Group #20298534 — subgroups

Subgroups — #20298534

Order 7232.

186 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 63 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×C₄⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14268626	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₁₁₃	C₂	1	2	yes	maximal
4	#16813529	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
7	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
8	C₄⋊Dic₁₁₃	—	2	2	no	—
9	#14268626	—	2	4	no	—
10	C₂²⋊Dic₁₁₃	—	2	4	no	—
11	C₂⁴×C₁₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
15	C₂³×C₁₁₃	—	2	1	no	—
16	C₂³×C₁₁₃	—	2	2	no	—
17	C₂³×C₁₁₃	—	2	2	no	—
18	C₂³×C₁₁₃	—	2	1	no	—
19	C₂³×C₁₁₃	Q₈	1	2	yes	—
20	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₁₁₃	—	2	2	no	—
22	C₂×C₄×C₁₁₃	—	2	1	no	—
23	C₂×C₄×C₁₁₃	—	2	2	no	—
24	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
25	C₂×Dic₁₁₃	—	2	2	no	—
26	C₂×Dic₁₁₃	—	2	2	no	—
27	C₂×Dic₁₁₃	—	2	2	no	—
28	C₂×Dic₁₁₃	—	2	2	no	—
29	C₂×Dic₁₁₃	—	2	2	no	—
30	C₂×Dic₁₁₃	D₄	1	4	yes	—
31	C₂²×C₁₁₃	—	2	4	no	—
32	C₂²×C₁₁₃	—	2	2	no	—
33	C₂²×C₁₁₃	—	4	1	no	—
34	C₂²×C₁₁₃	—	4	2	no	—
35	C₂²×C₁₁₃	—	4	1	no	—
36	C₂²×C₁₁₃	C₂×D₄	1	2	yes	—
37	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₁₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₁₁₃	#41	1	2	yes	—
40	C₂²×C₁₁₃	#41	1	1	yes	characteristic
41	C₄×C₁₁₃	—	2	2	no	—
42	C₄×C₁₁₃	—	4	1	no	—
43	Dic₁₁₃	—	2	4	no	—
44	Dic₁₁₃	—	4	2	no	—
45	C₂×C₁₁₃	—	4	2	no	—
46	C₂×C₁₁₃	C₂²⋊Q₈	1	2	yes	—
47	C₂×C₁₁₃	C₂²⋊Q₈	1	1	yes	characteristic
48	C₂×C₁₁₃	#123	1	1	yes	characteristic
49	C₂×C₁₁₃	#123	1	2	yes	—
50	C₂×C₁₁₃	#127	1	1	yes	characteristic
51	C₁₁₃	#399	1	1	yes	characteristic, complemented, Sylow
52	#399	—	113	1	no	maximal, Sylow
53	C₂×C₄⋊C₄	—	113	2	no	—
54	C₂×C₄⋊C₄	—	113	1	no	—
55	C₂×C₂²⋊C₄	—	113	2	no	—
56	C₂×C₂²⋊C₄	D₁₁₃	1	1	yes	characteristic
57	#95	—	113	1	no	—
58	C₂⁴	C₂×D₁₁₃	1	1	yes	characteristic
59	C₂²×C₄	—	113	2	no	—
60	C₂²×C₄	—	113	2	no	—
61	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
62	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
63	C₄⋊C₄	—	226	4	no	—
64	C₄⋊C₄	—	226	2	no	—
65	C₂²⋊C₄	—	2	2	no	—
66	C₂²⋊C₄	—	226	4	no	—
67	C₂³	—	2	2	no	—
68	C₂³	—	2	1	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	2	1	no	—
71	C₂³	Dic₂₂₆	1	2	yes	—
72	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	113	4	no	—
77	C₂×C₄	—	226	2	no	—
78	C₂×C₄	—	226	2	no	—
79	C₂×C₄	—	226	2	no	—
80	C₂×C₄	—	226	2	no	—
81	C₂×C₄	—	226	2	no	—
82	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	2	no	—
87	C₂²	—	4	1	no	—
88	C₂²	D₄×D₁₁₃	1	2	yes	—
89	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
90	C₂²	C₂×Dic₂₂₆	1	1	yes	characteristic
91	C₂²	#14268646	1	2	yes	—
92	C₂²	#14268647	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	4	1	no	—
95	C₄	—	452	2	no	—
96	C₄	—	226	4	no	—
97	C₂	—	4	2	no	—
98	C₂	#16813599	1	2	yes	—
99	C₂	#16813601	1	1	yes	characteristic
100	C₂	#16813603	1	1	yes	characteristic
101	C₂	#16813604	1	2	yes	—
102	C₂	#16813618	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₁₃

C₂

yes

maximal, characteristic, complemented

C₂×#14268626

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₁₃

C₂

yes

maximal

#16813529

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₁₁₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₁₃

C₂²

yes

—

C₂²×Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

—

#14268626

—

C₂²⋊Dic₁₁₃

—

C₂⁴×C₁₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

—

C₂³×C₁₁₃

Q₈

yes

—

C₂³×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

—

C₂×C₄×C₁₁₃

D₄

yes

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

D₄

yes

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×D₄

yes

—

C₂²×C₁₁₃

C₂×D₄

yes

characteristic

C₂²×C₁₁₃

C₂×Q₈

yes

characteristic

C₂²×C₁₁₃

#41

yes

—

C₂²×C₁₁₃

#41

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²⋊Q₈

yes

—

C₂×C₁₁₃

C₂²⋊Q₈

yes

C₂×C₁₁₃

yes

C₂×C₁₁₃

yes

—

C₂×C₁₁₃

#127

yes

characteristic

C₁₁₃

#399

yes

characteristic, complemented, Sylow

#399

—

113

maximal, Sylow

C₂×C₄⋊C₄

—

113

—

C₂×C₄⋊C₄

—

113

—

C₂×C₂²⋊C₄

—

113

—

C₂×C₂²⋊C₄

D₁₁₃

yes

characteristic

#95

—

113

—

C₂⁴

C₂×D₁₁₃

yes

characteristic

C₂²×C₄

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

—

226

—

C₄⋊C₄

—

226

—

C₂²⋊C₄

—

C₂²⋊C₄

—

226

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

Dic₂₂₆

yes

—

C₂³

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

113

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

—

226

—

C₂×C₄

C₂²⋊D₁₁₃

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₁₃

yes

—

C₂²

C₂×C₂²⋊D₁₁₃

yes

characteristic

C₂²

C₂×Dic₂₂₆

yes

characteristic

C₂²

#14268646

yes

—

C₂²

#14268647

yes

characteristic

C₄

—

C₄

—

C₄

—

452

—

C₄

—

226

—

C₂

—

C₂

#16813599

yes

—

C₂

#16813601

yes

characteristic

100

C₂

#16813603

yes

characteristic

101

C₂

#16813604

yes

—

102

C₂

#16813618

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₁₃
commutator, socle	C₂³×C₁₁₃	C₂³
Fitting	C₂×C₂²⋊C₄×C₁₁₃	C₂
hypercenter	C₂×C₂²⋊C₄	D₁₁₃
hypocenter	C₁₁₃	#399
LCS	G ▷ C₂³×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₁₃

commutator, socle

C₂³×C₁₁₃

C₂³

Fitting

C₂×C₂²⋊C₄×C₁₁₃

C₂

hypercenter

C₂×C₂²⋊C₄

D₁₁₃

C₁₁₃

G ▷ C₂³×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₁₁₃	C₂	semidirect
2. C₂×#14268626	C₂	semidirect
4. #16813529	C₂	semidirect
51. C₁₁₃	#399	semidirect

H ≅ G/N

type

1. C₂×C₄⋊Dic₁₁₃

C₂

semidirect

2. C₂×#14268626

C₂

semidirect

4. #16813529

C₂

semidirect

51. C₁₁₃

#399

semidirect

Subgroups — #20298534

Conjugacy classes of subgroups

Complements