Group #20299226 — subgroups

Subgroups — #20299226

Order 7232.

206 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 111 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	2	yes	maximal, complemented
2	#16813621	C₂	1	2	yes	maximal, complemented
3	#16813614	C₂	1	2	yes	maximal, complemented
4	#16813613	C₂	1	2	yes	maximal, complemented
5	#16813599	C₂	1	4	yes	maximal
6	#16813581	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
8	#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₂₆	C₂²	1	4	yes	—
10	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
11	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
12	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
13	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₁₃	C₂²	1	4	yes	—
16	#14268626	C₂²	1	4	yes	—
17	#14268626	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₁₃	C₂²	1	2	yes	—
19	C₄×Dic₁₁₃	C₂²	1	2	yes	—
20	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
24	C₄²×C₁₁₃	C₂²	1	2	yes	—
25	C₂³×C₁₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₁₁₃	Q₈	1	4	yes	complemented
27	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
28	C₂×C₄×C₁₁₃	C₂³	1	2	yes	—
29	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₁₃	—	4	2	no	—
32	C₂×Dic₁₁₃	C₂³	1	2	yes	—
33	C₂×Dic₁₁₃	C₂³	1	2	yes	—
34	C₂×Dic₁₁₃	C₂³	1	4	yes	—
35	Dic₂₂₆	—	4	4	no	—
36	C₂²×C₁₁₃	—	2	1	no	—
37	C₂²×C₁₁₃	—	2	1	no	—
38	C₂²×C₁₁₃	C₂×Q₈	1	2	yes	—
39	C₂²×C₁₁₃	C₂⁴	1	1	yes	characteristic
40	C₄×C₁₁₃	—	2	2	no	—
41	C₄×C₁₁₃	—	2	2	no	—
42	C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
43	C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
44	Dic₁₁₃	—	2	2	no	—
45	Dic₁₁₃	—	2	4	no	—
46	Dic₁₁₃	—	2	2	no	—
47	C₂×C₁₁₃	—	2	2	no	—
48	C₂×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
50	C₂×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
51	C₁₁₃	#544	1	1	yes	characteristic, complemented, Sylow
52	#544	—	113	1	no	maximal, Sylow
53	C₄⋊Q₈	—	113	2	no	—
54	C₄⋊Q₈	—	113	2	no	—
55	#125	—	113	2	no	—
56	#125	—	113	2	no	—
57	C₂²⋊Q₈	—	113	4	no	—
58	#117	—	113	1	no	—
59	#117	D₁₁₃	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	113	1	no	—
61	C₂×Q₈	—	113	4	no	—
62	C₂²×C₄	—	113	2	no	—
63	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
64	C₄⋊C₄	—	113	2	no	—
65	C₄⋊C₄	—	113	2	no	—
66	C₄⋊C₄	—	113	1	no	—
67	C₄⋊C₄	—	113	1	no	—
68	C₄⋊C₄	—	113	4	no	—
69	C₄⋊C₄	—	113	4	no	—
70	C₄⋊C₄	—	113	4	no	—
71	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
72	C₄⋊C₄	C₂×D₁₁₃	1	1	yes	characteristic
73	C₂²⋊C₄	—	113	2	no	—
74	C₂²⋊C₄	C₂×D₁₁₃	1	2	yes	—
75	C₄²	—	113	2	no	—
76	C₄²	C₂×D₁₁₃	1	2	yes	—
77	C₂³	C₂²×D₁₁₃	1	1	yes	characteristic
78	Q₈	—	452	4	no	—
79	C₂×C₄	—	113	2	no	—
80	C₂×C₄	—	113	2	no	—
81	C₂×C₄	—	113	4	no	—
82	C₂×C₄	—	452	2	no	—
83	C₂×C₄	Dic₂₂₆	1	4	yes	complemented
84	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
85	C₂×C₄	C₂²×D₁₁₃	1	2	yes	—
86	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	C₂×Dic₂₂₆	1	2	yes	—
91	C₂²	C₂³×D₁₁₃	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	226	4	no	—
95	C₄	—	226	2	no	—
96	C₄	—	226	2	no	—
97	C₄	C₂×Dic₂₂₆	1	2	yes	—
98	C₄	C₂×Dic₂₂₆	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	C₂²×Dic₂₂₆	1	1	yes	characteristic
101	C₂	#16813668	1	1	yes	characteristic
102	C₂	#16813669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16813622

C₂

yes

maximal, complemented

#16813621

C₂

yes

maximal, complemented

#16813614

C₂

yes

maximal, complemented

#16813613

C₂

yes

maximal, complemented

#16813599

C₂

yes

maximal

#16813581

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₁₃

C₂

yes

maximal, characteristic

#117×C₁₁₃

C₂

yes

maximal, characteristic

C₂×Dic₂₂₆

C₂²

yes

—

C₂²×Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

—

C₄⋊Dic₁₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₁₃

C₂²

yes

—

#14268626

C₂²

yes

—

#14268626

C₂²

yes

—

C₂²⋊Dic₁₁₃

C₂²

yes

—

C₄×Dic₁₁₃

C₂²

yes

—

C₂²×C₄×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₁₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₁₃

C₂²

yes

—

C₄²×C₁₁₃

C₂²

yes

—

C₂³×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

Q₈

yes

complemented

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

C₂³

yes

—

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₁₃

C₂³

yes

characteristic

C₂×Dic₁₁₃

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

C₂×Dic₁₁₃

C₂³

yes

—

Dic₂₂₆

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

—

C₂²×C₁₁₃

C₂×Q₈

yes

—

C₂²×C₁₁₃

C₂⁴

yes

characteristic

C₄×C₁₁₃

—

C₄×C₁₁₃

—

C₄×C₁₁₃

C₂×Q₈

yes

—

C₄×C₁₁₃

C₂×Q₈

yes

—

Dic₁₁₃

—

Dic₁₁₃

—

Dic₁₁₃

—

C₂×C₁₁₃

—

C₂×C₁₁₃

C₂²×Q₈

yes

characteristic

C₂×C₁₁₃

H(5,𝔽₂)

yes

characteristic

C₂×C₁₁₃

2₋¹⁺⁴

yes

characteristic

C₁₁₃

#544

yes

characteristic, complemented, Sylow

#544

—

113

maximal, Sylow

C₄⋊Q₈

—

113

—

C₄⋊Q₈

—

113

—

#125

—

113

—

#125

—

113

—

C₂²⋊Q₈

—

113

—

#117

—

113

—

#117

D₁₁₃

yes

characteristic

C₂×C₄⋊C₄

—

113

—

C₂×Q₈

—

113

—

C₂²×C₄

—

113

—

C₂²×C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

—

113

—

C₄⋊C₄

C₂×D₁₁₃

yes

characteristic

C₄⋊C₄

C₂×D₁₁₃

yes

characteristic

C₂²⋊C₄

—

113

—

C₂²⋊C₄

C₂×D₁₁₃

yes

—

C₄²

—

113

—

C₄²

C₂×D₁₁₃

yes

—

C₂³

C₂²×D₁₁₃

yes

characteristic

Q₈

—

452

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

113

—

C₂×C₄

—

452

—

C₂×C₄

Dic₂₂₆

yes

complemented

C₂×C₄

C₂²×D₁₁₃

yes

—

C₂×C₄

C₂²×D₁₁₃

yes

—

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂×C₄

C₂²×D₁₁₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₂₂₆

yes

—

C₂²

C₂³×D₁₁₃

yes

characteristic

C₄

—

C₄

—

C₄

—

226

—

C₄

—

226

—

C₄

—

226

—

C₄

C₂×Dic₂₂₆

yes

—

C₄

C₂×Dic₂₂₆

yes

—

C₂

—

100

C₂

C₂²×Dic₂₂₆

yes

characteristic

101

C₂

#16813668

yes

characteristic

102

C₂

#16813669

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂⁴
Fitting	#117×C₁₁₃	C₂
hypercenter	#117	D₁₁₃
hypocenter	C₁₁₃	#544
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₁₃

commutator, socle

C₂²×C₁₁₃

C₂⁴

Fitting

#117×C₁₁₃

C₂

D₁₁₃

C₁₁₃

G ▷ C₂²×C₁₁₃ ▷ C₁₁₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #16813622	C₂	semidirect
2. #16813621	C₂	semidirect
3. #16813614	C₂	semidirect
4. #16813613	C₂	semidirect
26. C₂×C₄×C₁₁₃	Q₈	semidirect
51. C₁₁₃	#544	semidirect
83. C₂×C₄	Dic₂₂₆	semidirect

H ≅ G/N

type

1. #16813622

C₂

semidirect

2. #16813621

C₂

semidirect

3. #16813614

C₂

semidirect

4. #16813613

C₂

semidirect

26. C₂×C₄×C₁₁₃

Q₈

semidirect

51. C₁₁₃

#544

semidirect

83. C₂×C₄

Dic₂₂₆

semidirect

Subgroups — #20299226

Conjugacy classes of subgroups

Complements