Group #26016169 — subgroups

Subgroups — C₈₉×#1695 #26016169

Order 11392.

154 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 68 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	#500×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#476×C₈₉	C₂	1	2	yes	maximal
3	#436×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#334×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#333×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×C₈₉	—	2	1	no	—
7	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
8	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
9	#107×C₈₉	—	2	1	no	—
10	#107×C₈₉	—	2	1	no	—
11	#106×C₈₉	—	2	2	no	—
12	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	#102×C₈₉	—	2	2	no	—
15	C₂²⋊C₈×C₈₉	—	2	1	no	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₈₉	—	2	2	no	—
18	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₈₉	—	2	1	no	—
20	M₁₆×C₈₉	—	4	2	no	—
21	C₂×C₈×C₈₉	—	2	1	no	—
22	C₂×C₈×C₈₉	—	2	2	no	—
23	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₄⋊C₄×C₈₉	—	2	2	no	—
26	C₄⋊C₄×C₈₉	—	2	2	no	—
27	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂²⋊C₄×C₈₉	—	2	1	no	—
29	C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	C₂³×C₈₉	—	2	1	no	—
31	Q₈×C₈₉	—	8	2	no	—
32	D₄×C₈₉	—	2	2	no	—
33	C₂×C₄×C₈₉	—	2	1	no	—
34	C₂×C₄×C₈₉	—	4	1	no	—
35	C₂×C₄×C₈₉	—	4	1	no	—
36	C₂×C₄×C₈₉	—	4	2	no	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
40	C₈×C₈₉	—	2	1	no	—
41	C₈×C₈₉	—	4	1	no	—
42	C₈×C₈₉	—	4	2	no	—
43	C₈×C₈₉	SD₁₆	1	2	yes	—
44	C₂²×C₈₉	—	4	1	no	—
45	C₂²×C₈₉	—	4	1	no	—
46	C₂²×C₈₉	—	4	1	no	—
47	C₂²×C₈₉	#121	1	1	yes	characteristic
48	C₄×C₈₉	—	2	1	no	—
49	C₄×C₈₉	—	4	1	no	—
50	C₄×C₈₉	—	8	2	no	—
51	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
52	C₄×C₈₉	#137	1	1	yes	characteristic
53	C₄×C₈₉	#137	1	2	yes	—
54	C₂×C₈₉	—	8	1	no	—
55	C₂×C₈₉	#461	1	1	yes	characteristic
56	C₂×C₈₉	#470	1	1	yes	characteristic
57	C₂×C₈₉	#472	1	1	yes	characteristic
58	#1695	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₈₉	#1695	1	1	yes	characteristic, direct product factor, Sylow
60	#500	C₂×C₈₉	1	1	yes	characteristic, complemented
61	#476	C₂×C₈₉	1	2	yes	—
62	#436	C₂×C₈₉	1	1	yes	characteristic
63	#334	C₂×C₈₉	1	1	yes	characteristic, complemented
64	#333	C₂×C₈₉	1	2	yes	complemented
65	C₂×M₁₆	—	2	1	no	—
66	C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
67	C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
68	#107	—	2	1	no	—
69	#107	—	2	1	no	—
70	#106	—	2	2	no	—
71	C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
72	C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
73	#102	—	2	2	no	—
74	C₂²⋊C₈	—	2	1	no	—
75	C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
76	C₂×Q₈	—	2	2	no	—
77	C₂×D₄	D₄×C₈₉	1	1	yes	characteristic
78	C₂²×C₄	—	2	1	no	—
79	M₁₆	—	4	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	D₄×C₈₉	1	1	yes	characteristic
83	C₂×C₈	D₄×C₈₉	1	1	yes	characteristic
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
87	C₂²⋊C₄	—	2	1	no	—
88	C₄²	C₂³×C₈₉	1	1	yes	characteristic
89	C₂³	—	2	1	no	—
90	Q₈	—	8	2	no	—
91	D₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
97	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
98	C₂×C₄	#41×C₈₉	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	4	2	no	—
102	C₈	SD₁₆×C₈₉	1	2	yes	—
103	C₂²	—	4	1	no	—
104	C₂²	—	4	1	no	—
105	C₂²	—	4	1	no	—
106	C₂²	#121×C₈₉	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	4	1	no	—
109	C₄	—	8	2	no	—
110	C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
111	C₄	#137×C₈₉	1	1	yes	characteristic
112	C₄	#137×C₈₉	1	2	yes	—
113	C₂	—	8	1	no	—
114	C₂	#461×C₈₉	1	1	yes	characteristic
115	C₂	#470×C₈₉	1	1	yes	characteristic
116	C₂	#472×C₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₈₉

C₂

yes

maximal, characteristic, complemented

#476×C₈₉

C₂

yes

maximal

#436×C₈₉

C₂

yes

maximal, characteristic

#334×C₈₉

C₂

yes

maximal, characteristic, complemented

#333×C₈₉

C₂

yes

maximal, complemented

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

C₄×D₄×C₈₉

C₂²

yes

characteristic

#107×C₈₉

—

#107×C₈₉

—

#106×C₈₉

—

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

#102×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

SD₁₆

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₈₉

#1695

yes

characteristic, direct product factor, Sylow

#500

C₂×C₈₉

yes

characteristic, complemented

#476

C₂×C₈₉

yes

—

#436

C₂×C₈₉

yes

characteristic

#334

C₂×C₈₉

yes

characteristic, complemented

#333

C₂×C₈₉

yes

complemented

C₂×M₁₆

—

C₄⋊Q₈

C₂²×C₈₉

yes

—

C₄×D₄

C₂²×C₈₉

yes

—

—

—

C₄⋊C₈

C₂²×C₈₉

yes

characteristic

C₄⋊C₈

C₂²×C₈₉

yes

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂²×C₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

D₄×C₈₉

yes

characteristic

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₈₉

yes

characteristic

C₂×C₈

D₄×C₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₈₉

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂³×C₈₉

yes

characteristic

C₂³

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

C₂×C₄

#41×C₈₉

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₈

SD₁₆×C₈₉

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#121×C₈₉

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

C₂×SD₁₆×C₈₉

yes

characteristic

111

C₄

#137×C₈₉

yes

characteristic

112

C₄

#137×C₈₉

yes

—

113

C₂

—

114

C₂

#461×C₈₉

yes

characteristic

115

C₂

#470×C₈₉

yes

characteristic

116

C₂

#472×C₈₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₈₉
prime radical	C₂×D₄×C₈₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₈₉

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₈₉

prime radical

C₂×D₄×C₈₉

D₄

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ G

N	H ≅ G/N	type
1. #500×C₈₉	C₂	semidirect
4. #334×C₈₉	C₂	semidirect
5. #333×C₈₉	C₂	semidirect
58. #1695	C₈₉	direct (N × H)
59. C₈₉	#1695	direct (N × H)
60. #500	C₂×C₈₉	semidirect
63. #334	C₂×C₈₉	semidirect
64. #333	C₂×C₈₉	semidirect

H ≅ G/N

type

1. #500×C₈₉

C₂

semidirect

4. #334×C₈₉

C₂

semidirect

5. #333×C₈₉

C₂

semidirect

58. #1695

C₈₉

direct (N × H)

59. C₈₉

#1695

direct (N × H)

60. #500

C₂×C₈₉

semidirect

63. #334

C₂×C₈₉

semidirect

64. #333

C₂×C₈₉

semidirect

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

subgroup	index
#500×C₈₉	2
#476×C₈₉	2
#436×C₈₉	2
#334×C₈₉	2
#333×C₈₉	2
#1695	89

subgroup

index

#500×C₈₉

#476×C₈₉

#436×C₈₉

#334×C₈₉

#333×C₈₉

#1695

Subgroups — C₈₉×#1695 #26016169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#1695 #26016169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#1695 #26016169