Group #36470847 — subgroups

Subgroups — #36470847

Order 22784.

342 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 159 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×#19151418	C₂	1	4	yes	maximal
2	#26022008	C₂	1	1	yes	maximal, characteristic
3	#26019938	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₂²	1	2	yes	—
6	C₂×#15667923	C₄	1	1	yes	characteristic
7	C₂×#15667922	C₄	1	2	yes	—
8	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
9	#19151444	C₄	1	1	yes	characteristic
10	#19151418	—	2	8	no	—
11	C₄⋊#13083549	—	2	2	no	—
12	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₈₉	—	2	2	no	—
15	C₂²×#10623494	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
17	#15667952	—	2	4	no	—
18	C₂×#13083549	—	2	2	no	—
19	#15667923	C₈	1	4	yes	—
20	#15667922	—	2	4	no	—
21	C₈×Dic₈₉	D₄	1	4	yes	—
22	C₈×Dic₈₉	Q₈	1	4	yes	—
23	#15667919	—	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	C₄⋊C₄	1	4	yes	—
33	C₂×#10623494	C₂×C₈	1	4	yes	—
34	C₂×#10623494	M₁₆	1	4	yes	—
35	C₄×Dic₈₉	—	2	2	no	—
36	C₄×Dic₈₉	C₄⋊C₄	1	4	yes	—
37	#13083549	—	4	2	no	—
38	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
39	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
43	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₄²×C₈₉	C₂×C₈	1	2	yes	—
48	C₁₆×C₈₉	—	4	2	no	—
49	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	4	2	no	—
51	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
56	C₂×Dic₈₉	—	2	4	no	—
57	C₂×Dic₈₉	—	4	4	no	—
58	C₈×C₈₉	—	2	2	no	—
59	C₈×C₈₉	#127	1	1	yes	characteristic
60	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
61	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
62	#10623494	C₄⋊C₈	1	8	yes	—
63	C₂²×C₈₉	—	2	2	no	—
64	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂²×C₈₉	#423	1	2	yes	—
66	C₄×C₈₉	—	2	4	no	—
67	C₄×C₈₉	#384	1	1	yes	characteristic
68	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
69	C₄×C₈₉	#423	1	1	yes	characteristic
70	Dic₈₉	—	4	4	no	—
71	#12901	—	89	1	no	maximal, Sylow
72	C₂×C₈₉	—	2	2	no	—
73	C₂×C₈₉	#1842	1	1	yes	characteristic
74	C₂×C₈₉	#2153	1	2	yes	—
75	C₂×#364	—	89	4	no	—
76	#2150	—	89	1	no	—
77	#2150	D₈₉	1	1	yes	characteristic
78	#1726	—	89	1	no	—
79	C₈₉	#12901	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	89	2	no	—
81	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
82	C₂×#97	—	89	1	no	—
83	C₂×#97	—	89	2	no	—
84	C₂×C₄×C₈	—	89	2	no	—
85	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
86	#376	—	89	1	no	—
87	#364	—	178	8	no	—
88	C₄⋊C₁₆	—	2	2	no	—
89	C₄⋊C₁₆	—	178	2	no	—
90	C₂²×C₈	—	89	2	no	—
91	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
93	C₂×C₄²	—	89	2	no	—
94	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	M₃₂	—	178	4	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	178	2	no	—
99	#97	—	89	4	no	—
100	#97	—	178	4	no	—
101	C₄×C₈	—	89	4	no	—
102	C₄×C₈	—	89	4	no	—
103	C₄×C₈	Dic₁₇₈	1	2	yes	—
104	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
105	#95	—	178	2	no	—
106	C₂²×C₄	—	178	2	no	—
107	C₂²×C₄	#13083557	1	2	yes	—
108	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	89	4	no	—
111	C₂×C₈	—	89	4	no	—
112	C₂×C₈	—	89	4	no	—
113	C₂×C₈	#13083553	1	2	yes	—
114	C₂×C₈	D₄×D₈₉	1	2	yes	—
115	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
116	C₂×C₈	Q₈×D₈₉	1	2	yes	—
117	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
118	C₄²	—	89	4	no	—
119	C₄²	—	178	2	no	—
120	C₄²	#13083553	1	2	yes	—
121	C₄²	C₈×D₈₉	1	2	yes	—
122	C₁₆	—	4	2	no	—
123	C₁₆	—	356	2	no	—
124	C₂³	C₂×#13083557	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	356	4	no	—
127	C₂×C₄	—	178	4	no	—
128	C₂×C₄	#15667941	1	4	yes	—
129	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
130	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
132	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	89	8	no	—
135	C₈	#15668008	1	1	yes	characteristic
136	C₈	#15668010	1	1	yes	characteristic
137	C₈	#15668011	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×#15667941	1	1	yes	characteristic
140	C₂²	#19151620	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	356	4	no	—
143	C₄	#19151467	1	1	yes	characteristic
144	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
145	C₄	#19151621	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#26020403	1	1	yes	characteristic
148	C₂	#26022017	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151418

C₂

yes

maximal

#26022008

C₂

yes

maximal, characteristic

#26019938

C₂

yes

maximal, characteristic

C₈₉×#2150

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₂×#15667923

C₄

yes

characteristic

C₂×#15667922

C₄

yes

—

C₂×C₈×Dic₈₉

C₂²

yes

—

#19151444

C₄

yes

characteristic

#19151418

—

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

yes

—

#15667952

—

C₂×#13083549

—

#15667923

C₈

yes

—

#15667922

—

C₈×Dic₈₉

D₄

yes

—

C₈×Dic₈₉

Q₈

yes

—

#15667919

—

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

C₄⋊C₄

yes

—

C₂×#10623494

C₂×C₈

yes

—

C₂×#10623494

M₁₆

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₄⋊C₄

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

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#127

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

—

#10623494

C₄⋊C₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₈₉

#423

yes

—

C₄×C₈₉

—

C₄×C₈₉

#384

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₈

yes

—

C₄×C₈₉

#423

yes

characteristic

Dic₈₉

—

#12901

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1842

yes

characteristic

C₂×C₈₉

#2153

yes

—

C₂×#364

—

#2150

—

#2150

D₈₉

yes

characteristic

#1726

—

C₈₉

#12901

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₉

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#376

—

#364

—

178

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

C₄×D₈₉

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₈₉

yes

characteristic

M₃₂

—

M₃₂

—

178

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

#97

—

100

#97

—

178

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

Dic₁₇₈

yes

—

104

C₄×C₈

C₂²⋊D₈₉

yes

—

105

#95

—

178

—

106

C₂²×C₄

—

178

—

107

C₂²×C₄

#13083557

yes

—

108

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

#13083553

yes

—

114

C₂×C₈

D₄×D₈₉

yes

—

115

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

116

C₂×C₈

Q₈×D₈₉

yes

—

117

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

118

C₄²

—

119

C₄²

—

178

—

120

C₄²

#13083553

yes

—

121

C₄²

C₈×D₈₉

yes

—

122

C₁₆

—

123

C₁₆

—

356

—

124

C₂³

C₂×#13083557

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

178

—

128

C₂×C₄

#15667941

yes

—

129

C₂×C₄

C₄⋊C₄×D₈₉

yes

—

130

C₂×C₄

C₂×#13083553

yes

characteristic

131

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

132

C₂×C₄

M₁₆×D₈₉

yes

—

133

C₈

—

134

C₈

—

135

C₈

#15668008

yes

characteristic

136

C₈

#15668010

yes

characteristic

137

C₈

#15668011

yes

—

138

C₂²

—

139

C₂²

C₂×#15667941

yes

characteristic

140

C₂²

#19151620

yes

—

141

C₄

—

142

C₄

—

356

—

143

C₄

#19151467

yes

characteristic

144

C₄

C₄⋊C₈×D₈₉

yes

—

145

C₄

#19151621

yes

characteristic

146

C₂

—

147

C₂

#26020403

yes

characteristic

148

C₂

#26022017

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	#12901
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
79. C₈₉	#12901	semidirect

H ≅ G/N

type

79. C₈₉

#12901

semidirect

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

subgroup	index
C₂×#19151418	2
#26022008	2
#26019938	2
C₈₉×#2150	2
#12901	89

subgroup

index

C₈₉×#2150

Subgroups — #36470847

Conjugacy classes of subgroups

Complements

Maximal subgroups