Group #28390730 — subgroups

Subgroups — #28390730

Order 14464.

280 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×#16813553	C₂	1	4	yes	maximal
2	C₂×C₄×#11744751	C₂	1	1	yes	maximal, characteristic
3	#20298468	C₂	1	1	yes	maximal, characteristic
4	#384×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11744751	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₁₃	—	2	2	no	—
7	C₂×C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
8	#16813553	—	2	8	no	—
9	C₄×#11744751	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₁₃	—	2	2	no	—
14	C₂×#11744751	C₂×C₄	1	4	yes	—
15	C₂×#11744751	D₄	1	4	yes	—
16	C₂×#11744751	Q₈	1	4	yes	—
17	C₄⋊Dic₁₁₃	—	2	4	no	—
18	C₄⋊Dic₁₁₃	—	4	4	no	—
19	C₂²×C₄×C₁₁₃	—	2	2	no	—
20	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₁₃	—	2	4	no	—
23	C₄⋊C₄×C₁₁₃	—	4	4	no	—
24	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
25	C₄²×C₁₁₃	Q₈	1	2	yes	—
26	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₁₃	—	4	2	no	—
28	C₂×C₄×C₁₁₃	—	4	4	no	—
29	C₂×C₄×C₁₁₃	C₄⋊C₄	1	4	yes	—
30	C₂×C₄×C₁₁₃	SD₁₆	1	4	yes	—
31	C₂×C₄×C₁₁₃	SD₁₆	1	4	yes	—
32	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×Dic₁₁₃	—	4	2	no	—
37	C₂×Dic₁₁₃	—	4	4	no	—
38	#11744751	C₄⋊C₄	1	8	yes	complemented
39	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₁₃	#127	1	1	yes	characteristic
41	C₂²×C₁₁₃	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₁₃	C₂×SD₁₆	1	2	yes	—
43	C₂²×C₁₁₃	C₂×SD₁₆	1	2	yes	—
44	C₄×C₁₁₃	—	8	2	no	—
45	C₄×C₁₁₃	#106	1	8	yes	complemented
46	C₄×C₁₁₃	C₂×C₄⋊C₄	1	2	yes	—
47	C₄×C₁₁₃	C₄⋊Q₈	1	2	yes	—
48	Dic₁₁₃	—	8	2	no	—
49	C₂×C₁₁₃	#384	1	1	yes	characteristic
50	C₂×C₁₁₃	C₂×#106	1	2	yes	—
51	C₂×C₁₁₃	#492	1	1	yes	characteristic
52	C₂×C₁₁₃	#492	1	1	yes	characteristic
53	C₂×C₁₁₃	#498	1	2	yes	—
54	#1844	—	113	1	no	maximal, Sylow
55	C₁₁₃	#1844	1	1	yes	characteristic, complemented, Sylow
56	C₂×#106	—	113	4	no	—
57	C₂×C₄×C₈	—	113	1	no	—
58	#384	—	113	1	no	—
59	#384	D₁₁₃	1	1	yes	characteristic
60	C₂²×C₈	—	113	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	113	2	no	—
63	C₂×C₄⋊C₄	—	226	2	no	—
64	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
65	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
66	#106	—	226	8	no	—
67	C₄×C₈	—	113	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	226	2	no	—
70	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
71	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
72	C₂×C₈	—	113	4	no	—
73	C₂×C₈	—	113	4	no	—
74	C₂×C₈	—	113	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	452	4	no	—
78	C₄⋊C₄	—	226	4	no	—
79	C₄²	Dic₂₂₆	1	2	yes	—
80	C₄²	C₄×D₁₁₃	1	2	yes	—
81	C₂³	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	452	2	no	—
85	C₂×C₄	—	452	4	no	—
86	C₂×C₄	#14268626	1	4	yes	—
87	C₂×C₄	#14268634	1	4	yes	—
88	C₂×C₄	#14268635	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
90	C₂×C₄	D₄×D₁₁₃	1	2	yes	—
91	C₂×C₄	Q₈×D₁₁₃	1	2	yes	—
92	C₂×C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
93	C₈	—	113	8	no	—
94	C₂²	C₂×#14268626	1	1	yes	characteristic
95	C₂²	#16813618	1	1	yes	characteristic
96	C₂²	#16813620	1	1	yes	characteristic
97	C₂²	C₂×#14268634	1	2	yes	—
98	C₂²	C₂×#14268635	1	2	yes	—
99	C₄	—	8	2	no	—
100	C₄	—	904	2	no	—
101	C₄	#16813553	1	8	yes	complemented
102	C₄	C₄⋊C₄×D₁₁₃	1	2	yes	—
103	C₄	#16813621	1	2	yes	—
104	C₂	#20298467	1	1	yes	characteristic
105	C₂	C₂×#16813553	1	2	yes	—
106	C₂	#20298970	1	1	yes	characteristic
107	C₂	#20298972	1	1	yes	characteristic
108	C₂	#20298994	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813553

C₂

yes

maximal

C₂×C₄×#11744751

C₂

yes

maximal, characteristic

#20298468

C₂

yes

maximal, characteristic

#384×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×#11744751

C₂²

yes

—

C₂×C₄⋊Dic₁₁₃

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

#16813553

—

C₄×#11744751

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₁₃

—

C₂×C₄⋊C₄×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

C₂²×Dic₁₁₃

—

C₂×#11744751

C₂×C₄

yes

—

C₂×#11744751

D₄

yes

—

C₂×#11744751

Q₈

yes

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₁₃

—

C₄⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

C₂×C₄

yes

—

C₄²×C₁₁₃

Q₈

yes

—

C₂³×C₁₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

SD₁₆

yes

—

C₂×C₄×C₁₁₃

SD₁₆

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₁₃

C₂×Q₈

yes

—

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

#11744751

C₄⋊C₄

yes

complemented

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₁₃

#127

yes

characteristic

C₂²×C₁₁₃

C₄⋊Q₈

yes

characteristic

C₂²×C₁₁₃

C₂×SD₁₆

yes

—

C₂²×C₁₁₃

C₂×SD₁₆

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#106

yes

complemented

C₄×C₁₁₃

C₂×C₄⋊C₄

yes

—

C₄×C₁₁₃

C₄⋊Q₈

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

#384

yes

characteristic

C₂×C₁₁₃

C₂×#106

yes

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

#1844

—

113

maximal, Sylow

C₁₁₃

#1844

yes

characteristic, complemented, Sylow

C₂×#106

—

113

—

C₂×C₄×C₈

—

113

—

#384

—

113

—

#384

D₁₁₃

yes

characteristic

C₂²×C₈

—

113

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

226

—

C₂×C₄⋊C₄

C₂×D₁₁₃

yes

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

#106

—

226

—

C₄×C₈

—

113

—

C₂²×C₄

—

C₂²×C₄

—

226

—

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

452

—

C₄⋊C₄

—

226

—

C₄²

Dic₂₂₆

yes

—

C₄²

C₄×D₁₁₃

yes

—

C₂³

C₂×C₂²⋊D₁₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

452

—

C₂×C₄

—

452

—

C₂×C₄

#14268626

yes

—

C₂×C₄

#14268634

yes

—

C₂×C₄

#14268635

yes

—

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

D₄×D₁₁₃

yes

—

C₂×C₄

Q₈×D₁₁₃

yes

—

C₂×C₄

C₂×Dic₂₂₆

yes

characteristic

C₈

—

113

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#14268635

yes

—

C₄

—

100

C₄

—

904

—

101

C₄

#16813553

yes

complemented

102

C₄

C₄⋊C₄×D₁₁₃

yes

—

103

C₄

#16813621

yes

—

104

C₂

#20298467

yes

characteristic

105

C₂

C₂×#16813553

yes

—

106

C₂

#20298970

yes

characteristic

107

C₂

#20298972

yes

characteristic

108

C₂

#20298994

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#384×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#384	D₁₁₃
hypocenter	C₁₁₃	#1844
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁₃

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#384×C₁₁₃

C₂

socle, prime radical

C₂³×C₁₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#11744751	C₄	semidirect
38. #11744751	C₄⋊C₄	semidirect
45. C₄×C₁₁₃	#106	semidirect
55. C₁₁₃	#1844	semidirect
101. C₄	#16813553	semidirect

H ≅ G/N

type

9. C₄×#11744751

C₄

semidirect

38. #11744751

C₄⋊C₄

semidirect

45. C₄×C₁₁₃

#106

semidirect

55. C₁₁₃

#1844

semidirect

101. C₄

#16813553

semidirect

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

subgroup	index
C₂×#16813553	2
C₂×C₄×#11744751	2
#20298468	2
#384×C₁₁₃	2
#1844	113

subgroup

index

#384×C₁₁₃

113

Subgroups — #28390730

Conjugacy classes of subgroups

Complements

Maximal subgroups