Group #26028326 — subgroups

Subgroups — #26028326

Order 11392.

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

normal subgroups only

This group has 456 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	2	yes	maximal, complemented
2	#19151528	C₂	1	3	yes	maximal, complemented
3	#19151520	C₂	1	6	yes	maximal, complemented
4	#19151515	C₂	1	3	yes	maximal
5	#384×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
7	#15668006	—	2	8	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
9	C₂×#13083553	C₂²	1	6	yes	—
10	C₂×#13083551	C₂²	1	6	yes	—
11	C₂×#13083551	C₂²	1	6	yes	—
12	#15667919	C₂²	1	3	yes	—
13	#15667919	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂³×D₈₉	C₂³	1	2	yes	—
18	C₂×D₃₅₆	—	4	3	no	—
19	C₂²×Dic₈₉	C₂³	1	6	yes	—
20	C₄⋊Dic₈₉	—	4	3	no	—
21	#13083553	—	2	12	no	—
22	#13083551	—	2	12	no	—
23	#13083551	—	4	6	no	—
24	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
27	C₄⋊C₄×C₈₉	—	4	3	no	—
28	C₄⋊C₄×C₈₉	—	4	3	no	—
29	C₄²×C₈₉	D₄	1	4	yes	complemented
30	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
31	C₂²×D₈₉	—	2	2	no	—
32	C₂²×D₈₉	—	2	6	no	—
33	C₂²×D₈₉	—	2	6	no	—
34	C₂×C₄×C₈₉	—	2	3	no	—
35	C₂×C₄×C₈₉	—	2	3	no	—
36	C₂×C₄×C₈₉	—	2	6	no	—
37	C₂×C₄×C₈₉	—	2	3	no	—
38	C₂×C₄×C₈₉	—	2	3	no	—
39	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
40	C₂×Dic₈₉	—	2	6	no	—
41	C₂×Dic₈₉	—	2	6	no	—
42	C₂×Dic₈₉	—	2	6	no	—
43	D₃₅₆	—	4	4	no	—
44	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
45	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₈₉	2₋¹⁺⁴	1	3	yes	—
47	C₂×D₈₉	—	4	6	no	—
48	C₂×D₈₉	—	4	2	no	—
49	C₂×D₈₉	—	4	6	no	—
50	C₄×C₈₉	—	2	6	no	—
51	C₄×C₈₉	—	4	1	no	—
52	C₄×C₈₉	—	4	3	no	—
53	Dic₈₉	—	4	6	no	—
54	C₂×C₈₉	#537	1	3	yes	—
55	C₂×C₈₉	#560	1	1	yes	characteristic
56	C₂×C₈₉	#563	1	3	yes	—
57	D₈₉	—	8	2	no	—
58	#2823	—	89	1	no	maximal, Sylow
59	C₈₉	#2823	1	1	yes	characteristic, complemented, Sylow
60	C₂×#126	—	89	2	no	—
61	#398	—	89	3	no	—
62	#396	—	89	6	no	—
63	#394	—	89	3	no	—
64	#384	D₈₉	1	1	yes	characteristic, complemented
65	C₂²×D₄	—	89	1	no	—
66	#126	—	178	8	no	—
67	C₂×C₄⋊C₄	—	89	3	no	—
68	C₂×C₄⋊C₄	—	89	6	no	—
69	C₂×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
70	C₂×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
71	C₂×C₂²⋊C₄	—	89	6	no	—
72	C₂×C₂²⋊C₄	—	89	6	no	—
73	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
74	#95	—	89	3	no	—
75	#95	—	89	3	no	—
76	C₂⁴	—	89	2	no	—
77	C₂×D₄	—	356	3	no	—
78	C₂²×C₄	—	89	6	no	—
79	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
80	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
81	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
82	C₄⋊C₄	—	4	3	no	—
83	C₄⋊C₄	—	4	3	no	—
84	C₄⋊C₄	—	356	3	no	—
85	C₄⋊C₄	—	178	12	no	—
86	C₂²⋊C₄	—	356	6	no	—
87	C₂²⋊C₄	—	178	12	no	—
88	C₄²	D₃₅₆	1	4	yes	complemented
89	C₂³	—	178	2	no	—
90	C₂³	—	178	6	no	—
91	C₂³	—	178	6	no	—
92	C₂³	C₂³×D₈₉	1	1	yes	characteristic
93	D₄	—	356	4	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	178	6	no	—
100	C₂×C₄	—	178	6	no	—
101	C₂×C₄	—	178	6	no	—
102	C₂×C₄	C₂×D₃₅₆	1	6	yes	—
103	C₂²	—	356	6	no	—
104	C₂²	—	356	2	no	—
105	C₂²	—	356	6	no	—
106	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
107	C₂²	#15668057	1	3	yes	—
108	C₂²	#15668060	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	3	no	—
112	C₄	—	356	6	no	—
113	C₂	—	712	2	no	—
114	C₂	#19152179	1	3	yes	—
115	C₂	#19152335	1	1	yes	characteristic
116	C₂	#19152347	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668006

C₂

yes

maximal, complemented

#19151528

C₂

yes

maximal, complemented

#19151520

C₂

yes

maximal, complemented

#19151515

C₂

yes

maximal

#384×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

#15668006

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

C₂³

yes

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083553

—

#13083551

—

#13083551

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

complemented

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×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

#537

yes

—

C₂×C₈₉

#560

yes

characteristic

C₂×C₈₉

#563

yes

—

D₈₉

—

#2823

—

maximal, Sylow

C₈₉

#2823

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

#126

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

#95

—

C₂⁴

—

C₂×D₄

—

356

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

356

—

C₂²⋊C₄

—

178

—

C₄²

D₃₅₆

yes

complemented

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

—

178

—

100

C₂×C₄

—

178

—

101

C₂×C₄

—

178

—

102

C₂×C₄

C₂×D₃₅₆

yes

—

103

C₂²

—

356

—

104

C₂²

—

356

—

105

C₂²

—

356

—

106

C₂²

C₂²×D₃₅₆

yes

characteristic

107

C₂²

#15668057

yes

—

108

C₂²

#15668060

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

356

—

113

C₂

—

712

—

114

C₂

#19152179

yes

—

115

C₂

#19152335

yes

characteristic

116

C₂

#19152347

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#384×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#384	D₈₉
hypocenter	C₈₉	#2823
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#384×C₈₉

C₂

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#15668006	C₂	semidirect
2. #19151528	C₂	semidirect
3. #19151520	C₂	semidirect
5. #384×C₈₉	C₂	semidirect
29. C₄²×C₈₉	D₄	semidirect
59. C₈₉	#2823	semidirect
64. #384	D₈₉	semidirect
88. C₄²	D₃₅₆	semidirect

H ≅ G/N

type

1. C₂×#15668006

C₂

semidirect

2. #19151528

C₂

semidirect

3. #19151520

C₂

semidirect

5. #384×C₈₉

C₂

semidirect

29. C₄²×C₈₉

D₄

semidirect

59. C₈₉

#2823

semidirect

64. #384

D₈₉

semidirect

88. C₄²

D₃₅₆

semidirect

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

subgroup	index
C₂×#15668006	2
#19151528	2
#19151520	2
#19151515	2
#384×C₈₉	2
#2823	89

subgroup

index

#384×C₈₉

Subgroups — #26028326

Conjugacy classes of subgroups

Complements

Maximal subgroups