Group #26019771 — subgroups

Subgroups — #26019771

Order 11392.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#19152003	C₂	1	1	yes	maximal, characteristic
2	#19151809	C₂	1	2	yes	maximal
3	#19151728	C₂	1	1	yes	maximal, characteristic
4	#19151345	C₂	1	2	yes	maximal
5	#334×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₅₆	—	2	2	no	—
7	#15668012	C₂²	1	2	yes	—
8	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	#15667946	—	2	1	no	—
10	#15667946	—	2	1	no	—
11	C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	#15667941	—	2	1	no	—
13	#15667937	—	2	2	no	—
14	#15667922	—	2	1	no	—
15	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
19	Dic₃₅₆	—	4	4	no	—
20	C₂×#10623494	—	2	1	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
24	#13083553	—	2	1	no	—
25	C₄×Dic₈₉	—	2	1	no	—
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
33	C₂×Dic₈₉	—	2	1	no	—
34	C₂×Dic₈₉	—	2	1	no	—
35	C₂×Dic₈₉	—	4	2	no	—
36	Dic₁₇₈	—	2	2	no	—
37	Dic₁₇₈	—	4	2	no	—
38	Dic₁₇₈	—	4	2	no	—
39	C₈×C₈₉	—	2	1	no	—
40	C₈×C₈₉	—	4	2	no	—
41	C₈×C₈₉	Q₁₆	1	2	yes	complemented
42	#10623494	—	4	1	no	—
43	C₂²×C₈₉	#121	1	1	yes	characteristic
44	C₄×C₈₉	—	2	1	no	—
45	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₄×C₈₉	#137	1	1	yes	characteristic
48	Dic₈₉	—	4	1	no	—
49	Dic₈₉	—	4	1	no	—
50	Dic₈₉	—	8	2	no	—
51	C₂×C₈₉	#462	1	1	yes	characteristic
52	C₂×C₈₉	#469	1	1	yes	characteristic
53	C₂×C₈₉	#473	1	1	yes	characteristic
54	#1694	—	89	1	no	maximal, Sylow
55	C₈₉	#1694	1	1	yes	characteristic, complemented, Sylow
56	#500	—	89	1	no	—
57	#462	—	89	2	no	—
58	#446	—	89	1	no	—
59	#334	D₈₉	1	1	yes	characteristic
60	#332	—	89	2	no	—
61	C₂×Q₁₆	—	178	2	no	—
62	C₄⋊Q₈	—	89	2	no	—
63	C₄×Q₈	—	89	1	no	—
64	#107	—	178	1	no	—
65	#107	—	178	1	no	—
66	C₄⋊C₈	—	89	1	no	—
67	C₄⋊C₈	—	178	1	no	—
68	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
69	#103	—	178	2	no	—
70	#97	—	178	1	no	—
71	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
72	C₂×Q₈	—	89	1	no	—
73	C₂×Q₈	—	178	2	no	—
74	Q₁₆	—	356	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	178	1	no	—
77	C₂×C₈	D₃₅₆	1	1	yes	characteristic
78	C₂×C₈	D₃₅₆	1	1	yes	characteristic
79	C₄⋊C₄	—	89	1	no	—
80	C₄⋊C₄	—	178	2	no	—
81	C₄⋊C₄	—	178	2	no	—
82	C₄⋊C₄	—	178	1	no	—
83	C₄²	—	178	1	no	—
84	C₄²	C₂²×D₈₉	1	1	yes	characteristic
85	Q₈	—	356	2	no	—
86	Q₈	—	356	2	no	—
87	Q₈	—	178	2	no	—
88	C₂×C₄	—	356	2	no	—
89	C₂×C₄	—	178	1	no	—
90	C₂×C₄	—	178	1	no	—
91	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
92	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
93	C₂×C₄	#13083575	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	356	1	no	—
97	C₈	Dic₃₅₆	1	2	yes	complemented
98	C₂²	#15667987	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	356	1	no	—
101	C₄	—	356	1	no	—
102	C₄	—	712	2	no	—
103	C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
104	C₄	#15668039	1	2	yes	—
105	C₄	#15668045	1	1	yes	characteristic
106	C₂	#19151809	1	1	yes	characteristic
107	C₂	#19151852	1	1	yes	characteristic
108	C₂	#19151874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152003

C₂

yes

maximal, characteristic

#19151809

C₂

yes

maximal

#19151728

C₂

yes

maximal, characteristic

#19151345

C₂

yes

maximal

#334×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

—

#15668012

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

—

C₂²

yes

—

—

—

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

characteristic

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

#13083553

—

C₄×Dic₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

Q₁₆

yes

complemented

#10623494

—

C₂²×C₈₉

#121

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₄×C₈₉

#137

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

#332

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

178

—

#107

—

178

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

—

#103

—

178

—

#97

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

Q₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

characteristic

C₂×C₈

D₃₅₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

Q₈

—

356

—

Q₈

—

356

—

Q₈

—

178

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

D₄×D₈₉

yes

characteristic

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

#13083575

yes

characteristic

C₈

—

C₈

—

C₈

—

356

—

C₈

Dic₃₅₆

yes

complemented

C₂²

#15667987

yes

characteristic

C₄

—

100

C₄

—

356

—

101

C₄

—

356

—

102

C₄

—

712

—

103

C₄

C₂×Dic₃₅₆

yes

characteristic

104

C₄

#15668039

yes

—

105

C₄

#15668045

yes

characteristic

106

C₂

#19151809

yes

characteristic

107

C₂

#19151852

yes

characteristic

108

C₂

#19151874

yes

characteristic

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#334×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#121
hypercenter	#334	D₈₉
hypocenter	C₈₉	#1694
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#334×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
41. C₈×C₈₉	Q₁₆	semidirect
55. C₈₉	#1694	semidirect
97. C₈	Dic₃₅₆	semidirect

H ≅ G/N

type

41. C₈×C₈₉

Q₁₆

semidirect

55. C₈₉

#1694

semidirect

97. C₈

Dic₃₅₆

semidirect

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

subgroup	index
#19152003	2
#19151809	2
#19151728	2
#19151345	2
#334×C₈₉	2
#1694	89

subgroup

index

#334×C₈₉

Subgroups — #26019771

Conjugacy classes of subgroups

Complements

Maximal subgroups