Group #36474564 — subgroups

Subgroups — #36474564

Order 22784.

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

normal subgroups only

This group has 234 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	#26021005	C₂	1	1	yes	maximal, characteristic
2	#26020427	C₂	1	1	yes	maximal, characteristic
3	#26018367	C₂	1	2	yes	maximal
4	#26018364	C₂	1	2	yes	maximal
5	C₈₉×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667948	—	2	1	no	—
7	C₂×#15667948	C₂²	1	1	yes	characteristic
8	C₂×#15667948	C₂²	1	2	yes	—
9	#19151636	C₂²	1	1	yes	characteristic
10	#19151624	—	2	1	no	—
11	#19151622	—	2	1	no	—
12	#19151358	—	2	1	no	—
13	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
14	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	2	no	—
16	C₂×#13083558	—	2	1	no	—
17	#15667971	—	2	1	no	—
18	#15667948	—	2	2	no	—
19	#15667948	—	2	4	no	—
20	#15667948	—	4	2	no	—
21	#15667946	—	2	1	no	—
22	#15667944	Q₈	1	2	yes	—
23	C₄⋊#10623494	—	4	2	no	—
24	#15667941	—	4	2	no	—
25	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
28	M₃₂×C₈₉	—	2	2	no	—
29	M₃₂×C₈₉	—	2	2	no	—
30	C₂×C₁₆×C₈₉	—	2	2	no	—
31	#97×C₈₉	—	2	2	no	—
32	#13083558	—	4	1	no	—
33	#13083558	—	4	2	no	—
34	#13083558	—	4	2	no	—
35	#13083558	—	4	1	no	—
36	C₂×#10623494	—	4	1	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₄⋊Dic₈₉	—	2	2	no	—
39	C₂²⋊Dic₈₉	—	4	1	no	—
40	C₄×Dic₈₉	—	4	1	no	—
41	C₂²×C₄×C₈₉	—	2	1	no	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	—	2	1	no	—
45	C₂×C₈×C₈₉	D₈	1	2	yes	—
46	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₉	#41	1	2	yes	—
48	C₂×C₈×C₈₉	#41	1	2	yes	—
49	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
50	C₄²×C₈₉	—	2	1	no	—
51	C₄²×C₈₉	SD₁₆	1	2	yes	—
52	C₁₆×C₈₉	—	4	2	no	—
53	C₂³×C₈₉	#137	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	—	8	1	no	—
56	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
62	C₂×Dic₈₉	—	4	2	no	—
63	C₈×C₈₉	—	2	2	no	—
64	C₈×C₈₉	#125	1	2	yes	—
65	C₈×C₈₉	#126	1	2	yes	—
66	#10623494	—	8	1	no	—
67	#10623494	—	8	2	no	—
68	C₂²×C₈₉	—	4	1	no	—
69	C₂²×C₈₉	#474	1	1	yes	characteristic
70	C₂²×C₈₉	#483	1	1	yes	characteristic
71	C₂²×C₈₉	#491	1	1	yes	characteristic
72	C₄×C₈₉	—	4	2	no	—
73	C₄×C₈₉	#400	1	1	yes	characteristic
74	C₄×C₈₉	#475	1	1	yes	characteristic
75	C₄×C₈₉	#480	1	1	yes	characteristic
76	C₄×C₈₉	#486	1	1	yes	characteristic
77	Dic₈₉	—	8	2	no	—
78	#13540	—	89	1	no	maximal, Sylow
79	C₂×C₈₉	—	2	1	no	—
80	C₂×C₈₉	—	2	1	no	—
81	C₂×C₈₉	—	2	1	no	—
82	C₂×C₈₉	#2099	1	1	yes	characteristic
83	#1946	—	89	1	no	—
84	#1849	—	89	1	no	—
85	#1389	—	89	2	no	—
86	#1388	—	89	2	no	—
87	#1375	D₈₉	1	1	yes	characteristic
88	C₈₉	#13540	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
90	C₂×#108	—	89	2	no	—
91	C₂×#108	—	89	1	no	—
92	C₂×#108	—	178	1	no	—
93	#427	—	89	1	no	—
94	#424	—	178	1	no	—
95	#423	—	178	1	no	—
96	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
97	#339	—	178	1	no	—
98	C₂×M₁₆	—	178	1	no	—
99	C₂×M₁₆	—	178	2	no	—
100	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
102	#117	—	178	1	no	—
103	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	#108	—	356	2	no	—
108	#108	—	178	2	no	—
109	#108	—	178	4	no	—
110	#107	—	178	1	no	—
111	#106	—	89	2	no	—
112	C₄⋊C₈	—	356	2	no	—
113	C₄⋊C₈	—	356	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
117	M₁₆	—	356	1	no	—
118	M₁₆	—	356	2	no	—
119	M₁₆	—	356	2	no	—
120	M₁₆	—	356	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	356	1	no	—
124	C₂×C₈	—	356	2	no	—
125	C₂×C₈	D₇₁₂	1	2	yes	—
126	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
127	C₂×C₈	#13083573	1	2	yes	—
128	C₂×C₈	#13083573	1	1	yes	characteristic
129	C₂×C₈	#13083574	1	2	yes	—
130	C₄⋊C₄	—	178	2	no	—
131	C₂²⋊C₄	—	356	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	356	1	no	—
134	C₄²	#13083563	1	2	yes	—
135	C₁₆	—	4	2	no	—
136	C₂³	#15668045	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	356	2	no	—
140	C₂×C₄	#15667992	1	1	yes	characteristic
141	C₂×C₄	#15667996	1	1	yes	characteristic
142	C₂×C₄	#15668000	1	1	yes	characteristic
143	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
144	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
145	C₂×C₄	#15668045	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	712	1	no	—
148	C₈	—	712	2	no	—
149	C₈	#15668002	1	2	yes	—
150	C₈	#15668005	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#19151879	1	1	yes	characteristic
153	C₂²	#19151929	1	1	yes	characteristic
154	C₂²	#19151967	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	712	2	no	—
157	C₄	#19151538	1	1	yes	characteristic
158	C₄	#19151884	1	1	yes	characteristic
159	C₄	#19151912	1	1	yes	characteristic
160	C₄	#19151945	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#26021811	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26021005

C₂

yes

maximal, characteristic

#26020427

C₂

yes

maximal, characteristic

#26018367

C₂

yes

maximal

#26018364

C₂

yes

maximal

C₈₉×#1375

C₂

yes

maximal, characteristic

C₂×#15667948

—

C₂×#15667948

C₂²

yes

characteristic

C₂×#15667948

C₂²

yes

—

#19151636

C₂²

yes

—

—

—

C₂×M₃₂×C₈₉

C₂²

yes

—

C₂×#97×C₈₉

C₂²

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₄⋊#10623494

—

#15667941

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

#97×C₈₉

—

—

—

—

—

—

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

#137

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊Q₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₉

#137

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#125

yes

—

C₈×C₈₉

#126

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

#13540

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#13540

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₈₉

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

—

178

—

#427

—

#424

—

178

—

#423

—

178

—

C₂×#97

C₂×D₈₉

yes

characteristic

#339

—

178

—

C₂×M₁₆

—

178

—

C₂×M₁₆

—

178

—

100

C₂²×C₈

D₃₅₆

yes

characteristic

101

C₂²×C₈

C₂²×D₈₉

yes

characteristic

102

#117

—

178

—

103

C₂×C₄²

D₃₅₆

yes

characteristic

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

356

—

108

#108

—

178

—

109

#108

—

178

—

110

#107

—

178

—

111

#106

—

112

C₄⋊C₈

—

356

—

113

C₄⋊C₈

—

356

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

117

M₁₆

—

356

—

118

M₁₆

—

356

—

119

M₁₆

—

356

—

120

M₁₆

—

356

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

356

—

124

C₂×C₈

—

356

—

125

C₂×C₈

D₇₁₂

yes

—

126

C₂×C₈

Q₈×D₈₉

yes

characteristic

127

C₂×C₈

#13083573

yes

—

128

C₂×C₈

#13083573

yes

characteristic

129

C₂×C₈

#13083574

yes

—

130

C₄⋊C₄

—

178

—

131

C₂²⋊C₄

—

356

—

132

C₄²

—

133

C₄²

—

356

—

134

C₄²

#13083563

yes

—

135

C₁₆

—

136

C₂³

#15668045

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

356

—

140

C₂×C₄

#15667992

yes

characteristic

141

C₂×C₄

#15667996

yes

characteristic

142

C₂×C₄

#15668000

yes

characteristic

143

C₂×C₄

C₂×D₇₁₂

yes

characteristic

144

C₂×C₄

C₂×#13083563

yes

characteristic

145

C₂×C₄

#15668045

yes

characteristic

146

C₈

—

147

C₈

—

712

—

148

C₈

—

712

—

149

C₈

#15668002

yes

—

150

C₈

#15668005

yes

—

151

C₂²

—

152

C₂²

#19151879

yes

characteristic

153

C₂²

#19151929

yes

characteristic

154

C₂²

#19151967

yes

characteristic

155

C₄

—

156

C₄

—

712

—

157

C₄

#19151538

yes

characteristic

158

C₄

#19151884

yes

characteristic

159

C₄

#19151912

yes

characteristic

160

C₄

#19151945

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#26021811

yes

characteristic

	subgroup	G/H
center	C₄	#19151945
commutator, cosocle	C₂²×C₈×C₈₉	C₂³
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1375	C₂
socle	C₂×C₈₉	#2099
prime radical	C₂³×C₈₉	#137
hypercenter	#1375	D₈₉
hypocenter	C₈₉	#13540
LCS	G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

subgroup

G/H

center

C₄

#19151945

commutator, cosocle

C₂²×C₈×C₈₉

C₂³

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1375

C₂

C₂×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

N	H ≅ G/N	type
88. C₈₉	#13540	semidirect

H ≅ G/N

type

88. C₈₉

#13540

semidirect

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

subgroup	index
#26021005	2
#26020427	2
#26018367	2
#26018364	2
C₈₉×#1375	2
#13540	89

subgroup

index

C₈₉×#1375

Subgroups — #36474564

Conjugacy classes of subgroups

Complements

Maximal subgroups