Group #36384991 — subgroups

Subgroups — C₈₉×#7143 #36384991

Order 22784.

234 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 102 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	2	yes	maximal
2	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
4	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
5	C₄³×C₈₉	C₄	1	1	yes	characteristic
6	C₄⋊C₁₆×C₈₉	—	2	2	no	—
7	C₂²⋊C₁₆×C₈₉	—	2	2	no	—
8	#346×C₈₉	C₄	1	4	yes	complemented
9	#336×C₈₉	—	2	2	no	—
10	C₂²×C₈×C₈₉	—	2	1	no	—
11	C₂²×C₈×C₈₉	D₄	1	2	yes	—
12	C₂×C₄²×C₈₉	—	2	1	no	—
13	C₂×C₄²×C₈₉	—	2	1	no	—
14	C₂×C₄²×C₈₉	C₈	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₈₉	—	2	4	no	—
17	C₄×C₈×C₈₉	—	2	1	no	—
18	C₄×C₈×C₈₉	C₈	1	2	yes	—
19	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
21	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₈₉	—	2	2	no	—
23	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	—	4	2	no	—
28	C₂×C₈×C₈₉	—	4	2	no	—
29	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
30	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
31	C₄²×C₈₉	—	2	4	no	—
32	C₄²×C₈₉	—	4	2	no	—
33	C₄²×C₈₉	—	4	2	no	—
34	C₄²×C₈₉	C₄²	1	1	yes	characteristic
35	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
38	C₁₆×C₈₉	—	4	4	no	—
39	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	—	2	4	no	—
44	C₂×C₄×C₈₉	—	4	2	no	—
45	C₂×C₄×C₈₉	—	4	2	no	—
46	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
52	C₈×C₈₉	—	2	2	no	—
53	C₈×C₈₉	—	2	2	no	—
54	C₈×C₈₉	—	4	2	no	—
55	C₂²×C₈₉	—	2	1	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	#336	1	1	yes	characteristic
58	C₂²×C₈₉	#349	1	2	yes	—
59	C₄×C₈₉	—	2	1	no	—
60	C₄×C₈₉	—	2	1	no	—
61	C₄×C₈₉	—	2	1	no	—
62	C₄×C₈₉	—	2	1	no	—
63	C₄×C₈₉	—	4	4	no	—
64	C₄×C₈₉	H(3,ℤ/4)	1	2	yes	—
65	C₄×C₈₉	#364	1	2	yes	—
66	#7143	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂×C₈₉	—	2	2	no	—
68	C₂×C₈₉	#1274	1	1	yes	characteristic
69	C₂×C₈₉	#1376	1	1	yes	characteristic
70	C₂×C₈₉	#1377	1	1	yes	characteristic
71	#2163	C₂×C₈₉	1	2	yes	—
72	#1744	C₂×C₈₉	1	1	yes	characteristic
73	C₈₉	#7143	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₄×C₈	C₄×C₈₉	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
76	C₄³	C₄×C₈₉	1	1	yes	characteristic
77	C₄⋊C₁₆	—	2	2	no	—
78	C₂²⋊C₁₆	—	2	2	no	—
79	#346	C₄×C₈₉	1	4	yes	complemented
80	#336	—	2	2	no	—
81	C₂²×C₈	—	2	1	no	—
82	C₂²×C₈	D₄×C₈₉	1	2	yes	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₈×C₈₉	1	2	yes	—
86	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	C₈×C₈₉	1	2	yes	—
90	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
91	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
92	C₄×C₈	Q₈×C₈₉	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
95	C₂²×C₄	C₂×C₈×C₈₉	1	1	yes	characteristic
96	C₂²×C₄	M₁₆×C₈₉	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
101	C₂×C₈	C₄⋊C₄×C₈₉	1	2	yes	—
102	C₄²	—	2	4	no	—
103	C₄²	—	4	2	no	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄²×C₈₉	1	1	yes	characteristic
106	C₄²	C₄⋊C₄×C₈₉	1	1	yes	characteristic
107	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
108	C₄²	M₁₆×C₈₉	1	1	yes	characteristic
109	C₁₆	—	4	4	no	—
110	C₂³	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	#95×C₈₉	1	1	yes	characteristic
118	C₂×C₄	C₄×C₈×C₈₉	1	1	yes	characteristic
119	C₂×C₄	#97×C₈₉	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
121	C₂×C₄	C₄⋊C₈×C₈₉	1	1	yes	characteristic
122	C₂×C₄	C₄⋊C₈×C₈₉	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	#336×C₈₉	1	1	yes	characteristic
129	C₂²	#349×C₈₉	1	2	yes	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	4	4	no	—
135	C₄	H(3,ℤ/4)×C₈₉	1	2	yes	—
136	C₄	#364×C₈₉	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₈₉×#1274	1	1	yes	characteristic
139	C₂	C₈₉×#1376	1	1	yes	characteristic
140	C₂	C₈₉×#1377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#2163

C₂

yes

maximal

C₈₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₈₉

C₄

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄³×C₈₉

C₄

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₂²⋊C₁₆×C₈₉

—

#346×C₈₉

C₄

yes

complemented

#336×C₈₉

—

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₈

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₈

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

Q₈

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

M₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₄²

yes

characteristic

C₄²×C₈₉

C₄⋊C₄

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#95

yes

characteristic

C₂×C₄×C₈₉

C₄×C₈

yes

characteristic

C₂×C₄×C₈₉

#97

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₈

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#336

yes

characteristic

C₂²×C₈₉

#349

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

H(3,ℤ/4)

yes

—

C₄×C₈₉

#364

yes

—

#7143

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#1744

C₂×C₈₉

yes

characteristic

C₈₉

#7143

yes

characteristic, direct product factor, Sylow

C₂×C₄×C₈

C₄×C₈₉

yes

characteristic

C₂×C₄×C₈

C₂²×C₈₉

yes

characteristic

C₄³

C₄×C₈₉

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

C₄×C₈₉

yes

complemented

#336

—

C₂²×C₈

—

C₂²×C₈

D₄×C₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₈×C₈₉

yes

—

C₂×C₄²

C₂×C₄×C₈₉

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

C₈×C₈₉

yes

—

C₄×C₈

C₂×C₄×C₈₉

yes

—

C₄×C₈

D₄×C₈₉

yes

characteristic

C₄×C₈

Q₈×C₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

characteristic

C₂²×C₄

C₂×C₈×C₈₉

yes

characteristic

C₂²×C₄

M₁₆×C₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂²⋊C₄×C₈₉

yes

—

101

C₂×C₈

C₄⋊C₄×C₈₉

yes

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₄²×C₈₉

yes

characteristic

106

C₄²

C₄⋊C₄×C₈₉

yes

characteristic

107

C₄²

C₂×C₈×C₈₉

yes

characteristic

108

C₄²

M₁₆×C₈₉

yes

characteristic

109

C₁₆

—

110

C₂³

C₂²⋊C₈×C₈₉

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#95×C₈₉

yes

characteristic

118

C₂×C₄

C₄×C₈×C₈₉

yes

characteristic

119

C₂×C₄

#97×C₈₉

yes

characteristic

120

C₂×C₄

C₂²⋊C₈×C₈₉

yes

characteristic

121

C₂×C₄

C₄⋊C₈×C₈₉

yes

characteristic

122

C₂×C₄

C₄⋊C₈×C₈₉

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

—

127

C₂²

—

128

C₂²

#336×C₈₉

yes

characteristic

129

C₂²

#349×C₈₉

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

H(3,ℤ/4)×C₈₉

yes

—

136

C₄

#364×C₈₉

yes

—

137

C₂

—

138

C₂

C₈₉×#1274

yes

characteristic

139

C₂

C₈₉×#1376

yes

characteristic

140

C₂

C₈₉×#1377

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₈₉	#95
commutator	C₂×C₄	C₄×C₈×C₈₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₈₉
socle	C₂²×C₈₉	#336
prime radical	C₂³×C₈₉	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₈₉

#95

commutator

C₂×C₄

C₄×C₈×C₈₉

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₈₉

socle

C₂²×C₈₉

#336

prime radical

C₂³×C₈₉

C₂²⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

N	H ≅ G/N	type
8. #346×C₈₉	C₄	semidirect
66. #7143	C₈₉	direct (N × H)
73. C₈₉	#7143	direct (N × H)
79. #346	C₄×C₈₉	semidirect

H ≅ G/N

type

8. #346×C₈₉

C₄

semidirect

66. #7143

C₈₉

direct (N × H)

73. C₈₉

#7143

direct (N × H)

79. #346

C₄×C₈₉

semidirect

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

subgroup	index
C₈₉×#2163	2
C₈₉×#1744	2
#7143	89

subgroup

index

C₈₉×#2163

C₈₉×#1744

#7143

Subgroups — C₈₉×#7143 #36384991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7143 #36384991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7143 #36384991