Group #28407704 — subgroups

Subgroups — #28407704

Order 14464.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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×#11744762	C₂	1	2	yes	maximal
2	#20299687	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11744762	C₂	1	2	yes	maximal
4	#20299682	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#90673	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11744762	—	2	2	no	—
7	C₄⋊#11744762	C₂²	1	2	yes	—
8	C₄×#11744762	—	2	2	no	—
9	C₄×#11744762	C₂²	1	1	yes	characteristic
10	C₄×#11744762	C₂²	1	1	yes	characteristic
11	Q₈×#90673	C₂²	1	2	yes	—
12	#16813691	C₄	1	1	yes	characteristic
13	#16813685	C₄	1	2	yes	—
14	C₈×#90673	C₂²	1	1	yes	characteristic
15	Q₁₆×D₁₁₃	C₄	1	1	yes	characteristic
16	C₂×#11744762	—	2	2	no	—
17	C₂×#11744762	—	2	1	no	—
18	C₂×#11744762	D₄	1	2	yes	—
19	#14268655	C₂×C₄	1	1	yes	characteristic
20	C₄⋊#90673	—	2	2	no	—
21	C₄⋊#90673	C₂×C₄	1	2	yes	—
22	C₄×#90673	—	2	2	no	—
23	C₄×#90673	C₂³	1	1	yes	characteristic
24	Q₈×D₁₁₃	C₂×C₄	1	2	yes	—
25	Dic₄₅₂	C₈	1	1	yes	characteristic, complemented
26	#14268637	C₈	1	2	yes	complemented
27	C₈×D₁₁₃	C₂×C₄	1	1	yes	characteristic
28	Q₁₆×C₁₁₃	C₈	1	1	yes	characteristic, complemented
29	C₂×#90673	—	2	2	no	—
30	C₂×#90673	C₂×D₄	1	1	yes	characteristic
31	C₂×#90673	#41	1	1	yes	characteristic
32	#11744762	—	2	2	no	—
33	#11744762	—	4	1	no	—
34	#11744762	—	4	2	no	—
35	Q₈×C₁₁₃	C₂×C₈	1	2	yes	—
36	C₄×D₁₁₃	—	2	2	no	—
37	C₄×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
38	Dic₂₂₆	—	2	2	no	—
39	Dic₂₂₆	C₂×C₈	1	2	yes	—
40	#11744753	—	2	1	no	—
41	C₈×C₁₁₃	C₂×C₈	1	1	yes	characteristic
42	#11744751	C₂×C₈	1	1	yes	characteristic
43	C₂×D₁₁₃	C₄×D₄	1	1	yes	characteristic
44	#90673	—	2	1	no	—
45	#90673	—	4	2	no	—
46	#90673	#137	1	2	yes	—
47	C₄×C₁₁₃	—	2	2	no	—
48	C₄×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
49	Dic₁₁₃	—	2	2	no	—
50	Dic₁₁₃	#131	1	1	yes	characteristic
51	C₂×C₁₁₃	C₈×D₄	1	1	yes	characteristic
52	D₁₁₃	#441	1	1	yes	characteristic
53	D₁₁₃	#444	1	1	yes	characteristic
54	#1579	—	113	1	no	maximal, Sylow
55	C₁₁₃	#1579	1	1	yes	characteristic, complemented, Sylow
56	C₈×Q₈	—	113	2	no	—
57	C₄×Q₁₆	—	113	1	no	—
58	#334	—	113	1	no	—
59	Q₈⋊C₈	—	113	2	no	—
60	#322	—	113	1	no	—
61	C₂×Q₁₆	—	113	1	no	—
62	C₄×Q₈	—	113	2	no	—
63	#107	—	113	1	no	—
64	C₄⋊C₈	—	113	2	no	—
65	C₄⋊C₈	—	226	2	no	—
66	#103	—	113	2	no	—
67	C₄×C₈	—	113	1	no	—
68	C₄×C₈	—	113	1	no	—
69	C₄×C₈	—	113	1	no	—
70	C₄×C₈	—	226	2	no	—
71	C₂×Q₈	—	113	2	no	—
72	Q₁₆	—	113	2	no	—
73	Q₁₆	—	113	1	no	—
74	Q₁₆	#11744762	1	1	yes	characteristic, complemented
75	C₂×C₈	—	113	1	no	—
76	C₂×C₈	—	113	2	no	—
77	C₂×C₈	—	113	1	no	—
78	C₂×C₈	—	226	1	no	—
79	C₂×C₈	—	226	2	no	—
80	C₄⋊C₄	—	113	2	no	—
81	C₄⋊C₄	—	226	2	no	—
82	C₄²	—	113	1	no	—
83	C₄²	—	226	2	no	—
84	Q₈	—	113	2	no	—
85	Q₈	—	226	2	no	—
86	Q₈	C₂×#11744762	1	2	yes	—
87	C₂×C₄	—	113	1	no	—
88	C₂×C₄	—	113	1	no	—
89	C₂×C₄	—	113	1	no	—
90	C₂×C₄	—	226	2	no	—
91	C₂×C₄	—	226	2	no	—
92	C₈	—	113	1	no	—
93	C₈	—	452	2	no	—
94	C₈	—	452	1	no	—
95	C₈	—	226	1	no	—
96	C₈	—	226	2	no	—
97	C₈	C₂×#11744762	1	1	yes	characteristic
98	C₂²	—	113	1	no	—
99	C₄	—	2	2	no	—
100	C₄	—	113	2	no	—
101	C₄	—	113	1	no	—
102	C₄	—	452	2	no	—
103	C₄	—	226	1	no	—
104	C₄	—	226	2	no	—
105	C₄	C₂²×#11744762	1	1	yes	characteristic
106	C₂	—	113	1	no	—
107	C₂	—	113	1	no	—
108	C₂	D₄×#11744762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11744762

C₂

yes

maximal

#20299687

C₂

yes

maximal, characteristic

Q₈⋊#11744762

C₂

yes

maximal

#20299682

C₂

yes

maximal, characteristic

Q₁₆×#90673

C₂

yes

maximal, characteristic

C₄⋊#11744762

—

C₄⋊#11744762

C₂²

yes

—

C₄×#11744762

—

C₄×#11744762

C₂²

yes

characteristic

C₄×#11744762

C₂²

yes

characteristic

Q₈×#90673

C₂²

yes

—

#16813691

C₄

yes

characteristic

#16813685

C₄

yes

—

C₈×#90673

C₂²

yes

characteristic

Q₁₆×D₁₁₃

C₄

yes

—

—

D₄

yes

—

#14268655

C₂×C₄

yes

characteristic

C₄⋊#90673

—

C₄⋊#90673

C₂×C₄

yes

—

C₄×#90673

—

C₄×#90673

C₂³

yes

characteristic

Q₈×D₁₁₃

C₂×C₄

yes

—

Dic₄₅₂

C₈

yes

characteristic, complemented

#14268637

C₈

yes

complemented

C₈×D₁₁₃

C₂×C₄

yes

characteristic

Q₁₆×C₁₁₃

C₈

yes

characteristic, complemented

C₂×#90673

—

C₂×#90673

C₂×D₄

yes

yes

—

—

—

Q₈×C₁₁₃

C₂×C₈

yes

—

C₄×D₁₁₃

—

C₄×D₁₁₃

C₂²×C₄

yes

characteristic

Dic₂₂₆

—

Dic₂₂₆

C₂×C₈

yes

—

#11744753

—

C₈×C₁₁₃

C₂×C₈

yes

characteristic

#11744751

C₂×C₈

yes

characteristic

C₂×D₁₁₃

C₄×D₄

yes

—

—

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂²×C₈

yes

characteristic

Dic₁₁₃

—

Dic₁₁₃

#131

yes

characteristic

C₂×C₁₁₃

C₈×D₄

yes

D₁₁₃

yes

D₁₁₃

yes

—

113

maximal, Sylow

C₁₁₃

#1579

yes

characteristic, complemented, Sylow

C₈×Q₈

—

113

—

C₄×Q₁₆

—

113

—

#334

—

113

—

Q₈⋊C₈

—

113

—

#322

—

113

—

C₂×Q₁₆

—

113

—

C₄×Q₈

—

113

—

#107

—

113

—

C₄⋊C₈

—

113

—

C₄⋊C₈

—

226

—

#103

—

113

—

C₄×C₈

—

113

—

C₄×C₈

—

113

—

C₄×C₈

—

113

—

C₄×C₈

—

226

—

C₂×Q₈

—

113

—

Q₁₆

—

113

—

Q₁₆

—

113

—

Q₁₆

#11744762

yes

characteristic, complemented

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

226

—

C₄²

—

113

—

C₄²

—

226

—

Q₈

—

113

—

Q₈

—

226

—

Q₈

C₂×#11744762

yes

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₈

—

113

—

C₈

—

452

—

C₈

—

452

—

C₈

—

226

—

C₈

—

226

—

C₈

C₂×#11744762

yes

characteristic

C₂²

—

113

—

C₄

—

100

C₄

—

113

—

101

C₄

—

113

—

102

C₄

—

452

—

103

C₄

—

226

—

104

C₄

—

226

—

105

C₄

C₂²×#11744762

yes

characteristic

106

C₂

—

113

—

107

C₂

—

113

—

108

C₂

D₄×#11744762

yes

characteristic

	subgroup	G/H
center	C₂	D₄×#11744762
commutator	C₄×C₁₁₃	C₂²×C₈
Frattini	C₄	C₂²×#11744762
Fitting	Q₁₆×C₁₁₃	C₈
socle	C₂×C₁₁₃	C₈×D₄
prime radical	C₂×D₁₁₃	C₄×D₄
cosocle	C₄×#90673	C₂³
hypercenter	Q₁₆	#11744762
hypocenter	C₁₁₃	#1579
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

subgroup

G/H

center

C₂

D₄×#11744762

commutator

C₄×C₁₁₃

C₂²×C₈

Frattini

C₄

C₂²×#11744762

Fitting

Q₁₆×C₁₁₃

C₈

socle

C₂×C₁₁₃

C₈×D₄

prime radical

C₂×D₁₁₃

C₄×D₄

C₂³

Q₁₆

C₁₁₃

G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

N	H ≅ G/N	type
25. Dic₄₅₂	C₈	semidirect
26. #14268637	C₈	semidirect
28. Q₁₆×C₁₁₃	C₈	semidirect
55. C₁₁₃	#1579	semidirect
74. Q₁₆	#11744762	semidirect

H ≅ G/N

type

25. Dic₄₅₂

C₈

semidirect

26. #14268637

C₈

semidirect

28. Q₁₆×C₁₁₃

C₈

semidirect

55. C₁₁₃

#1579

semidirect

74. Q₁₆

#11744762

semidirect

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

subgroup	index
Q₈×#11744762	2
#20299687	2
Q₈⋊#11744762	2
#20299682	2
Q₁₆×#90673	2
#1579	113

subgroup

index

Q₁₆×#90673

113

Subgroups — #28407704

Conjugacy classes of subgroups

Complements

Maximal subgroups