Group #36475057 — subgroups

Subgroups — #36475057

Order 22784.

166 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26018447	C₂	1	2	yes	maximal
3	C₄×#15667918	C₂	1	1	yes	maximal, characteristic
4	C₃₂×Dic₈₉	C₂	1	2	yes	maximal
5	C₈₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic
7	C₂×#15667918	C₂²	1	1	yes	characteristic
8	C₂×#15667918	C₂²	1	1	yes	characteristic
9	C₄⋊#13083549	C₄	1	1	yes	characteristic
10	#19151416	C₄	1	2	yes	—
11	C₁₆×Dic₈₉	C₂²	1	2	yes	—
12	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₇₈	C₈	1	1	yes	characteristic
15	C₂×#13083549	C₂×C₄	1	2	yes	—
16	C₄⋊#10623494	C₈	1	1	yes	characteristic
17	#15667941	C₈	1	2	yes	—
18	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
19	#15667918	—	2	1	no	—
20	#15667918	Q₈	1	2	yes	complemented
21	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₃₂×C₈₉	—	2	2	no	—
25	C₂×Dic₁₇₈	C₁₆	1	1	yes	characteristic
26	C₂×#10623494	C₂×C₈	1	2	yes	—
27	C₄⋊Dic₈₉	C₁₆	1	1	yes	characteristic
28	#13083553	C₁₆	1	2	yes	—
29	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
30	#13083549	—	2	2	no	—
31	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₈₉	—	2	1	no	—
35	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₈₉	#41	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
42	Dic₁₇₈	C₃₂	1	4	yes	complemented
43	C₈×C₈₉	—	2	1	no	—
44	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₈₉	#131	1	1	yes	characteristic
46	#10623494	—	2	2	no	—
47	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
48	C₄×C₈₉	—	2	1	no	—
49	C₄×C₈₉	C₂×C₃₂	1	2	yes	—
50	C₄×C₈₉	C₈×Q₈	1	1	yes	characteristic
51	C₄×C₈₉	#504	1	1	yes	characteristic
52	Dic₈₉	—	2	2	no	—
53	Dic₈₉	C₂×C₃₂	1	4	yes	—
54	Q₈×C₃₂	—	89	1	no	maximal, Sylow
55	C₂×C₈₉	Q₈×C₁₆	1	1	yes	characteristic
56	C₂×C₈₉	C₂²×C₃₂	1	1	yes	characteristic
57	C₂×C₈₉	#2258	1	1	yes	characteristic
58	Q₈×C₁₆	—	89	1	no	—
59	C₄⋊C₃₂	—	89	2	no	—
60	C₄⋊C₃₂	D₈₉	1	1	yes	characteristic
61	C₄×C₃₂	—	89	2	no	—
62	C₄×C₃₂	—	89	1	no	—
63	C₈₉	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
64	C₈×Q₈	—	89	1	no	—
65	C₂×C₃₂	—	89	1	no	—
66	C₂×C₃₂	—	89	1	no	—
67	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
68	C₄⋊C₁₆	—	89	2	no	—
69	C₄⋊C₁₆	—	89	1	no	—
70	C₄×C₁₆	—	89	2	no	—
71	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
72	C₄×Q₈	—	89	1	no	—
73	C₂×C₁₆	—	89	2	no	—
74	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
76	C₄⋊C₈	—	89	2	no	—
77	C₄⋊C₈	—	89	1	no	—
78	C₄×C₈	—	89	2	no	—
79	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
80	C₃₂	—	2	2	no	—
81	C₃₂	—	89	2	no	—
82	C₃₂	—	178	1	no	—
83	C₂×Q₈	—	89	1	no	—
84	C₂×C₈	—	89	2	no	—
85	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
87	C₄⋊C₄	—	89	2	no	—
88	C₄⋊C₄	—	89	1	no	—
89	C₄²	—	89	2	no	—
90	C₄²	C₈×D₈₉	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	178	2	no	—
93	C₁₆	Q₈×D₈₉	1	1	yes	characteristic
94	C₁₆	#13083573	1	1	yes	characteristic
95	Q₈	—	89	4	no	—
96	C₂×C₄	—	89	2	no	—
97	C₂×C₄	—	89	2	no	—
98	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
99	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	178	2	no	—
103	C₈	#15667977	1	1	yes	characteristic
104	C₈	#15668020	1	1	yes	characteristic
105	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	89	4	no	—
108	C₄	—	178	2	no	—
109	C₄	C₃₂×D₈₉	1	2	yes	—
110	C₄	#19151722	1	1	yes	characteristic
111	C₄	#19152016	1	1	yes	characteristic
112	C₂	#26022154	1	1	yes	characteristic
113	C₂	C₂×C₃₂×D₈₉	1	1	yes	characteristic
114	C₂	#26022547	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₇₈

C₂

yes

maximal, characteristic

#26018447

C₂

yes

maximal

C₄×#15667918

C₂

yes

maximal, characteristic

C₃₂×Dic₈₉

C₂

yes

maximal

C₈₉×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₈×Dic₁₇₈

C₄

yes

characteristic

C₂×#15667918

C₂²

yes

characteristic

C₂×#15667918

C₂²

yes

characteristic

C₄⋊#13083549

C₄

yes

characteristic

#19151416

C₄

yes

—

C₁₆×Dic₈₉

C₂²

yes

—

C₂×C₃₂×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×Dic₁₇₈

C₈

yes

characteristic

C₂×#13083549

C₂×C₄

yes

—

C₄⋊#10623494

C₈

yes

characteristic

#15667941

C₈

yes

—

C₈×Dic₈₉

C₂×C₄

yes

—

#15667918

—

#15667918

Q₈

yes

complemented

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

—

C₂×Dic₁₇₈

C₁₆

yes

characteristic

C₂×#10623494

C₂×C₈

yes

—

C₄⋊Dic₈₉

C₁₆

yes

characteristic

#13083553

C₁₆

yes

—

C₄×Dic₈₉

C₂×C₈

yes

—

#13083549

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×Q₈

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₁₆

yes

—

C₂×Dic₈₉

C₂×C₁₆

yes

—

Dic₁₇₈

C₃₂

yes

complemented

C₈×C₈₉

—

C₈×C₈₉

C₄×Q₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂²×C₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×C₃₂

yes

—

C₄×C₈₉

C₈×Q₈

yes

characteristic

C₄×C₈₉

#504

yes

characteristic

Dic₈₉

—

Dic₈₉

C₂×C₃₂

yes

—

Q₈×C₃₂

—

maximal, Sylow

C₂×C₈₉

Q₈×C₁₆

yes

characteristic

C₂×C₈₉

C₂²×C₃₂

yes

characteristic

C₂×C₈₉

#2258

yes

characteristic

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

D₈₉

yes

characteristic

C₄×C₃₂

—

C₄×C₃₂

—

C₈₉

Q₈×C₃₂

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₉

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₈₉

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

178

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₈₉

yes

characteristic

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₈₉

yes

characteristic

C₁₆

—

C₁₆

—

178

—

C₁₆

Q₈×D₈₉

yes

characteristic

C₁₆

#13083573

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×D₈₉

yes

characteristic

C₂×C₄

C₁₆×D₈₉

yes

characteristic

100

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

101

C₈

—

102

C₈

—

178

—

103

C₈

#15667977

yes

characteristic

104

C₈

#15668020

yes

characteristic

105

C₂²

C₂×C₁₆×D₈₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

178

—

109

C₄

C₃₂×D₈₉

yes

—

110

C₄

#19151722

yes

characteristic

111

C₄

#19152016

yes

characteristic

112

C₂

#26022154

yes

characteristic

113

C₂

C₂×C₃₂×D₈₉

yes

characteristic

114

C₂

#26022547

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₃₂
Fitting	C₈₉×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₈₉	C₂²×C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₄⋊C₃₂	D₈₉
hypocenter	C₈₉	Q₈×C₃₂
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₈₉

commutator

C₂×C₈₉

C₂²×C₃₂

Fitting

C₈₉×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₈₉

C₂²×C₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

C₄⋊C₃₂

D₈₉

hypocenter

C₈₉

Q₈×C₃₂

LCS

G ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
20. #15667918	Q₈	semidirect
42. Dic₁₇₈	C₃₂	semidirect
63. C₈₉	Q₈×C₃₂	semidirect

H ≅ G/N

type

20. #15667918

Q₈

semidirect

42. Dic₁₇₈

C₃₂

semidirect

63. C₈₉

Q₈×C₃₂

semidirect

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

subgroup	index
C₁₆×Dic₁₇₈	2
#26018447	2
C₄×#15667918	2
C₃₂×Dic₈₉	2
C₈₉×C₄⋊C₃₂	2
Q₈×C₃₂	89

subgroup

index

C₁₆×Dic₁₇₈

#26018447

C₄×#15667918

C₃₂×Dic₈₉

C₈₉×C₄⋊C₃₂

Q₈×C₃₂

Subgroups — #36475057

Conjugacy classes of subgroups

Complements

Maximal subgroups