Group #36475064 — subgroups

Subgroups — #36475064

Order 22784.

158 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₄⋊#15667918	C₂	1	2	yes	maximal
2	C₄⋊#15667918	C₂	1	1	yes	maximal, characteristic
3	#26018386	C₂	1	2	yes	maximal
4	C₄×#15667918	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667918	C₂²	1	1	yes	characteristic
7	C₂×#15667918	C₂²	1	2	yes	—
8	C₂×#15667918	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₈₉	C₄	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	#15667918	—	2	1	no	—
15	#15667918	—	2	2	no	—
16	#15667918	Q₈	1	2	yes	complemented
17	C₄×Q₈×C₈₉	C₈	1	1	yes	characteristic
18	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
21	C₄⋊C₈×C₈₉	C₈	1	2	yes	—
22	C₄⋊C₈×C₈₉	C₈	1	1	yes	characteristic
23	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
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	—	89	1	no	maximal, 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	D₈₉	1	1	yes	characteristic
59	C₄⋊C₃₂	—	89	1	no	—
60	C₄⋊C₃₂	—	89	2	no	—
61	#1397	—	89	2	no	—
62	C₄×C₃₂	—	89	1	no	—
63	C₈₉	#13660	1	1	yes	characteristic, complemented, Sylow
64	C₈×Q₈	Dic₈₉	1	1	yes	characteristic
65	C₂×C₃₂	—	89	1	no	—
66	C₂×C₃₂	—	89	1	no	—
67	C₂×C₃₂	—	89	2	no	—
68	C₄⋊C₁₆	Dic₈₉	1	1	yes	characteristic
69	C₄⋊C₁₆	Dic₈₉	1	2	yes	—
70	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
71	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
72	C₄×Q₈	#10623494	1	1	yes	characteristic
73	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
74	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
75	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
76	C₄⋊C₈	#10623494	1	2	yes	—
77	C₄⋊C₈	#10623494	1	1	yes	characteristic
78	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
79	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
80	C₃₂	—	89	2	no	—
81	C₃₂	—	178	1	no	—
82	C₃₂	—	178	2	no	—
83	C₂×Q₈	#13083549	1	1	yes	characteristic
84	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
85	C₂×C₈	C₂×#10623494	1	2	yes	—
86	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
87	C₄⋊C₄	#13083549	1	2	yes	—
88	C₄⋊C₄	#13083549	1	1	yes	characteristic
89	C₄²	C₂×#10623494	1	2	yes	—
90	C₄²	C₂×#10623494	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	2	2	no	—
93	C₁₆	Q₈×D₈₉	1	1	yes	characteristic
94	C₁₆	#13083575	1	1	yes	characteristic
95	Q₈	—	2	2	no	—
96	C₂×C₄	C₂×#13083549	1	2	yes	—
97	C₂×C₄	C₂×#13083549	1	1	yes	characteristic
98	C₂×C₄	C₂×#13083549	1	2	yes	—
99	C₂×C₄	C₂×#13083549	1	1	yes	characteristic
100	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	2	no	—
103	C₈	Q₈×Dic₈₉	1	1	yes	characteristic
104	C₈	#15668019	1	1	yes	characteristic
105	C₂²	C₂²×#13083549	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	#19151432	1	2	yes	—
110	C₄	Q₈×#10623494	1	1	yes	characteristic
111	C₄	#19152015	1	1	yes	characteristic
112	C₂	Q₈×#13083549	1	1	yes	characteristic
113	C₂	C₂×#19151432	1	1	yes	characteristic
114	C₂	#26022546	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#15667918

C₂

yes

maximal

C₄⋊#15667918

C₂

yes

maximal, characteristic

#26018386

C₂

yes

maximal

C₄×#15667918

C₂

yes

maximal, characteristic

Q₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667918

C₂²

yes

characteristic

C₂×#15667918

C₂²

yes

—

C₂×#15667918

C₂²

yes

characteristic

C₈×Q₈×C₈₉

C₄

yes

characteristic

C₄⋊C₁₆×C₈₉

C₄

yes

characteristic

C₄⋊C₁₆×C₈₉

C₄

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

—

#15667918

—

#15667918

—

#15667918

Q₈

yes

complemented

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

—

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

—

C₂×C₈₉

Q₈×C₁₆

yes

characteristic

C₂×C₈₉

C₂×M₆₄

yes

characteristic

C₂×C₈₉

#2258

yes

characteristic

Q₈×C₁₆

D₈₉

yes

characteristic

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

—

C₈₉

#13660

yes

characteristic, complemented, Sylow

C₈×Q₈

Dic₈₉

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

Dic₈₉

yes

characteristic

C₄⋊C₁₆

Dic₈₉

yes

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

C₂×D₈₉

yes

—

C₄×Q₈

#10623494

yes

characteristic

C₂×C₁₆

C₂×Dic₈₉

yes

characteristic

C₂×C₁₆

C₂×Dic₈₉

yes

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

#10623494

yes

—

C₄⋊C₈

#10623494

yes

characteristic

C₄×C₈

C₂×Dic₈₉

yes

characteristic

C₄×C₈

C₂×Dic₈₉

yes

—

C₃₂

—

C₃₂

—

178

—

C₃₂

—

178

—

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂²×Dic₈₉

yes

characteristic

C₄⋊C₄

#13083549

yes

—

C₄⋊C₄

#13083549

yes

characteristic

C₄²

C₂×#10623494

yes

—

C₄²

C₂×#10623494

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

Q₈×D₈₉

yes

characteristic

C₁₆

#13083575

yes

characteristic

Q₈

—

C₂×C₄

C₂×#13083549

yes

—

C₂×C₄

C₂×#13083549

yes

characteristic

C₂×C₄

C₂×#13083549

yes

—

C₂×C₄

C₂×#13083549

yes

characteristic

100

C₂×C₄

C₂²×#10623494

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

Q₈×Dic₈₉

yes

characteristic

104

C₈

#15668019

yes

characteristic

105

C₂²

C₂²×#13083549

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#19151432

yes

—

110

C₄

Q₈×#10623494

yes

characteristic

111

C₄

#19152015

yes

characteristic

112

C₂

Q₈×#13083549

yes

characteristic

113

C₂

C₂×#19151432

yes

characteristic

114

C₂

#26022546

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₉
commutator, socle, prime radical	C₂²×C₈₉	C₂²×C₁₆
Fitting	Q₈×C₁₆×C₈₉	C₂
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#13660
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₈₉

commutator, socle, prime radical

C₂²×C₈₉

C₂²×C₁₆

Fitting

Q₈×C₁₆×C₈₉

C₂

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

Q₈×C₁₆

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

N	H ≅ G/N	type
16. #15667918	Q₈	semidirect
63. C₈₉	#13660	semidirect

H ≅ G/N

type

16. #15667918

Q₈

semidirect

63. C₈₉

#13660

semidirect

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

subgroup	index
C₄⋊#15667918	2
#26018386	2
C₄×#15667918	2
Q₈×C₁₆×C₈₉	2
#13660	89

subgroup

index

Q₈×C₁₆×C₈₉

Subgroups — #36475064

Conjugacy classes of subgroups

Complements

Maximal subgroups