Group #26026212 — subgroups

Subgroups — #26026212

Order 11392.

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

normal subgroups only

This group has 420 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	#19151527	C₂	1	6	yes	maximal, complemented
2	#19151522	C₂	1	2	yes	maximal, complemented
3	#19151500	C₂	1	3	yes	maximal
4	#19151496	C₂	1	3	yes	maximal, complemented
5	#376×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
8	C₂×#13083553	C₂²	1	6	yes	—
9	C₂×#13083551	C₂²	1	6	yes	—
10	C₂×#13083551	C₂²	1	6	yes	—
11	#15667919	C₂²	1	6	yes	—
12	C₂×C₄²×C₈₉	C₂²	1	3	yes	—
13	#95×C₈₉	C₂²	1	1	yes	characteristic
14	#95×C₈₉	C₂²	1	3	yes	—
15	C₂³×D₈₉	C₂³	1	2	yes	—
16	C₂×D₃₅₆	—	2	6	no	—
17	C₂²×Dic₈₉	C₂³	1	6	yes	—
18	C₄⋊Dic₈₉	—	2	6	no	—
19	#13083553	—	4	6	no	—
20	#13083551	—	4	6	no	—
21	#13083551	—	4	6	no	—
22	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
23	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
24	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
25	C₄²×C₈₉	—	2	6	no	—
26	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
27	C₂²×D₈₉	—	2	6	no	—
28	C₂²×D₈₉	—	2	6	no	—
29	C₂²×D₈₉	—	2	2	no	—
30	C₂×C₄×C₈₉	—	2	3	no	—
31	C₂×C₄×C₈₉	—	2	3	no	—
32	C₂×C₄×C₈₉	—	2	3	no	—
33	C₂×C₄×C₈₉	—	2	3	no	—
34	C₂×C₄×C₈₉	—	2	3	no	—
35	C₂×C₄×C₈₉	—	2	3	no	—
36	C₂×C₄×C₈₉	#41	1	6	yes	—
37	C₂×Dic₈₉	—	2	6	no	—
38	C₂×Dic₈₉	—	2	6	no	—
39	C₂×Dic₈₉	—	2	6	no	—
40	D₃₅₆	—	4	4	no	—
41	C₂²×C₈₉	C₂×#41	1	3	yes	—
42	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
43	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
44	C₂×D₈₉	—	4	2	no	—
45	C₂×D₈₉	—	4	6	no	—
46	C₂×D₈₉	—	4	6	no	—
47	C₄×C₈₉	—	4	3	no	—
48	C₄×C₈₉	—	4	3	no	—
49	C₄×C₈₉	#126	1	4	yes	—
50	Dic₈₉	—	4	6	no	—
51	C₂×C₈₉	C₂×#126	1	1	yes	characteristic
52	C₂×C₈₉	#540	1	3	yes	—
53	C₂×C₈₉	#542	1	3	yes	—
54	D₈₉	—	8	2	no	—
55	#2644	—	89	1	no	maximal, Sylow
56	C₈₉	#2644	1	1	yes	characteristic, complemented, Sylow
57	#397	—	89	6	no	—
58	#396	—	89	2	no	—
59	#390	—	89	3	no	—
60	#389	—	89	3	no	—
61	#376	D₈₉	1	1	yes	characteristic, complemented
62	C₂²×D₄	—	89	1	no	—
63	C₂×C₄⋊C₄	—	89	6	no	—
64	C₂×C₄⋊C₄	—	89	3	no	—
65	C₂×C₂²⋊C₄	—	89	6	no	—
66	C₂×C₂²⋊C₄	—	89	6	no	—
67	C₂×C₄²	C₂×D₈₉	1	3	yes	—
68	#95	—	89	6	no	—
69	#95	C₂×D₈₉	1	1	yes	characteristic
70	#95	C₂×D₈₉	1	3	yes	—
71	C₂⁴	—	89	2	no	—
72	C₂×D₄	—	178	6	no	—
73	C₂²×C₄	—	89	6	no	—
74	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
75	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
76	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
77	C₄⋊C₄	—	356	6	no	—
78	C₄⋊C₄	—	178	6	no	—
79	C₂²⋊C₄	—	356	6	no	—
80	C₂²⋊C₄	—	356	6	no	—
81	C₄²	—	2	6	no	—
82	C₂³	—	178	6	no	—
83	C₂³	—	178	6	no	—
84	C₂³	—	178	2	no	—
85	C₂³	C₂³×D₈₉	1	1	yes	characteristic
86	D₄	—	356	4	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	178	6	no	—
94	C₂×C₄	—	178	6	no	—
95	C₂×C₄	—	178	6	no	—
96	C₂×C₄	#13083574	1	6	yes	—
97	C₂²	—	356	6	no	—
98	C₂²	—	356	6	no	—
99	C₂²	—	356	2	no	—
100	C₂²	C₂×#13083574	1	3	yes	—
101	C₂²	#15668058	1	3	yes	—
102	C₂²	#15668059	1	1	yes	characteristic
103	C₄	—	4	3	no	—
104	C₄	—	4	3	no	—
105	C₄	—	356	6	no	—
106	C₄	#15668006	1	4	yes	—
107	C₂	—	712	2	no	—
108	C₂	C₂×#15668006	1	1	yes	characteristic
109	C₂	#19152203	1	3	yes	—
110	C₂	#19152219	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151527

C₂

yes

maximal, complemented

#19151522

C₂

yes

maximal, complemented

#19151500

C₂

yes

maximal

#19151496

C₂

yes

maximal, complemented

#376×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083553

—

#13083551

—

#13083551

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#126

yes

—

Dic₈₉

—

C₂×C₈₉

C₂×#126

yes

characteristic

C₂×C₈₉

#540

yes

—

C₂×C₈₉

#542

yes

—

D₈₉

—

#2644

—

maximal, Sylow

C₈₉

#2644

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

—

#95

—

#95

C₂×D₈₉

yes

characteristic

#95

C₂×D₈₉

yes

—

C₂⁴

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

356

—

C₂²⋊C₄

—

356

—

C₄²

—

C₂³

—

178

—

C₂³

—

178

—

C₂³

—

178

—

C₂³

C₂³×D₈₉

yes

characteristic

D₄

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

#13083574

yes

—

C₂²

—

356

—

C₂²

—

356

—

C₂²

—

356

—

100

C₂²

C₂×#13083574

yes

—

101

C₂²

#15668058

yes

—

102

C₂²

#15668059

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

356

—

106

C₄

#15668006

yes

—

107

C₂

—

712

—

108

C₂

C₂×#15668006

yes

characteristic

109

C₂

#19152203

yes

—

110

C₂

#19152219

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#376×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#376	D₈₉
hypocenter	C₈₉	#2644
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#376×C₈₉

C₂

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
1. #19151527	C₂	semidirect
2. #19151522	C₂	semidirect
4. #19151496	C₂	semidirect
5. #376×C₈₉	C₂	semidirect
56. C₈₉	#2644	semidirect
61. #376	D₈₉	semidirect

H ≅ G/N

type

1. #19151527

C₂

semidirect

2. #19151522

C₂

semidirect

4. #19151496

C₂

semidirect

5. #376×C₈₉

C₂

semidirect

56. C₈₉

#2644

semidirect

61. #376

D₈₉

semidirect

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

subgroup	index
#19151527	2
#19151522	2
#19151500	2
#19151496	2
#376×C₈₉	2
#2644	89

subgroup

index

#376×C₈₉

Subgroups — #26026212

Conjugacy classes of subgroups

Complements

Maximal subgroups