Group #22562443 — subgroups

Subgroups — #22562443

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	#16863780	C₂	1	1	yes	maximal, characteristic
2	#16863199	C₂	1	2	yes	maximal, complemented
3	#16863196	C₂	1	2	yes	maximal
4	C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×#11761617	—	2	1	no	—
8	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
9	#14524031	—	2	1	no	—
10	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#125×C₆₇	—	2	1	no	—
12	#107×C₆₇	C₂²	1	2	yes	—
13	#106×C₆₇	—	2	1	no	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₃₄	—	2	1	no	—
16	C₂×D₂₆₈	—	2	1	no	—
17	Dic₂₆₈	—	2	2	no	—
18	#11761617	—	4	2	no	—
19	D₅₃₆	—	2	2	no	—
20	(C₂×D₆₇)⋊C₄	—	4	2	no	—
21	C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₆₇	—	2	2	no	—
25	C₄⋊C₄×C₆₇	—	4	2	no	—
26	C₄²×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×D₆₇	—	4	1	no	—
28	C₂×C₄×C₆₇	—	4	2	no	—
29	C₂×C₄×C₆₇	D₈	1	2	yes	—
30	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₆₇	—	4	1	no	—
32	D₂₆₈	—	4	1	no	—
33	D₂₆₈	—	4	1	no	—
34	Dic₁₃₄	—	4	1	no	—
35	Dic₁₃₄	—	4	1	no	—
36	C₈×C₆₇	SD₁₆	1	2	yes	—
37	C₈×C₆₇	#41	1	2	yes	—
38	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
39	C₂×D₆₇	—	8	1	no	—
40	C₂×D₆₇	—	8	2	no	—
41	C₄×C₆₇	—	2	2	no	—
42	C₄×C₆₇	—	8	2	no	—
43	C₄×C₆₇	#124	1	1	yes	characteristic
44	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
45	Dic₆₇	—	8	1	no	—
46	C₂×C₆₇	#486	1	1	yes	characteristic
47	C₂×C₆₇	#508	1	2	yes	—
48	D₆₇	—	16	1	no	—
49	#2242	—	67	1	no	maximal, Sylow
50	C₆₇	#2242	1	1	yes	characteristic, complemented, Sylow
51	#499	D₆₇	1	1	yes	characteristic, complemented
52	#495	—	67	1	no	—
53	#358	—	67	2	no	—
54	#357	—	67	2	no	—
55	C₄×C₁₆	—	67	1	no	—
56	C₂×Q₁₆	—	67	1	no	—
57	C₂×SD₁₆	—	134	1	no	—
58	C₂×D₈	—	67	1	no	—
59	#125	—	2	1	no	—
60	#124	—	134	1	no	—
61	C₂×C₁₆	—	67	2	no	—
62	#107	C₂×D₆₇	1	2	yes	—
63	#106	—	2	1	no	—
64	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
65	C₂×Q₈	—	134	1	no	—
66	C₂×D₄	—	134	1	no	—
67	Q₁₆	—	134	2	no	—
68	SD₁₆	—	268	2	no	—
69	D₈	—	134	2	no	—
70	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₂²⋊C₄	—	268	2	no	—
75	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
76	C₁₆	—	134	2	no	—
77	C₂³	—	268	1	no	—
78	Q₈	—	268	1	no	—
79	Q₈	—	268	1	no	—
80	D₄	—	268	1	no	—
81	D₄	—	268	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	268	1	no	—
84	C₂×C₄	#11761613	1	2	yes	—
85	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
86	C₈	#11761615	1	2	yes	—
87	C₈	#11761629	1	2	yes	—
88	C₂²	—	536	1	no	—
89	C₂²	—	536	2	no	—
90	C₂²	C₂×#11761613	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	8	2	no	—
93	C₄	—	536	1	no	—
94	C₄	#14524032	1	1	yes	characteristic
95	C₄	C₂×#11761615	1	1	yes	characteristic
96	C₂	—	1072	1	no	—
97	C₂	#16863744	1	1	yes	characteristic
98	C₂	#16863826	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863780

C₂

yes

maximal, characteristic

#16863199

C₂

yes

maximal, complemented

#16863196

C₂

yes

maximal

C₄×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#499×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₈

C₂²

yes

characteristic

C₂×#11761617

—

C₂×D₅₃₆

C₂²

yes

characteristic

#14524031

—

C₂×C₆₇⋊C₁₆

C₂²

yes

—

#125×C₆₇

—

#107×C₆₇

C₂²

yes

—

#106×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

Dic₂₆₈

—

#11761617

—

D₅₃₆

—

(C₂×D₆₇)⋊C₄

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

D₈

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

SD₁₆

yes

—

C₈×C₆₇

#41

yes

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#124

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

C₂×C₆₇

#486

yes

characteristic

C₂×C₆₇

#508

yes

—

D₆₇

—

#2242

—

maximal, Sylow

C₆₇

#2242

yes

characteristic, complemented, Sylow

#499

D₆₇

yes

characteristic, complemented

#495

—

#358

—

#357

—

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

134

—

C₂×D₈

—

#125

—

#124

—

134

—

C₂×C₁₆

—

#107

C₂×D₆₇

yes

—

#106

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

134

—

C₂×D₄

—

134

—

Q₁₆

—

134

—

SD₁₆

—

268

—

D₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

268

—

C₄²

C₂²⋊D₆₇

yes

characteristic

C₁₆

—

134

—

C₂³

—

268

—

Q₈

—

268

—

Q₈

—

268

—

D₄

—

268

—

D₄

—

268

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

#11761613

yes

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₈

#11761615

yes

—

C₈

#11761629

yes

—

C₂²

—

536

—

C₂²

—

536

—

C₂²

C₂×#11761613

yes

characteristic

C₄

—

C₄

—

C₄

—

536

—

C₄

yes

C₄

yes

C₂

—

1072

—

C₂

#16863744

yes

characteristic

C₂

#16863826

yes

—

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#499×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#499	D₆₇
hypocenter	C₆₇	#2242
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

subgroup

G/H

center

C₂²

C₂×#11761613

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#499×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

N	H ≅ G/N	type
2. #16863199	C₂	semidirect
4. C₄×C₆₇⋊C₁₆	C₂	semidirect
5. #499×C₆₇	C₂	semidirect
50. C₆₇	#2242	semidirect
51. #499	D₆₇	semidirect

H ≅ G/N

type

2. #16863199

C₂

semidirect

4. C₄×C₆₇⋊C₁₆

C₂

semidirect

5. #499×C₆₇

C₂

semidirect

50. C₆₇

#2242

semidirect

51. #499

D₆₇

semidirect

Subgroups — #22562443

Conjugacy classes of subgroups

Complements