Group #36391545 — subgroups

Subgroups — C₈₉×#13697 #36391545

Order 22784.

210 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 74 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×D₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2215	C₂	1	2	yes	maximal, complemented
4	C₈₉×#1424	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1415	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆×C₈₉	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
8	#466×C₈₉	—	2	2	no	—
9	D₃₂×C₈₉	—	2	2	no	—
10	M₆₄×C₈₉	—	2	2	no	—
11	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
13	#366×C₈₉	C₂²	1	2	yes	—
14	#357×C₈₉	—	2	2	no	—
15	C₂×D₈×C₈₉	—	2	2	no	—
16	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
17	#121×C₈₉	—	4	2	no	—
18	D₁₆×C₈₉	—	2	2	no	—
19	D₁₆×C₈₉	—	2	2	no	—
20	C₂×C₁₆×C₈₉	—	2	1	no	—
21	C₂×C₁₆×C₈₉	—	2	1	no	—
22	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
24	#107×C₈₉	—	2	2	no	—
25	#102×C₈₉	—	4	2	no	—
26	C₃₂×C₈₉	—	2	1	no	—
27	C₃₂×C₈₉	D₄	1	2	yes	complemented
28	C₂×D₄×C₈₉	—	4	2	no	—
29	C₂×D₄×C₈₉	—	8	2	no	—
30	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
31	D₈×C₈₉	—	4	2	no	—
32	D₈×C₈₉	—	4	2	no	—
33	C₂×C₈×C₈₉	—	2	1	no	—
34	C₂×C₈×C₈₉	—	2	1	no	—
35	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
36	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₈₉	—	4	2	no	—
38	C₂²⋊C₄×C₈₉	—	8	2	no	—
39	C₁₆×C₈₉	—	2	1	no	—
40	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	#41	1	1	yes	characteristic
42	C₂³×C₈₉	—	8	2	no	—
43	C₂³×C₈₉	D₁₆	1	1	yes	characteristic
44	D₄×C₈₉	—	8	2	no	—
45	D₄×C₈₉	—	8	2	no	—
46	D₄×C₈₉	—	16	4	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	—	8	2	no	—
50	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
52	C₈×C₈₉	—	2	1	no	—
53	C₈×C₈₉	#121	1	1	yes	characteristic
54	C₈×C₈₉	#135	1	1	yes	characteristic
55	C₂²×C₈₉	—	2	1	no	—
56	C₂²×C₈₉	—	2	2	no	—
57	C₂²×C₈₉	—	16	2	no	—
58	C₂²×C₈₉	—	16	2	no	—
59	C₂²×C₈₉	—	16	2	no	—
60	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
61	C₄×C₈₉	—	2	1	no	—
62	C₄×C₈₉	—	16	2	no	—
63	C₄×C₈₉	#466	1	1	yes	characteristic
64	C₄×C₈₉	#508	1	1	yes	characteristic
65	#13697	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₈₉	—	4	1	no	—
67	C₂×C₈₉	—	32	2	no	—
68	C₂×C₈₉	#2215	1	1	yes	characteristic
69	C₂×C₈₉	#2263	1	2	yes	—
70	C₂×D₃₂	C₂×C₈₉	1	1	yes	characteristic, complemented
71	C₂×M₆₄	C₂×C₈₉	1	1	yes	characteristic, complemented
72	#2215	C₂×C₈₉	1	2	yes	complemented
73	#1424	C₂×C₈₉	1	1	yes	characteristic, complemented
74	#1415	C₂×C₈₉	1	2	yes	complemented
75	C₈₉	#13697	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×D₁₆	C₂²×C₈₉	1	2	yes	—
77	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
78	#466	—	2	2	no	—
79	D₃₂	—	2	2	no	—
80	M₆₄	—	2	2	no	—
81	C₂×C₃₂	C₂²×C₈₉	1	1	yes	characteristic
82	C₂×C₃₂	C₂²×C₈₉	1	1	yes	characteristic
83	#366	C₂²×C₈₉	1	2	yes	—
84	#357	—	2	2	no	—
85	C₂×D₈	—	2	2	no	—
86	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
87	#121	—	4	2	no	—
88	D₁₆	—	2	2	no	—
89	D₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	D₄×C₈₉	1	1	yes	characteristic
93	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
94	#107	—	2	2	no	—
95	#102	—	4	2	no	—
96	C₃₂	—	2	1	no	—
97	C₃₂	D₄×C₈₉	1	2	yes	complemented
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	8	2	no	—
100	C₂²×C₄	D₈×C₈₉	1	1	yes	characteristic
101	D₈	—	4	2	no	—
102	D₈	—	4	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	D₈×C₈₉	1	1	yes	characteristic
106	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
107	C₄⋊C₄	—	4	2	no	—
108	C₂²⋊C₄	—	8	2	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	C₂×D₄×C₈₉	1	1	yes	characteristic
111	C₁₆	#41×C₈₉	1	1	yes	characteristic
112	C₂³	—	8	2	no	—
113	C₂³	D₁₆×C₈₉	1	1	yes	characteristic
114	D₄	—	8	2	no	—
115	D₄	—	8	2	no	—
116	D₄	—	16	4	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	D₁₆×C₈₉	1	1	yes	characteristic
121	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	#121×C₈₉	1	1	yes	characteristic
124	C₈	#135×C₈₉	1	1	yes	characteristic
125	C₂²	—	2	1	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	16	2	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	16	2	no	—
130	C₂²	C₂×D₁₆×C₈₉	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	16	2	no	—
133	C₄	#466×C₈₉	1	1	yes	characteristic
134	C₄	#508×C₈₉	1	1	yes	characteristic
135	C₂	—	4	1	no	—
136	C₂	—	32	2	no	—
137	C₂	C₈₉×#2215	1	1	yes	characteristic
138	C₂	C₈₉×#2263	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₃₂×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×M₆₄×C₈₉

C₂

yes

maximal, characteristic, complemented

C₈₉×#2215

C₂

yes

maximal, complemented

C₈₉×#1424

C₂

yes

maximal, characteristic, complemented

C₈₉×#1415

C₂

yes

maximal, complemented

C₂×D₁₆×C₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

#466×C₈₉

—

D₃₂×C₈₉

—

M₆₄×C₈₉

—

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#366×C₈₉

C₂²

yes

—

#357×C₈₉

—

C₂×D₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

#121×C₈₉

—

D₁₆×C₈₉

—

D₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

#102×C₈₉

—

C₃₂×C₈₉

—

C₃₂×C₈₉

D₄

yes

complemented

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

D₁₆

yes

characteristic

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#2215

yes

characteristic

C₂×C₈₉

#2263

yes

—

C₂×D₃₂

C₂×C₈₉

yes

characteristic, complemented

C₂×M₆₄

C₂×C₈₉

yes

characteristic, complemented

#2215

C₂×C₈₉

yes

complemented

#1424

C₂×C₈₉

yes

characteristic, complemented

#1415

C₂×C₈₉

yes

complemented

C₈₉

#13697

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

C₂²×C₈₉

yes

—

C₂²×C₁₆

C₂²×C₈₉

yes

characteristic

#466

—

D₃₂

—

M₆₄

—

C₂×C₃₂

C₂²×C₈₉

yes

characteristic

C₂×C₃₂

C₂²×C₈₉

yes

characteristic

#366

C₂²×C₈₉

yes

—

#357

—

C₂×D₈

—

C₂²×C₈

D₄×C₈₉

yes

characteristic

#121

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₈₉

yes

characteristic

C₂×C₁₆

C₂³×C₈₉

yes

characteristic

#107

—

#102

—

C₃₂

—

C₃₂

D₄×C₈₉

yes

complemented

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

D₈×C₈₉

yes

characteristic

101

D₈

—

102

D₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

D₈×C₈₉

yes

characteristic

106

C₂×C₈

C₂×D₄×C₈₉

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₁₆

—

110

C₁₆

C₂×D₄×C₈₉

yes

characteristic

111

C₁₆

#41×C₈₉

yes

characteristic

112

C₂³

—

113

C₂³

D₁₆×C₈₉

yes

characteristic

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₁₆×C₈₉

yes

characteristic

121

C₂×C₄

C₂×D₈×C₈₉

yes

characteristic

122

C₈

—

123

C₈

#121×C₈₉

yes

characteristic

124

C₈

#135×C₈₉

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×D₁₆×C₈₉

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

#466×C₈₉

yes

characteristic

134

C₄

#508×C₈₉

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

C₈₉×#2215

yes

characteristic

138

C₂

C₈₉×#2263

yes

—

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₈₉
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂²×C₈×C₈₉ ◁ C₂²×C₁₆×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₈₉

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂²×C₈×C₈₉ ◁ C₂²×C₁₆×C₈₉ ◁ G

N	H ≅ G/N	type
1. C₂×D₃₂×C₈₉	C₂	semidirect
2. C₂×M₆₄×C₈₉	C₂	semidirect
3. C₈₉×#2215	C₂	semidirect
4. C₈₉×#1424	C₂	semidirect
5. C₈₉×#1415	C₂	semidirect
27. C₃₂×C₈₉	D₄	semidirect
65. #13697	C₈₉	direct (N × H)
70. C₂×D₃₂	C₂×C₈₉	semidirect
71. C₂×M₆₄	C₂×C₈₉	semidirect
72. #2215	C₂×C₈₉	semidirect
73. #1424	C₂×C₈₉	semidirect
74. #1415	C₂×C₈₉	semidirect
75. C₈₉	#13697	direct (N × H)
97. C₃₂	D₄×C₈₉	semidirect

H ≅ G/N

type

1. C₂×D₃₂×C₈₉

C₂

semidirect

2. C₂×M₆₄×C₈₉

C₂

semidirect

3. C₈₉×#2215

C₂

semidirect

4. C₈₉×#1424

C₂

semidirect

5. C₈₉×#1415

C₂

semidirect

27. C₃₂×C₈₉

D₄

semidirect

65. #13697

C₈₉

direct (N × H)

70. C₂×D₃₂

C₂×C₈₉

semidirect

71. C₂×M₆₄

C₂×C₈₉

semidirect

72. #2215

C₂×C₈₉

semidirect

73. #1424

C₂×C₈₉

semidirect

74. #1415

C₂×C₈₉

semidirect

75. C₈₉

#13697

direct (N × H)

97. C₃₂

D₄×C₈₉

semidirect

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

subgroup	index
C₂×D₃₂×C₈₉	2
C₂×M₆₄×C₈₉	2
C₈₉×#2215	2
C₈₉×#1424	2
C₈₉×#1415	2
#13697	89

subgroup

index

C₂×D₃₂×C₈₉

C₂×M₆₄×C₈₉

C₈₉×#2215

C₈₉×#1424

C₈₉×#1415

#13697

Subgroups — C₈₉×#13697 #36391545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#13697 #36391545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#13697 #36391545