Group #28390328 — subgroups

Subgroups — #28390328

Order 14464.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20298556	C₂	1	1	yes	maximal, characteristic
2	#20298554	C₂	1	2	yes	maximal
3	#20298356	C₂	1	1	yes	maximal, characteristic
4	#20298354	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14268631	C₂²	1	2	yes	—
7	C₂²×#11744751	C₂²	1	1	yes	characteristic
8	#16813581	C₂²	1	1	yes	characteristic
9	#16813533	C₄	1	4	yes	complemented
10	#16813532	C₄	1	2	yes	complemented
11	C₄×#11744751	C₄	1	4	yes	complemented
12	C₈×Dic₁₁₃	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
14	#117×C₁₁₃	C₂²	1	2	yes	—
15	#14268631	—	2	2	no	—
16	#14268631	—	2	2	no	—
17	C₂×#11744751	C₂×C₄	1	4	yes	—
18	C₂×#11744751	C₂×C₄	1	1	yes	characteristic
19	C₂×#11744751	C₂×C₄	1	2	yes	—
20	C₂×#11744751	C₂×C₄	1	1	yes	characteristic
21	C₂×#11744751	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₁₃	—	2	1	no	—
23	C₂²⋊Dic₁₁₃	—	2	1	no	—
24	C₄×Dic₁₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₁₃	—	2	2	no	—
27	C₂×C₈×C₁₁₃	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₁₃	—	2	2	no	—
29	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
30	C₄²×C₁₁₃	C₂×C₄	1	4	yes	—
31	C₂³×C₁₁₃	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₁₃	—	2	4	no	—
33	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₁₃	#41	1	2	yes	—
36	C₂×C₄×C₁₁₃	#41	1	1	yes	characteristic
37	C₂×Dic₁₁₃	—	2	2	no	—
38	C₈×C₁₁₃	—	2	2	no	—
39	#11744751	—	2	4	no	—
40	#11744751	C₄²	1	1	yes	characteristic
41	#11744751	C₄²	1	1	yes	characteristic
42	#11744751	C₄²	1	2	yes	—
43	C₂²×C₁₁₃	—	4	1	no	—
44	C₂²×C₁₁₃	#117	1	2	yes	—
45	C₂²×C₁₁₃	#117	1	1	yes	characteristic
46	C₄×C₁₁₃	—	4	4	no	—
47	C₄×C₁₁₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₁₃	#117	1	1	yes	characteristic
49	C₄×C₁₁₃	#117	1	2	yes	—
50	Dic₁₁₃	—	4	2	no	—
51	C₂×C₁₁₃	—	2	1	no	—
52	C₂×C₁₁₃	—	2	2	no	—
53	C₂×C₁₁₃	#376	1	1	yes	characteristic
54	#1747	—	113	1	no	maximal, Sylow
55	C₁₁₃	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	113	1	no	—
57	#405	—	113	2	no	—
58	#338	—	113	1	no	—
59	H(3,ℤ/4)	—	113	2	no	—
60	H(3,ℤ/4)	D₁₁₃	1	1	yes	characteristic
61	C₂×M₁₆	—	113	2	no	—
62	C₂×M₁₆	C₂×D₁₁₃	1	1	yes	characteristic
63	C₂²×C₈	—	113	1	no	—
64	#117	—	113	1	no	—
65	#117	C₂×D₁₁₃	1	2	yes	—
66	#97	—	113	2	no	—
67	#97	—	113	4	no	—
68	C₄×C₈	—	113	2	no	—
69	C₄×C₈	—	113	4	no	—
70	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	226	2	no	—
73	M₁₆	—	226	2	no	—
74	C₂×C₈	—	113	1	no	—
75	C₂×C₈	—	113	2	no	—
76	C₂×C₈	—	113	1	no	—
77	C₂×C₈	—	113	2	no	—
78	C₂×C₈	—	113	4	no	—
79	C₂×C₈	C₄×D₁₁₃	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	226	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	226	1	no	—
84	C₄²	—	113	2	no	—
85	C₄²	C₄×D₁₁₃	1	4	yes	—
86	C₂³	#14268646	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	226	2	no	—
89	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₁₃	1	2	yes	—
91	C₂×C₄	#14268646	1	2	yes	—
92	C₂×C₄	#14268648	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	113	1	no	—
95	C₈	—	113	1	no	—
96	C₈	—	113	2	no	—
97	C₈	—	226	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#16813578	1	2	yes	—
100	C₂²	#16813580	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	452	2	no	—
103	C₄	C₄²×D₁₁₃	1	1	yes	characteristic
104	C₄	#16813578	1	1	yes	characteristic
105	C₄	#16813580	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#20298443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20298556

C₂

yes

maximal, characteristic

#20298554

C₂

yes

maximal

#20298356

C₂

yes

maximal, characteristic

#20298354

C₂

yes

maximal

H(3,ℤ/4)×C₁₁₃

C₂

yes

maximal, characteristic

C₂×#14268631

C₂²

yes

—

C₂²×#11744751

C₂²

yes

characteristic

#16813581

C₂²

yes

characteristic

#16813533

C₄

yes

complemented

#16813532

C₄

yes

complemented

C₄×#11744751

C₄

yes

complemented

C₈×Dic₁₁₃

C₄

yes

complemented

C₂×M₁₆×C₁₁₃

C₂²

yes

characteristic

#117×C₁₁₃

C₂²

yes

—

#14268631

—

#14268631

—

C₂×#11744751

C₂×C₄

yes

—

C₂×#11744751

C₂×C₄

yes

characteristic

C₂×#11744751

C₂×C₄

yes

—

C₂×#11744751

C₂×C₄

yes

characteristic

C₂×#11744751

C₂×C₄

yes

—

C₄⋊Dic₁₁₃

—

C₂²⋊Dic₁₁₃

—

C₄×Dic₁₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

M₁₆×C₁₁₃

—

C₂×C₈×C₁₁₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₁₃

—

C₂²⋊C₄×C₁₁₃

—

C₄²×C₁₁₃

C₂×C₄

yes

—

C₂³×C₁₁₃

#41

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

C₂×C₄×C₁₁₃

#41

yes

characteristic

C₂×Dic₁₁₃

—

C₈×C₁₁₃

—

#11744751

—

#11744751

C₄²

yes

characteristic

#11744751

C₄²

yes

characteristic

#11744751

C₄²

yes

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

#117

yes

—

C₂²×C₁₁₃

#117

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×C₄²

yes

C₄×C₁₁₃

yes

C₄×C₁₁₃

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

#376

yes

characteristic

#1747

—

113

maximal, Sylow

C₁₁₃

#1747

yes

characteristic, complemented, Sylow

#405

—

113

—

#405

—

113

—

#338

—

113

—

H(3,ℤ/4)

—

113

—

H(3,ℤ/4)

D₁₁₃

yes

characteristic

C₂×M₁₆

—

113

—

C₂×M₁₆

C₂×D₁₁₃

yes

characteristic

C₂²×C₈

—

113

—

#117

—

113

—

#117

C₂×D₁₁₃

yes

—

#97

—

113

—

#97

—

113

—

C₄×C₈

—

113

—

C₄×C₈

—

113

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

M₁₆

—

M₁₆

—

226

—

M₁₆

—

226

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

—

113

—

C₂×C₈

C₄×D₁₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

226

—

C₂²⋊C₄

—

C₂²⋊C₄

—

226

—

C₄²

—

113

—

C₄²

C₄×D₁₁₃

yes

—

C₂³

#14268646

yes

characteristic

C₂×C₄

—

C₂×C₄

—

226

—

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

C₂×C₄×D₁₁₃

yes

—

C₂×C₄

#14268646

yes

—

C₂×C₄

#14268648

yes

characteristic

C₈

—

C₈

—

113

—

C₈

—

113

—

C₈

—

113

—

C₈

—

226

—

C₂²

—

C₂²

#16813578

yes

—

100

C₂²

#16813580

yes

characteristic

101

C₄

—

102

C₄

—

452

—

103

C₄

C₄²×D₁₁₃

yes

characteristic

104

C₄

#16813578

yes

characteristic

105

C₄

#16813580

yes

—

106

C₂

—

107

C₂

—

108

C₂

#20298443

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₁₃
commutator	C₄×C₁₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	H(3,ℤ/4)×C₁₁₃	C₂
socle	C₂×C₁₁₃	#376
prime radical	C₂³×C₁₁₃	#41
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	H(3,ℤ/4)	D₁₁₃
hypocenter	C₁₁₃	#1747
LCS	G ▷ C₄×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₁₃

commutator

C₄×C₁₁₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

H(3,ℤ/4)×C₁₁₃

C₂

C₂×C₁₁₃

C₂³×C₁₁₃

C₂²×C₄×C₁₁₃

C₂³

D₁₁₃

C₁₁₃

G ▷ C₄×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #16813533	C₄	semidirect
10. #16813532	C₄	semidirect
11. C₄×#11744751	C₄	semidirect
12. C₈×Dic₁₁₃	C₄	semidirect
55. C₁₁₃	#1747	semidirect

H ≅ G/N

type

9. #16813533

C₄

semidirect

10. #16813532

C₄

semidirect

11. C₄×#11744751

C₄

semidirect

12. C₈×Dic₁₁₃

C₄

semidirect

55. C₁₁₃

#1747

semidirect

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

subgroup	index
#20298556	2
#20298554	2
#20298356	2
#20298354	2
H(3,ℤ/4)×C₁₁₃	2
#1747	113

subgroup

index

H(3,ℤ/4)×C₁₁₃

113

Subgroups — #28390328

Conjugacy classes of subgroups

Complements

Maximal subgroups