Group #26028293 — subgroups

Subgroups — #26028293

Order 11392.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15668006	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151512	C₂	1	6	yes	maximal, complemented
3	#19151506	C₂	1	6	yes	maximal, complemented
4	#19151466	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊D₈₉	C₂²	1	4	yes	—
7	#15668006	—	2	4	no	—
8	C₂×#13083553	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₈₉	C₂²	1	12	yes	—
10	C₂×#13083551	C₂²	1	3	yes	—
11	#15667919	C₂²	1	6	yes	—
12	C₂²×D₄×C₈₉	C₂²	1	6	yes	—
13	#127×C₈₉	—	2	4	no	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×D₈₉	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₈₉	—	4	12	no	—
17	C₂²×Dic₈₉	C₂³	1	4	yes	—
18	C₂²×Dic₈₉	C₂³	1	3	yes	—
19	#13083553	—	2	6	no	—
20	C₂²⋊Dic₈₉	—	4	12	no	—
21	#13083551	—	2	6	no	—
22	C₂⁴×C₈₉	C₂³	1	4	yes	—
23	C₂×D₄×C₈₉	—	2	12	no	—
24	C₂×D₄×C₈₉	—	4	12	no	—
25	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
26	C₄²×C₈₉	D₄	1	4	yes	complemented
27	C₂³×C₈₉	—	2	4	no	—
28	C₂³×C₈₉	—	2	12	no	—
29	C₂³×C₈₉	—	2	12	no	—
30	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
31	C₂²×D₈₉	—	2	1	no	—
32	C₂²×D₈₉	—	2	6	no	—
33	D₄×C₈₉	—	4	24	no	—
34	C₂×C₄×C₈₉	—	2	6	no	—
35	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
36	C₂²⋊D₈₉	—	4	16	no	—
37	C₂×Dic₈₉	—	2	3	no	—
38	C₂×Dic₈₉	—	2	6	no	—
39	C₂×Dic₈₉	—	2	12	no	—
40	C₂²×C₈₉	—	4	12	no	—
41	C₂²×C₈₉	—	4	4	no	—
42	C₂²×C₈₉	—	4	12	no	—
43	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
44	C₂²×C₈₉	H(5,𝔽₂)	1	6	yes	—
45	C₂×D₈₉	—	4	3	no	—
46	C₂×D₈₉	—	4	4	no	—
47	C₄×C₈₉	—	2	6	no	—
48	Dic₈₉	—	4	3	no	—
49	Dic₈₉	—	4	4	no	—
50	C₂×C₈₉	—	8	4	no	—
51	C₂×C₈₉	#535	1	3	yes	—
52	C₂×C₈₉	#560	1	4	yes	—
53	D₈₉	—	8	1	no	—
54	#2818	—	89	1	no	maximal, Sylow
55	C₈₉	#2818	1	1	yes	characteristic, complemented, Sylow
56	C₂×#127	D₈₉	1	1	yes	characteristic, complemented
57	C₂×#126	—	89	1	no	—
58	#394	—	89	6	no	—
59	#392	—	89	6	no	—
60	#383	—	89	1	no	—
61	C₂²×D₄	—	89	4	no	—
62	C₂²×D₄	C₂×D₈₉	1	6	yes	—
63	#127	—	2	4	no	—
64	#126	—	178	4	no	—
65	C₂×C₄⋊C₄	—	89	3	no	—
66	C₂×C₂²⋊C₄	—	89	12	no	—
67	C₂×C₂²⋊C₄	—	89	3	no	—
68	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
69	#95	—	89	6	no	—
70	C₂⁴	—	89	1	no	—
71	C₂⁴	C₂²×D₈₉	1	4	yes	—
72	C₂×D₄	—	2	12	no	—
73	C₂×D₄	—	4	12	no	—
74	C₂×D₄	—	356	12	no	—
75	C₂²×C₄	—	89	4	no	—
76	C₂²×C₄	—	89	3	no	—
77	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
78	C₄⋊C₄	—	178	6	no	—
79	C₂²⋊C₄	—	356	12	no	—
80	C₂²⋊C₄	—	178	6	no	—
81	C₄²	C₂²⋊D₈₉	1	4	yes	complemented
82	C₂³	—	2	4	no	—
83	C₂³	—	2	12	no	—
84	C₂³	—	2	12	no	—
85	C₂³	—	178	1	no	—
86	C₂³	—	178	6	no	—
87	C₂³	C₂³×D₈₉	1	1	yes	characteristic
88	D₄	—	4	24	no	—
89	D₄	—	356	16	no	—
90	C₂×C₄	—	2	6	no	—
91	C₂×C₄	—	178	3	no	—
92	C₂×C₄	—	178	6	no	—
93	C₂×C₄	—	178	12	no	—
94	C₂×C₄	C₂×C₂²⋊D₈₉	1	6	yes	—
95	C₂²	—	4	12	no	—
96	C₂²	—	4	4	no	—
97	C₂²	—	4	12	no	—
98	C₂²	—	356	3	no	—
99	C₂²	—	356	4	no	—
100	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
101	C₂²	#15668057	1	6	yes	—
102	C₄	—	2	6	no	—
103	C₄	—	356	3	no	—
104	C₄	—	356	4	no	—
105	C₂	—	8	4	no	—
106	C₂	—	712	1	no	—
107	C₂	#19152165	1	3	yes	—
108	C₂	#19152335	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668006

C₂

yes

maximal, characteristic, complemented

#19151512

C₂

yes

maximal, complemented

#19151506

C₂

yes

maximal, complemented

#19151466

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₈₉

C₂²

yes

—

#15668006

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×D₄×C₈₉

C₂²

yes

—

#127×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

C₂³

yes

characteristic

C₂×C₂²⋊D₈₉

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

#13083551

—

C₂⁴×C₈₉

C₂³

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

D₄

yes

complemented

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#535

yes

—

C₂×C₈₉

#560

yes

—

D₈₉

—

#2818

—

maximal, Sylow

C₈₉

#2818

yes

characteristic, complemented, Sylow

C₂×#127

D₈₉

yes

characteristic, complemented

—

—

—

—

C₂²×D₄

—

C₂²×D₄

C₂×D₈₉

yes

—

#127

—

#126

—

178

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

C₂⁴

—

C₂⁴

C₂²×D₈₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

356

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

356

—

C₂²⋊C₄

—

178

—

C₄²

C₂²⋊D₈₉

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

178

—

C₂³

—

178

—

C₂³

C₂³×D₈₉

yes

characteristic

D₄

—

D₄

—

356

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

356

—

C₂²

—

356

—

100

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

101

C₂²

#15668057

yes

—

102

C₄

—

103

C₄

—

356

—

104

C₄

—

356

—

105

C₂

—

106

C₂

—

712

—

107

C₂

#19152165

yes

—

108

C₂

#19152335

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×#127×C₈₉	C₂
hypercenter	C₂×#127	D₈₉
hypocenter	C₈₉	#2818
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×#127×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#15668006	C₂	semidirect
2. #19151512	C₂	semidirect
3. #19151506	C₂	semidirect
4. #19151466	C₂	semidirect
5. C₂×#127×C₈₉	C₂	semidirect
26. C₄²×C₈₉	D₄	semidirect
55. C₈₉	#2818	semidirect
56. C₂×#127	D₈₉	semidirect
81. C₄²	C₂²⋊D₈₉	semidirect

H ≅ G/N

type

1. C₂×#15668006

C₂

semidirect

2. #19151512

C₂

semidirect

3. #19151506

C₂

semidirect

4. #19151466

C₂

semidirect

5. C₂×#127×C₈₉

C₂

semidirect

26. C₄²×C₈₉

D₄

semidirect

55. C₈₉

#2818

semidirect

56. C₂×#127

D₈₉

semidirect

81. C₄²

C₂²⋊D₈₉

semidirect

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

subgroup	index
C₂×#15668006	2
#19151512	2
#19151506	2
#19151466	2
C₂×#127×C₈₉	2
#2818	89

subgroup

index

C₂×#127×C₈₉

Subgroups — #26028293

Conjugacy classes of subgroups

Complements

Maximal subgroups