Group #26025971 — subgroups

Subgroups — #26025971

Order 11392.

782 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 181 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×#15667995	C₂	1	3	yes	maximal
2	#19151517	C₂	1	4	yes	maximal, complemented
3	#19151509	C₂	1	4	yes	maximal, complemented
4	#19151461	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
7	#15667995	—	2	12	no	—
8	C₂×#13083553	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₈₉	C₂²	1	3	yes	—
10	C₂×#13083551	C₂²	1	6	yes	—
11	#15667919	C₂²	1	12	yes	—
12	C₂²×Q₈×C₈₉	—	2	12	no	—
13	C₂²×Q₈×C₈₉	C₂²	1	4	yes	—
14	C₂³×C₄×C₈₉	C₂²	1	3	yes	—
15	C₂³×D₈₉	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₈₉	—	2	6	no	—
17	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₈₉	C₂³	1	6	yes	—
19	#13083553	—	2	12	no	—
20	C₂²⋊Dic₈₉	—	2	6	no	—
21	#13083551	—	2	12	no	—
22	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
23	C₂×Q₈×C₈₉	—	2	32	no	—
24	C₂×Q₈×C₈₉	—	2	24	no	—
25	C₂²×C₄×C₈₉	—	2	12	no	—
26	C₂²×C₄×C₈₉	D₄	1	12	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	6	yes	—
28	C₂³×C₈₉	—	2	4	no	—
29	C₂³×C₈₉	C₂×D₄	1	6	yes	—
30	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
31	C₂²×D₈₉	—	2	3	no	—
32	C₂²×D₈₉	—	2	4	no	—
33	Q₈×C₈₉	—	2	32	no	—
34	C₂×C₄×C₈₉	—	2	24	no	—
35	C₂×C₄×C₈₉	—	2	12	no	—
36	C₂×C₄×C₈₉	C₂×D₄	1	12	yes	—
37	C₂²⋊D₈₉	—	4	4	no	—
38	C₂×Dic₈₉	—	2	6	no	—
39	C₂×Dic₈₉	—	2	3	no	—
40	C₂×Dic₈₉	—	2	12	no	—
41	C₂²×C₈₉	—	2	12	no	—
42	C₂²×C₈₉	#120	1	4	yes	—
43	C₂²×C₈₉	C₂²×D₄	1	3	yes	—
44	C₂²×C₈₉	2₋¹⁺⁴	1	4	yes	—
45	C₂×D₈₉	—	4	1	no	—
46	C₂×D₈₉	—	4	6	no	—
47	C₄×C₈₉	—	2	12	no	—
48	Dic₈₉	—	4	1	no	—
49	Dic₈₉	—	4	6	no	—
50	C₂×C₈₉	—	2	4	no	—
51	C₂×C₈₉	C₂×#120	1	1	yes	characteristic
52	C₂×C₈₉	#536	1	6	yes	—
53	D₈₉	—	8	1	no	—
54	#2614	—	89	1	no	maximal, Sylow
55	C₈₉	#2614	1	1	yes	characteristic, complemented, Sylow
56	C₂³×Q₈	D₈₉	1	1	yes	characteristic, complemented
57	C₂×#123	—	89	3	no	—
58	#395	—	89	4	no	—
59	#393	—	89	4	no	—
60	#381	—	89	3	no	—
61	C₂²×Q₈	—	2	12	no	—
62	C₂²×Q₈	C₂×D₈₉	1	4	yes	—
63	C₂²×D₄	—	89	1	no	—
64	C₂³×C₄	C₂×D₈₉	1	3	yes	—
65	#123	—	178	12	no	—
66	C₂×C₄⋊C₄	—	89	6	no	—
67	C₂×C₂²⋊C₄	—	89	3	no	—
68	C₂×C₂²⋊C₄	—	89	6	no	—
69	#95	—	89	12	no	—
70	C₂⁴	—	89	1	no	—
71	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
72	C₂×Q₈	—	2	32	no	—
73	C₂×Q₈	—	2	24	no	—
74	C₂×D₄	—	178	6	no	—
75	C₂²×C₄	—	2	12	no	—
76	C₂²×C₄	—	89	6	no	—
77	C₂²×C₄	—	89	1	no	—
78	C₂²×C₄	C₂²⋊D₈₉	1	12	yes	—
79	C₂²×C₄	C₂²×D₈₉	1	6	yes	—
80	C₄⋊C₄	—	178	12	no	—
81	C₂²⋊C₄	—	178	6	no	—
82	C₂²⋊C₄	—	178	12	no	—
83	C₂³	—	2	4	no	—
84	C₂³	—	178	3	no	—
85	C₂³	—	178	4	no	—
86	C₂³	C₂×C₂²⋊D₈₉	1	6	yes	—
87	C₂³	C₂³×D₈₉	1	1	yes	characteristic
88	Q₈	—	2	32	no	—
89	D₄	—	356	4	no	—
90	C₂×C₄	—	2	24	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	178	6	no	—
93	C₂×C₄	—	178	3	no	—
94	C₂×C₄	—	178	12	no	—
95	C₂×C₄	C₂×C₂²⋊D₈₉	1	12	yes	—
96	C₂²	—	2	12	no	—
97	C₂²	—	356	6	no	—
98	C₂²	—	356	1	no	—
99	C₂²	#15667981	1	4	yes	—
100	C₂²	C₂²×C₂²⋊D₈₉	1	3	yes	—
101	C₂²	#15668060	1	4	yes	—
102	C₄	—	2	12	no	—
103	C₄	—	356	6	no	—
104	C₄	—	356	1	no	—
105	C₂	—	2	4	no	—
106	C₂	—	712	1	no	—
107	C₂	C₂×#15667981	1	1	yes	characteristic
108	C₂	#19152172	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667995

C₂

yes

maximal

#19151517

C₂

yes

maximal, complemented

#19151509

C₂

yes

maximal, complemented

#19151461

C₂

yes

maximal, complemented

C₂³×Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₈₉

C₂²

yes

characteristic

#15667995

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

#13083551

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

—

C₂²×C₈₉

C₂²×D₄

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#120

yes

characteristic

C₂×C₈₉

#536

yes

—

D₈₉

—

#2614

—

maximal, Sylow

C₈₉

#2614

yes

characteristic, complemented, Sylow

C₂³×Q₈

D₈₉

yes

characteristic, complemented

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₉

yes

—

C₂²×D₄

—

C₂³×C₄

C₂×D₈₉

yes

—

#123

—

178

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₂³

—

C₂³

—

178

—

C₂³

—

178

—

C₂³

C₂×C₂²⋊D₈₉

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

D₄

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

C₂²

—

C₂²

—

356

—

C₂²

—

356

—

C₂²

#15667981

yes

—

100

C₂²

C₂²×C₂²⋊D₈₉

yes

—

101

C₂²

#15668060

yes

—

102

C₄

—

103

C₄

—

356

—

104

C₄

—

356

—

105

C₂

—

106

C₂

—

712

—

107

C₂

C₂×#15667981

yes

characteristic

108

C₂

#19152172

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂³×Q₈×C₈₉	C₂
prime radical	C₂²×C₂²⋊D₈₉	C₂²
hypercenter	C₂³×Q₈	D₈₉
hypocenter	C₈₉	#2614
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂³×Q₈×C₈₉

C₂

prime radical

C₂²×C₂²⋊D₈₉

C₂²

hypercenter

C₂³×Q₈

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #19151517	C₂	semidirect
3. #19151509	C₂	semidirect
4. #19151461	C₂	semidirect
5. C₂³×Q₈×C₈₉	C₂	semidirect
55. C₈₉	#2614	semidirect
56. C₂³×Q₈	D₈₉	semidirect

H ≅ G/N

type

2. #19151517

C₂

semidirect

3. #19151509

C₂

semidirect

4. #19151461

C₂

semidirect

5. C₂³×Q₈×C₈₉

C₂

semidirect

55. C₈₉

#2614

semidirect

56. C₂³×Q₈

D₈₉

semidirect

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

subgroup	index
C₂×#15667995	2
#19151517	2
#19151509	2
#19151461	2
C₂³×Q₈×C₈₉	2
#2614	89

subgroup

index

C₂³×Q₈×C₈₉

Subgroups — #26025971

Conjugacy classes of subgroups

Complements

Maximal subgroups