Group #26034746 — subgroups

Subgroups — #26034746

Order 11392.

890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 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⋊#564	C₂	1	3	yes	maximal, complemented
2	#19152325	C₂	1	12	yes	maximal, complemented
3	#19152207	C₂	1	12	yes	maximal
4	#19152154	C₂	1	3	yes	maximal
5	#564×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668011	C₂²	1	6	yes	—
7	#15668011	C₂²	1	12	yes	—
8	#15668007	C₂²	1	12	yes	—
9	#15668006	C₂²	1	4	yes	—
10	#15668004	C₂²	1	3	yes	—
11	#15668003	C₂²	1	12	yes	—
12	#15668002	C₂²	1	12	yes	—
13	#15668002	C₂²	1	12	yes	—
14	#15668002	C₂²	1	12	yes	—
15	#15667990	C₂²	1	12	yes	—
16	C₄×Dic₁₇₈	C₂²	1	12	yes	—
17	#15667977	C₂²	1	12	yes	—
18	#15667977	C₂²	1	12	yes	—
19	#15667970	C₂²	1	6	yes	—
20	C₄²×D₈₉	C₂²	1	1	yes	characteristic
21	#125×C₈₉	C₂²	1	3	yes	—
22	#125×C₈₉	C₂²	1	12	yes	—
23	C₂×Dic₁₇₈	C₂³	1	12	yes	—
24	C₂×C₄×D₈₉	C₂³	1	3	yes	—
25	C₄⋊Dic₈₉	C₂³	1	6	yes	—
26	C₄⋊Dic₈₉	C₂³	1	12	yes	—
27	#13083553	C₂³	1	12	yes	—
28	#13083553	C₂³	1	12	yes	—
29	#13083553	C₂³	1	12	yes	—
30	#13083553	C₂³	1	12	yes	—
31	#13083551	C₂³	1	12	yes	—
32	C₄×Dic₈₉	C₂³	1	12	yes	—
33	C₄×Dic₈₉	C₂³	1	3	yes	—
34	C₄×Dic₈₉	C₂³	1	12	yes	—
35	C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
36	C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
37	C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
38	C₄²×C₈₉	C₂³	1	4	yes	—
39	C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₂²×D₈₉	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₂⁴	1	12	yes	—
42	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
43	C₂×Dic₈₉	C₂⁴	1	12	yes	—
44	C₂×Dic₈₉	C₂⁴	1	3	yes	—
45	C₄×D₈₉	—	2	6	no	—
46	Dic₁₇₈	—	2	24	no	—
47	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
48	C₂×D₈₉	—	2	3	no	—
49	C₄×C₈₉	—	2	12	no	—
50	C₄×C₈₉	—	2	3	no	—
51	Dic₈₉	—	2	12	no	—
52	Dic₈₉	2₋¹⁺⁴	1	6	yes	—
53	C₂×C₈₉	C₂×2₋¹⁺⁴	1	3	yes	—
54	D₈₉	—	4	1	no	—
55	#3556	—	89	1	no	maximal, Sylow
56	C₈₉	#3556	1	1	yes	characteristic, complemented, Sylow
57	#564	—	89	3	no	—
58	#564	D₈₉	1	1	yes	characteristic, complemented
59	#557	—	89	12	no	—
60	#541	—	89	12	no	—
61	#533	—	89	3	no	—
62	C₄⋊Q₈	—	89	6	no	—
63	C₄⋊Q₈	—	89	12	no	—
64	#126	—	89	12	no	—
65	#126	—	89	4	no	—
66	#125	—	89	12	no	—
67	#125	—	89	12	no	—
68	#125	—	89	12	no	—
69	#125	—	89	12	no	—
70	#125	—	89	3	no	—
71	#125	C₂×D₈₉	1	3	yes	—
72	#125	C₂×D₈₉	1	12	yes	—
73	C₂²⋊Q₈	—	89	12	no	—
74	C₄×Q₈	—	89	12	no	—
75	C₄×Q₈	—	89	12	no	—
76	C₄×Q₈	—	89	12	no	—
77	#117	—	89	6	no	—
78	C₂×C₄²	—	89	1	no	—
79	C₂×Q₈	—	89	12	no	—
80	C₂²×C₄	—	89	3	no	—
81	C₄⋊C₄	—	89	6	no	—
82	C₄⋊C₄	—	89	12	no	—
83	C₄⋊C₄	—	89	12	no	—
84	C₄⋊C₄	—	89	12	no	—
85	C₄⋊C₄	—	89	12	no	—
86	C₄⋊C₄	—	89	12	no	—
87	C₄⋊C₄	C₂²×D₈₉	1	6	yes	—
88	C₄⋊C₄	C₂²×D₈₉	1	12	yes	—
89	C₄⋊C₄	C₂²×D₈₉	1	12	yes	—
90	C₂²⋊C₄	—	89	12	no	—
91	C₄²	—	89	3	no	—
92	C₄²	—	89	12	no	—
93	C₄²	—	89	12	no	—
94	C₄²	C₂²×D₈₉	1	1	yes	characteristic
95	C₄²	C₂²×D₈₉	1	4	yes	—
96	C₂³	—	89	1	no	—
97	Q₈	—	178	24	no	—
98	C₂×C₄	—	89	3	no	—
99	C₂×C₄	—	89	12	no	—
100	C₂×C₄	—	178	6	no	—
101	C₂×C₄	C₂³×D₈₉	1	12	yes	—
102	C₂×C₄	C₂³×D₈₉	1	3	yes	—
103	C₂²	—	178	3	no	—
104	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
105	C₄	—	2	3	no	—
106	C₄	—	2	12	no	—
107	C₄	—	89	6	no	—
108	C₄	—	178	12	no	—
109	C₂	—	356	1	no	—
110	C₂	2₋¹⁺⁴×D₈₉	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉⋊#564

C₂

yes

maximal, complemented

#19152325

C₂

yes

maximal, complemented

#19152207

C₂

yes

maximal

#19152154

C₂

yes

maximal

#564×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668011

C₂²

yes

—

#15668011

C₂²

yes

—

#15668007

C₂²

yes

—

#15668006

C₂²

yes

—

#15668004

C₂²

yes

—

#15668003

C₂²

yes

—

#15668002

C₂²

yes

—

#15668002

C₂²

yes

—

#15668002

C₂²

yes

—

#15667990

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667977

C₂²

yes

—

#15667977

C₂²

yes

—

#15667970

C₂²

yes

—

C₄²×D₈₉

C₂²

yes

characteristic

#125×C₈₉

C₂²

yes

—

#125×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂×C₄×D₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

C₂³

yes

—

#13083553

C₂³

yes

—

#13083553

C₂³

yes

—

#13083553

C₂³

yes

—

#13083551

C₂³

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

—

C₂×Dic₈₉

C₂⁴

yes

—

C₂×Dic₈₉

C₂⁴

yes

—

C₄×D₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂⁵

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

2₋¹⁺⁴

yes

—

C₂×C₈₉

C₂×2₋¹⁺⁴

yes

—

D₈₉

—

#3556

—

maximal, Sylow

C₈₉

#3556

yes

characteristic, complemented, Sylow

#564

—

#564

D₈₉

yes

characteristic, complemented

#557

—

#541

—

#533

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

—

—

C₂×D₈₉

yes

—

#125

C₂×D₈₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

C₂²×D₈₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₈₉

yes

characteristic

C₄²

C₂²×D₈₉

yes

—

C₂³

—

Q₈

—

178

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

178

—

101

C₂×C₄

C₂³×D₈₉

yes

—

102

C₂×C₄

C₂³×D₈₉

yes

—

103

C₂²

—

178

—

104

C₂²

C₂⁴×D₈₉

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

178

—

109

C₂

—

356

—

110

C₂

2₋¹⁺⁴×D₈₉

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#564×C₈₉	C₂
prime radical	C₂²×D₈₉	C₂⁴
hypercenter	#564	D₈₉
hypocenter	C₈₉	#3556
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₈₉

commutator, socle, cosocle

C₂²×C₈₉

C₂⁵

Fitting

#564×C₈₉

C₂

prime radical

C₂²×D₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. C₈₉⋊#564	C₂	semidirect
2. #19152325	C₂	semidirect
5. #564×C₈₉	C₂	semidirect
56. C₈₉	#3556	semidirect
58. #564	D₈₉	semidirect

H ≅ G/N

type

1. C₈₉⋊#564

C₂

semidirect

2. #19152325

C₂

semidirect

5. #564×C₈₉

C₂

semidirect

56. C₈₉

#3556

semidirect

58. #564

D₈₉

semidirect

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

subgroup	index
C₈₉⋊#564	2
#19152325	2
#19152207	2
#19152154	2
#564×C₈₉	2
#3556	89

subgroup

index

C₈₉⋊#564

#564×C₈₉

Subgroups — #26034746

Conjugacy classes of subgroups

Complements

Maximal subgroups