Group #36385175 — subgroups

Subgroups — C₈₉×#7327 #36385175

Order 22784.

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

normal subgroups only

This group has 244 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×#1818	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1464	C₂	1	2	yes	maximal
3	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
4	#432×C₈₉	—	2	2	no	—
5	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
6	#336×C₈₉	—	2	2	no	—
7	#324×C₈₉	C₄	1	4	yes	—
8	#323×C₈₉	—	2	2	no	—
9	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₈₉	—	2	2	no	—
11	#117×C₈₉	—	4	2	no	—
12	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
13	C₂×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₈₉	Q₈	1	1	yes	characteristic
15	C₂×C₄²×C₈₉	D₄	1	2	yes	—
16	C₄⋊C₈×C₈₉	—	4	2	no	—
17	C₂²⋊C₈×C₈₉	—	2	4	no	—
18	C₂²⋊C₈×C₈₉	—	4	2	no	—
19	#97×C₈₉	—	4	2	no	—
20	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₉	—	2	2	no	—
22	C₂²×C₄×C₈₉	—	4	2	no	—
23	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₈₉	—	4	4	no	—
28	C₂×C₈×C₈₉	—	4	4	no	—
29	C₂×C₈×C₈₉	—	4	2	no	—
30	C₄⋊C₄×C₈₉	—	4	2	no	—
31	C₂²⋊C₄×C₈₉	—	4	1	no	—
32	C₂²⋊C₄×C₈₉	—	4	1	no	—
33	C₄²×C₈₉	—	2	2	no	—
34	C₄²×C₈₉	—	2	2	no	—
35	C₂³×C₈₉	—	2	1	no	—
36	C₂³×C₈₉	—	2	2	no	—
37	C₂³×C₈₉	—	4	2	no	—
38	C₂³×C₈₉	#95	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	4	4	no	—
45	C₂×C₄×C₈₉	—	4	2	no	—
46	C₂×C₄×C₈₉	—	4	2	no	—
47	C₂×C₄×C₈₉	#99	1	2	yes	—
48	C₈×C₈₉	—	8	2	no	—
49	C₈×C₈₉	—	8	4	no	—
50	C₂²×C₈₉	—	2	2	no	—
51	C₂²×C₈₉	—	4	1	no	—
52	C₂²×C₈₉	—	4	2	no	—
53	C₂²×C₈₉	—	8	2	no	—
54	C₂²×C₈₉	#342	1	1	yes	characteristic
55	C₂²×C₈₉	#344	1	2	yes	—
56	C₄×C₈₉	—	2	2	no	—
57	C₄×C₈₉	—	4	1	no	—
58	C₄×C₈₉	—	4	2	no	—
59	C₄×C₈₉	—	4	2	no	—
60	#7327	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₈₉	—	2	2	no	—
62	C₂×C₈₉	—	8	1	no	—
63	C₂×C₈₉	#1306	1	1	yes	characteristic
64	C₂×C₈₉	#1395	1	2	yes	—
65	#1818	C₂×C₈₉	1	1	yes	characteristic
66	#1464	C₂×C₈₉	1	2	yes	—
67	C₈₉	#7327	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
69	#432	—	2	2	no	—
70	C₂×C₂²⋊C₈	C₄×C₈₉	1	2	yes	—
71	#336	—	2	2	no	—
72	#324	C₄×C₈₉	1	4	yes	—
73	#323	—	2	2	no	—
74	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
75	C₂²×C₈	—	2	2	no	—
76	#117	—	4	2	no	—
77	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	2	yes	—
78	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
79	C₂×C₂²⋊C₄	Q₈×C₈₉	1	1	yes	characteristic
80	C₂×C₄²	D₄×C₈₉	1	2	yes	—
81	C₄⋊C₈	—	4	2	no	—
82	C₂²⋊C₈	—	2	4	no	—
83	C₂²⋊C₈	—	4	2	no	—
84	#97	—	4	2	no	—
85	C₂⁴	C₄⋊C₄×C₈₉	1	1	yes	characteristic
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	C₄²×C₈₉	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
90	C₂²×C₄	C₂²⋊C₄×C₈₉	1	2	yes	—
91	C₂²×C₄	C₄⋊C₄×C₈₉	1	2	yes	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	—	4	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₂²⋊C₄	—	4	1	no	—
97	C₂²⋊C₄	—	4	1	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	—	2	1	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	4	2	no	—
103	C₂³	#95×C₈₉	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	#99×C₈₉	1	2	yes	—
113	C₈	—	8	2	no	—
114	C₈	—	8	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	4	1	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	8	2	no	—
119	C₂²	#342×C₈₉	1	1	yes	characteristic
120	C₂²	#344×C₈₉	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	2	no	—
125	C₂	—	2	2	no	—
126	C₂	—	8	1	no	—
127	C₂	C₈₉×#1306	1	1	yes	characteristic
128	C₂	C₈₉×#1395	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#1818

C₂

yes

maximal, characteristic

C₈₉×#1464

C₂

yes

maximal

C₂×#117×C₈₉

C₂²

yes

characteristic

#432×C₈₉

—

C₂×C₂²⋊C₈×C₈₉

C₄

yes

—

#336×C₈₉

—

#324×C₈₉

C₄

yes

—

#323×C₈₉

—

C₂³×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

—

#117×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₈₉

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

#97×C₈₉

—

C₂⁴×C₈₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₄²

yes

characteristic

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#95

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

#99

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#342

yes

characteristic

C₂²×C₈₉

#344

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

#7327

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1306

yes

characteristic

C₂×C₈₉

#1395

yes

—

#1818

C₂×C₈₉

yes

characteristic

#1464

C₂×C₈₉

yes

—

C₈₉

#7327

yes

characteristic, direct product factor, Sylow

C₂×#117

C₂²×C₈₉

yes

characteristic

#432

—

C₂×C₂²⋊C₈

C₄×C₈₉

yes

—

#336

—

#324

C₄×C₈₉

yes

—

#323

—

C₂³×C₄

C₂×C₄×C₈₉

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

C₂×C₄×C₈₉

yes

—

C₂×C₂²⋊C₄

D₄×C₈₉

yes

characteristic

C₂×C₂²⋊C₄

Q₈×C₈₉

yes

characteristic

C₂×C₄²

D₄×C₈₉

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂⁴

C₄⋊C₄×C₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄²×C₈₉

yes

characteristic

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

characteristic

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

—

C₂²×C₄

C₄⋊C₄×C₈₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

#95×C₈₉

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#99×C₈₉

yes

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#342×C₈₉

yes

characteristic

120

C₂²

#344×C₈₉

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₂

—

126

C₂

—

127

C₂

C₈₉×#1306

yes

characteristic

128

C₂

C₈₉×#1395

yes

—

	subgroup	G/H
center, socle	C₂²×C₈₉	#342
commutator	C₂²×C₄	C₄²×C₈₉
Frattini, cosocle	C₂×#117	C₂²×C₈₉
prime radical	C₂⁴×C₈₉	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

#342

commutator

C₂²×C₄

C₄²×C₈₉

Frattini, cosocle

C₂×#117

C₂²×C₈₉

prime radical

C₂⁴×C₈₉

C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

N	H ≅ G/N	type
60. #7327	C₈₉	direct (N × H)
67. C₈₉	#7327	direct (N × H)

H ≅ G/N

type

60. #7327

C₈₉

direct (N × H)

67. C₈₉

#7327

direct (N × H)

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

subgroup	index
C₈₉×#1818	2
C₈₉×#1464	2
#7327	89

subgroup

index

C₈₉×#1818

C₈₉×#1464

#7327

Subgroups — C₈₉×#7327 #36385175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7327 #36385175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7327 #36385175