Group #12019936 — subgroups

Subgroups — #12019936

Order 1280.

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	C₅×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#3926	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#3926	C₂	1	2	yes	maximal
4	#10597999	C₂	1	2	yes	maximal
5	C₄×#3926	C₂	1	1	yes	maximal, characteristic
6	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
7	C₅×C₄⋊C₁₆	C₄	1	1	yes	characteristic
8	C₅×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₅×C₁₆	C₂²	1	2	yes	—
11	C₂×#3926	C₂²	1	1	yes	characteristic
12	C₂×#3926	C₂²	1	2	yes	—
13	C₂×#3926	C₂²	1	1	yes	characteristic
14	#13660	—	5	1	no	maximal, Sylow
15	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
16	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
19	C₅×C₄⋊C₈	C₈	1	2	yes	—
20	C₅×C₄⋊C₈	C₈	1	1	yes	characteristic
21	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
23	#3926	—	2	1	no	—
24	#3926	—	2	2	no	—
25	#3926	Q₈	1	2	yes	complemented
26	Q₈×C₁₆	D₅	1	1	yes	characteristic
27	C₄⋊C₃₂	—	5	1	no	—
28	C₄⋊C₃₂	—	5	2	no	—
29	#1397	—	5	2	no	—
30	C₄×C₃₂	—	5	1	no	—
31	C₂×C₅×Q₈	C₁₆	1	1	yes	characteristic
32	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
34	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₅×C₄⋊C₄	C₁₆	1	2	yes	—
36	C₅×C₄⋊C₄	C₁₆	1	1	yes	characteristic
37	C₄²×C₅	C₂×C₈	1	2	yes	—
38	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
39	C₅×C₁₆	—	2	1	no	—
40	C₅×C₁₆	—	2	2	no	—
41	C₅×C₁₆	C₂×Q₈	1	1	yes	characteristic
42	C₅×C₁₆	#41	1	1	yes	characteristic
43	C₈×Q₈	Dic₅	1	1	yes	characteristic
44	C₂×C₃₂	—	5	1	no	—
45	C₂×C₃₂	—	5	1	no	—
46	C₂×C₃₂	—	5	2	no	—
47	C₄⋊C₁₆	Dic₅	1	1	yes	characteristic
48	C₄⋊C₁₆	Dic₅	1	2	yes	—
49	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
50	C₄×C₁₆	C₂×D₅	1	2	yes	—
51	C₅×Q₈	—	2	2	no	—
52	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
57	C₅×C₈	—	2	1	no	—
58	C₅×C₈	—	2	2	no	—
59	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
60	C₅×C₈	#131	1	1	yes	characteristic
61	C₄×Q₈	#168	1	1	yes	characteristic
62	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
63	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
64	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
65	C₄⋊C₈	#168	1	2	yes	—
66	C₄⋊C₈	#168	1	1	yes	characteristic
67	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
68	C₄×C₈	C₂×Dic₅	1	2	yes	—
69	C₃₂	—	5	2	no	—
70	C₃₂	—	10	1	no	—
71	C₃₂	—	10	2	no	—
72	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
73	C₄×C₅	—	2	2	no	—
74	C₄×C₅	—	2	2	no	—
75	C₄×C₅	—	2	1	no	—
76	C₄×C₅	M₆₄	1	2	yes	—
77	C₄×C₅	C₈×Q₈	1	1	yes	characteristic
78	C₄×C₅	#504	1	1	yes	characteristic
79	C₂×Q₈	#672	1	1	yes	characteristic
80	C₂×C₈	C₂×#168	1	1	yes	characteristic
81	C₂×C₈	C₂×#168	1	2	yes	—
82	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
83	C₄⋊C₄	#672	1	2	yes	—
84	C₄⋊C₄	#672	1	1	yes	characteristic
85	C₄²	C₂×#168	1	2	yes	—
86	C₄²	C₂×#168	1	1	yes	characteristic
87	C₁₆	—	2	1	no	—
88	C₁₆	—	2	2	no	—
89	C₁₆	Q₈×D₅	1	1	yes	characteristic
90	C₁₆	#713	1	1	yes	characteristic
91	C₂×C₅	Q₈×C₁₆	1	1	yes	characteristic
92	C₂×C₅	C₂×M₆₄	1	1	yes	characteristic
93	C₂×C₅	#2258	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	C₂×C₄	C₂×#672	1	2	yes	—
96	C₂×C₄	C₂×#672	1	1	yes	characteristic
97	C₂×C₄	C₂×#672	1	2	yes	—
98	C₂×C₄	C₂×#672	1	1	yes	characteristic
99	C₂×C₄	C₂²×#168	1	1	yes	characteristic
100	C₈	—	2	1	no	—
101	C₈	—	2	2	no	—
102	C₈	Q₈×Dic₅	1	1	yes	characteristic
103	C₈	#4094	1	1	yes	characteristic
104	C₅	#13660	1	1	yes	characteristic, complemented, Sylow
105	C₂²	C₂²×#672	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	1	no	—
108	C₄	—	2	2	no	—
109	C₄	#64719	1	2	yes	—
110	C₄	Q₈×#168	1	1	yes	characteristic
111	C₄	#65429	1	1	yes	characteristic
112	C₂	Q₈×#672	1	1	yes	characteristic
113	C₂	C₂×#64719	1	1	yes	characteristic
114	C₂	#10602244	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×Q₈×C₁₆

C₂

yes

maximal, characteristic

C₄⋊#3926

C₂

yes

maximal, characteristic

C₄⋊#3926

C₂

yes

maximal

#10597999

C₂

yes

maximal

C₄×#3926

C₂

yes

maximal, characteristic

C₅×C₈×Q₈

C₄

yes

characteristic

C₅×C₄⋊C₁₆

C₄

yes

characteristic

C₅×C₄⋊C₁₆

C₄

yes

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

C₄×C₅×C₁₆

C₂²

yes

—

C₂×#3926

C₂²

yes

characteristic

C₂×#3926

C₂²

yes

—

C₂×#3926

C₂²

yes

characteristic

#13660

—

maximal, Sylow

C₄×C₅×Q₈

C₈

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₂×C₅×C₁₆

C₂³

yes

characteristic

C₅×C₄⋊C₈

C₈

yes

—

C₅×C₄⋊C₈

C₈

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

—

#3926

—

#3926

—

#3926

Q₈

yes

complemented

Q₈×C₁₆

D₅

yes

characteristic

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

—

C₂×C₅×Q₈

C₁₆

yes

characteristic

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂×C₈

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₅×C₄⋊C₄

C₁₆

yes

—

C₅×C₄⋊C₄

C₁₆

yes

characteristic

C₄²×C₅

C₂×C₈

yes

—

C₄²×C₅

C₂×C₈

yes

characteristic

C₅×C₁₆

—

C₅×C₁₆

—

C₅×C₁₆

C₂×Q₈

yes

characteristic

C₅×C₁₆

#41

yes

characteristic

C₈×Q₈

Dic₅

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

Dic₅

yes

characteristic

C₄⋊C₁₆

Dic₅

yes

—

C₄×C₁₆

C₂×D₅

yes

characteristic

C₄×C₁₆

C₂×D₅

yes

—

C₅×Q₈

—

C₂×C₄×C₅

C₂×C₁₆

yes

—

C₂×C₄×C₅

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅

C₂×C₁₆

yes

—

C₂×C₄×C₅

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₄×Q₈

yes

C₅×C₈

yes

C₄×Q₈

yes

C₂×C₁₆

C₂×Dic₅

yes

characteristic

C₂×C₁₆

C₂×Dic₅

yes

—

C₂×C₁₆

C₂²×D₅

yes

characteristic

C₄⋊C₈

#168

yes

—

C₄⋊C₈

#168

yes

characteristic

C₄×C₈

C₂×Dic₅

yes

characteristic

C₄×C₈

C₂×Dic₅

yes

—

C₃₂

—

C₃₂

—

C₃₂

—

C₂²×C₅

C₂²×C₁₆

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

M₆₄

yes

—

C₄×C₅

C₈×Q₈

yes

C₄×C₅

yes

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂²×Dic₅

yes

characteristic

C₄⋊C₄

#672

yes

—

C₄⋊C₄

#672

yes

characteristic

C₄²

C₂×#168

yes

—

C₄²

C₂×#168

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

Q₈×D₅

yes

characteristic

C₁₆

#713

yes

characteristic

C₂×C₅

Q₈×C₁₆

yes

characteristic

C₂×C₅

C₂×M₆₄

yes

characteristic

C₂×C₅

#2258

yes

characteristic

Q₈

—

C₂×C₄

C₂×#672

yes

—

C₂×C₄

C₂×#672

yes

characteristic

C₂×C₄

C₂×#672

yes

—

C₂×C₄

C₂×#672

yes

characteristic

C₂×C₄

C₂²×#168

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

Q₈×Dic₅

yes

characteristic

103

C₈

#4094

yes

characteristic

104

C₅

#13660

yes

characteristic, complemented, Sylow

105

C₂²

C₂²×#672

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#64719

yes

—

110

C₄

Q₈×#168

yes

characteristic

111

C₄

#65429

yes

characteristic

112

C₂

Q₈×#672

yes

characteristic

113

C₂

C₂×#64719

yes

characteristic

114

C₂

#10602244

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅
commutator, socle, prime radical	C₂²×C₅	C₂²×C₁₆
Fitting	C₅×Q₈×C₁₆	C₂
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	Q₈×C₁₆	D₅
hypocenter	C₅	#13660
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅

commutator, socle, prime radical

C₂²×C₅

C₂²×C₁₆

Fitting

C₅×Q₈×C₁₆

C₂

cosocle

C₂×C₅×C₁₆

C₂³

hypercenter

Q₈×C₁₆

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

N	H ≅ G/N	type
25. #3926	Q₈	semidirect
104. C₅	#13660	semidirect

H ≅ G/N

type

25. #3926

Q₈

semidirect

104. C₅

#13660

semidirect

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

subgroup	index
C₅×Q₈×C₁₆	2
C₄⋊#3926	2
#10597999	2
C₄×#3926	2
#13660	5

subgroup

index

C₅×Q₈×C₁₆

Subgroups — #12019936

Conjugacy classes of subgroups

Complements

Maximal subgroups