Group #26019622 — subgroups

Subgroups — #26019622

Order 11392.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#19151978	C₂	1	1	yes	maximal, characteristic
2	#19151883	C₂	1	2	yes	maximal
3	#19151351	C₂	1	1	yes	maximal, characteristic
4	#19151345	C₂	1	2	yes	maximal
5	C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₅₆	—	2	1	no	—
7	C₂×Dic₃₅₆	—	2	1	no	—
8	#15668012	C₂²	1	2	yes	—
9	#15667943	—	2	2	no	—
10	C₄⋊#10623494	C₂²	1	2	yes	—
11	#15667936	—	2	2	no	—
12	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₈₉	—	2	1	no	—
14	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₈₉	—	2	1	no	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	2	no	—
18	Dic₃₅₆	—	4	2	no	—
19	Dic₃₅₆	—	4	2	no	—
20	C₂×#10623494	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₈₉	—	2	1	no	—
25	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₈₉	—	2	1	no	—
28	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₄²×C₈₉	—	2	1	no	—
30	C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	Q₈×C₈₉	—	2	1	no	—
32	Q₈×C₈₉	Q₁₆	1	2	yes	complemented
33	C₂×C₄×C₈₉	—	2	1	no	—
34	C₂×C₄×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
38	C₂×Dic₈₉	—	4	2	no	—
39	Dic₁₇₈	—	4	2	no	—
40	Dic₁₇₈	—	4	2	no	—
41	C₈×C₈₉	—	2	1	no	—
42	C₈×C₈₉	—	4	1	no	—
43	C₈×C₈₉	SD₁₆	1	2	yes	—
44	#10623494	—	4	2	no	—
45	C₂²×C₈₉	#121	1	1	yes	characteristic
46	C₄×C₈₉	—	2	2	no	—
47	C₄×C₈₉	—	2	1	no	—
48	C₄×C₈₉	—	4	1	no	—
49	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₈₉	#135	1	1	yes	characteristic
52	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₈₉	—	8	2	no	—
54	C₂×C₈₉	#460	1	1	yes	characteristic
55	C₂×C₈₉	#467	1	1	yes	characteristic
56	C₂×C₈₉	#473	1	1	yes	characteristic
57	#1670	—	89	1	no	maximal, Sylow
58	C₈₉	#1670	1	1	yes	characteristic, complemented, Sylow
59	#494	—	89	1	no	—
60	#475	—	89	2	no	—
61	C₈×Q₈	D₈₉	1	1	yes	characteristic
62	#335	—	89	1	no	—
63	#332	—	89	2	no	—
64	C₂×Q₁₆	—	178	1	no	—
65	C₂×Q₁₆	—	178	1	no	—
66	C₄⋊Q₈	—	89	2	no	—
67	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
68	#106	—	178	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	89	2	no	—
71	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
72	#103	—	178	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
75	C₂×Q₈	—	178	2	no	—
76	C₂×Q₈	D₃₅₆	1	1	yes	characteristic
77	Q₁₆	—	356	2	no	—
78	Q₁₆	—	356	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	178	2	no	—
81	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	178	2	no	—
85	C₄⋊C₄	—	178	2	no	—
86	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₈₉	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	356	2	no	—
91	Q₈	—	356	2	no	—
92	Q₈	Dic₃₅₆	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	356	2	no	—
96	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
97	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
98	C₂×C₄	#13083574	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	356	2	no	—
102	C₈	#13083561	1	2	yes	—
103	C₂²	#15667986	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	—	4	1	no	—
107	C₄	—	712	2	no	—
108	C₄	C₂×#13083561	1	1	yes	characteristic
109	C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
110	C₄	#15668034	1	1	yes	characteristic
111	C₄	#15668037	1	1	yes	characteristic
112	C₂	#19151798	1	1	yes	characteristic
113	C₂	#19151839	1	1	yes	characteristic
114	C₂	#19151873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151978

C₂

yes

maximal, characteristic

#19151883

C₂

yes

maximal

#19151351

C₂

yes

maximal, characteristic

#19151345

C₂

yes

maximal

C₈×Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

C₂²

yes

—

#15667943

—

C₄⋊#10623494

C₂²

yes

—

#15667936

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

SD₁₆

yes

—

#10623494

—

C₂²×C₈₉

#121

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₄×C₈₉

C₂×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#494

—

#475

—

C₈×Q₈

D₈₉

yes

characteristic

#335

—

#332

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₈₉

yes

characteristic

#106

—

178

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

characteristic

#103

—

178

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

178

—

C₂×Q₈

D₃₅₆

yes

characteristic

Q₁₆

—

356

—

Q₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

D₃₅₆

yes

characteristic

C₄²

—

C₄²

C₂²×D₈₉

yes

characteristic

Q₈

—

Q₈

—

356

—

Q₈

—

356

—

Q₈

Dic₃₅₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

#13083574

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

356

—

102

C₈

#13083561

yes

—

103

C₂²

#15667986

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

712

—

108

C₄

C₂×#13083561

yes

characteristic

109

C₄

C₂×Dic₃₅₆

yes

characteristic

110

C₄

#15668034

yes

characteristic

111

C₄

#15668037

yes

characteristic

112

C₂

#19151798

yes

characteristic

113

C₂

#19151839

yes

characteristic

114

C₂

#19151873

yes

characteristic

	subgroup	G/H
center	C₂²	#15667986
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#121
hypercenter	C₈×Q₈	D₈₉
hypocenter	C₈₉	#1670
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#15667986

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

C₈×Q₈×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

#121

hypercenter

C₈×Q₈

D₈₉

hypocenter

C₈₉

#1670

LCS

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
32. Q₈×C₈₉	Q₁₆	semidirect
58. C₈₉	#1670	semidirect
92. Q₈	Dic₃₅₆	semidirect

H ≅ G/N

type

32. Q₈×C₈₉

Q₁₆

semidirect

58. C₈₉

#1670

semidirect

92. Q₈

Dic₃₅₆

semidirect

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

subgroup	index
#19151978	2
#19151883	2
#19151351	2
#19151345	2
C₈×Q₈×C₈₉	2
#1670	89

subgroup

index

C₈×Q₈×C₈₉

Subgroups — #26019622

Conjugacy classes of subgroups

Complements

Maximal subgroups