Group #36448130 — subgroups

Subgroups — #36448130

Order 22784.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#26020390	C₂	1	1	yes	maximal, characteristic
2	#26018544	C₂	1	1	yes	maximal, characteristic
3	#26018110	C₂	1	2	yes	maximal
4	#26018110	C₂	1	2	yes	maximal
5	C₈₉×#1846	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	#19151622	C₂²	1	1	yes	characteristic
8	#19151622	C₂²	1	1	yes	characteristic
9	#19151363	—	2	1	no	—
10	#19151363	—	2	1	no	—
11	#19151320	C₄	1	4	yes	complemented
12	C₂×#107×C₈₉	—	2	1	no	—
13	C₂×#106×C₈₉	—	2	1	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#382×C₈₉	C₂²	1	2	yes	—
16	C₂×#13083558	—	2	1	no	—
17	C₂×#13083558	—	2	1	no	—
18	C₄⋊#10623494	—	2	2	no	—
19	C₄⋊#10623494	—	2	2	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₂²⋊#10623494	—	2	2	no	—
22	C₄×#10623494	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
26	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
27	#107×C₈₉	—	4	2	no	—
28	#106×C₈₉	—	4	2	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₈₉	Q₈	1	2	yes	—
31	#95×C₈₉	—	2	4	no	—
32	#13083558	—	4	1	no	—
33	#13083558	—	4	1	no	—
34	#13083558	—	4	1	no	—
35	#13083558	—	4	1	no	—
36	C₂×#10623494	—	2	4	no	—
37	C₂×#10623494	—	4	1	no	—
38	C₂×#10623494	—	4	1	no	—
39	C₂²×C₄×C₈₉	—	4	2	no	—
40	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	#41	1	2	yes	—
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	—	2	1	no	—
45	C₂×C₈×C₈₉	—	2	1	no	—
46	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₄⋊C₄×C₈₉	—	4	2	no	—
49	C₄⋊C₄×C₈₉	—	4	2	no	—
50	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₈₉	#41	1	2	yes	—
53	C₂³×C₈₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	4	2	no	—
55	C₂×C₄×C₈₉	—	8	4	no	—
56	C₂×C₄×C₈₉	—	8	2	no	—
57	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#117	1	2	yes	—
59	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#125	1	2	yes	—
61	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
65	C₈×C₈₉	—	2	2	no	—
66	C₈×C₈₉	—	2	2	no	—
67	#10623494	—	4	4	no	—
68	#10623494	—	8	1	no	—
69	#10623494	—	8	1	no	—
70	C₂²×C₈₉	—	2	2	no	—
71	C₂²×C₈₉	#382	1	1	yes	characteristic
72	C₂²×C₈₉	#490	1	1	yes	characteristic
73	C₂²×C₈₉	#491	1	1	yes	characteristic
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	—	16	2	no	—
77	C₄×C₈₉	#428	1	1	yes	characteristic
78	C₄×C₈₉	#428	1	1	yes	characteristic
79	C₄×C₈₉	#489	1	1	yes	characteristic
80	C₄×C₈₉	#489	1	1	yes	characteristic
81	#8795	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	—	2	1	no	—
83	C₂×C₈₉	—	2	1	no	—
84	C₂×C₈₉	#1833	1	2	yes	—
85	C₂×C₈₉	#1836	1	1	yes	characteristic
86	#1846	D₈₉	1	1	yes	characteristic
87	#1837	—	89	1	no	—
88	#1452	—	89	1	no	—
89	#1286	—	89	2	no	—
90	#1286	—	89	2	no	—
91	C₈₉	#8795	1	1	yes	characteristic, complemented, Sylow
92	#431	—	89	2	no	—
93	C₂×#107	—	2	1	no	—
94	C₂×#106	—	2	1	no	—
95	#423	—	89	1	no	—
96	#423	—	89	1	no	—
97	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
98	#382	C₂×D₈₉	1	2	yes	—
99	#341	—	178	1	no	—
100	#341	—	178	1	no	—
101	#322	—	89	4	no	—
102	C₂×M₁₆	—	178	1	no	—
103	C₂×M₁₆	—	178	1	no	—
104	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
108	#107	—	4	2	no	—
109	#106	—	4	2	no	—
110	C₄⋊C₈	—	178	2	no	—
111	C₄⋊C₈	—	178	2	no	—
112	C₄⋊C₈	—	178	2	no	—
113	C₂²⋊C₈	—	178	2	no	—
114	C₄×C₈	—	89	4	no	—
115	C₄×C₈	Dic₁₇₈	1	2	yes	—
116	C₄×C₈	C₄×D₈₉	1	2	yes	—
117	#95	—	2	4	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
120	C₂²×C₄	#13083573	1	2	yes	—
121	M₁₆	—	356	1	no	—
122	M₁₆	—	356	1	no	—
123	M₁₆	—	356	1	no	—
124	M₁₆	—	356	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	356	1	no	—
130	C₂×C₈	—	356	1	no	—
131	C₂×C₈	—	178	4	no	—
132	C₂×C₈	#13083553	1	2	yes	—
133	C₂×C₈	#13083553	1	2	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
137	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
138	C₄²	#13083575	1	2	yes	—
139	C₂³	#15667999	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	4	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
144	C₂×C₄	#15667970	1	2	yes	—
145	C₂×C₄	#15668001	1	1	yes	characteristic
146	C₂×C₄	#15668003	1	2	yes	—
147	C₂×C₄	#15668038	1	1	yes	characteristic
148	C₂×C₄	#15668038	1	1	yes	characteristic
149	C₂×C₄	#15668043	1	1	yes	characteristic
150	C₂×C₄	#15668043	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	356	4	no	—
154	C₈	—	712	1	no	—
155	C₈	—	712	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#19151463	1	1	yes	characteristic
158	C₂²	#19151962	1	1	yes	characteristic
159	C₂²	#19151966	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	16	2	no	—
163	C₄	#19151638	1	1	yes	characteristic
164	C₄	#19151638	1	1	yes	characteristic
165	C₄	#19151960	1	1	yes	characteristic
166	C₄	#19151960	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#26020375	1	2	yes	—
170	C₂	#26020387	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020390

C₂

yes

maximal, characteristic

#26018544

C₂

yes

maximal, characteristic

#26018110

C₂

yes

maximal

#26018110

C₂

yes

maximal

C₈₉×#1846

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

—

#19151622

C₂²

yes

characteristic

#19151622

C₂²

yes

—

—

C₄

yes

complemented

C₂×#107×C₈₉

—

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#382×C₈₉

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

#106×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#95×C₈₉

—

—

—

—

—

—

—

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

#124

yes

characteristic

C₂×C₄×C₈₉

#125

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1833

yes

—

C₂×C₈₉

#1836

yes

characteristic

#1846

D₈₉

yes

—

—

—

—

C₈₉

#8795

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#382

C₂×D₈₉

yes

—

#341

—

178

—

100

#341

—

178

—

101

#322

—

102

C₂×M₁₆

—

178

—

103

C₂×M₁₆

—

178

—

104

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

105

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₈₉

yes

characteristic

108

#107

—

109

#106

—

110

C₄⋊C₈

—

178

—

111

C₄⋊C₈

—

178

—

112

C₄⋊C₈

—

178

—

113

C₂²⋊C₈

—

178

—

114

C₄×C₈

—

115

C₄×C₈

Dic₁₇₈

yes

—

116

C₄×C₈

C₄×D₈₉

yes

—

117

#95

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

120

C₂²×C₄

#13083573

yes

—

121

M₁₆

—

356

—

122

M₁₆

—

356

—

123

M₁₆

—

356

—

124

M₁₆

—

356

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

356

—

130

C₂×C₈

—

356

—

131

C₂×C₈

—

178

—

132

C₂×C₈

#13083553

yes

—

133

C₂×C₈

#13083553

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

C₂×C₄×D₈₉

yes

characteristic

137

C₄²

C₂×Dic₁₇₈

yes

characteristic

138

C₄²

#13083575

yes

—

139

C₂³

#15667999

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂×#13083553

yes

characteristic

144

C₂×C₄

#15667970

yes

—

145

C₂×C₄

#15668001

yes

characteristic

146

C₂×C₄

#15668003

yes

—

147

C₂×C₄

#15668038

yes

characteristic

148

C₂×C₄

#15668038

yes

characteristic

149

C₂×C₄

#15668043

yes

characteristic

150

C₂×C₄

#15668043

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

356

—

154

C₈

—

712

—

155

C₈

—

712

—

156

C₂²

—

157

C₂²

#19151463

yes

characteristic

158

C₂²

#19151962

yes

characteristic

159

C₂²

#19151966

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#19151638

yes

characteristic

164

C₄

#19151638

yes

characteristic

165

C₄

#19151960

yes

characteristic

166

C₄

#19151960

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#26020375

yes

—

170

C₂

#26020387

yes

characteristic

	subgroup	G/H
center	C₂²	#19151463
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1846	C₂
socle	C₂²×C₈₉	#382
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1846	D₈₉
hypocenter	C₈₉	#8795
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

subgroup

G/H

center

C₂²

#19151463

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1846

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1846

N	H ≅ G/N	type
11. #19151320	C₄	semidirect
91. C₈₉	#8795	semidirect

H ≅ G/N

type

11. #19151320

C₄

semidirect

91. C₈₉

#8795

semidirect

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

subgroup	index
#26020390	2
#26018544	2
#26018110	2
C₈₉×#1846	2
#8795	89

subgroup

index

C₈₉×#1846

Subgroups — #36448130

Conjugacy classes of subgroups

Complements

Maximal subgroups