Group #36385315 — subgroups

Subgroups — C₈₉×#7467 #36385315

Order 22784.

190 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 118 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×M₆₄×C₈₉	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₈₉	C₄	1	1	yes	characteristic
4	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
6	M₆₄×C₈₉	—	2	2	no	—
7	M₆₄×C₈₉	C₄	1	4	yes	complemented
8	C₂×C₃₂×C₈₉	—	2	2	no	—
9	C₄×C₁₆×C₈₉	—	2	2	no	—
10	C₂²×C₈×C₈₉	C₈	1	1	yes	characteristic
11	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄²×C₈₉	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₈₉	—	2	1	no	—
14	C₂×C₁₆×C₈₉	—	2	1	no	—
15	C₂×C₁₆×C₈₉	C₈	1	2	yes	—
16	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
18	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
20	C₄×C₈×C₈₉	—	2	1	no	—
21	C₄×C₈×C₈₉	—	2	1	no	—
22	C₃₂×C₈₉	—	2	4	no	—
23	C₂²×C₄×C₈₉	—	2	1	no	—
24	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₈₉	—	2	1	no	—
26	C₂×C₈×C₈₉	—	2	1	no	—
27	C₂×C₈×C₈₉	C₁₆	1	2	yes	—
28	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
34	C₄²×C₈₉	—	2	1	no	—
35	C₄²×C₈₉	C₁₆	1	2	yes	—
36	C₁₆×C₈₉	—	2	2	no	—
37	C₁₆×C₈₉	C₂²⋊C₄	1	2	yes	—
38	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
39	C₂³×C₈₉	M₃₂	1	1	yes	characteristic
40	C₂×C₄×C₈₉	—	2	4	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₈₉	M₃₂	1	1	yes	characteristic
48	C₈×C₈₉	—	2	2	no	—
49	C₈×C₈₉	#95	1	1	yes	characteristic
50	C₈×C₈₉	#97	1	1	yes	characteristic
51	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
52	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
53	C₂²×C₈₉	—	2	2	no	—
54	C₂²×C₈₉	#346	1	1	yes	characteristic
55	C₂²×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₂²×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
57	C₄×C₈₉	—	2	4	no	—
58	C₄×C₈₉	#336	1	1	yes	characteristic
59	C₄×C₈₉	C₄×C₁₆	1	1	yes	characteristic
60	C₄×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
61	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
62	#7467	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₈₉	—	2	1	no	—
64	C₂×C₈₉	—	2	1	no	—
65	C₂×C₈₉	#1374	1	1	yes	characteristic
66	C₂×C₈₉	#1422	1	2	yes	—
67	C₂×M₆₄	C₂×C₈₉	1	2	yes	—
68	C₂×C₄×C₁₆	C₂×C₈₉	1	1	yes	characteristic
69	C₈₉	#7467	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×C₁₆	C₄×C₈₉	1	1	yes	characteristic
71	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
72	C₂×C₄×C₈	C₄×C₈₉	1	1	yes	characteristic
73	M₆₄	—	2	2	no	—
74	M₆₄	C₄×C₈₉	1	4	yes	complemented
75	C₂×C₃₂	—	2	2	no	—
76	C₄×C₁₆	—	2	2	no	—
77	C₂²×C₈	C₈×C₈₉	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
79	C₂×C₄²	C₈×C₈₉	1	1	yes	characteristic
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	C₈×C₈₉	1	2	yes	—
83	C₂×C₁₆	C₂×C₄×C₈₉	1	2	yes	—
84	C₂×C₁₆	D₄×C₈₉	1	1	yes	characteristic
85	C₂×C₁₆	D₄×C₈₉	1	2	yes	—
86	C₂×C₁₆	Q₈×C₈₉	1	1	yes	characteristic
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₃₂	—	2	4	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	C₂×C₈×C₈₉	1	1	yes	characteristic
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	C₁₆×C₈₉	1	2	yes	—
95	C₂×C₈	C₄²×C₈₉	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
98	C₂×C₈	C₂×C₈×C₈₉	1	1	yes	characteristic
99	C₂×C₈	M₁₆×C₈₉	1	1	yes	characteristic
100	C₂×C₈	M₁₆×C₈₉	1	1	yes	characteristic
101	C₄²	—	2	1	no	—
102	C₄²	C₁₆×C₈₉	1	2	yes	—
103	C₁₆	—	2	2	no	—
104	C₁₆	C₂²⋊C₄×C₈₉	1	2	yes	—
105	C₁₆	C₄⋊C₄×C₈₉	1	2	yes	—
106	C₂³	M₃₂×C₈₉	1	1	yes	characteristic
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	C₄×C₈×C₈₉	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
111	C₂×C₄	C₄⋊C₈×C₈₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₁₆×C₈₉	1	1	yes	characteristic
113	C₂×C₄	C₂×C₁₆×C₈₉	1	1	yes	characteristic
114	C₂×C₄	M₃₂×C₈₉	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	#95×C₈₉	1	1	yes	characteristic
117	C₈	#97×C₈₉	1	1	yes	characteristic
118	C₈	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
119	C₈	C₄⋊C₈×C₈₉	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#346×C₈₉	1	1	yes	characteristic
122	C₂²	C₂²⋊C₁₆×C₈₉	1	1	yes	characteristic
123	C₂²	C₄⋊C₁₆×C₈₉	1	1	yes	characteristic
124	C₄	—	2	4	no	—
125	C₄	#336×C₈₉	1	1	yes	characteristic
126	C₄	C₄×C₁₆×C₈₉	1	1	yes	characteristic
127	C₄	C₂²⋊C₁₆×C₈₉	1	1	yes	characteristic
128	C₄	C₄⋊C₁₆×C₈₉	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	C₈₉×#1374	1	1	yes	characteristic
132	C₂	C₈₉×#1422	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₆₄×C₈₉

C₂

yes

maximal

C₂×C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₈₉

C₄

yes

characteristic

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₄

yes

characteristic

M₆₄×C₈₉

—

M₆₄×C₈₉

C₄

yes

complemented

C₂×C₃₂×C₈₉

—

C₄×C₁₆×C₈₉

—

C₂²×C₈×C₈₉

C₈

yes

characteristic

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₈

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₈

yes

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

Q₈

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₃₂×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₁₆

yes

—

C₂×C₈×C₈₉

C₄²

yes

characteristic

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

M₁₆

yes

characteristic

C₂×C₈×C₈₉

M₁₆

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₁₆

yes

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂²⋊C₄

yes

—

C₁₆×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

M₃₂

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

M₃₂

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

C₂²⋊C₈

yes

characteristic

C₈×C₈₉

C₄⋊C₈

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

#346

yes

characteristic

C₂²×C₈₉

C₂²⋊C₁₆

yes

characteristic

C₂²×C₈₉

C₄⋊C₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#336

yes

characteristic

C₄×C₈₉

C₄×C₁₆

yes

characteristic

C₄×C₈₉

C₂²⋊C₁₆

yes

characteristic

C₄×C₈₉

C₄⋊C₁₆

yes

characteristic

#7467

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1374

yes

characteristic

C₂×C₈₉

#1422

yes

—

C₂×M₆₄

C₂×C₈₉

yes

—

C₂×C₄×C₁₆

C₂×C₈₉

yes

characteristic

C₈₉

#7467

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₄×C₈₉

yes

characteristic

C₂²×C₁₆

C₂²×C₈₉

yes

characteristic

C₂×C₄×C₈

C₄×C₈₉

yes

characteristic

M₆₄

—

M₆₄

C₄×C₈₉

yes

complemented

C₂×C₃₂

—

C₄×C₁₆

—

C₂²×C₈

C₈×C₈₉

yes

characteristic

C₂²×C₈

C₂×C₄×C₈₉

yes

characteristic

C₂×C₄²

C₈×C₈₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₈×C₈₉

yes

—

C₂×C₁₆

C₂×C₄×C₈₉

yes

—

C₂×C₁₆

D₄×C₈₉

yes

characteristic

C₂×C₁₆

D₄×C₈₉

yes

—

C₂×C₁₆

Q₈×C₈₉

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₂×C₈×C₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₁₆×C₈₉

yes

—

C₂×C₈

C₄²×C₈₉

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₈₉

yes

characteristic

C₂×C₈

C₄⋊C₄×C₈₉

yes

characteristic

C₂×C₈

C₂×C₈×C₈₉

yes

characteristic

C₂×C₈

M₁₆×C₈₉

yes

characteristic

100

C₂×C₈

M₁₆×C₈₉

yes

characteristic

101

C₄²

—

102

C₄²

C₁₆×C₈₉

yes

—

103

C₁₆

—

104

C₁₆

C₂²⋊C₄×C₈₉

yes

—

105

C₁₆

C₄⋊C₄×C₈₉

yes

—

106

C₂³

M₃₂×C₈₉

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄×C₈×C₈₉

yes

characteristic

110

C₂×C₄

C₂²⋊C₈×C₈₉

yes

characteristic

111

C₂×C₄

C₄⋊C₈×C₈₉

yes

characteristic

112

C₂×C₄

C₂×C₁₆×C₈₉

yes

characteristic

113

C₂×C₄

C₂×C₁₆×C₈₉

yes

characteristic

114

C₂×C₄

M₃₂×C₈₉

yes

characteristic

115

C₈

—

116

C₈

#95×C₈₉

yes

characteristic

117

C₈

#97×C₈₉

yes

characteristic

118

C₈

C₂²⋊C₈×C₈₉

yes

characteristic

119

C₈

C₄⋊C₈×C₈₉

yes

characteristic

120

C₂²

—

121

C₂²

#346×C₈₉

yes

characteristic

122

C₂²

C₂²⋊C₁₆×C₈₉

yes

characteristic

123

C₂²

C₄⋊C₁₆×C₈₉

yes

characteristic

124

C₄

—

125

C₄

#336×C₈₉

yes

characteristic

126

C₄

C₄×C₁₆×C₈₉

yes

characteristic

127

C₄

C₂²⋊C₁₆×C₈₉

yes

characteristic

128

C₄

C₄⋊C₁₆×C₈₉

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

C₈₉×#1374

yes

characteristic

132

C₂

C₈₉×#1422

yes

—

	subgroup	G/H
center	C₂×C₁₆×C₈₉	D₄
commutator	C₄	C₄×C₁₆×C₈₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₈₉
socle	C₂²×C₈₉	C₄⋊C₁₆
prime radical	C₂³×C₈₉	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₈₉ ◁ C₂²×C₁₆×C₈₉ ◁ G

subgroup

G/H

center

C₂×C₁₆×C₈₉

D₄

commutator

C₄

C₄×C₁₆×C₈₉

Frattini, cosocle

C₂²×C₁₆

C₂²×C₈₉

socle

C₂²×C₈₉

C₄⋊C₁₆

prime radical

C₂³×C₈₉

M₃₂

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₆×C₈₉ ◁ C₂²×C₁₆×C₈₉ ◁ G

N	H ≅ G/N	type
7. M₆₄×C₈₉	C₄	semidirect
62. #7467	C₈₉	direct (N × H)
69. C₈₉	#7467	direct (N × H)
74. M₆₄	C₄×C₈₉	semidirect

H ≅ G/N

type

7. M₆₄×C₈₉

C₄

semidirect

62. #7467

C₈₉

direct (N × H)

69. C₈₉

#7467

direct (N × H)

74. M₆₄

C₄×C₈₉

semidirect

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

subgroup	index
C₂×M₆₄×C₈₉	2
C₂×C₄×C₁₆×C₈₉	2
#7467	89

subgroup

index

C₂×M₆₄×C₈₉

C₂×C₄×C₁₆×C₈₉

#7467

Subgroups — C₈₉×#7467 #36385315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7467 #36385315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7467 #36385315