Group #36385142 — subgroups

Subgroups — C₈₉×#7294 #36385142

Order 22784.

190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₈₉×#2163	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#432×C₈₉	—	2	1	no	—
5	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
6	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
7	C₄⋊C₁₆×C₈₉	—	2	1	no	—
8	C₂²⋊C₁₆×C₈₉	C₄	1	2	yes	—
9	#346×C₈₉	—	2	1	no	—
10	#335×C₈₉	—	2	1	no	—
11	#334×C₈₉	—	2	1	no	—
12	#334×C₈₉	C₄	1	2	yes	—
13	#334×C₈₉	C₄	1	2	yes	—
14	C₂²×C₈×C₈₉	—	4	1	no	—
15	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₈₉	—	2	2	no	—
19	C₄⋊C₈×C₈₉	—	2	2	no	—
20	C₄⋊C₈×C₈₉	—	2	2	no	—
21	C₄⋊C₈×C₈₉	—	4	1	no	—
22	C₂²⋊C₈×C₈₉	—	4	1	no	—
23	#97×C₈₉	—	4	1	no	—
24	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	—	4	1	no	—
33	C₂×C₈×C₈₉	—	4	2	no	—
34	C₂×C₈×C₈₉	—	4	2	no	—
35	C₂×C₈×C₈₉	—	8	2	no	—
36	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₈₉	D₈	1	1	yes	characteristic
43	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
44	C₁₆×C₈₉	—	4	2	no	—
45	C₂³×C₈₉	#106	1	1	yes	characteristic
46	C₂×C₄×C₈₉	—	2	2	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
56	C₈×C₈₉	—	2	1	no	—
57	C₈×C₈₉	—	2	1	no	—
58	C₈×C₈₉	—	2	1	no	—
59	C₈×C₈₉	—	8	2	no	—
60	C₈×C₈₉	—	8	2	no	—
61	C₈×C₈₉	#108	1	2	yes	—
62	C₂²×C₈₉	—	2	1	no	—
63	C₂²×C₈₉	—	2	1	no	—
64	C₂²×C₈₉	#340	1	1	yes	characteristic
65	C₂²×C₈₉	#365	1	2	yes	—
66	C₄×C₈₉	—	2	1	no	—
67	C₄×C₈₉	—	2	1	no	—
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	2	1	no	—
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	#341	1	1	yes	characteristic
72	C₄×C₈₉	#344	1	1	yes	characteristic
73	#7294	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₈₉	—	2	2	no	—
75	C₂×C₈₉	#1301	1	1	yes	characteristic
76	C₂×C₈₉	#1383	1	1	yes	characteristic
77	C₂×C₈₉	#1389	1	1	yes	characteristic
78	#2163	C₂×C₈₉	1	1	yes	characteristic
79	#1568	C₂×C₈₉	1	1	yes	characteristic
80	C₂×#334	C₂×C₈₉	1	1	yes	characteristic
81	C₈₉	#7294	1	1	yes	characteristic, direct product factor, Sylow
82	#432	—	2	1	no	—
83	C₂×C₄⋊C₈	—	2	1	no	—
84	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
85	C₄⋊C₁₆	—	2	1	no	—
86	C₂²⋊C₁₆	C₄×C₈₉	1	2	yes	—
87	#346	—	2	1	no	—
88	#335	—	2	1	no	—
89	#334	—	2	1	no	—
90	#334	C₄×C₈₉	1	2	yes	—
91	#334	C₄×C₈₉	1	2	yes	—
92	C₂²×C₈	—	4	1	no	—
93	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
94	C₂²×C₈	Q₈×C₈₉	1	1	yes	characteristic
95	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
96	C₂×C₁₆	—	2	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	4	1	no	—
100	C₂²⋊C₈	—	4	1	no	—
101	#97	—	4	1	no	—
102	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
103	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
104	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
105	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
106	C₂²×C₄	C₄⋊C₄×C₈₉	1	1	yes	characteristic
107	C₂²×C₄	SD₁₆×C₈₉	1	1	yes	characteristic
108	C₂²×C₄	SD₁₆×C₈₉	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	8	2	no	—
114	C₂×C₈	C₄²×C₈₉	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
116	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
117	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
118	C₄²	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
119	C₄²	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
120	C₄²	D₈×C₈₉	1	1	yes	characteristic
121	C₄²	Q₁₆×C₈₉	1	1	yes	characteristic
122	C₁₆	—	4	2	no	—
123	C₂³	#106×C₈₉	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#95×C₈₉	1	1	yes	characteristic
129	C₂×C₄	#102×C₈₉	1	1	yes	characteristic
130	C₂×C₄	#102×C₈₉	1	1	yes	characteristic
131	C₂×C₄	#103×C₈₉	1	1	yes	characteristic
132	C₂×C₄	#103×C₈₉	1	1	yes	characteristic
133	C₂×C₄	#107×C₈₉	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	2	no	—
138	C₈	—	8	2	no	—
139	C₈	#108×C₈₉	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#340×C₈₉	1	1	yes	characteristic
143	C₂²	#365×C₈₉	1	2	yes	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#341×C₈₉	1	1	yes	characteristic
150	C₄	#344×C₈₉	1	1	yes	characteristic
151	C₂	—	2	2	no	—
152	C₂	C₈₉×#1301	1	1	yes	characteristic
153	C₂	C₈₉×#1383	1	1	yes	characteristic
154	C₂	C₈₉×#1389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#2163

C₂

yes

maximal, characteristic

C₈₉×#1568

C₂

yes

maximal, characteristic

C₂×#334×C₈₉

C₂

yes

maximal, characteristic

#432×C₈₉

—

C₂×C₄⋊C₈×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₂²⋊C₁₆×C₈₉

C₄

yes

—

#346×C₈₉

—

#335×C₈₉

—

#334×C₈₉

—

#334×C₈₉

C₄

yes

—

#334×C₈₉

C₄

yes

—

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

SD₁₆

yes

characteristic

C₂²×C₄×C₈₉

SD₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄²

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₄²×C₈₉

D₈

yes

characteristic

C₄²×C₈₉

Q₁₆

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

#106

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#108

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#340

yes

characteristic

C₂²×C₈₉

#365

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

characteristic

#1568

C₂×C₈₉

yes

characteristic

C₂×#334

C₂×C₈₉

yes

characteristic

C₈₉

#7294

yes

characteristic, direct product factor, Sylow

#432

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂²×C₈₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₄×C₈₉

yes

—

—

—

—

C₄×C₈₉

yes

—

#334

C₄×C₈₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₈₉

yes

characteristic

C₂²×C₈

Q₈×C₈₉

yes

characteristic

C₂×C₄²

D₄×C₈₉

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

#97

—

102

C₄×C₈

C₂×C₄×C₈₉

yes

characteristic

103

C₄×C₈

C₂×C₄×C₈₉

yes

characteristic

104

C₄×C₈

D₄×C₈₉

yes

characteristic

105

C₄×C₈

D₄×C₈₉

yes

characteristic

106

C₂²×C₄

C₄⋊C₄×C₈₉

yes

characteristic

107

C₂²×C₄

SD₁₆×C₈₉

yes

characteristic

108

C₂²×C₄

SD₁₆×C₈₉

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄²×C₈₉

yes

characteristic

115

C₂×C₈

C₂²⋊C₄×C₈₉

yes

characteristic

116

C₂×C₈

C₄⋊C₄×C₈₉

yes

characteristic

117

C₂×C₈

C₄⋊C₄×C₈₉

yes

characteristic

118

C₄²

C₂²⋊C₄×C₈₉

yes

characteristic

119

C₄²

C₂²⋊C₄×C₈₉

yes

characteristic

120

C₄²

D₈×C₈₉

yes

characteristic

121

C₄²

Q₁₆×C₈₉

yes

characteristic

122

C₁₆

—

123

C₂³

#106×C₈₉

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#95×C₈₉

yes

characteristic

129

C₂×C₄

#102×C₈₉

yes

characteristic

130

C₂×C₄

#102×C₈₉

yes

characteristic

131

C₂×C₄

#103×C₈₉

yes

characteristic

132

C₂×C₄

#103×C₈₉

yes

characteristic

133

C₂×C₄

#107×C₈₉

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

#108×C₈₉

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

#340×C₈₉

yes

characteristic

143

C₂²

#365×C₈₉

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#341×C₈₉

yes

characteristic

150

C₄

#344×C₈₉

yes

characteristic

151

C₂

—

152

C₂

C₈₉×#1301

yes

characteristic

153

C₂

C₈₉×#1383

yes

characteristic

154

C₂

C₈₉×#1389

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#340
commutator	C₂×C₈	C₄²×C₈₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₈₉
prime radical	C₂³×C₈₉	#106
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

#340

commutator

C₂×C₈

C₄²×C₈₉

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₈₉

prime radical

C₂³×C₈₉

#106

LCS

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

N	H ≅ G/N	type
73. #7294	C₈₉	direct (N × H)
81. C₈₉	#7294	direct (N × H)

H ≅ G/N

type

73. #7294

C₈₉

direct (N × H)

81. C₈₉

#7294

direct (N × H)

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

subgroup	index
C₈₉×#2163	2
C₈₉×#1568	2
C₂×#334×C₈₉	2
#7294	89

subgroup

index

C₈₉×#2163

C₈₉×#1568

C₂×#334×C₈₉

#7294

Subgroups — C₈₉×#7294 #36385142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7294 #36385142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7294 #36385142