Group #26028731 — subgroups

Subgroups — #26028731

Order 11392.

756 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 289 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	#19152365	C₂	1	2	yes	maximal, complemented
2	#19151582	C₂	1	6	yes	maximal, complemented
3	C₂×#15667929	C₂	1	6	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#15668019	—	2	8	no	—
6	C₂×#13083558	C₂²	1	12	yes	complemented
7	#15667931	C₂²	1	4	yes	complemented
8	#15667929	C₂²	1	12	yes	complemented
9	H(5,𝔽₂)×C₈₉	—	2	8	no	—
10	C₂×#41×C₈₉	C₄	1	2	yes	—
11	C₂×#41×C₈₉	C₂²	1	3	yes	—
12	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₈₉	C₄	1	6	yes	—
14	C₂²×D₄×C₈₉	C₂²	1	3	yes	—
15	#13083558	—	2	12	no	—
16	#13083558	C₂³	1	8	yes	—
17	C₂×#10623494	—	2	12	no	—
18	C₂⁴×C₈₉	C₂×C₄	1	6	yes	—
19	#41×C₈₉	—	2	8	no	—
20	#41×C₈₉	—	2	12	no	—
21	#41×C₈₉	D₄	1	8	yes	—
22	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
24	C₂×D₄×C₈₉	—	2	24	no	—
25	C₂×D₄×C₈₉	—	2	12	no	—
26	C₂×D₄×C₈₉	C₂×C₄	1	6	yes	—
27	C₂×D₄×C₈₉	C₂×C₄	1	6	yes	—
28	C₂×D₄×C₈₉	C₂×C₄	1	3	yes	—
29	C₂×D₄×C₈₉	C₂³	1	3	yes	—
30	C₂²×C₄×C₈₉	C₂×C₄	1	6	yes	—
31	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
32	C₂³×C₈₉	—	2	12	no	—
33	C₂³×C₈₉	—	8	6	no	—
34	C₂³×C₈₉	C₂²×C₄	1	6	yes	—
35	C₂³×C₈₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₈₉	—	2	2	no	—
37	Q₈×C₈₉	C₂²⋊C₄	1	4	yes	—
38	D₄×C₈₉	—	2	12	no	—
39	D₄×C₈₉	—	2	12	no	—
40	D₄×C₈₉	—	2	6	no	—
41	D₄×C₈₉	C₂²⋊C₄	1	12	yes	—
42	C₂×C₄×C₈₉	—	2	12	no	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₈₉	C₂×D₄	1	12	yes	—
46	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
47	#10623494	—	2	8	no	—
48	C₂²×C₈₉	—	2	6	no	—
49	C₂²×C₈₉	—	4	3	no	—
50	C₂²×C₈₉	—	4	6	no	—
51	C₂²×C₈₉	—	8	12	no	—
52	C₂²×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
53	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
56	C₄×C₈₉	C₂²×D₄	1	2	yes	—
57	C₂×C₈₉	—	2	1	no	—
58	C₂×C₈₉	—	2	6	no	—
59	C₂×C₈₉	—	4	6	no	—
60	C₂×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
61	#2888	—	89	1	no	maximal, Sylow
62	C₈₉	#2888	1	1	yes	characteristic, complemented, Sylow
63	C₂×H(5,𝔽₂)	D₈₉	1	1	yes	characteristic
64	#568	—	89	2	no	—
65	#413	—	89	6	no	—
66	C₂×#100	—	89	6	no	—
67	H(5,𝔽₂)	—	2	8	no	—
68	C₂×#41	Dic₈₉	1	2	yes	—
69	C₂×#41	C₂×D₈₉	1	3	yes	—
70	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
71	C₂²×D₄	Dic₈₉	1	6	yes	—
72	C₂²×D₄	C₂×D₈₉	1	3	yes	—
73	#131	—	178	8	no	—
74	C₂×M₁₆	—	89	12	no	—
75	#101	—	89	4	no	—
76	#100	—	89	12	no	—
77	C₂⁴	C₂×Dic₈₉	1	6	yes	—
78	#41	—	2	8	no	—
79	#41	—	2	12	no	—
80	#41	C₂²⋊D₈₉	1	8	yes	—
81	C₂×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
82	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
83	C₂×D₄	—	2	24	no	—
84	C₂×D₄	—	2	12	no	—
85	C₂×D₄	C₂×Dic₈₉	1	6	yes	—
86	C₂×D₄	C₂×Dic₈₉	1	6	yes	—
87	C₂×D₄	C₂×Dic₈₉	1	3	yes	—
88	C₂×D₄	C₂²×D₈₉	1	3	yes	—
89	C₂²×C₄	C₂×Dic₈₉	1	6	yes	—
90	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
91	M₁₆	—	89	8	no	—
92	M₁₆	—	178	12	no	—
93	C₂×C₈	—	178	12	no	—
94	C₂³	—	2	12	no	—
95	C₂³	—	8	6	no	—
96	C₂³	C₂²×Dic₈₉	1	6	yes	—
97	C₂³	C₂²×Dic₈₉	1	3	yes	—
98	Q₈	—	2	2	no	—
99	Q₈	C₂²⋊Dic₈₉	1	4	yes	—
100	D₄	—	2	12	no	—
101	D₄	—	2	6	no	—
102	D₄	—	2	12	no	—
103	D₄	C₂²⋊Dic₈₉	1	12	yes	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
106	C₂×C₄	C₂²×Dic₈₉	1	3	yes	—
107	C₂×C₄	C₂×C₂²⋊D₈₉	1	12	yes	—
108	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
109	C₈	—	178	8	no	—
110	C₂²	—	2	6	no	—
111	C₂²	—	4	3	no	—
112	C₂²	—	4	6	no	—
113	C₂²	—	8	12	no	—
114	C₂²	C₂×C₂²⋊Dic₈₉	1	6	yes	—
115	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	C₂×C₂²⋊Dic₈₉	1	6	yes	—
118	C₄	C₂²×C₂²⋊D₈₉	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	6	no	—
121	C₂	—	4	6	no	—
122	C₂	C₂²×C₂²⋊Dic₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152365

C₂

yes

maximal, complemented

#19151582

C₂

yes

maximal, complemented

C₂×#15667929

C₂

yes

maximal, complemented

C₂×H(5,𝔽₂)×C₈₉

C₂

yes

maximal, characteristic

#15668019

—

C₂×#13083558

C₂²

yes

complemented

#15667931

C₂²

yes

complemented

#15667929

C₂²

yes

complemented

H(5,𝔽₂)×C₈₉

—

C₂×#41×C₈₉

C₄

yes

—

C₂×#41×C₈₉

C₂²

yes

—

C₂×#41×C₈₉

C₂²

yes

characteristic

C₂²×D₄×C₈₉

C₄

yes

—

C₂²×D₄×C₈₉

C₂²

yes

—

#13083558

—

#13083558

C₂³

yes

—

C₂×#10623494

—

C₂⁴×C₈₉

C₂×C₄

yes

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

D₄

yes

—

C₂×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×Q₈×C₈₉

C₂³

yes

characteristic

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²×C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

—

Q₈×C₈₉

—

Q₈×C₈₉

C₂²⋊C₄

yes

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×C₂²⋊C₄

yes

—

C₄×C₈₉

C₂²×D₄

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×C₂²⋊C₄

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×H(5,𝔽₂)

D₈₉

yes

—

—

—

—

C₂×#41

Dic₈₉

yes

—

C₂×#41

C₂×D₈₉

yes

—

C₂×#41

C₂×D₈₉

yes

characteristic

C₂²×D₄

Dic₈₉

yes

—

C₂²×D₄

C₂×D₈₉

yes

—

#131

—

178

—

C₂×M₁₆

—

#101

—

#100

—

C₂⁴

C₂×Dic₈₉

yes

—

#41

—

#41

—

#41

C₂²⋊D₈₉

yes

—

C₂×Q₈

C₂×Dic₈₉

yes

characteristic

C₂×Q₈

C₂²×D₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₈₉

yes

—

C₂×D₄

C₂×Dic₈₉

yes

—

C₂×D₄

C₂×Dic₈₉

yes

—

C₂×D₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

178

—

C₂³

—

C₂³

—

C₂³

C₂²×Dic₈₉

yes

—

C₂³

C₂²×Dic₈₉

yes

—

Q₈

—

Q₈

C₂²⋊Dic₈₉

yes

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

C₂²⋊Dic₈₉

yes

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₈₉

yes

—

106

C₂×C₄

C₂²×Dic₈₉

yes

—

107

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

108

C₂×C₄

C₂³×D₈₉

yes

characteristic

109

C₈

—

178

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊Dic₈₉

yes

—

115

C₂²

C₂³×Dic₈₉

yes

characteristic

116

C₄

—

117

C₄

C₂×C₂²⋊Dic₈₉

yes

—

118

C₄

C₂²×C₂²⋊D₈₉

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

C₂²×C₂²⋊Dic₈₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₈₉
commutator	C₂²×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting, prime radical	C₂×H(5,𝔽₂)×C₈₉	C₂
socle	C₂×C₈₉	C₂²×C₂²⋊C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×H(5,𝔽₂)	D₈₉
hypocenter	C₈₉	#2888
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊Dic₈₉

commutator

C₂²×C₈₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₈₉

Fitting, prime radical

C₂×H(5,𝔽₂)×C₈₉

C₂

socle

C₂×C₈₉

C₂²×C₂²⋊C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

hypercenter

C₂×H(5,𝔽₂)

D₈₉

hypocenter

C₈₉

#2888

LCS

G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

N	H ≅ G/N	type
1. #19152365	C₂	semidirect
2. #19151582	C₂	semidirect
3. C₂×#15667929	C₂	semidirect
6. C₂×#13083558	C₂²	semidirect
7. #15667931	C₂²	semidirect
8. #15667929	C₂²	semidirect
62. C₈₉	#2888	semidirect

H ≅ G/N

type

1. #19152365

C₂

semidirect

2. #19151582

C₂

semidirect

3. C₂×#15667929

C₂

semidirect

6. C₂×#13083558

C₂²

semidirect

7. #15667931

C₂²

semidirect

8. #15667929

C₂²

semidirect

62. C₈₉

#2888

semidirect

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

subgroup	index
#19152365	2
#19151582	2
C₂×#15667929	2
C₂×H(5,𝔽₂)×C₈₉	2
#2888	89

subgroup

index

C₂×H(5,𝔽₂)×C₈₉

Subgroups — #26028731

Conjugacy classes of subgroups

Complements

Maximal subgroups