

Q₈

—

712

—

137

D₄

—

1424

—

138

D₄

—

712

—

139

D₄

—

712

—

140

D₄

—

712

—

141

D₄

—

712

—

142

C₂×C₄

—

712

—

143

C₂×C₄

D₁₄₂₄

yes

—

144

C₂×C₄

C₂×D₇₁₂

yes

characteristic

145

C₈

D₁₄₂₄

yes

—

146

C₈

#15668009

yes

characteristic

147

C₈

C₂×D₇₁₂

yes

characteristic

148

C₂²

—

1424

—

149

C₂²

—

1424

—

150

C₂²

—

1424

—

151

C₂²

—

1424

—

152

C₂²

C₂×D₁₄₂₄

yes

characteristic

153

C₄

—

154

C₄

—

1424

—

155

C₄

#19151975

yes

characteristic

156

C₄

C₂×D₁₄₂₄

yes

characteristic

157

C₂

—

2848

—

158

C₂

—

2848

—

159

C₂

#26022502

yes

characteristic

160

C₂

#26022567

yes

—

	subgroup	G/H
center	C₂²	C₂×D₁₄₂₄
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1397	C₂
socle	C₂²×C₈₉	C₂×D₁₆
hypercenter	#1397	D₈₉
hypocenter	C₈₉	#13729
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₁₄₂₄

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1397

C₂

socle

C₂²×C₈₉

C₂×D₁₆

D₈₉

C₈₉

G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
1. C₂×#19151437	C₂	semidirect
2. C₂×D₂₈₄₈	C₂	semidirect
3. #26022513	C₂	semidirect
4. #26022502	C₂	semidirect
5. C₈₉×#1397	C₂	semidirect
33. C₃₂×C₈₉	D₄	semidirect
85. #1397	D₈₉	semidirect
86. C₈₉	#13729	semidirect
114. C₃₂	D₃₅₆	semidirect

H ≅ G/N

type

1. C₂×#19151437

C₂

semidirect

2. C₂×D₂₈₄₈

C₂

semidirect

3. #26022513

C₂

semidirect

4. #26022502

C₂

semidirect

5. C₈₉×#1397

C₂

semidirect

33. C₃₂×C₈₉

D₄

semidirect

85. #1397

D₈₉

semidirect

86. C₈₉

#13729

semidirect

114. C₃₂

D₃₅₆

semidirect

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

subgroup	index
C₂×#19151437	2
C₂×D₂₈₄₈	2
#26022513	2
#26022502	2
C₈₉×#1397	2
#13729	89

subgroup

index

C₂×D₂₈₄₈

C₈₉×#1397

Subgroups — #36475439

Conjugacy classes of subgroups

Complements

Maximal subgroups