Group #26019150 — subgroups

Subgroups — #26019150

Order 11392.

152 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19151728	C₂	1	2	yes	maximal
2	C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#19151351	C₂	1	1	yes	maximal, characteristic
4	#19151330	C₂	1	2	yes	maximal
5	C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₅₆	C₄	1	1	yes	characteristic
7	C₄×Dic₁₇₈	C₂²	1	2	yes	—
8	#15667946	C₄	1	1	yes	characteristic
9	C₄⋊#10623494	C₂²	1	2	yes	—
10	#15667941	—	2	2	no	—
11	#15667937	C₄	1	2	yes	—
12	#15667922	—	2	2	no	—
13	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
17	Dic₃₅₆	—	2	2	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
20	#13083553	—	2	2	no	—
21	C₄×Dic₈₉	—	2	2	no	—
22	C₂×C₈×C₈₉	—	2	1	no	—
23	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₄²×C₈₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
31	C₂×Dic₈₉	—	2	2	no	—
32	C₂×Dic₈₉	—	2	2	no	—
33	Dic₁₇₈	—	2	2	no	—
34	Dic₁₇₈	—	2	2	no	—
35	C₈×C₈₉	—	2	2	no	—
36	C₈×C₈₉	—	2	1	no	—
37	C₈×C₈₉	—	2	1	no	—
38	C₈×C₈₉	M₁₆	1	2	yes	—
39	C₈×C₈₉	Q₁₆	1	2	yes	complemented
40	#10623494	—	4	2	no	—
41	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
42	C₄×C₈₉	—	2	1	no	—
43	C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₈₉	#131	1	1	yes	characteristic
45	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₈₉	#135	1	1	yes	characteristic
47	Dic₈₉	—	4	2	no	—
48	Dic₈₉	—	4	2	no	—
49	C₂×C₈₉	#436	1	1	yes	characteristic
50	C₂×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
51	C₂×C₈₉	#443	1	1	yes	characteristic
52	#1591	—	89	1	no	maximal, Sylow
53	C₈₉	#1591	1	1	yes	characteristic, complemented, Sylow
54	#446	—	89	2	no	—
55	C₄×Q₁₆	—	89	1	no	—
56	#335	—	89	1	no	—
57	Q₈⋊C₈	—	89	2	no	—
58	C₈²	D₈₉	1	1	yes	characteristic
59	C₂×Q₁₆	—	89	1	no	—
60	C₄×Q₈	—	89	2	no	—
61	#107	—	89	1	no	—
62	C₄⋊C₈	—	89	2	no	—
63	C₄⋊C₈	—	178	2	no	—
64	#103	—	89	2	no	—
65	#97	—	178	2	no	—
66	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
67	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
68	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
69	C₂×Q₈	—	89	2	no	—
70	Q₁₆	—	178	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	178	2	no	—
73	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
74	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
75	C₂×C₈	D₃₅₆	1	1	yes	characteristic
76	C₂×C₈	D₃₅₆	1	1	yes	characteristic
77	C₄⋊C₄	—	89	2	no	—
78	C₄⋊C₄	—	178	2	no	—
79	C₄²	—	178	2	no	—
80	C₄²	C₂²×D₈₉	1	1	yes	characteristic
81	Q₈	—	178	2	no	—
82	Q₈	—	178	2	no	—
83	C₂×C₄	—	178	2	no	—
84	C₂×C₄	—	178	2	no	—
85	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
87	C₂×C₄	#13083574	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	356	2	no	—
92	C₈	#13083557	1	2	yes	—
93	C₈	Dic₃₅₆	1	2	yes	complemented
94	C₂²	C₄×D₃₅₆	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	356	2	no	—
97	C₄	—	356	2	no	—
98	C₄	C₂×#13083557	1	1	yes	characteristic
99	C₄	#15668018	1	1	yes	characteristic
100	C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
101	C₄	#15668034	1	1	yes	characteristic
102	C₂	#19151677	1	1	yes	characteristic
103	C₂	C₄×Dic₃₅₆	1	1	yes	characteristic
104	C₂	#19151716	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151728

C₂

yes

maximal

C₄×Dic₃₅₆

C₂

yes

maximal, characteristic

#19151351

C₂

yes

maximal, characteristic

#19151330

C₂

yes

maximal

C₈²×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₄

yes

characteristic

C₄×Dic₁₇₈

C₂²

yes

—

#15667946

C₄

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

#15667941

—

#15667937

C₄

yes

—

#15667922

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

C₂×C₄

yes

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083553

—

C₄×Dic₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

M₁₆

yes

—

C₈×C₈₉

Q₁₆

yes

complemented

#10623494

—

C₂²×C₈₉

C₄×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×M₁₆

yes

characteristic

C₄×C₈₉

#131

yes

characteristic

C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₄×C₈₉

#135

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#436

yes

characteristic

C₂×C₈₉

C₄×Q₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#446

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

D₈₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

#103

—

#97

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₄×C₈

C₂×D₈₉

yes

characteristic

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

D₃₅₆

yes

characteristic

C₂×C₈

D₃₅₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

Q₈

—

178

—

Q₈

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

#13083574

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

356

—

C₈

#13083557

yes

—

C₈

Dic₃₅₆

yes

complemented

C₂²

C₄×D₃₅₆

yes

characteristic

C₄

—

C₄

—

356

—

C₄

—

356

—

C₄

yes

C₄

yes

100

C₄

C₂×Dic₃₅₆

yes

characteristic

101

C₄

#15668034

yes

characteristic

102

C₂

#19151677

yes

characteristic

103

C₂

C₄×Dic₃₅₆

yes

characteristic

104

C₂

#19151716

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₄²	C₂²×D₈₉
Fitting	C₈²×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂³
hypercenter	C₈²	D₈₉
hypocenter	C₈₉	#1591
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₄²

C₂²×D₈₉

Fitting

C₈²×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₄×D₄

cosocle

C₄²×C₈₉

C₂³

hypercenter

C₈²

D₈₉

hypocenter

C₈₉

#1591

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
39. C₈×C₈₉	Q₁₆	semidirect
53. C₈₉	#1591	semidirect
93. C₈	Dic₃₅₆	semidirect

H ≅ G/N

type

39. C₈×C₈₉

Q₁₆

semidirect

53. C₈₉

#1591

semidirect

93. C₈

Dic₃₅₆

semidirect

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

subgroup	index
#19151728	2
C₄×Dic₃₅₆	2
#19151351	2
#19151330	2
C₈²×C₈₉	2
#1591	89

subgroup

index

C₄×Dic₃₅₆

C₈²×C₈₉

Subgroups — #26019150

Conjugacy classes of subgroups

Complements

Maximal subgroups