Group #28390707 — subgroups

Subgroups — #28390707

Order 14464.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16813533	C₂	1	1	yes	maximal, characteristic
2	#20298468	C₂	1	1	yes	maximal, characteristic
3	#20298360	C₂	1	4	yes	maximal
4	#382×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	#16813533	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
11	#95×C₁₁₃	—	2	2	no	—
12	C₂²×Dic₁₁₃	—	2	2	no	—
13	C₂×#11744751	—	2	2	no	—
14	C₂×#11744751	—	2	2	no	—
15	C₂×#11744751	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₁₃	—	2	4	no	—
17	C₄⋊Dic₁₁₃	—	4	4	no	—
18	C₂²×C₄×C₁₁₃	—	2	2	no	—
19	C₂²×C₄×C₁₁₃	D₄	1	2	yes	—
20	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₁₃	—	2	4	no	—
22	C₄²×C₁₁₃	C₂×C₄	1	2	yes	—
23	C₄²×C₁₁₃	Q₈	1	2	yes	—
24	C₂³×C₁₁₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₁₃	—	2	2	no	—
26	C₂×C₄×C₁₁₃	—	2	2	no	—
27	C₂×C₄×C₁₁₃	—	4	2	no	—
28	C₂×C₄×C₁₁₃	—	4	2	no	—
29	C₂×C₄×C₁₁₃	—	4	2	no	—
30	C₂×C₄×C₁₁₃	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₁₃	#41	1	2	yes	—
34	C₂×C₄×C₁₁₃	#41	1	2	yes	—
35	C₂×Dic₁₁₃	—	4	2	no	—
36	C₂×Dic₁₁₃	—	4	4	no	—
37	#11744751	—	2	4	no	—
38	C₂²×C₁₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₁₃	#124	1	1	yes	characteristic
40	C₂²×C₁₁₃	#124	1	1	yes	characteristic
41	C₂²×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₁₃	#137	1	2	yes	—
43	C₄×C₁₁₃	—	2	4	no	—
44	C₄×C₁₁₃	—	8	2	no	—
45	C₄×C₁₁₃	#117	1	2	yes	—
46	C₄×C₁₁₃	#125	1	2	yes	—
47	Dic₁₁₃	—	8	2	no	—
48	C₂×C₁₁₃	#382	1	1	yes	characteristic
49	C₂×C₁₁₃	#428	1	2	yes	—
50	C₂×C₁₁₃	#489	1	2	yes	—
51	C₂×C₁₁₃	#490	1	1	yes	characteristic
52	C₂×C₁₁₃	#491	1	1	yes	characteristic
53	#1836	—	113	1	no	maximal, Sylow
54	C₁₁₃	#1836	1	1	yes	characteristic, complemented, Sylow
55	C₂×#97	—	113	1	no	—
56	#384	—	113	1	no	—
57	#382	D₁₁₃	1	1	yes	characteristic
58	#340	—	113	4	no	—
59	C₂²×C₈	—	113	2	no	—
60	C₂×C₄⋊C₄	—	113	2	no	—
61	C₂×C₄⋊C₄	—	226	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
63	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
64	#97	—	113	4	no	—
65	#95	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	226	2	no	—
68	C₂²×C₄	C₂²⋊D₁₁₃	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
70	C₂×C₈	—	113	4	no	—
71	C₂×C₈	—	226	2	no	—
72	C₂×C₈	—	226	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	452	4	no	—
75	C₄⋊C₄	—	226	4	no	—
76	C₄²	Dic₂₂₆	1	2	yes	—
77	C₄²	C₄×D₁₁₃	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	452	2	no	—
85	C₂×C₄	—	452	4	no	—
86	C₂×C₄	#14268626	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₂₆	1	1	yes	characteristic
89	C₂×C₄	#14268646	1	2	yes	—
90	C₂×C₄	#14268648	1	2	yes	—
91	C₈	—	226	4	no	—
92	C₂²	C₂×#14268626	1	1	yes	characteristic
93	C₂²	#16813609	1	1	yes	characteristic
94	C₂²	#16813611	1	1	yes	characteristic
95	C₂²	#16813647	1	2	yes	—
96	C₂²	#16813652	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	904	2	no	—
100	C₄	#16813580	1	2	yes	—
101	C₄	#16813613	1	2	yes	—
102	C₂	#20298463	1	1	yes	characteristic
103	C₂	#20298641	1	2	yes	—
104	C₂	#20298957	1	2	yes	—
105	C₂	#20298964	1	1	yes	characteristic
106	C₂	#20298968	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16813533

C₂

yes

maximal, characteristic

#20298468

C₂

yes

maximal, characteristic

#20298360

C₂

yes

maximal

#382×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×#11744751

C₂²

yes

—

C₂×C₄⋊Dic₁₁₃

—

C₂×C₄⋊Dic₁₁₃

C₂²

yes

—

#16813533

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₁₃

C₂²

yes

—

C₂×C₄²×C₁₁₃

C₂²

yes

characteristic

#95×C₁₁₃

—

C₂²×Dic₁₁₃

—

C₂×#11744751

—

C₂×#11744751

—

C₂×#11744751

C₂×C₄

yes

—

C₄⋊Dic₁₁₃

—

C₄⋊Dic₁₁₃

—

C₂²×C₄×C₁₁₃

—

C₂²×C₄×C₁₁₃

D₄

yes

—

C₂²×C₄×C₁₁₃

C₂³

yes

characteristic

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

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₁₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₁₃

#41

yes

—

C₂×C₄×C₁₁₃

#41

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

#11744751

—

C₂²×C₁₁₃

C₂×C₄⋊C₄

yes

C₂²×C₁₁₃

yes

C₂²×C₁₁₃

yes

C₂²×C₁₁₃

yes

—

C₂²×C₁₁₃

#137

yes

—

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#117

yes

—

C₄×C₁₁₃

#125

yes

—

Dic₁₁₃

—

C₂×C₁₁₃

#382

yes

characteristic

C₂×C₁₁₃

#428

yes

—

C₂×C₁₁₃

#489

yes

—

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

113

maximal, Sylow

C₁₁₃

#1836

yes

characteristic, complemented, Sylow

C₂×#97

—

113

—

#384

—

113

—

#382

D₁₁₃

yes

characteristic

#340

—

113

—

C₂²×C₈

—

113

—

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

226

—

C₂×C₄⋊C₄

C₂×D₁₁₃

yes

—

C₂×C₄²

C₂×D₁₁₃

yes

characteristic

#97

—

113

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

226

—

C₂²×C₄

C₂²⋊D₁₁₃

yes

—

C₂²×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₈

—

113

—

C₂×C₈

—

226

—

C₂×C₈

—

226

—

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

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

452

—

C₂×C₄

—

452

—

C₂×C₄

#14268626

yes

—

C₂×C₄

C₂×C₄×D₁₁₃

yes

characteristic

C₂×C₄

C₂×Dic₂₂₆

yes

characteristic

C₂×C₄

#14268646

yes

—

C₂×C₄

#14268648

yes

—

C₈

—

226

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#16813652

yes

—

C₄

—

C₄

—

C₄

—

904

—

100

C₄

#16813580

yes

—

101

C₄

#16813613

yes

—

102

C₂

#20298463

yes

characteristic

103

C₂

#20298641

yes

—

104

C₂

#20298957

yes

—

105

C₂

#20298964

yes

characteristic

106

C₂

#20298968

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₁₃
commutator	C₂×C₄×C₁₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₁₃
Fitting	#382×C₁₁₃	C₂
socle, prime radical	C₂³×C₁₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₁₃	C₂³
hypercenter	#382	D₁₁₃
hypocenter	C₁₁₃	#1836
LCS	G ▷ C₂×C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₁₃

commutator

C₂×C₄×C₁₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₁₃

Fitting

#382×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 ◁ #382

N	H ≅ G/N	type
8. #16813533	C₄	semidirect
54. C₁₁₃	#1836	semidirect

H ≅ G/N

type

8. #16813533

C₄

semidirect

54. C₁₁₃

#1836

semidirect

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

subgroup	index
C₂×#16813533	2
#20298468	2
#20298360	2
#382×C₁₁₃	2
#1836	113

subgroup

index

#382×C₁₁₃

113

Subgroups — #28390707

Conjugacy classes of subgroups

Complements

Maximal subgroups