Group #36391508 — subgroups

Subgroups — C₈₉×#13660 #36391508

Order 22784.

158 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 126 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	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₈₉×C₄⋊C₃₂	C₂	1	2	yes	maximal
4	C₈₉×#1397	C₂	1	2	yes	maximal
5	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
8	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
9	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆×C₈₉	C₄	1	1	yes	characteristic
11	C₄⋊C₁₆×C₈₉	C₄	1	2	yes	—
12	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₄×Q₈×C₈₉	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
18	C₄⋊C₈×C₈₉	C₈	1	2	yes	—
19	C₄⋊C₈×C₈₉	C₈	1	1	yes	characteristic
20	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
22	C₃₂×C₈₉	—	2	1	no	—
23	C₃₂×C₈₉	—	2	2	no	—
24	C₃₂×C₈₉	Q₈	1	2	yes	complemented
25	C₂×Q₈×C₈₉	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	C₁₆	1	2	yes	—
30	C₄⋊C₄×C₈₉	C₁₆	1	1	yes	characteristic
31	C₄²×C₈₉	C₂×C₈	1	2	yes	—
32	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
33	C₁₆×C₈₉	—	2	1	no	—
34	C₁₆×C₈₉	—	2	2	no	—
35	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₈₉	#41	1	1	yes	characteristic
37	Q₈×C₈₉	—	2	2	no	—
38	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
39	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
41	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
43	C₈×C₈₉	—	2	1	no	—
44	C₈×C₈₉	—	2	2	no	—
45	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
46	C₈×C₈₉	#131	1	1	yes	characteristic
47	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
48	C₄×C₈₉	—	2	2	no	—
49	C₄×C₈₉	—	2	2	no	—
50	C₄×C₈₉	—	2	1	no	—
51	C₄×C₈₉	M₆₄	1	2	yes	—
52	C₄×C₈₉	C₈×Q₈	1	1	yes	characteristic
53	C₄×C₈₉	#504	1	1	yes	characteristic
54	#13660	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₈₉	Q₈×C₁₆	1	1	yes	characteristic
56	C₂×C₈₉	C₂×M₆₄	1	1	yes	characteristic
57	C₂×C₈₉	#2258	1	1	yes	characteristic
58	Q₈×C₁₆	C₂×C₈₉	1	1	yes	characteristic
59	C₄⋊C₃₂	C₂×C₈₉	1	1	yes	characteristic
60	C₄⋊C₃₂	C₂×C₈₉	1	2	yes	—
61	#1397	C₂×C₈₉	1	2	yes	—
62	C₄×C₃₂	C₂×C₈₉	1	1	yes	characteristic
63	C₈₉	#13660	1	1	yes	characteristic, direct product factor, Sylow
64	C₈×Q₈	C₄×C₈₉	1	1	yes	characteristic
65	C₂×C₃₂	C₂²×C₈₉	1	1	yes	characteristic
66	C₂×C₃₂	C₂²×C₈₉	1	2	yes	—
67	C₂×C₃₂	C₂²×C₈₉	1	1	yes	characteristic
68	C₄⋊C₁₆	C₄×C₈₉	1	1	yes	characteristic
69	C₄⋊C₁₆	C₄×C₈₉	1	2	yes	—
70	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
71	C₄×C₁₆	C₂²×C₈₉	1	2	yes	—
72	C₄×Q₈	C₈×C₈₉	1	1	yes	characteristic
73	C₂×C₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic
74	C₂×C₁₆	C₂×C₄×C₈₉	1	2	yes	—
75	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
76	C₄⋊C₈	C₈×C₈₉	1	2	yes	—
77	C₄⋊C₈	C₈×C₈₉	1	1	yes	characteristic
78	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
79	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
80	C₃₂	—	2	1	no	—
81	C₃₂	—	2	2	no	—
82	C₃₂	Q₈×C₈₉	1	2	yes	complemented
83	C₂×Q₈	C₁₆×C₈₉	1	1	yes	characteristic
84	C₂×C₈	C₂×C₈×C₈₉	1	1	yes	characteristic
85	C₂×C₈	C₂×C₈×C₈₉	1	2	yes	—
86	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
87	C₄⋊C₄	C₁₆×C₈₉	1	2	yes	—
88	C₄⋊C₄	C₁₆×C₈₉	1	1	yes	characteristic
89	C₄²	C₂×C₈×C₈₉	1	2	yes	—
90	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	2	2	no	—
93	C₁₆	C₂×Q₈×C₈₉	1	1	yes	characteristic
94	C₁₆	#41×C₈₉	1	1	yes	characteristic
95	Q₈	—	2	2	no	—
96	C₂×C₄	C₂×C₁₆×C₈₉	1	2	yes	—
97	C₂×C₄	C₂×C₁₆×C₈₉	1	1	yes	characteristic
98	C₂×C₄	C₂×C₁₆×C₈₉	1	2	yes	—
99	C₂×C₄	C₂×C₁₆×C₈₉	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	2	no	—
103	C₈	C₄×Q₈×C₈₉	1	1	yes	characteristic
104	C₈	#131×C₈₉	1	1	yes	characteristic
105	C₂²	C₂²×C₁₆×C₈₉	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	M₆₄×C₈₉	1	2	yes	—
110	C₄	C₈×Q₈×C₈₉	1	1	yes	characteristic
111	C₄	#504×C₈₉	1	1	yes	characteristic
112	C₂	Q₈×C₁₆×C₈₉	1	1	yes	characteristic
113	C₂	C₂×M₆₄×C₈₉	1	1	yes	characteristic
114	C₂	C₈₉×#2258	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₈₉×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₈₉×C₄⋊C₃₂

C₂

yes

maximal

C₈₉×#1397

C₂

yes

maximal

C₄×C₃₂×C₈₉

C₂

yes

maximal, characteristic

C₈×Q₈×C₈₉

C₄

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

—

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

C₄

yes

characteristic

C₄⋊C₁₆×C₈₉

C₄

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

—

C₄×Q₈×C₈₉

C₈

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

C₈

yes

—

C₄⋊C₈×C₈₉

C₈

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₃₂×C₈₉

—

C₃₂×C₈₉

—

C₃₂×C₈₉

Q₈

yes

complemented

C₂×Q₈×C₈₉

C₁₆

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₈₉

C₁₆

yes

—

C₄⋊C₄×C₈₉

C₁₆

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×Q₈

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

C₂×C₁₆

yes

—

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

—

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄×Q₈

yes

characteristic

C₈×C₈₉

#131

yes

characteristic

C₂²×C₈₉

C₂²×C₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

M₆₄

yes

—

C₄×C₈₉

C₈×Q₈

yes

C₄×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

Q₈×C₁₆

yes

characteristic

C₂×C₈₉

C₂×M₆₄

yes

characteristic

C₂×C₈₉

#2258

yes

characteristic

Q₈×C₁₆

C₂×C₈₉

yes

characteristic

C₄⋊C₃₂

C₂×C₈₉

yes

characteristic

C₄⋊C₃₂

C₂×C₈₉

yes

—

#1397

C₂×C₈₉

yes

—

C₄×C₃₂

C₂×C₈₉

yes

characteristic

C₈₉

#13660

yes

characteristic, direct product factor, Sylow

C₈×Q₈

C₄×C₈₉

yes

characteristic

C₂×C₃₂

C₂²×C₈₉

yes

characteristic

C₂×C₃₂

C₂²×C₈₉

yes

—

C₂×C₃₂

C₂²×C₈₉

yes

characteristic

C₄⋊C₁₆

C₄×C₈₉

yes

characteristic

C₄⋊C₁₆

C₄×C₈₉

yes

—

C₄×C₁₆

C₂²×C₈₉

yes

characteristic

C₄×C₁₆

C₂²×C₈₉

yes

—

C₄×Q₈

C₈×C₈₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₈₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₈₉

yes

—

C₂×C₁₆

C₂³×C₈₉

yes

characteristic

C₄⋊C₈

C₈×C₈₉

yes

—

C₄⋊C₈

C₈×C₈₉

yes

characteristic

C₄×C₈

C₂×C₄×C₈₉

yes

characteristic

C₄×C₈

C₂×C₄×C₈₉

yes

—

C₃₂

—

C₃₂

—

C₃₂

Q₈×C₈₉

yes

complemented

C₂×Q₈

C₁₆×C₈₉

yes

characteristic

C₂×C₈

C₂×C₈×C₈₉

yes

characteristic

C₂×C₈

C₂×C₈×C₈₉

yes

—

C₂×C₈

C₂²×C₄×C₈₉

yes

characteristic

C₄⋊C₄

C₁₆×C₈₉

yes

—

C₄⋊C₄

C₁₆×C₈₉

yes

characteristic

C₄²

C₂×C₈×C₈₉

yes

—

C₄²

C₂×C₈×C₈₉

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₂×Q₈×C₈₉

yes

characteristic

C₁₆

#41×C₈₉

yes

characteristic

Q₈

—

C₂×C₄

C₂×C₁₆×C₈₉

yes

—

C₂×C₄

C₂×C₁₆×C₈₉

yes

characteristic

C₂×C₄

C₂×C₁₆×C₈₉

yes

—

C₂×C₄

C₂×C₁₆×C₈₉

yes

characteristic

100

C₂×C₄

C₂²×C₈×C₈₉

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

C₄×Q₈×C₈₉

yes

characteristic

104

C₈

#131×C₈₉

yes

characteristic

105

C₂²

C₂²×C₁₆×C₈₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

M₆₄×C₈₉

yes

—

110

C₄

C₈×Q₈×C₈₉

yes

characteristic

111

C₄

#504×C₈₉

yes

characteristic

112

C₂

Q₈×C₁₆×C₈₉

yes

characteristic

113

C₂

C₂×M₆₄×C₈₉

yes

characteristic

114

C₂

C₈₉×#2258

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆×C₈₉	C₂³
commutator	C₂²	C₂²×C₁₆×C₈₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₈₉
socle, prime radical	C₂²×C₈₉	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₈₉ ◁ G

subgroup

G/H

center

C₂×C₁₆×C₈₉

C₂³

commutator

C₂²

C₂²×C₁₆×C₈₉

Frattini, cosocle

C₂×C₁₆

C₂³×C₈₉

socle, prime radical

C₂²×C₈₉

C₂²×C₁₆

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₁₆×C₈₉ ◁ G

N	H ≅ G/N	type
24. C₃₂×C₈₉	Q₈	semidirect
54. #13660	C₈₉	direct (N × H)
63. C₈₉	#13660	direct (N × H)
82. C₃₂	Q₈×C₈₉	semidirect

H ≅ G/N

type

24. C₃₂×C₈₉

Q₈

semidirect

54. #13660

C₈₉

direct (N × H)

63. C₈₉

#13660

direct (N × H)

82. C₃₂

Q₈×C₈₉

semidirect

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

subgroup	index
Q₈×C₁₆×C₈₉	2
C₈₉×C₄⋊C₃₂	2
C₈₉×#1397	2
C₄×C₃₂×C₈₉	2
#13660	89

subgroup

index

Q₈×C₁₆×C₈₉

C₈₉×C₄⋊C₃₂

C₈₉×#1397

C₄×C₃₂×C₈₉

#13660

Subgroups — C₈₉×#13660 #36391508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#13660 #36391508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#13660 #36391508