Group #26021245 — subgroups

Subgroups — #26021245

Order 11392.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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	1	yes	maximal, characteristic
2	#19151622	C₂	1	1	yes	maximal, characteristic
3	#19151366	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13083558	C₂²	1	2	yes	—
6	#15668004	—	2	2	no	—
7	#15668004	—	2	1	no	—
8	#15668004	—	2	1	no	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×#13083553	C₄	1	4	yes	—
11	C₄⋊#10623494	—	2	2	no	—
12	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	#97×C₈₉	—	2	1	no	—
15	C₄×C₈×C₈₉	—	2	1	no	—
16	C₂²×Dic₈₉	C₂×C₄	1	4	yes	—
17	#13083558	—	2	4	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₄⋊Dic₈₉	—	2	2	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	#13083553	—	2	4	no	—
22	#13083553	—	2	4	no	—
23	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₈₉	—	2	2	no	—
26	M₁₆×C₈₉	Q₈	1	4	yes	—
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₄²×C₈₉	D₄	1	2	yes	—
29	C₄²×C₈₉	D₄	1	1	yes	characteristic
30	C₄²×C₈₉	D₄	1	1	yes	characteristic
31	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	4	2	no	—
33	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₈₉	#41	1	2	yes	—
39	C₂×Dic₈₉	—	2	4	no	—
40	C₂×Dic₈₉	—	4	4	no	—
41	C₈×C₈₉	—	2	4	no	—
42	#10623494	—	4	2	no	—
43	C₂²×C₈₉	—	2	2	no	—
44	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₈₉	C₄×Q₈	1	2	yes	—
46	C₄×C₈₉	—	2	2	no	—
47	C₄×C₈₉	—	2	2	no	—
48	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₈₉	#124	1	1	yes	characteristic
50	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₈₉	—	4	4	no	—
52	C₂×C₈₉	—	2	2	no	—
53	C₂×C₈₉	#391	1	1	yes	characteristic
54	C₂×C₈₉	#413	1	2	yes	—
55	#1998	—	89	1	no	maximal, Sylow
56	C₈₉	#1998	1	1	yes	characteristic, complemented, Sylow
57	C₂×#125	—	89	1	no	—
58	#423	—	89	1	no	—
59	C₄×M₁₆	D₈₉	1	1	yes	characteristic
60	#343	—	89	4	no	—
61	C₂×M₁₆	—	89	2	no	—
62	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
63	#125	—	178	2	no	—
64	#125	—	178	1	no	—
65	#125	—	178	1	no	—
66	C₂×C₄⋊C₄	—	89	4	no	—
67	C₂×C₄⋊C₄	—	89	2	no	—
68	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
69	C₄⋊C₈	—	178	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	89	4	no	—
73	C₂²×C₄	C₄×D₈₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	178	4	no	—
77	M₁₆	Dic₁₇₈	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	178	2	no	—
80	C₄⋊C₄	—	178	4	no	—
81	C₄⋊C₄	—	178	4	no	—
82	C₄⋊C₄	—	178	2	no	—
83	C₄⋊C₄	—	178	2	no	—
84	C₄²	D₃₅₆	1	2	yes	—
85	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
86	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
87	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	356	4	no	—
90	C₂×C₄	—	178	4	no	—
91	C₂×C₄	#13083551	1	2	yes	—
92	C₂×C₄	#13083551	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
94	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
96	C₂×C₄	#13083574	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	356	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×#13083551	1	1	yes	characteristic
101	C₂²	C₄×Dic₁₇₈	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	356	4	no	—
105	C₄	#15667991	1	2	yes	—
106	C₄	#15668000	1	1	yes	characteristic
107	C₄	#15668012	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#19151504	1	1	yes	characteristic
110	C₂	#19151580	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal, characteristic

#19151622

C₂

yes

maximal, characteristic

#19151366

C₂

yes

maximal

C₄×M₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

#15668004

—

#15668004

—

#15668004

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₄

yes

—

C₄⋊#10623494

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#97×C₈₉

—

C₄×C₈×C₈₉

—

C₂²×Dic₈₉

C₂×C₄

yes

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

Q₈

yes

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

C₄×Q₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#124

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#391

yes

characteristic

C₂×C₈₉

#413

yes

—

#1998

—

maximal, Sylow

C₈₉

#1998

yes

characteristic, complemented, Sylow

C₂×#125

—

#423

—

C₄×M₁₆

D₈₉

yes

characteristic

#343

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

#125

—

178

—

#125

—

178

—

#125

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

178

—

#97

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

178

—

M₁₆

Dic₁₇₈

yes

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

D₃₅₆

yes

—

C₄²

C₂²⋊D₈₉

yes

characteristic

C₄²

C₂²⋊D₈₉

yes

characteristic

C₂³

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

#13083551

yes

—

C₂×C₄

#13083551

yes

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

#13083574

yes

—

C₈

—

C₈

—

356

—

C₂²

—

100

C₂²

C₂×#13083551

yes

characteristic

101

C₂²

C₄×Dic₁₇₈

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

356

—

105

C₄

#15667991

yes

—

106

C₄

#15668000

yes

characteristic

107

C₄

#15668012

yes

characteristic

108

C₂

—

109

C₂

#19151504

yes

characteristic

110

C₂

#19151580

yes

—

	subgroup	G/H
center	C₂²	C₂×#13083551
commutator, prime radical	C₂³×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₄×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₄×M₁₆	D₈₉
hypocenter	C₈₉	#1998
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#13083551

commutator, prime radical

C₂³×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₄×M₁₆×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₄×M₁₆

D₈₉

hypocenter

C₈₉

#1998

LCS

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
56. C₈₉	#1998	semidirect

H ≅ G/N

type

56. C₈₉

#1998

semidirect

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

subgroup	index
C₂×#15668004	2
#19151622	2
#19151366	2
C₄×M₁₆×C₈₉	2
#1998	89

subgroup

index

C₄×M₁₆×C₈₉

Subgroups — #26021245

Conjugacy classes of subgroups

Complements

Maximal subgroups