Group #36385219 — subgroups

Subgroups — C₈₉×#7371 #36385219

Order 22784.

190 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 50 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×#1844	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1325	C₂	1	2	yes	maximal
3	C₂×#106×C₈₉	—	2	2	no	—
4	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
5	#384×C₈₉	C₄	1	1	yes	characteristic
6	#384×C₈₉	C₂²	1	1	yes	characteristic
7	#324×C₈₉	—	2	2	no	—
8	C₂²×C₈×C₈₉	—	2	1	no	—
9	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
10	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
11	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
12	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
13	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
14	#106×C₈₉	—	4	4	no	—
15	C₂²⋊C₈×C₈₉	—	4	2	no	—
16	C₄×C₈×C₈₉	—	2	1	no	—
17	C₄×C₈×C₈₉	C₈	1	2	yes	complemented
18	C₂²×C₄×C₈₉	—	2	1	no	—
19	C₂²×C₄×C₈₉	—	2	2	no	—
20	C₂²×C₄×C₈₉	—	4	1	no	—
21	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₈₉	—	2	2	no	—
23	C₂×C₈×C₈₉	—	2	2	no	—
24	C₂×C₈×C₈₉	—	2	2	no	—
25	C₂×C₈×C₈₉	—	8	2	no	—
26	C₄⋊C₄×C₈₉	—	4	2	no	—
27	C₄⋊C₄×C₈₉	—	4	2	no	—
28	C₄⋊C₄×C₈₉	—	8	2	no	—
29	C₄⋊C₄×C₈₉	—	8	2	no	—
30	C₄²×C₈₉	—	2	1	no	—
31	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
33	C₂³×C₈₉	#99	1	1	yes	characteristic
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	—	2	2	no	—
36	C₂×C₄×C₈₉	—	2	1	no	—
37	C₂×C₄×C₈₉	—	2	1	no	—
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	4	2	no	—
41	C₂×C₄×C₈₉	—	8	2	no	—
42	C₂×C₄×C₈₉	—	8	2	no	—
43	C₂×C₄×C₈₉	—	8	1	no	—
44	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₈₉	#101	1	1	yes	characteristic
46	C₈×C₈₉	—	2	4	no	—
47	C₈×C₈₉	—	16	2	no	—
48	C₂²×C₈₉	—	2	1	no	—
49	C₂²×C₈₉	—	2	1	no	—
50	C₂²×C₈₉	#324	1	1	yes	characteristic
51	C₂²×C₈₉	#353	1	1	yes	characteristic
52	C₂²×C₈₉	#356	1	1	yes	characteristic
53	C₄×C₈₉	—	2	2	no	—
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	—	2	1	no	—
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	—	8	2	no	—
58	C₄×C₈₉	—	16	1	no	—
59	#7371	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₈₉	—	2	1	no	—
61	C₂×C₈₉	—	2	1	no	—
62	C₂×C₈₉	#1325	1	1	yes	characteristic
63	C₂×C₈₉	#1409	1	1	yes	characteristic
64	C₂×C₈₉	#1410	1	1	yes	characteristic
65	#1844	C₂×C₈₉	1	1	yes	characteristic
66	#1325	C₂×C₈₉	1	2	yes	—
67	C₈₉	#7371	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×#106	—	2	2	no	—
69	C₂×C₄×C₈	C₄×C₈₉	1	1	yes	characteristic
70	#384	C₄×C₈₉	1	1	yes	characteristic
71	#384	C₂²×C₈₉	1	1	yes	characteristic
72	#324	—	2	2	no	—
73	C₂²×C₈	—	2	1	no	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	4	1	no	—
77	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
78	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
79	#106	—	4	4	no	—
80	C₂²⋊C₈	—	4	2	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	C₈×C₈₉	1	2	yes	complemented
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	8	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	8	2	no	—
94	C₄⋊C₄	—	8	2	no	—
95	C₄²	—	2	1	no	—
96	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
97	C₄²	M₁₆×C₈₉	1	1	yes	characteristic
98	C₂³	#99×C₈₉	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	8	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	8	1	no	—
109	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
110	C₂×C₄	#101×C₈₉	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	16	2	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	#324×C₈₉	1	1	yes	characteristic
116	C₂²	#353×C₈₉	1	1	yes	characteristic
117	C₂²	#356×C₈₉	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	8	2	no	—
123	C₄	—	16	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	C₈₉×#1325	1	1	yes	characteristic
127	C₂	C₈₉×#1409	1	1	yes	characteristic
128	C₂	C₈₉×#1410	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#1844

C₂

yes

maximal, characteristic

C₈₉×#1325

C₂

yes

maximal

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₄

yes

characteristic

#384×C₈₉

C₄

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

#324×C₈₉

—

C₂²×C₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

#106×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₈

yes

complemented

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

—

C₂×C₈×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

C₂³×C₈₉

#99

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

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

#101

yes

characteristic

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

—

C₄×C₈₉

—

#7371

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

#1325

C₂×C₈₉

yes

—

C₈₉

#7371

yes

characteristic, direct product factor, Sylow

C₂×#106

—

C₂×C₄×C₈

C₄×C₈₉

yes

characteristic

#384

C₄×C₈₉

yes

characteristic

#384

C₂²×C₈₉

yes

characteristic

#324

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₈₉

yes

—

C₂×C₄²

C₂×C₄×C₈₉

yes

characteristic

#106

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₈×C₈₉

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×C₈×C₈₉

yes

characteristic

C₄²

M₁₆×C₈₉

yes

characteristic

C₂³

#99×C₈₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊C₈×C₈₉

yes

characteristic

110

C₂×C₄

#101×C₈₉

yes

characteristic

111

C₈

—

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

#324×C₈₉

yes

characteristic

116

C₂²

#353×C₈₉

yes

characteristic

117

C₂²

#356×C₈₉

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₂

—

125

C₂

—

126

C₂

C₈₉×#1325

yes

characteristic

127

C₂

C₈₉×#1409

yes

characteristic

128

C₂

C₈₉×#1410

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#353
commutator	C₄²	C₂×C₈×C₈₉
Frattini, cosocle	#384	C₂²×C₈₉
prime radical	C₂³×C₈₉	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #384×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

#353

commutator

C₄²

C₂×C₈×C₈₉

C₂²×C₈₉

C₂³×C₈₉

G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #384×C₈₉ ◁ G

N	H ≅ G/N	type
17. C₄×C₈×C₈₉	C₈	semidirect
59. #7371	C₈₉	direct (N × H)
67. C₈₉	#7371	direct (N × H)
82. C₄×C₈	C₈×C₈₉	semidirect

H ≅ G/N

type

17. C₄×C₈×C₈₉

C₈

semidirect

59. #7371

C₈₉

direct (N × H)

67. C₈₉

#7371

direct (N × H)

82. C₄×C₈

C₈×C₈₉

semidirect

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

subgroup	index
C₈₉×#1844	2
C₈₉×#1325	2
#7371	89

subgroup

index

C₈₉×#1844

C₈₉×#1325

#7371

Subgroups — C₈₉×#7371 #36385219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7371 #36385219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7371 #36385219