Group #26022048 — subgroups

Subgroups — #26022048

Order 11392.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	#19151416	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#13083549	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₈₉	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₈₉	C₄	1	1	yes	characteristic
8	C₂×#13083549	C₂²	1	2	yes	—
9	C₈×Dic₈₉	C₄	1	1	yes	characteristic
10	C₈×Dic₈₉	C₄	1	1	yes	characteristic
11	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
12	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂²×Dic₈₉	C₈	1	1	yes	characteristic
16	C₂²×Dic₈₉	C₈	1	1	yes	characteristic
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
19	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₈₉	C₈	1	1	yes	characteristic
21	C₄×Dic₈₉	C₈	1	1	yes	characteristic
22	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
24	#13083549	—	2	2	no	—
25	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₁₆×C₈₉	—	2	2	no	—
30	C₂³×C₈₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₈₉	—	2	1	no	—
32	C₂×C₄×C₈₉	—	2	1	no	—
33	C₂×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₈₉	—	2	2	no	—
36	C₂×Dic₈₉	C₁₆	1	4	yes	complemented
37	C₂×Dic₈₉	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₈₉	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₈₉	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₈₉	C₂×C₈	1	1	yes	characteristic
41	C₈×C₈₉	—	2	1	no	—
42	C₈×C₈₉	#41	1	2	yes	—
43	#10623494	—	2	1	no	—
44	#10623494	M₁₆	1	2	yes	—
45	C₂²×C₈₉	—	2	1	no	—
46	C₂²×C₈₉	—	2	1	no	—
47	C₂²×C₈₉	C₂×C₁₆	1	2	yes	—
48	C₂²×C₈₉	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₈₉	—	2	1	no	—
50	C₄×C₈₉	#117	1	1	yes	characteristic
51	C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
52	Dic₈₉	—	2	1	no	—
53	Dic₈₉	C₂×C₁₆	1	2	yes	—
54	Dic₈₉	C₂×C₁₆	1	2	yes	—
55	Dic₈₉	M₃₂	1	2	yes	—
56	C₂×C₈₉	—	2	2	no	—
57	C₂×C₈₉	#431	1	1	yes	characteristic
58	C₂×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
59	C₂×C₈₉	C₂×M₃₂	1	1	yes	characteristic
60	#2162	—	89	1	no	maximal, Sylow
61	C₈₉	#2162	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄×C₈	—	89	1	no	—
63	C₄⋊C₁₆	—	89	2	no	—
64	C₂²⋊C₁₆	—	89	1	no	—
65	C₂²⋊C₁₆	D₈₉	1	1	yes	characteristic
66	C₄×C₁₆	—	89	2	no	—
67	C₂²×C₈	—	89	1	no	—
68	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
69	C₂×C₄²	—	89	1	no	—
70	C₂×C₁₆	—	89	2	no	—
71	C₂×C₁₆	C₂×D₈₉	1	2	yes	—
72	C₄×C₈	—	89	1	no	—
73	C₄×C₈	—	89	1	no	—
74	C₄×C₈	—	89	1	no	—
75	C₄×C₈	—	89	1	no	—
76	C₂²×C₄	—	89	1	no	—
77	C₂²×C₄	—	89	1	no	—
78	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	89	1	no	—
81	C₂×C₈	—	89	1	no	—
82	C₂×C₈	—	178	2	no	—
83	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
85	C₄²	—	89	1	no	—
86	C₄²	—	89	1	no	—
87	C₄²	—	89	1	no	—
88	C₄²	—	89	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	178	2	no	—
91	C₂³	C₈×D₈₉	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	89	1	no	—
95	C₂×C₄	—	89	1	no	—
96	C₂×C₄	—	89	1	no	—
97	C₂×C₄	—	89	4	no	—
98	C₂×C₄	—	89	1	no	—
99	C₂×C₄	—	178	2	no	—
100	C₂×C₄	C₈×D₈₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	89	2	no	—
104	C₈	—	178	1	no	—
105	C₈	#13083573	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	C₁₆×D₈₉	1	2	yes	—
109	C₂²	C₂×C₈×D₈₉	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	89	2	no	—
112	C₄	—	89	2	no	—
113	C₄	—	89	2	no	—
114	C₄	—	178	1	no	—
115	C₄	#15667968	1	1	yes	characteristic
116	C₄	M₁₆×D₈₉	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#19151648	1	1	yes	characteristic
119	C₂	C₂×C₁₆×D₈₉	1	1	yes	characteristic
120	C₂	M₃₂×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₈₉

C₂

yes

maximal, characteristic

#19151416

C₂

yes

maximal, complemented

C₂²⋊#13083549

C₂

yes

maximal, characteristic

C₁₆×Dic₈₉

C₂

yes

maximal, complemented

C₂²⋊C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₄

yes

characteristic

C₂×C₄×Dic₈₉

C₄

yes

characteristic

C₂×#13083549

C₂²

yes

—

C₈×Dic₈₉

C₄

yes

characteristic

C₈×Dic₈₉

C₄

yes

characteristic

C₈×Dic₈₉

C₂²

yes

characteristic

C₈×Dic₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₁₆×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

C₈

yes

characteristic

C₂²×Dic₈₉

C₈

yes

characteristic

C₂×#10623494

—

C₂×#10623494

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

characteristic

C₄×Dic₈₉

C₈

yes

characteristic

C₄×Dic₈₉

C₈

yes

characteristic

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₄×Dic₈₉

C₂×C₄

yes

characteristic

#13083549

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₁₆

yes

complemented

C₂×Dic₈₉

C₂×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₈

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

#41

yes

—

#10623494

—

#10623494

M₁₆

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₁₆

yes

—

C₂²×C₈₉

C₂²×C₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#117

yes

characteristic

C₄×C₈₉

C₂×M₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

C₂×C₁₆

yes

—

Dic₈₉

C₂×C₁₆

yes

—

Dic₈₉

M₃₂

yes

—

C₂×C₈₉

—

C₂×C₈₉

#431

yes

characteristic

C₂×C₈₉

C₂²×C₁₆

yes

characteristic

C₂×C₈₉

C₂×M₃₂

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₈₉

yes

characteristic

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

178

—

C₂³

C₈×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

100

C₂×C₄

C₈×D₈₉

yes

characteristic

101

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

178

—

105

C₈

#13083573

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₁₆×D₈₉

yes

—

109

C₂²

C₂×C₈×D₈₉

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

178

—

115

C₄

#15667968

yes

characteristic

116

C₄

M₁₆×D₈₉

yes

characteristic

117

C₂

—

118

C₂

#19151648

yes

characteristic

119

C₂

C₂×C₁₆×D₈₉

yes

characteristic

120

C₂

M₃₂×D₈₉

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₁₆
Fitting	C₂²⋊C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₈
prime radical	C₂³×C₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂³
hypercenter	C₂²⋊C₁₆	D₈₉
hypocenter	C₈₉	#2162
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₈₉

commutator

C₂×C₈₉

C₂²×C₁₆

Fitting

C₂²⋊C₁₆×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×C₈

prime radical

C₂³×C₈₉

C₂×C₈

cosocle

C₂×C₈×C₈₉

C₂³

hypercenter

C₂²⋊C₁₆

D₈₉

C₈₉

G ▷ C₂×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
2. #19151416	C₂	semidirect
4. C₁₆×Dic₈₉	C₂	semidirect
36. C₂×Dic₈₉	C₁₆	semidirect
61. C₈₉	#2162	semidirect

H ≅ G/N

type

2. #19151416

C₂

semidirect

4. C₁₆×Dic₈₉

C₂

semidirect

36. C₂×Dic₈₉

C₁₆

semidirect

61. C₈₉

#2162

semidirect

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

subgroup	index
C₂×C₈×Dic₈₉	2
#19151416	2
C₂²⋊#13083549	2
C₁₆×Dic₈₉	2
C₂²⋊C₁₆×C₈₉	2
#2162	89

subgroup

index

C₂×C₈×Dic₈₉

#19151416

C₂²⋊#13083549

C₁₆×Dic₈₉

C₂²⋊C₁₆×C₈₉

#2162

Subgroups — #26022048

Conjugacy classes of subgroups

Complements

Maximal subgroups