Group #36384978 — subgroups

Subgroups — C₈₉×#7130 #36384978

Order 22784.

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

normal subgroups only

This group has 206 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	C₈₉×#1992	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1375	C₂	1	2	yes	maximal
3	C₂×#125×C₈₉	C₄	1	1	yes	characteristic
4	C₂×M₃₂×C₈₉	—	2	2	no	—
5	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
6	#384×C₈₉	C₄	1	1	yes	characteristic
7	#340×C₈₉	—	2	2	no	—
8	C₂²×C₈×C₈₉	D₄	1	2	yes	—
9	#125×C₈₉	—	2	1	no	—
10	#125×C₈₉	—	4	1	no	—
11	#125×C₈₉	C₈	1	2	yes	—
12	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
13	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
15	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
16	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
17	M₃₂×C₈₉	—	4	2	no	—
18	M₃₂×C₈₉	—	4	2	no	—
19	C₂×C₁₆×C₈₉	—	2	4	no	—
20	#97×C₈₉	—	4	1	no	—
21	C₂²×C₄×C₈₉	—	2	1	no	—
22	C₂²×C₄×C₈₉	—	4	1	no	—
23	C₂²×C₄×C₈₉	—	4	1	no	—
24	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₉	—	2	2	no	—
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
28	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
29	C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₄⋊C₄×C₈₉	—	4	2	no	—
31	C₄⋊C₄×C₈₉	—	4	2	no	—
32	C₄⋊C₄×C₈₉	—	4	1	no	—
33	C₄⋊C₄×C₈₉	—	4	2	no	—
34	C₄⋊C₄×C₈₉	—	8	2	no	—
35	C₄²×C₈₉	—	2	1	no	—
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	#100	1	1	yes	characteristic
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈₉	—	4	1	no	—
43	C₂×C₄×C₈₉	—	4	1	no	—
44	C₂×C₄×C₈₉	—	4	2	no	—
45	C₂×C₄×C₈₉	—	8	1	no	—
46	C₂×C₄×C₈₉	—	8	1	no	—
47	C₂×C₄×C₈₉	—	8	2	no	—
48	C₂×C₄×C₈₉	—	8	1	no	—
49	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#103	1	2	yes	—
52	C₂×C₄×C₈₉	#104	1	2	yes	—
53	C₈×C₈₉	—	2	2	no	—
54	C₈×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	—	2	1	no	—
56	C₂²×C₈₉	—	2	1	no	—
57	C₂²×C₈₉	#323	1	1	yes	characteristic
58	C₂²×C₈₉	#329	1	1	yes	characteristic
59	C₂²×C₈₉	#333	1	1	yes	characteristic
60	C₄×C₈₉	—	4	1	no	—
61	C₄×C₈₉	—	4	1	no	—
62	C₄×C₈₉	—	8	2	no	—
63	C₄×C₈₉	—	16	1	no	—
64	C₄×C₈₉	Q₈⋊C₈	1	2	yes	—
65	C₄×C₈₉	#328	1	2	yes	—
66	#7130	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂×C₈₉	—	2	1	no	—
68	C₂×C₈₉	—	2	1	no	—
69	C₂×C₈₉	#1272	1	1	yes	characteristic
70	C₂×C₈₉	#1355	1	1	yes	characteristic
71	C₂×C₈₉	#1356	1	1	yes	characteristic
72	#1992	C₂×C₈₉	1	1	yes	characteristic
73	#1375	C₂×C₈₉	1	2	yes	—
74	C₈₉	#7130	1	1	yes	characteristic, direct product factor, Sylow
75	C₂×#125	C₄×C₈₉	1	1	yes	characteristic
76	C₂×M₃₂	—	2	2	no	—
77	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
78	#384	C₄×C₈₉	1	1	yes	characteristic
79	#340	—	2	2	no	—
80	C₂²×C₈	D₄×C₈₉	1	2	yes	—
81	#125	—	2	1	no	—
82	#125	—	4	1	no	—
83	#125	C₈×C₈₉	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
89	M₃₂	—	4	2	no	—
90	M₃₂	—	4	2	no	—
91	C₂×C₁₆	—	2	4	no	—
92	#97	—	4	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	—	4	1	no	—
96	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	SD₁₆×C₈₉	1	2	yes	—
100	C₂×C₈	Q₁₆×C₈₉	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
109	C₄²	M₁₆×C₈₉	1	1	yes	characteristic
110	C₁₆	—	4	4	no	—
111	C₂³	#100×C₈₉	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	1	no	—
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
122	C₂×C₄	#99×C₈₉	1	1	yes	characteristic
123	C₂×C₄	#103×C₈₉	1	2	yes	—
124	C₂×C₄	#104×C₈₉	1	2	yes	—
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#323×C₈₉	1	1	yes	characteristic
130	C₂²	#329×C₈₉	1	1	yes	characteristic
131	C₂²	#333×C₈₉	1	1	yes	characteristic
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	8	2	no	—
135	C₄	—	16	1	no	—
136	C₄	Q₈⋊C₈×C₈₉	1	2	yes	—
137	C₄	#328×C₈₉	1	2	yes	—
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	C₈₉×#1272	1	1	yes	characteristic
141	C₂	C₈₉×#1355	1	1	yes	characteristic
142	C₂	C₈₉×#1356	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#1992

C₂

yes

maximal, characteristic

C₈₉×#1375

C₂

yes

maximal

C₂×#125×C₈₉

C₄

yes

characteristic

C₂×M₃₂×C₈₉

—

C₂×#97×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₄

yes

characteristic

#340×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

#125×C₈₉

—

#125×C₈₉

—

#125×C₈₉

C₈

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

M₃₂×C₈₉

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

#97×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

#100

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

#104

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

Q₈⋊C₈

yes

—

C₄×C₈₉

#328

yes

—

#7130

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

characteristic

#1375

C₂×C₈₉

yes

—

C₈₉

#7130

yes

characteristic, direct product factor, Sylow

C₂×#125

C₄×C₈₉

yes

characteristic

C₂×M₃₂

—

C₂×#97

C₂²×C₈₉

yes

characteristic

#384

C₄×C₈₉

yes

characteristic

#340

—

C₂²×C₈

D₄×C₈₉

yes

—

#125

—

#125

—

#125

C₈×C₈₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄×C₈₉

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆×C₈₉

yes

—

100

C₂×C₈

Q₁₆×C₈₉

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

C₂×C₈×C₈₉

yes

characteristic

109

C₄²

M₁₆×C₈₉

yes

characteristic

110

C₁₆

—

111

C₂³

#100×C₈₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊C₈×C₈₉

yes

characteristic

122

C₂×C₄

#99×C₈₉

yes

characteristic

123

C₂×C₄

#103×C₈₉

yes

—

124

C₂×C₄

#104×C₈₉

yes

—

125

C₈

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

#323×C₈₉

yes

characteristic

130

C₂²

#329×C₈₉

yes

characteristic

131

C₂²

#333×C₈₉

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

Q₈⋊C₈×C₈₉

yes

—

137

C₄

#328×C₈₉

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₈₉×#1272

yes

characteristic

141

C₂

C₈₉×#1355

yes

characteristic

142

C₂

C₈₉×#1356

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#329
commutator	C₄²	C₂×C₈×C₈₉
Frattini, cosocle	C₂×#97	C₂²×C₈₉
prime radical	C₂³×C₈₉	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂×#97×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

#329

commutator

C₄²

C₂×C₈×C₈₉

Frattini, cosocle

C₂×#97

C₂²×C₈₉

prime radical

C₂³×C₈₉

#100

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂×#97×C₈₉ ◁ G

N	H ≅ G/N	type
66. #7130	C₈₉	direct (N × H)
74. C₈₉	#7130	direct (N × H)

H ≅ G/N

type

66. #7130

C₈₉

direct (N × H)

74. C₈₉

#7130

direct (N × H)

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

subgroup	index
C₈₉×#1992	2
C₈₉×#1375	2
#7130	89

subgroup

index

C₈₉×#1992

C₈₉×#1375

#7130

Subgroups — C₈₉×#7130 #36384978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7130 #36384978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7130 #36384978