Group #26026354 — subgroups

Subgroups — #26026354

Order 11392.

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

normal subgroups only

This group has 804 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×#15667986	C₂	1	6	yes	maximal, complemented
2	#19151535	C₂	1	4	yes	maximal, complemented
3	#19151508	C₂	1	4	yes	maximal, complemented
4	C₂×#95×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₅₆	C₂²	1	6	yes	—
6	C₂²×C₂²⋊D₈₉	C₂²	1	4	yes	—
7	#15667986	—	2	24	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	6	yes	—
9	C₂×#13083551	C₂²	1	12	yes	—
10	C₂³×C₄×C₈₉	C₂²	1	3	yes	—
11	#95×C₈₉	C₂²	1	4	yes	—
12	C₂³×D₈₉	C₂³	1	4	yes	—
13	C₂×D₃₅₆	—	2	12	no	—
14	C₂×D₃₅₆	—	4	12	no	—
15	C₂×C₂²⋊D₈₉	—	2	24	no	—
16	C₂²×Dic₈₉	C₂³	1	4	yes	—
17	C₄⋊Dic₈₉	—	2	12	no	—
18	#13083551	—	2	24	no	—
19	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₈₉	—	4	6	no	—
21	C₂²×C₄×C₈₉	D₄	1	12	yes	complemented
22	C₂²×C₄×C₈₉	C₂³	1	6	yes	—
23	C₂³×C₈₉	—	2	3	no	—
24	C₂³×C₈₉	—	2	1	no	—
25	C₂³×C₈₉	C₂×D₄	1	6	yes	—
26	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
27	C₂²×D₈₉	—	2	12	no	—
28	C₂²×D₈₉	—	2	12	no	—
29	C₂²×D₈₉	—	2	4	no	—
30	C₂×C₄×C₈₉	—	2	12	no	—
31	C₂×C₄×C₈₉	—	4	12	no	—
32	C₂×C₄×C₈₉	C₂×D₄	1	12	yes	—
33	C₂²⋊D₈₉	—	4	16	no	—
34	C₂×Dic₈₉	—	2	12	no	—
35	D₃₅₆	—	4	24	no	—
36	C₂²×C₈₉	—	2	6	no	—
37	C₂²×C₈₉	—	2	6	no	—
38	C₂²×C₈₉	#127	1	4	yes	—
39	C₂²×C₈₉	C₂²×D₄	1	3	yes	—
40	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
41	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
42	C₂×D₈₉	—	4	4	no	—
43	C₂×D₈₉	—	4	12	no	—
44	C₂×D₈₉	—	4	12	no	—
45	C₄×C₈₉	—	2	12	no	—
46	Dic₈₉	—	4	4	no	—
47	C₂×C₈₉	—	2	4	no	—
48	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
49	C₂×C₈₉	#534	1	3	yes	—
50	C₂×C₈₉	#537	1	3	yes	—
51	D₈₉	—	8	4	no	—
52	#2656	—	89	1	no	maximal, Sylow
53	C₈₉	#2656	1	1	yes	characteristic, complemented, Sylow
54	C₂×#121	—	89	6	no	—
55	#399	—	89	4	no	—
56	#392	—	89	4	no	—
57	C₂×#95	D₈₉	1	1	yes	characteristic, complemented
58	C₂²×D₄	—	89	4	no	—
59	C₂²×D₄	—	89	6	no	—
60	C₂³×C₄	C₂×D₈₉	1	3	yes	—
61	#121	—	178	24	no	—
62	C₂×C₄⋊C₄	—	89	6	no	—
63	C₂×C₂²⋊C₄	—	89	12	no	—
64	#95	C₂×D₈₉	1	4	yes	—
65	C₂⁴	—	89	4	no	—
66	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
67	C₂×D₄	—	356	12	no	—
68	C₂×D₄	—	178	24	no	—
69	C₂×D₄	—	178	12	no	—
70	C₂²×C₄	—	4	6	no	—
71	C₂²×C₄	—	89	4	no	—
72	C₂²×C₄	D₃₅₆	1	12	yes	complemented
73	C₂²×C₄	C₂²×D₈₉	1	6	yes	—
74	C₄⋊C₄	—	178	12	no	—
75	C₂²⋊C₄	—	178	24	no	—
76	C₂³	—	2	3	no	—
77	C₂³	—	2	1	no	—
78	C₂³	—	178	12	no	—
79	C₂³	—	178	12	no	—
80	C₂³	—	178	4	no	—
81	C₂³	C₂×D₃₅₆	1	6	yes	—
82	C₂³	C₂³×D₈₉	1	1	yes	characteristic
83	D₄	—	356	24	no	—
84	D₄	—	356	16	no	—
85	C₂×C₄	—	2	12	no	—
86	C₂×C₄	—	4	12	no	—
87	C₂×C₄	—	178	12	no	—
88	C₂×C₄	C₂×D₃₅₆	1	12	yes	—
89	C₂²	—	2	6	no	—
90	C₂²	—	2	6	no	—
91	C₂²	—	356	4	no	—
92	C₂²	—	356	12	no	—
93	C₂²	—	356	12	no	—
94	C₂²	#15668009	1	4	yes	—
95	C₂²	C₂²×D₃₅₆	1	3	yes	—
96	C₂²	#15668057	1	3	yes	—
97	C₂²	#15668060	1	1	yes	characteristic
98	C₄	—	2	12	no	—
99	C₄	—	356	4	no	—
100	C₂	—	2	4	no	—
101	C₂	—	712	4	no	—
102	C₂	C₂×#15668009	1	1	yes	characteristic
103	C₂	#19152160	1	3	yes	—
104	C₂	#19152179	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667986

C₂

yes

maximal, complemented

#19151535

C₂

yes

maximal, complemented

#19151508

C₂

yes

maximal, complemented

C₂×#95×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

—

C₂²×C₂²⋊D₈₉

C₂²

yes

—

#15667986

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083551

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

complemented

C₂²×C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#127

yes

—

C₂²×C₈₉

C₂²×D₄

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#127

yes

characteristic

C₂×C₈₉

#534

yes

—

C₂×C₈₉

#537

yes

—

D₈₉

—

#2656

—

maximal, Sylow

C₈₉

#2656

yes

characteristic, complemented, Sylow

C₂×#121

—

#399

—

#392

—

C₂×#95

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

C₂×D₈₉

yes

—

#121

—

178

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

#95

C₂×D₈₉

yes

—

C₂⁴

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₃₅₆

yes

complemented

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₂³

—

C₂³

—

C₂³

—

178

—

C₂³

—

178

—

C₂³

—

178

—

C₂³

C₂×D₃₅₆

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

D₄

—

356

—

D₄

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂²

—

C₂²

—

C₂²

—

356

—

C₂²

—

356

—

C₂²

—

356

—

C₂²

#15668009

yes

—

C₂²

C₂²×D₃₅₆

yes

—

C₂²

#15668057

yes

—

C₂²

#15668060

yes

characteristic

C₄

—

C₄

—

356

—

100

C₂

—

101

C₂

—

712

—

102

C₂

C₂×#15668009

yes

characteristic

103

C₂

#19152160

yes

—

104

C₂

#19152179

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×#95×C₈₉	C₂
hypercenter	C₂×#95	D₈₉
hypocenter	C₈₉	#2656
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#95

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×#95×C₈₉

C₂

hypercenter

C₂×#95

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#95

N	H ≅ G/N	type
1. C₂×#15667986	C₂	semidirect
2. #19151535	C₂	semidirect
3. #19151508	C₂	semidirect
4. C₂×#95×C₈₉	C₂	semidirect
21. C₂²×C₄×C₈₉	D₄	semidirect
53. C₈₉	#2656	semidirect
57. C₂×#95	D₈₉	semidirect
72. C₂²×C₄	D₃₅₆	semidirect

H ≅ G/N

type

1. C₂×#15667986

C₂

semidirect

2. #19151535

C₂

semidirect

3. #19151508

C₂

semidirect

4. C₂×#95×C₈₉

C₂

semidirect

21. C₂²×C₄×C₈₉

D₄

semidirect

53. C₈₉

#2656

semidirect

57. C₂×#95

D₈₉

semidirect

72. C₂²×C₄

D₃₅₆

semidirect

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

subgroup	index
C₂×#15667986	2
#19151535	2
#19151508	2
C₂×#95×C₈₉	2
#2656	89

subgroup

index

C₂×#95×C₈₉

Subgroups — #26026354

Conjugacy classes of subgroups

Complements

Maximal subgroups