Group #20299318 — subgroups

Subgroups — #20299318

Order 7232.

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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	#16813622	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813615	C₂	1	4	yes	maximal, complemented
3	#16813609	C₂	1	2	yes	maximal
4	#16813601	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813587	C₂	1	2	yes	maximal, complemented
7	#16813581	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₂₆	C₂²	1	2	yes	—
10	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
13	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
14	#14268626	C₂²	1	4	yes	—
15	#14268626	C₂²	1	2	yes	—
16	C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
17	C₂²⋊Dic₁₁₃	C₂²	1	4	yes	—
18	C₄×Dic₁₁₃	C₂²	1	4	yes	—
19	C₄×Dic₁₁₃	C₂²	1	2	yes	—
20	C₂×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₁₃	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
24	C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
25	C₂³×C₁₁₃	C₂³	1	2	yes	—
26	D₄×C₁₁₃	—	2	2	no	—
27	C₂×C₄×C₁₁₃	—	2	4	no	—
28	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
30	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₁₃	C₂³	1	2	yes	—
33	C₂×Dic₁₁₃	C₂³	1	2	yes	—
34	C₂×Dic₁₁₃	C₂³	1	4	yes	—
35	Dic₂₂₆	—	2	2	no	—
36	Dic₂₂₆	—	2	4	no	—
37	C₂²×C₁₁₃	—	2	2	no	—
38	C₂²×C₁₁₃	—	2	2	no	—
39	C₂²×C₁₁₃	—	2	2	no	—
40	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
41	C₄×C₁₁₃	—	2	2	no	—
42	C₄×C₁₁₃	—	2	1	no	—
43	C₄×C₁₁₃	#41	1	2	yes	—
44	C₄×C₁₁₃	#41	1	2	yes	—
45	Dic₁₁₃	—	2	2	no	—
46	Dic₁₁₃	—	2	4	no	—
47	Dic₁₁₃	—	2	2	no	—
48	C₂×C₁₁₃	—	4	2	no	—
49	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
50	C₂×C₁₁₃	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
52	C₁₁₃	#556	1	1	yes	characteristic, complemented, Sylow
53	#556	—	113	1	no	maximal, Sylow
54	C₄⋊Q₈	—	113	1	no	—
55	#126	—	113	4	no	—
56	#124	—	113	2	no	—
57	C₂²⋊Q₈	—	113	2	no	—
58	C₄×Q₈	—	113	2	no	—
59	C₄×Q₈	—	113	1	no	—
60	C₄×D₄	D₁₁₃	1	1	yes	characteristic
61	#117	—	113	2	no	—
62	C₂×Q₈	—	113	1	no	—
63	C₂×Q₈	—	113	2	no	—
64	C₂×D₄	C₂×D₁₁₃	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₁₁₃	1	2	yes	—
66	C₄⋊C₄	—	113	1	no	—
67	C₄⋊C₄	—	113	2	no	—
68	C₄⋊C₄	—	113	4	no	—
69	C₄⋊C₄	—	113	2	no	—
70	C₄⋊C₄	—	113	2	no	—
71	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
72	C₂²⋊C₄	—	113	4	no	—
73	C₂²⋊C₄	—	113	4	no	—
74	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
75	C₄²	—	113	4	no	—
76	C₄²	—	113	2	no	—
77	C₄²	C₂×D₁₁₃	1	1	yes	characteristic
78	C₂³	C₂²×D₁₁₃	1	2	yes	—
79	Q₈	—	226	2	no	—
80	Q₈	—	226	4	no	—
81	D₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	113	2	no	—
84	C₂×C₄	—	113	2	no	—
85	C₂×C₄	—	113	4	no	—
86	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
87	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	—	2	2	no	—
92	C₂²	—	2	2	no	—
93	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	226	2	no	—
97	C₄	—	226	4	no	—
98	C₄	—	226	2	no	—
99	C₄	#14268646	1	2	yes	—
100	C₄	#14268647	1	2	yes	—
101	C₂	—	4	2	no	—
102	C₂	C₂×#14268646	1	1	yes	characteristic
103	C₂	C₂×#14268647	1	1	yes	characteristic
104	C₂	#16813669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16813622

C₂

yes

maximal, characteristic, complemented

#16813615

C₂

yes

maximal, complemented

#16813609

C₂

yes

maximal

#16813601

C₂

yes

maximal, complemented

C₄×Dic₂₂₆

C₂

yes

maximal, characteristic, complemented

#16813587

C₂

yes

maximal, complemented

#16813581

C₂

yes

maximal, complemented

C₄×D₄×C₁₁₃

C₂

yes

maximal, characteristic

C₂×Dic₂₂₆

C₂²

yes

—

C₂×Dic₂₂₆

C₂²

yes

characteristic

C₄⋊Dic₁₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

—

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

C₂²⋊Dic₁₁₃

C₂²

yes

—

C₂²⋊Dic₁₁₃

C₂²

yes

—

C₄×Dic₁₁₃

C₂²

yes

—

C₄×Dic₁₁₃

C₂²

yes

—

C₂×D₄×C₁₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₁₃

C₂²

yes

—

C₄⋊C₄×C₁₁₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₁₃

C₂²

yes

—

C₄²×C₁₁₃

C₂²

yes

characteristic

C₂³×C₁₁₃

C₂³

yes

—

D₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

Dic₂₂₆

—

Dic₂₂₆

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂⁴

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

#41

yes

—

C₄×C₁₁₃

#41

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂×#41

yes

characteristic

C₂×C₁₁₃

C₂×#41

yes

characteristic

C₂×C₁₁₃

2₋¹⁺⁴

yes

characteristic

C₁₁₃

#556

yes

characteristic, complemented, Sylow

#556

—

113

maximal, Sylow

C₄⋊Q₈

—

113

—

#126

—

113

—

#124

—

113

—

C₂²⋊Q₈

—

113

—

C₄×Q₈

—

113

—

C₄×Q₈

—

113

—

C₄×D₄

D₁₁₃

yes

characteristic

#117

—

113

—

C₂×Q₈

—

113

—

C₂×Q₈

—

113

—

C₂×D₄

C₂×D₁₁₃

yes

characteristic

C₂²×C₄

C₂×D₁₁₃

yes

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

C₂×D₁₁₃

yes

characteristic

C₂²⋊C₄

—

113

—

C₂²⋊C₄

—

113

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₄²

—

113

—

C₄²

—

113

—

C₄²

C₂×D₁₁₃

yes

characteristic

C₂³

C₂²×D₁₁₃

yes

—

Q₈

—

226

—

Q₈

—

226

—

D₄

—

C₂×C₄

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

C₂²×D₁₁₃

yes

—

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×D₁₁₃

yes

characteristic

C₄

—

C₄

—

C₄

—

226

—

C₄

—

226

—

C₄

—

226

—

C₄

#14268646

yes

—

100

C₄

#14268647

yes

—

101

C₂

—

102

C₂

C₂×#14268646

yes

characteristic

103

C₂

C₂×#14268647

yes

characteristic

104

C₂

#16813669

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	C₄×D₄×C₁₁₃	C₂
hypercenter	C₄×D₄	D₁₁₃
hypocenter	C₁₁₃	#556
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂⁴

Fitting

C₄×D₄×C₁₁₃

C₂

hypercenter

C₄×D₄

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #16813622	C₂	semidirect
2. #16813615	C₂	semidirect
4. #16813601	C₂	semidirect
5. C₄×Dic₂₂₆	C₂	semidirect
6. #16813587	C₂	semidirect
7. #16813581	C₂	semidirect
52. C₁₁₃	#556	semidirect

H ≅ G/N

type

1. #16813622

C₂

semidirect

2. #16813615

C₂

semidirect

4. #16813601

C₂

semidirect

5. C₄×Dic₂₂₆

C₂

semidirect

6. #16813587

C₂

semidirect

7. #16813581

C₂

semidirect

52. C₁₁₃

#556

semidirect

Subgroups — #20299318

Conjugacy classes of subgroups

Complements