Group #26018160 — subgroups

Subgroups — #26018160

Order 11392.

192 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151562	C₂	1	2	yes	maximal
2	#381×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#13083558	—	2	2	no	—
4	C₂²⋊#10623494	—	2	2	no	—
5	C₂²⋊#10623494	C₄	1	4	yes	complemented
6	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₈₉	C₄	1	2	yes	—
8	#95×C₈₉	—	2	2	no	—
9	#13083558	—	4	4	no	—
10	C₂×#10623494	—	2	4	no	—
11	C₂⁴×C₈₉	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₈₉	—	2	2	no	—
13	C₂²×C₄×C₈₉	—	4	2	no	—
14	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₈₉	D₄	1	2	yes	—
16	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₈₉	Q₈	1	1	yes	characteristic
18	C₂²⋊C₄×C₈₉	—	4	2	no	—
19	C₂³×C₈₉	—	2	2	no	—
20	C₂³×C₈₉	—	4	2	no	—
21	C₂³×C₈₉	C₄²	1	1	yes	characteristic
22	C₂³×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂³×C₈₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₈₉	—	2	4	no	—
25	C₂×C₄×C₈₉	—	4	2	no	—
26	C₂×C₄×C₈₉	—	4	1	no	—
27	C₂×C₄×C₈₉	—	4	1	no	—
28	C₂×C₄×C₈₉	—	4	4	no	—
29	C₂×C₄×C₈₉	—	4	2	no	—
30	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
31	C₂×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
32	#10623494	—	4	4	no	—
33	C₂²×C₈₉	—	2	2	no	—
34	C₂²×C₈₉	—	2	1	no	—
35	C₂²×C₈₉	—	2	1	no	—
36	C₂²×C₈₉	—	2	2	no	—
37	C₂²×C₈₉	—	2	2	no	—
38	C₂²×C₈₉	—	4	4	no	—
39	C₂²×C₈₉	#95	1	1	yes	characteristic
40	C₂²×C₈₉	#99	1	2	yes	—
41	C₄×C₈₉	—	2	2	no	—
42	C₄×C₈₉	—	2	2	no	—
43	C₄×C₈₉	—	8	2	no	—
44	C₂×C₈₉	—	2	2	no	—
45	C₂×C₈₉	—	4	1	no	—
46	C₂×C₈₉	—	4	1	no	—
47	C₂×C₈₉	#342	1	1	yes	characteristic
48	C₂×C₈₉	#344	1	2	yes	—
49	#1306	—	89	1	no	maximal, Sylow
50	C₈₉	#1306	1	1	yes	characteristic, complemented, Sylow
51	#407	—	89	2	no	—
52	#381	D₈₉	1	1	yes	characteristic
53	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
54	C₂×M₁₆	—	178	2	no	—
55	C₂×C₂²⋊C₄	Dic₈₉	1	2	yes	—
56	C₂²⋊C₈	—	89	4	no	—
57	C₂²⋊C₈	—	178	2	no	—
58	#95	—	2	2	no	—
59	C₂⁴	C₂×Dic₈₉	1	1	yes	characteristic
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	Dic₁₇₈	1	1	yes	characteristic
63	C₂²×C₄	C₄×D₈₉	1	2	yes	—
64	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
65	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
66	M₁₆	—	356	4	no	—
67	C₂×C₈	—	178	4	no	—
68	C₂²⋊C₄	—	4	2	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₄×Dic₈₉	1	1	yes	characteristic
72	C₂³	C₂²⋊Dic₈₉	1	1	yes	characteristic
73	C₂³	C₄⋊Dic₈₉	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	#13083551	1	2	yes	—
81	C₂×C₄	#13083553	1	2	yes	—
82	C₈	—	356	4	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	#15667919	1	1	yes	characteristic
90	C₂²	#15667927	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	2	no	—
93	C₄	—	8	2	no	—
94	C₂	—	2	2	no	—
95	C₂	—	4	1	no	—
96	C₂	—	4	1	no	—
97	C₂	#19151365	1	1	yes	characteristic
98	C₂	#19151368	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151562

C₂

yes

maximal

#381×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

—

C₂²⋊#10623494

—

C₂²⋊#10623494

C₄

yes

complemented

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₉

C₄

yes

—

#95×C₈₉

—

#13083558

—

C₂×#10623494

—

C₂⁴×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

Q₈

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₄²

yes

characteristic

C₂³×C₈₉

C₂²⋊C₄

yes

characteristic

C₂³×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#95

yes

characteristic

C₂²×C₈₉

#99

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#342

yes

characteristic

C₂×C₈₉

#344

yes

—

#1306

—

maximal, Sylow

C₈₉

#1306

yes

characteristic, complemented, Sylow

#407

—

#381

D₈₉

yes

characteristic

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

178

—

C₂×C₂²⋊C₄

Dic₈₉

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

178

—

#95

—

C₂⁴

C₂×Dic₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₇₈

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

—

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

—

M₁₆

—

356

—

C₂×C₈

—

178

—

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

C₄×Dic₈₉

yes

characteristic

C₂³

C₂²⋊Dic₈₉

yes

characteristic

C₂³

C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13083551

yes

—

C₂×C₄

#13083553

yes

—

C₈

—

356

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#15667919

yes

characteristic

C₂²

#15667927

yes

—

C₄

—

C₄

—

C₄

—

C₂

—

C₂

—

C₂

—

C₂

#19151365

yes

characteristic

C₂

#19151368

yes

—

	subgroup	G/H
center	C₂²	#15667919
commutator	C₂³×C₈₉	C₄²
Frattini	C₂³×C₄	C₂×D₈₉
Fitting	#381×C₈₉	C₂
socle	C₂²×C₈₉	#95
prime radical	C₂⁴×C₈₉	C₂×C₄
cosocle	C₂³×C₄×C₈₉	C₂²
hypercenter	#381	D₈₉
hypocenter	C₈₉	#1306
LCS	G ▷ C₂³×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

subgroup

G/H

center

C₂²

#15667919

commutator

C₂³×C₈₉

C₄²

Frattini

C₂³×C₄

C₂×D₈₉

Fitting

#381×C₈₉

C₂

socle

C₂²×C₈₉

#95

prime radical

C₂⁴×C₈₉

C₂×C₄

cosocle

C₂³×C₄×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

N	H ≅ G/N	type
5. C₂²⋊#10623494	C₄	semidirect
50. C₈₉	#1306	semidirect

H ≅ G/N

type

5. C₂²⋊#10623494

C₄

semidirect

50. C₈₉

#1306

semidirect

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

subgroup	index
#19151562	2
#381×C₈₉	2
#1306	89

subgroup

index

#19151562

#381×C₈₉

#1306

Subgroups — #26018160

Conjugacy classes of subgroups

Complements

Maximal subgroups