Group #36470788 — subgroups

Subgroups — #36470788

Order 22784.

342 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 143 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	#26022008	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018333	C₂	1	4	yes	maximal
4	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₂²	1	2	yes	—
6	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
8	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
9	C₄²×Dic₈₉	C₄	1	1	yes	characteristic
10	C₄⋊#13083549	—	2	2	no	—
11	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₈₉	—	2	2	no	—
14	C₂²×#10623494	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₈₉	—	2	2	no	—
16	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
17	#15667952	—	2	4	no	—
18	C₂×#13083549	—	2	2	no	—
19	C₄×#10623494	C₈	1	2	yes	—
20	C₄×#10623494	C₈	1	2	yes	—
21	C₈×Dic₈₉	—	2	4	no	—
22	C₈×Dic₈₉	—	2	2	no	—
23	C₈×Dic₈₉	—	2	2	no	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₈₉	—	2	4	no	—
28	C₂×C₁₆×C₈₉	—	2	2	no	—
29	C₄×C₈×C₈₉	D₄	1	2	yes	—
30	C₄×C₈×C₈₉	Q₈	1	2	yes	—
31	C₂²×Dic₈₉	—	2	2	no	—
32	C₂×#10623494	—	2	2	no	—
33	C₂×#10623494	—	2	2	no	—
34	C₂×#10623494	C₂×C₈	1	4	yes	—
35	C₄×Dic₈₉	—	2	8	no	—
36	C₄×Dic₈₉	—	2	2	no	—
37	C₄×Dic₈₉	M₁₆	1	4	yes	—
38	#13083549	—	4	2	no	—
39	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₈₉	#41	1	2	yes	—
46	C₂×C₈×C₈₉	#41	1	2	yes	—
47	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
50	C₁₆×C₈₉	—	4	2	no	—
51	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	2	4	no	—
53	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#117	1	2	yes	—
57	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
59	C₂×Dic₈₉	—	2	8	no	—
60	C₂×Dic₈₉	—	2	4	no	—
61	C₈×C₈₉	—	2	2	no	—
62	C₈×C₈₉	#124	1	1	yes	characteristic
63	C₈×C₈₉	#124	1	1	yes	characteristic
64	C₈×C₈₉	#125	1	2	yes	—
65	#10623494	—	2	4	no	—
66	C₂²×C₈₉	—	2	2	no	—
67	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₈₉	#432	1	2	yes	—
69	C₄×C₈₉	—	2	2	no	—
70	C₄×C₈₉	#364	1	4	yes	—
71	C₄×C₈₉	#382	1	1	yes	characteristic
72	C₄×C₈₉	#423	1	1	yes	characteristic
73	C₄×C₈₉	#431	1	2	yes	—
74	Dic₈₉	—	2	8	no	—
75	#12883	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	—	2	2	no	—
77	C₂×C₈₉	#1831	1	1	yes	characteristic
78	C₂×C₈₉	C₂×#364	1	2	yes	—
79	#2150	—	89	1	no	—
80	#2150	D₈₉	1	1	yes	characteristic
81	C₄²×C₈	—	89	1	no	—
82	#1376	—	89	4	no	—
83	C₈₉	#12883	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₃₂	—	89	2	no	—
85	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
86	C₂×C₄×C₈	—	89	1	no	—
87	C₂×C₄×C₈	—	89	2	no	—
88	C₂×C₄×C₈	—	89	2	no	—
89	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
90	C₄³	—	89	1	no	—
91	C₄⋊C₁₆	—	2	2	no	—
92	C₄⋊C₁₆	—	178	2	no	—
93	C₂²×C₈	—	89	2	no	—
94	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
96	C₂×C₄²	—	89	2	no	—
97	C₂×C₄²	—	178	2	no	—
98	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
99	M₃₂	—	2	4	no	—
100	M₃₂	—	178	4	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	178	2	no	—
103	C₄×C₈	—	89	2	no	—
104	C₄×C₈	—	89	2	no	—
105	C₄×C₈	—	178	4	no	—
106	C₄×C₈	—	178	2	no	—
107	C₄×C₈	—	178	2	no	—
108	C₄×C₈	Dic₁₇₈	1	2	yes	—
109	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
110	C₂²×C₄	—	178	2	no	—
111	C₂²×C₄	#13083557	1	2	yes	—
112	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	89	4	no	—
115	C₂×C₈	—	178	2	no	—
116	C₂×C₈	—	178	2	no	—
117	C₂×C₈	#13083553	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
119	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
120	C₂×C₈	#13083573	1	2	yes	—
121	C₂×C₈	#13083575	1	2	yes	—
122	C₄²	—	89	4	no	—
123	C₄²	—	178	8	no	—
124	C₄²	—	178	2	no	—
125	C₄²	#13083553	1	2	yes	—
126	C₄²	C₈×D₈₉	1	1	yes	characteristic
127	C₄²	C₈×D₈₉	1	1	yes	characteristic
128	C₁₆	—	4	2	no	—
129	C₁₆	—	356	2	no	—
130	C₂³	C₂×#13083557	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	178	8	no	—
133	C₂×C₄	—	178	4	no	—
134	C₂×C₄	#15667941	1	2	yes	—
135	C₂×C₄	#15667941	1	2	yes	—
136	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
137	C₂×C₄	#15667970	1	2	yes	—
138	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
139	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
140	C₈	—	2	2	no	—
141	C₈	—	178	4	no	—
142	C₈	#15667999	1	1	yes	characteristic
143	C₈	#15668001	1	1	yes	characteristic
144	C₈	#15668003	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂×#15667941	1	1	yes	characteristic
147	C₂²	#19151654	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	178	8	no	—
150	C₄	#19151418	1	4	yes	—
151	C₄	#19151463	1	1	yes	characteristic
152	C₄	#19151621	1	1	yes	characteristic
153	C₄	#19151650	1	2	yes	—
154	C₂	—	2	2	no	—
155	C₂	#26020368	1	1	yes	characteristic
156	C₂	C₂×#19151418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022008

C₂

yes

maximal, characteristic

C₄×C₈×Dic₈₉

C₂

yes

maximal, characteristic

#26018333

C₂

yes

maximal

C₈₉×#2150

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₂×C₄×#10623494

C₄

yes

characteristic

C₂×C₈×Dic₈₉

C₄

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

—

C₄²×Dic₈₉

C₄

yes

characteristic

C₄⋊#13083549

—

C₂×M₃₂×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₂²×#10623494

C₂×C₄

yes

—

C₂×C₄×Dic₈₉

—

C₂×C₄×Dic₈₉

C₂×C₄

yes

—

#15667952

—

C₂×#13083549

—

C₄×#10623494

C₈

yes

—

C₄×#10623494

C₈

yes

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

C₈×Dic₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

C₂×C₈

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

M₁₆

yes

—

#13083549

—

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₈₉

#432

yes

—

C₄×C₈₉

—

C₄×C₈₉

#364

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

#12883

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1831

yes

characteristic

C₂×C₈₉

C₂×#364

yes

—

#2150

—

#2150

D₈₉

yes

characteristic

C₄²×C₈

—

#1376

—

C₈₉

#12883

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

C₄×D₈₉

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

178

—

C₂×C₄²

C₄×D₈₉

yes

characteristic

M₃₂

—

100

M₃₂

—

178

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

178

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

178

—

106

C₄×C₈

—

178

—

107

C₄×C₈

—

178

—

108

C₄×C₈

Dic₁₇₈

yes

—

109

C₄×C₈

C₂²⋊D₈₉

yes

—

110

C₂²×C₄

—

178

—

111

C₂²×C₄

#13083557

yes

—

112

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

178

—

116

C₂×C₈

—

178

—

117

C₂×C₈

#13083553

yes

—

118

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

119

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

120

C₂×C₈

#13083573

yes

—

121

C₂×C₈

#13083575

yes

—

122

C₄²

—

123

C₄²

—

178

—

124

C₄²

—

178

—

125

C₄²

#13083553

yes

—

126

C₄²

C₈×D₈₉

yes

characteristic

127

C₄²

C₈×D₈₉

yes

characteristic

128

C₁₆

—

129

C₁₆

—

356

—

130

C₂³

C₂×#13083557

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

178

—

133

C₂×C₄

—

178

—

134

C₂×C₄

#15667941

yes

—

135

C₂×C₄

#15667941

yes

—

136

C₂×C₄

C₂×#13083553

yes

characteristic

137

C₂×C₄

#15667970

yes

—

138

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

139

C₂×C₄

M₁₆×D₈₉

yes

—

140

C₈

—

141

C₈

—

178

—

142

C₈

#15667999

yes

characteristic

143

C₈

#15668001

yes

characteristic

144

C₈

#15668003

yes

—

145

C₂²

—

146

C₂²

C₂×#15667941

yes

characteristic

147

C₂²

#19151654

yes

—

148

C₄

—

149

C₄

—

178

—

150

C₄

#19151418

yes

—

151

C₄

#19151463

yes

characteristic

152

C₄

#19151621

yes

characteristic

153

C₄

#19151650

yes

—

154

C₂

—

155

C₂

#26020368

yes

characteristic

156

C₂

C₂×#19151418

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2150	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#12883
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#2150

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₈

prime radical

C₂³×C₈₉

C₂×M₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
83. C₈₉	#12883	semidirect

H ≅ G/N

type

83. C₈₉

#12883

semidirect

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

subgroup	index
#26022008	2
C₄×C₈×Dic₈₉	2
#26018333	2
C₈₉×#2150	2
#12883	89

subgroup

index

#26022008

C₄×C₈×Dic₈₉

#26018333

C₈₉×#2150

#12883

Subgroups — #36470788

Conjugacy classes of subgroups

Complements

Maximal subgroups