Group #36384977 — subgroups

Subgroups — C₈₉×#7129 #36384977

Order 22784.

210 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 70 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×#1995	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1274	C₂	1	2	yes	maximal
3	#533×C₈₉	C₄	1	1	yes	characteristic
4	#431×C₈₉	—	2	2	no	—
5	#423×C₈₉	C₄	1	1	yes	characteristic
6	#376×C₈₉	C₂²	1	1	yes	characteristic
7	#339×C₈₉	—	2	2	no	—
8	C₂×M₁₆×C₈₉	—	2	1	no	—
9	C₄⋊Q₈×C₈₉	—	2	1	no	—
10	#125×C₈₉	C₈	1	2	yes	complemented
11	C₂²⋊Q₈×C₈₉	—	4	1	no	—
12	C₄×Q₈×C₈₉	—	4	1	no	—
13	#117×C₈₉	—	2	1	no	—
14	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₈₉	D₄	1	2	yes	—
16	C₄⋊C₈×C₈₉	—	4	1	no	—
17	C₄⋊C₈×C₈₉	—	4	2	no	—
18	C₂²⋊C₈×C₈₉	—	4	2	no	—
19	C₄×C₈×C₈₉	—	2	4	no	—
20	#95×C₈₉	—	4	1	no	—
21	C₂×Q₈×C₈₉	—	4	1	no	—
22	C₂²×C₄×C₈₉	—	2	2	no	—
23	C₂²×C₄×C₈₉	—	2	1	no	—
24	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
25	M₁₆×C₈₉	—	4	2	no	—
26	C₂×C₈×C₈₉	—	4	4	no	—
27	C₂×C₈×C₈₉	—	4	1	no	—
28	C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₄⋊C₄×C₈₉	—	4	1	no	—
30	C₄⋊C₄×C₈₉	—	4	2	no	—
31	C₂²⋊C₄×C₈₉	—	4	1	no	—
32	C₄²×C₈₉	—	2	1	no	—
33	C₄²×C₈₉	—	2	2	no	—
34	C₄²×C₈₉	—	4	1	no	—
35	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
37	C₄²×C₈₉	SD₁₆	1	2	yes	—
38	C₄²×C₈₉	Q₁₆	1	2	yes	—
39	C₂³×C₈₉	#99	1	1	yes	characteristic
40	Q₈×C₈₉	—	8	2	no	—
41	C₂×C₄×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	4	2	no	—
46	C₂×C₄×C₈₉	—	8	2	no	—
47	C₂×C₄×C₈₉	—	8	1	no	—
48	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#103	1	2	yes	—
51	C₂×C₄×C₈₉	#104	1	2	yes	—
52	C₈×C₈₉	—	8	4	no	—
53	C₈×C₈₉	—	8	1	no	—
54	C₂²×C₈₉	—	4	1	no	—
55	C₂²×C₈₉	#323	1	1	yes	characteristic
56	C₂²×C₈₉	#328	1	2	yes	—
57	C₄×C₈₉	—	4	2	no	—
58	C₄×C₈₉	—	4	1	no	—
59	C₄×C₈₉	—	4	2	no	—
60	C₄×C₈₉	—	4	1	no	—
61	C₄×C₈₉	—	8	2	no	—
62	C₄×C₈₉	Q₈⋊C₈	1	2	yes	—
63	C₄×C₈₉	#329	1	1	yes	characteristic
64	C₄×C₈₉	#333	1	1	yes	characteristic
65	#7129	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₈₉	—	2	1	no	—
67	C₂×C₈₉	—	2	2	no	—
68	C₂×C₈₉	#1272	1	1	yes	characteristic
69	#1995	C₂×C₈₉	1	1	yes	characteristic
70	#1274	C₂×C₈₉	1	2	yes	—
71	C₈₉	#7129	1	1	yes	characteristic, direct product factor, Sylow
72	#533	C₄×C₈₉	1	1	yes	characteristic
73	#431	—	2	2	no	—
74	#423	C₄×C₈₉	1	1	yes	characteristic
75	#376	C₂²×C₈₉	1	1	yes	characteristic
76	#339	—	2	2	no	—
77	C₂×M₁₆	—	2	1	no	—
78	C₄⋊Q₈	—	2	1	no	—
79	#125	C₈×C₈₉	1	2	yes	complemented
80	C₂²⋊Q₈	—	4	1	no	—
81	C₄×Q₈	—	4	1	no	—
82	#117	—	2	1	no	—
83	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
84	C₂×C₄²	D₄×C₈₉	1	2	yes	—
85	C₄⋊C₈	—	4	1	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₂²⋊C₈	—	4	2	no	—
88	C₄×C₈	—	2	4	no	—
89	#95	—	4	1	no	—
90	C₂×Q₈	—	4	1	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
94	M₁₆	—	4	2	no	—
95	C₂×C₈	—	4	4	no	—
96	C₂×C₈	—	4	1	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₂²⋊C₄	—	4	1	no	—
101	C₄²	—	2	1	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	4	1	no	—
104	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
105	C₄²	M₁₆×C₈₉	1	1	yes	characteristic
106	C₄²	SD₁₆×C₈₉	1	2	yes	—
107	C₄²	Q₁₆×C₈₉	1	2	yes	—
108	C₂³	#99×C₈₉	1	1	yes	characteristic
109	Q₈	—	8	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	—	8	1	no	—
117	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
118	C₂×C₄	#100×C₈₉	1	1	yes	characteristic
119	C₂×C₄	#103×C₈₉	1	2	yes	—
120	C₂×C₄	#104×C₈₉	1	2	yes	—
121	C₈	—	8	4	no	—
122	C₈	—	8	1	no	—
123	C₂²	—	4	1	no	—
124	C₂²	#323×C₈₉	1	1	yes	characteristic
125	C₂²	#328×C₈₉	1	2	yes	—
126	C₄	—	4	2	no	—
127	C₄	—	4	1	no	—
128	C₄	—	4	1	no	—
129	C₄	—	4	2	no	—
130	C₄	—	8	2	no	—
131	C₄	Q₈⋊C₈×C₈₉	1	2	yes	—
132	C₄	#329×C₈₉	1	1	yes	characteristic
133	C₄	#333×C₈₉	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	2	2	no	—
136	C₂	C₈₉×#1272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#1995

C₂

yes

maximal, characteristic

C₈₉×#1274

C₂

yes

maximal

#533×C₈₉

C₄

yes

characteristic

#431×C₈₉

—

#423×C₈₉

C₄

yes

characteristic

#376×C₈₉

C₂²

yes

characteristic

#339×C₈₉

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

#125×C₈₉

C₈

yes

complemented

C₂²⋊Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

#117×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

#95×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

characteristic

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₂³×C₈₉

#99

yes

characteristic

Q₈×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

#323

yes

characteristic

C₂²×C₈₉

#328

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

Q₈⋊C₈

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1272

yes

characteristic

#1995

C₂×C₈₉

yes

characteristic

#1274

C₂×C₈₉

yes

—

C₈₉

#7129

yes

characteristic, direct product factor, Sylow

#533

C₄×C₈₉

yes

characteristic

#431

—

#423

C₄×C₈₉

yes

characteristic

#376

C₂²×C₈₉

yes

characteristic

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

C₈×C₈₉

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×C₄×C₈₉

yes

characteristic

C₂×C₄²

D₄×C₈₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×C₈×C₈₉

yes

characteristic

105

C₄²

M₁₆×C₈₉

yes

characteristic

106

C₄²

SD₁₆×C₈₉

yes

—

107

C₄²

Q₁₆×C₈₉

yes

—

108

C₂³

#99×C₈₉

yes

characteristic

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊C₈×C₈₉

yes

characteristic

118

C₂×C₄

#100×C₈₉

yes

characteristic

119

C₂×C₄

#103×C₈₉

yes

—

120

C₂×C₄

#104×C₈₉

yes

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

#323×C₈₉

yes

characteristic

125

C₂²

#328×C₈₉

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

Q₈⋊C₈×C₈₉

yes

—

132

C₄

#329×C₈₉

yes

characteristic

133

C₄

#333×C₈₉

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

C₈₉×#1272

yes

characteristic

	subgroup	G/H
center	C₄×C₈₉	#329
commutator	C₄²	C₂×C₈×C₈₉
Frattini, cosocle	#376	C₂²×C₈₉
socle	C₂×C₈₉	#1272
prime radical	C₂³×C₈₉	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #376×C₈₉ ◁ G

subgroup

G/H

center

C₄×C₈₉

#329

commutator

C₄²

C₂×C₈×C₈₉

C₂²×C₈₉

C₂×C₈₉

C₂³×C₈₉

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #376×C₈₉ ◁ G

N	H ≅ G/N	type
10. #125×C₈₉	C₈	semidirect
65. #7129	C₈₉	direct (N × H)
71. C₈₉	#7129	direct (N × H)
79. #125	C₈×C₈₉	semidirect

H ≅ G/N

type

10. #125×C₈₉

C₈

semidirect

65. #7129

C₈₉

direct (N × H)

71. C₈₉

#7129

direct (N × H)

79. #125

C₈×C₈₉

semidirect

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

subgroup	index
C₈₉×#1995	2
C₈₉×#1274	2
#7129	89

subgroup

index

C₈₉×#1995

C₈₉×#1274

#7129

Subgroups — C₈₉×#7129 #36384977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7129 #36384977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7129 #36384977