Group #36391543 — subgroups

Subgroups — C₈₉×#13695 #36391543

Order 22784.

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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×Q₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2216	C₂	1	2	yes	maximal
4	C₈₉×#1423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1416	C₂	1	2	yes	maximal, complemented
6	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
8	#467×C₈₉	—	2	2	no	—
9	Q₆₄×C₈₉	—	2	2	no	—
10	C₂×C₃₂×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	#358×C₈₉	—	2	2	no	—
15	C₂×Q₁₆×C₈₉	—	2	2	no	—
16	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
17	C₂²⋊Q₈×C₈₉	—	4	2	no	—
18	Q₃₂×C₈₉	—	2	2	no	—
19	Q₃₂×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	#103×C₈₉	—	4	2	no	—
26	C₃₂×C₈₉	—	2	1	no	—
27	C₃₂×C₈₉	D₄	1	2	yes	—
28	C₂×Q₈×C₈₉	—	4	2	no	—
29	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
30	Q₁₆×C₈₉	—	4	2	no	—
31	Q₁₆×C₈₉	—	4	2	no	—
32	C₂×C₈×C₈₉	—	2	1	no	—
33	C₂×C₈×C₈₉	—	2	1	no	—
34	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	—	4	2	no	—
37	C₄⋊C₄×C₈₉	—	8	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	D₁₆	1	1	yes	characteristic
43	Q₈×C₈₉	—	8	2	no	—
44	Q₈×C₈₉	—	8	2	no	—
45	C₂×C₄×C₈₉	—	2	1	no	—
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	—	8	2	no	—
48	C₂×C₄×C₈₉	—	8	2	no	—
49	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
51	C₈×C₈₉	—	2	1	no	—
52	C₈×C₈₉	#121	1	1	yes	characteristic
53	C₈×C₈₉	#135	1	1	yes	characteristic
54	C₂²×C₈₉	—	2	1	no	—
55	C₂²×C₈₉	—	2	1	no	—
56	C₂²×C₈₉	Q₆₄	1	2	yes	complemented
57	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
58	C₄×C₈₉	—	2	1	no	—
59	C₄×C₈₉	—	16	2	no	—
60	C₄×C₈₉	—	16	2	no	—
61	C₄×C₈₉	#466	1	1	yes	characteristic
62	C₄×C₈₉	#508	1	1	yes	characteristic
63	#13695	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₈₉	—	2	2	no	—
65	C₂×C₈₉	#2215	1	1	yes	characteristic
66	C₂×C₈₉	C₂×Q₆₄	1	1	yes	characteristic
67	C₂×C₈₉	#2262	1	1	yes	characteristic
68	C₂×Q₆₄	C₂×C₈₉	1	1	yes	characteristic, complemented
69	C₂²×C₃₂	C₂×C₈₉	1	1	yes	characteristic
70	#2216	C₂×C₈₉	1	2	yes	—
71	#1423	C₂×C₈₉	1	1	yes	characteristic, complemented
72	#1416	C₂×C₈₉	1	2	yes	complemented
73	C₈₉	#13695	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×Q₃₂	C₂²×C₈₉	1	2	yes	—
75	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
76	#467	—	2	2	no	—
77	Q₆₄	—	2	2	no	—
78	C₂×C₃₂	—	2	2	no	—
79	C₂×C₃₂	C₂²×C₈₉	1	1	yes	characteristic
80	C₂×C₃₂	C₂²×C₈₉	1	1	yes	characteristic
81	#366	C₂²×C₈₉	1	2	yes	—
82	#358	—	2	2	no	—
83	C₂×Q₁₆	—	2	2	no	—
84	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
85	C₂²⋊Q₈	—	4	2	no	—
86	Q₃₂	—	2	2	no	—
87	Q₃₂	—	2	2	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	D₄×C₈₉	1	1	yes	characteristic
91	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
92	#107	—	2	2	no	—
93	#103	—	4	2	no	—
94	C₃₂	—	2	1	no	—
95	C₃₂	D₄×C₈₉	1	2	yes	—
96	C₂×Q₈	—	4	2	no	—
97	C₂²×C₄	D₈×C₈₉	1	1	yes	characteristic
98	Q₁₆	—	4	2	no	—
99	Q₁₆	—	4	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	D₈×C₈₉	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₂²⋊C₄	—	8	2	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	C₂×D₄×C₈₉	1	1	yes	characteristic
109	C₁₆	#41×C₈₉	1	1	yes	characteristic
110	C₂³	D₁₆×C₈₉	1	1	yes	characteristic
111	Q₈	—	8	2	no	—
112	Q₈	—	8	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	D₁₆×C₈₉	1	1	yes	characteristic
118	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	#121×C₈₉	1	1	yes	characteristic
121	C₈	#135×C₈₉	1	1	yes	characteristic
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	Q₆₄×C₈₉	1	2	yes	complemented
125	C₂²	C₂×D₁₆×C₈₉	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	16	2	no	—
128	C₄	—	16	2	no	—
129	C₄	#466×C₈₉	1	1	yes	characteristic
130	C₄	#508×C₈₉	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	C₈₉×#2215	1	1	yes	characteristic
133	C₂	C₂×Q₆₄×C₈₉	1	1	yes	characteristic
134	C₂	C₈₉×#2262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₆₄×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₃₂×C₈₉

C₂

yes

maximal, characteristic

C₈₉×#2216

C₂

yes

maximal

C₈₉×#1423

C₂

yes

maximal, characteristic, complemented

C₈₉×#1416

C₂

yes

maximal, complemented

C₂×Q₃₂×C₈₉

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

#467×C₈₉

—

Q₆₄×C₈₉

—

C₂×C₃₂×C₈₉

—

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

#366×C₈₉

C₂²

yes

—

#358×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²⋊Q₈×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

#103×C₈₉

—

C₃₂×C₈₉

—

C₃₂×C₈₉

D₄

yes

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

C₂³×C₈₉

D₁₆

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×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

Q₆₄

yes

complemented

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

#2215

yes

characteristic

C₂×C₈₉

C₂×Q₆₄

yes

characteristic

C₂×C₈₉

#2262

yes

characteristic

C₂×Q₆₄

C₂×C₈₉

yes

characteristic, complemented

C₂²×C₃₂

C₂×C₈₉

yes

characteristic

#2216

C₂×C₈₉

yes

—

#1423

C₂×C₈₉

yes

characteristic, complemented

#1416

C₂×C₈₉

yes

complemented

C₈₉

#13695

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂²×C₈₉

yes

—

C₂²×C₁₆

C₂²×C₈₉

yes

characteristic

#467

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂²×C₈₉

yes

characteristic

C₂×C₃₂

C₂²×C₈₉

yes

characteristic

#366

C₂²×C₈₉

yes

—

#358

—

C₂×Q₁₆

—

C₂²×C₈

D₄×C₈₉

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₈₉

yes

characteristic

C₂×C₁₆

C₂³×C₈₉

yes

characteristic

#107

—

#103

—

C₃₂

—

C₃₂

D₄×C₈₉

yes

—

C₂×Q₈

—

C₂²×C₄

D₈×C₈₉

yes

characteristic

Q₁₆

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

D₈×C₈₉

yes

characteristic

103

C₂×C₈

C₂×D₄×C₈₉

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₁₆

—

108

C₁₆

C₂×D₄×C₈₉

yes

characteristic

109

C₁₆

#41×C₈₉

yes

characteristic

110

C₂³

D₁₆×C₈₉

yes

characteristic

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₁₆×C₈₉

yes

characteristic

118

C₂×C₄

C₂×D₈×C₈₉

yes

characteristic

119

C₈

—

120

C₈

#121×C₈₉

yes

characteristic

121

C₈

#135×C₈₉

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

Q₆₄×C₈₉

yes

complemented

125

C₂²

C₂×D₁₆×C₈₉

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#466×C₈₉

yes

characteristic

130

C₄

#508×C₈₉

yes

characteristic

131

C₂

—

132

C₂

C₈₉×#2215

yes

characteristic

133

C₂

C₂×Q₆₄×C₈₉

yes

characteristic

134

C₂

C₈₉×#2262

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₈₉
prime radical	C₂³×C₈₉	D₁₆
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

prime radical

C₂³×C₈₉

D₁₆

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×Q₆₄×C₈₉	C₂	semidirect
4. C₈₉×#1423	C₂	semidirect
5. C₈₉×#1416	C₂	semidirect
56. C₂²×C₈₉	Q₆₄	semidirect
63. #13695	C₈₉	direct (N × H)
68. C₂×Q₆₄	C₂×C₈₉	semidirect
71. #1423	C₂×C₈₉	semidirect
72. #1416	C₂×C₈₉	semidirect
73. C₈₉	#13695	direct (N × H)
124. C₂²	Q₆₄×C₈₉	semidirect

H ≅ G/N

type

1. C₂×Q₆₄×C₈₉

C₂

semidirect

4. C₈₉×#1423

C₂

semidirect

5. C₈₉×#1416

C₂

semidirect

56. C₂²×C₈₉

Q₆₄

semidirect

63. #13695

C₈₉

direct (N × H)

68. C₂×Q₆₄

C₂×C₈₉

semidirect

71. #1423

C₂×C₈₉

semidirect

72. #1416

C₂×C₈₉

semidirect

73. C₈₉

#13695

direct (N × H)

124. C₂²

Q₆₄×C₈₉

semidirect

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

subgroup	index
C₂×Q₆₄×C₈₉	2
C₂²×C₃₂×C₈₉	2
C₈₉×#2216	2
C₈₉×#1423	2
C₈₉×#1416	2
#13695	89

subgroup

index

C₂×Q₆₄×C₈₉

C₂²×C₃₂×C₈₉

C₈₉×#2216

C₈₉×#1423

C₈₉×#1416

#13695

Subgroups — C₈₉×#13695 #36391543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#13695 #36391543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#13695 #36391543