Group #36448106 — subgroups

Subgroups — #36448106

Order 22784.

342 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 147 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	#26020369	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	#26018086	C₂	1	4	yes	maximal
4	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
6	C₂×C₄×#10623494	C₂²	1	2	yes	—
7	#19151353	—	2	2	no	—
8	#19151353	C₄	1	2	yes	—
9	#19151353	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
11	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
12	C₄³×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×#10623494	—	2	2	no	—
14	C₂²×#10623494	C₂×C₄	1	4	yes	—
15	C₄⋊#10623494	—	2	4	no	—
16	C₄×#10623494	—	2	2	no	—
17	C₄×#10623494	—	2	2	no	—
18	C₂²×C₈×C₈₉	—	2	2	no	—
19	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₈₉	D₄	1	2	yes	—
22	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
23	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₈₉	—	2	4	no	—
25	C₄⋊C₈×C₈₉	—	4	4	no	—
26	C₂×#10623494	—	2	4	no	—
27	C₂×#10623494	—	2	4	no	—
28	C₂×#10623494	—	2	4	no	—
29	C₂×#10623494	—	4	2	no	—
30	C₂×#10623494	—	4	2	no	—
31	C₂×#10623494	—	4	2	no	—
32	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₈₉	—	4	2	no	—
38	C₂×C₈×C₈₉	—	4	4	no	—
39	C₄²×C₈₉	—	2	4	no	—
40	C₄²×C₈₉	—	2	2	no	—
41	C₄²×C₈₉	—	2	2	no	—
42	C₄²×C₈₉	—	2	2	no	—
43	C₄²×C₈₉	—	2	2	no	—
44	C₄²×C₈₉	#41	1	2	yes	—
45	C₄²×C₈₉	#41	1	2	yes	—
46	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	—	2	2	no	—
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	#108	1	4	yes	—
54	C₂×C₄×C₈₉	#117	1	2	yes	—
55	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#125	1	2	yes	—
58	C₂×C₄×C₈₉	#131	1	2	yes	—
59	C₂×C₄×C₈₉	#131	1	2	yes	—
60	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₄×C₈₉	#137	1	2	yes	—
62	C₈×C₈₉	—	8	2	no	—
63	#10623494	—	4	4	no	—
64	#10623494	—	8	2	no	—
65	C₂²×C₈₉	#382	1	1	yes	characteristic
66	C₂²×C₈₉	#424	1	1	yes	characteristic
67	C₂²×C₈₉	#424	1	1	yes	characteristic
68	C₂²×C₈₉	#428	1	2	yes	—
69	C₂²×C₈₉	C₂×#108	1	2	yes	—
70	C₄×C₈₉	—	2	4	no	—
71	C₄×C₈₉	—	2	2	no	—
72	C₄×C₈₉	—	4	2	no	—
73	C₄×C₈₉	#433	1	4	yes	—
74	C₄×C₈₉	#489	1	2	yes	—
75	C₄×C₈₉	#490	1	1	yes	characteristic
76	C₄×C₈₉	#491	1	1	yes	characteristic
77	#8789	—	89	1	no	maximal, Sylow
78	C₂×C₈₉	#1573	1	2	yes	—
79	C₂×C₈₉	#1574	1	2	yes	—
80	C₂×C₈₉	#1832	1	1	yes	characteristic
81	C₂×C₈₉	#1836	1	1	yes	characteristic
82	C₂×C₈₉	#1837	1	1	yes	characteristic
83	#1842	D₈₉	1	1	yes	characteristic
84	#1831	—	89	1	no	—
85	#1744	—	89	1	no	—
86	#1277	—	89	4	no	—
87	C₈₉	#8789	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊C₈	—	2	2	no	—
89	C₂×C₄⋊C₈	—	89	2	no	—
90	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
91	C₂×C₄×C₈	—	89	2	no	—
92	C₄³	C₂×D₈₉	1	1	yes	characteristic
93	#336	—	89	2	no	—
94	#336	—	89	2	no	—
95	#336	—	178	2	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	89	4	no	—
98	C₂²×C₈	—	178	2	no	—
99	C₂×C₄²	Dic₁₇₈	1	2	yes	—
100	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
101	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
103	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	4	4	no	—
106	C₄⋊C₈	—	178	4	no	—
107	C₄×C₈	—	178	2	no	—
108	C₄×C₈	—	178	2	no	—
109	C₂²×C₄	#13083553	1	2	yes	—
110	C₂²×C₄	#13083553	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	4	no	—
116	C₂×C₈	—	356	2	no	—
117	C₂×C₈	—	356	2	no	—
118	C₂×C₈	—	356	2	no	—
119	C₂×C₈	—	178	4	no	—
120	C₂×C₈	—	178	4	no	—
121	C₂×C₈	—	178	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	4	no	—
127	C₄²	#13083573	1	2	yes	—
128	C₄²	#13083575	1	2	yes	—
129	C₂³	C₂×#13083553	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#15667947	1	4	yes	—
137	C₂×C₄	#15667970	1	2	yes	—
138	C₂×C₄	#15667999	1	1	yes	characteristic
139	C₂×C₄	#15668001	1	1	yes	characteristic
140	C₂×C₄	#15668003	1	2	yes	—
141	C₂×C₄	#15668018	1	2	yes	—
142	C₂×C₄	#15668020	1	2	yes	—
143	C₂×C₄	#15668038	1	2	yes	—
144	C₂×C₄	#15668043	1	2	yes	—
145	C₈	—	8	2	no	—
146	C₈	—	356	4	no	—
147	C₈	—	712	2	no	—
148	C₂²	#19151463	1	1	yes	characteristic
149	C₂²	#19151624	1	1	yes	characteristic
150	C₂²	#19151626	1	1	yes	characteristic
151	C₂²	#19151638	1	2	yes	—
152	C₂²	C₂×#15667947	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	4	no	—
155	C₄	—	4	2	no	—
156	C₄	#19151658	1	4	yes	—
157	C₄	#19151960	1	2	yes	—
158	C₄	#19151962	1	1	yes	characteristic
159	C₄	#19151966	1	1	yes	characteristic
160	C₂	#26019035	1	2	yes	—
161	C₂	#26019043	1	2	yes	—
162	C₂	#26020371	1	1	yes	characteristic
163	C₂	#26020387	1	1	yes	characteristic
164	C₂	#26020389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020369

C₂

yes

maximal, characteristic

#26019998

C₂

yes

maximal, characteristic

#26018086

C₂

yes

maximal

C₈₉×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

—

#19151353

—

#19151353

C₄

yes

—

#19151353

C₄

yes

—

C₂×C₄⋊C₈×C₈₉

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

C₂²×#10623494

—

C₂²×#10623494

C₂×C₄

yes

—

C₄⋊#10623494

—

C₄×#10623494

—

C₄×#10623494

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

—

—

—

—

—

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

#41

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#108

yes

—

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₉

#137

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

C₂×#108

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#433

yes

—

C₄×C₈₉

#489

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

yes

—

C₂×C₈₉

#1574

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

—

—

—

C₈₉

#8789

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₈₉

yes

—

—

—

178

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

178

—

C₂×C₄²

Dic₁₇₈

yes

—

100

C₂×C₄²

C₄×D₈₉

yes

characteristic

101

C₂×C₄²

C₄×D₈₉

yes

characteristic

102

C₂×C₄²

C₂²⋊D₈₉

yes

—

103

C₂×C₄²

C₂²×D₈₉

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

178

—

107

C₄×C₈

—

178

—

108

C₄×C₈

—

178

—

109

C₂²×C₄

#13083553

yes

—

110

C₂²×C₄

#13083553

yes

—

111

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

112

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

113

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

356

—

117

C₂×C₈

—

356

—

118

C₂×C₈

—

356

—

119

C₂×C₈

—

178

—

120

C₂×C₈

—

178

—

121

C₂×C₈

—

178

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#13083573

yes

—

128

C₄²

#13083575

yes

—

129

C₂³

C₂×#13083553

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#15667947

yes

—

137

C₂×C₄

#15667970

yes

—

138

C₂×C₄

#15667999

yes

characteristic

139

C₂×C₄

#15668001

yes

characteristic

140

C₂×C₄

#15668003

yes

—

141

C₂×C₄

#15668018

yes

—

142

C₂×C₄

#15668020

yes

—

143

C₂×C₄

#15668038

yes

—

144

C₂×C₄

#15668043

yes

—

145

C₈

—

146

C₈

—

356

—

147

C₈

—

712

—

148

C₂²

#19151463

yes

characteristic

149

C₂²

#19151624

yes

characteristic

150

C₂²

#19151626

yes

characteristic

151

C₂²

#19151638

yes

—

152

C₂²

C₂×#15667947

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#19151658

yes

—

157

C₄

#19151960

yes

—

158

C₄

#19151962

yes

characteristic

159

C₄

#19151966

yes

characteristic

160

C₂

#26019035

yes

—

161

C₂

#26019043

yes

—

162

C₂

#26020371

yes

characteristic

163

C₂

#26020387

yes

characteristic

164

C₂

#26020389

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#8789
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₈₉

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1842

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
87. C₈₉	#8789	semidirect

H ≅ G/N

type

87. C₈₉

#8789

semidirect

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

subgroup	index
#26020369	2
#26019998	2
#26018086	2
C₈₉×#1842	2
#8789	89

subgroup

index

C₈₉×#1842

Subgroups — #36448106

Conjugacy classes of subgroups

Complements

Maximal subgroups