Group #22562440 — subgroups

Subgroups — #22562440

Order 8576.

138 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16863798	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863200	C₂	1	2	yes	maximal, complemented
3	#16863197	C₂	1	2	yes	maximal
4	C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#495×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524035	—	2	1	no	—
7	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
8	#14523977	C₂²	1	2	yes	—
9	#14523975	—	2	1	no	—
10	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₆₇	—	2	1	no	—
12	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
13	#124×C₆₇	—	2	1	no	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₆₇	—	2	2	no	—
16	Dic₆₇⋊C₄	—	4	2	no	—
17	C₆₇⋊C₁₆	—	2	2	no	—
18	C₂×Q₈×C₆₇	—	2	1	no	—
19	C₂×D₄×C₆₇	—	2	1	no	—
20	Q₁₆×C₆₇	—	2	2	no	—
21	SD₁₆×C₆₇	—	4	2	no	—
22	D₈×C₆₇	—	2	2	no	—
23	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₂²⋊C₄×C₆₇	—	4	2	no	—
26	C₄²×C₆₇	D₄	1	1	yes	characteristic
27	C₂³×C₆₇	—	4	1	no	—
28	Q₈×C₆₇	—	4	1	no	—
29	Q₈×C₆₇	—	4	1	no	—
30	D₄×C₆₇	—	4	1	no	—
31	D₄×C₆₇	—	4	1	no	—
32	C₂×C₄×C₆₇	—	4	1	no	—
33	C₂×C₄×C₆₇	D₈	1	2	yes	—
34	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₆₇	—	4	2	no	—
36	C₈×C₆₇	SD₁₆	1	2	yes	—
37	C₈×C₆₇	#41	1	2	yes	—
38	C₂²×C₆₇	—	8	1	no	—
39	C₂²×C₆₇	—	8	2	no	—
40	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
41	C₄×C₆₇	—	2	2	no	—
42	C₄×C₆₇	—	8	1	no	—
43	C₄×C₆₇	#124	1	1	yes	characteristic
44	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
45	Dic₆₇	—	8	2	no	—
46	C₂×C₆₇	—	16	1	no	—
47	C₂×C₆₇	#486	1	1	yes	characteristic
48	C₂×C₆₇	#508	1	2	yes	—
49	#2242	—	67	1	no	maximal, Sylow
50	C₆₇	#2242	1	1	yes	characteristic, complemented, Sylow
51	#499	—	67	1	no	—
52	#495	D₆₇	1	1	yes	characteristic
53	#358	—	67	2	no	—
54	#357	—	67	2	no	—
55	C₄×C₁₆	—	67	1	no	—
56	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
57	C₂×SD₁₆	—	2	1	no	—
58	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
59	#125	—	134	1	no	—
60	#124	—	2	1	no	—
61	C₂×C₁₆	—	67	2	no	—
62	#107	—	67	2	no	—
63	#106	—	134	1	no	—
64	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
65	C₂×Q₈	—	2	1	no	—
66	C₂×D₄	—	2	1	no	—
67	Q₁₆	—	2	2	no	—
68	SD₁₆	—	4	2	no	—
69	D₈	—	2	2	no	—
70	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
72	C₄⋊C₄	—	268	2	no	—
73	C₄⋊C₄	—	134	2	no	—
74	C₂²⋊C₄	—	4	2	no	—
75	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
76	C₁₆	—	134	2	no	—
77	C₂³	—	4	1	no	—
78	Q₈	—	4	1	no	—
79	Q₈	—	4	1	no	—
80	D₄	—	4	1	no	—
81	D₄	—	4	1	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	268	2	no	—
84	C₂×C₄	#11761613	1	2	yes	—
85	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
86	C₈	#11761616	1	2	yes	—
87	C₈	#11761627	1	2	yes	—
88	C₂²	—	8	2	no	—
89	C₂²	—	8	1	no	—
90	C₂²	C₂×#11761613	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	8	1	no	—
93	C₄	—	536	2	no	—
94	C₄	#14524030	1	1	yes	characteristic
95	C₄	C₂×#11761616	1	1	yes	characteristic
96	C₂	—	16	1	no	—
97	C₂	#16863742	1	1	yes	characteristic
98	C₂	#16863826	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863798

C₂

yes

maximal, characteristic, complemented

#16863200

C₂

yes

maximal, complemented

#16863197

C₂

yes

maximal

C₄×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#495×C₆₇

C₂

yes

maximal, characteristic

#14524035

—

C₂×C₆₇⋊C₁₆

C₂²

yes

—

#14523977

C₂²

yes

—

#14523975

—

C₂×Q₁₆×C₆₇

C₂²

yes

characteristic

C₂×SD₁₆×C₆₇

—

C₂×D₈×C₆₇

C₂²

yes

characteristic

#124×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

Q₁₆×C₆₇

—

SD₁₆×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

D₈

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₈×C₆₇

SD₁₆

yes

—

C₈×C₆₇

#41

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#124

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#486

yes

characteristic

C₂×C₆₇

#508

yes

—

#2242

—

maximal, Sylow

C₆₇

#2242

yes

characteristic, complemented, Sylow

#499

—

#495

D₆₇

yes

characteristic

#358

—

#357

—

C₄×C₁₆

—

C₂×Q₁₆

C₂×D₆₇

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂×D₆₇

yes

characteristic

#125

—

134

—

#124

—

C₂×C₁₆

—

#107

—

#106

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₆₇

yes

characteristic

C₁₆

—

134

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

#11761613

yes

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₈

#11761616

yes

—

C₈

#11761627

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#11761613

yes

characteristic

C₄

—

C₄

—

C₄

—

536

—

C₄

yes

C₄

yes

C₂

—

C₂

#16863742

yes

characteristic

C₂

#16863826

yes

—

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#495×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#495	D₆₇
hypocenter	C₆₇	#2242
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

C₂×#11761613

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#495×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. #16863798	C₂	semidirect
2. #16863200	C₂	semidirect
4. C₄×C₆₇⋊C₁₆	C₂	semidirect
50. C₆₇	#2242	semidirect

H ≅ G/N

type

1. #16863798

C₂

semidirect

2. #16863200

C₂

semidirect

4. C₄×C₆₇⋊C₁₆

C₂

semidirect

50. C₆₇

#2242

semidirect

Subgroups — #22562440

Conjugacy classes of subgroups

Complements