Group #26028320 — subgroups

Subgroups — #26028320

Order 11392.

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

normal subgroups only

This group has 380 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×#15668004	C₂	1	2	yes	maximal
2	#19151538	C₂	1	4	yes	maximal
3	#19151530	C₂	1	4	yes	maximal
4	#19151528	C₂	1	4	yes	maximal
5	#382×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668004	—	2	8	no	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
10	C₂×#13083553	C₂²	1	4	yes	—
11	C₂×#13083553	C₂²	1	4	yes	—
12	#15667919	C₂²	1	8	yes	—
13	#15667919	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	#95×C₈₉	C₂²	1	4	yes	—
17	C₂²×Dic₈₉	C₂³	1	4	yes	—
18	C₂²×Dic₈₉	C₂³	1	4	yes	—
19	C₄⋊Dic₈₉	—	2	4	no	—
20	C₄⋊Dic₈₉	—	2	4	no	—
21	C₄⋊Dic₈₉	—	4	4	no	—
22	#13083553	—	2	8	no	—
23	#13083553	—	2	8	no	—
24	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
25	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₈₉	—	4	2	no	—
28	C₄²×C₈₉	D₄	1	4	yes	—
29	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₈₉	—	2	1	no	—
31	C₂×C₄×C₈₉	—	2	4	no	—
32	C₂×C₄×C₈₉	—	2	1	no	—
33	C₂×C₄×C₈₉	—	2	4	no	—
34	C₂×C₄×C₈₉	—	2	8	no	—
35	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
37	C₂×Dic₈₉	—	2	4	no	—
38	C₂×Dic₈₉	—	2	4	no	—
39	C₂×Dic₈₉	—	2	4	no	—
40	C₂×Dic₈₉	—	2	8	no	—
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
43	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
44	C₂²×C₈₉	2₋¹⁺⁴	1	4	yes	—
45	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
46	C₄×C₈₉	—	2	4	no	—
47	C₄×C₈₉	—	2	2	no	—
48	C₄×C₈₉	—	4	4	no	—
49	Dic₈₉	—	4	4	no	—
50	Dic₈₉	—	4	4	no	—
51	C₂×C₈₉	#536	1	2	yes	—
52	C₂×C₈₉	#537	1	1	yes	characteristic
53	C₂×C₈₉	#563	1	2	yes	—
54	C₂×C₈₉	#564	1	2	yes	—
55	#2822	—	89	1	no	maximal, Sylow
56	C₈₉	#2822	1	1	yes	characteristic, complemented, Sylow
57	C₂×#125	—	89	2	no	—
58	#400	—	89	4	no	—
59	#398	—	89	4	no	—
60	#398	—	89	4	no	—
61	#382	D₈₉	1	1	yes	characteristic
62	#125	—	178	8	no	—
63	C₂×C₄⋊C₄	—	89	4	no	—
64	C₂×C₄⋊C₄	—	89	4	no	—
65	C₂×C₄⋊C₄	—	89	2	no	—
66	C₂×C₄⋊C₄	—	89	4	no	—
67	C₂×C₄⋊C₄	—	89	2	no	—
68	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
69	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
70	#95	—	89	8	no	—
71	#95	—	89	4	no	—
72	#95	C₂×D₈₉	1	4	yes	—
73	C₂²×C₄	—	89	4	no	—
74	C₂²×C₄	—	89	4	no	—
75	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
76	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
77	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	356	4	no	—
80	C₄⋊C₄	—	178	8	no	—
81	C₄⋊C₄	—	178	8	no	—
82	C₄⋊C₄	—	178	4	no	—
83	C₄⋊C₄	—	178	4	no	—
84	C₄²	D₃₅₆	1	4	yes	—
85	C₂³	C₂³×D₈₉	1	1	yes	characteristic
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	178	4	no	—
92	C₂×C₄	—	178	4	no	—
93	C₂×C₄	—	178	4	no	—
94	C₂×C₄	—	178	8	no	—
95	C₂×C₄	—	178	4	no	—
96	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
97	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
98	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
99	C₂²	#15668057	1	1	yes	characteristic
100	C₂²	#15668059	1	4	yes	—
101	C₂²	#15668060	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	—	356	4	no	—
106	C₄	—	356	4	no	—
107	C₂	#19152174	1	2	yes	—
108	C₂	#19152179	1	1	yes	characteristic
109	C₂	#19152348	1	2	yes	—
110	C₂	C₈₉⋊#564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal

#19151538

C₂

yes

maximal

#19151530

C₂

yes

maximal

#19151528

C₂

yes

maximal

#382×C₈₉

C₂

yes

maximal, characteristic

#15668004

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#536

yes

—

C₂×C₈₉

#537

yes

characteristic

C₂×C₈₉

#563

yes

—

C₂×C₈₉

#564

yes

—

#2822

—

maximal, Sylow

C₈₉

#2822

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic

#125

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

—

—

C₂×D₈₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

D₃₅₆

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂²

C₂²×D₃₅₆

yes

characteristic

C₂²

#15668057

yes

characteristic

100

C₂²

#15668059

yes

—

101

C₂²

#15668060

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

356

—

106

C₄

—

356

—

107

C₂

#19152174

yes

—

108

C₂

#19152179

yes

characteristic

109

C₂

#19152348

yes

—

110

C₂

C₈₉⋊#564

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#382×C₈₉	C₂
hypercenter	#382	D₈₉
hypocenter	C₈₉	#2822
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#382×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
56. C₈₉	#2822	semidirect

H ≅ G/N

type

56. C₈₉

#2822

semidirect

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

subgroup	index
C₂×#15668004	2
#19151538	2
#19151530	2
#19151528	2
#382×C₈₉	2
#2822	89

subgroup

index

#382×C₈₉

Subgroups — #26028320

Conjugacy classes of subgroups

Complements

Maximal subgroups