Group #22560114 — subgroups

Subgroups — #22560114

Order 8576.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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×#14523978	C₂	1	2	yes	maximal
2	M₁₆×Dic₆₇	C₂	1	2	yes	maximal
3	C₂×#14523953	C₂	1	1	yes	maximal, characteristic
4	#16863161	C₂	1	1	yes	maximal, characteristic
5	#341×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	2	yes	—
7	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
9	#14523978	C₄	1	8	yes	complemented
10	#14523953	—	2	2	no	—
11	#14523953	C₄	1	4	yes	—
12	C₈×Dic₆₇	—	2	2	no	—
13	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₆₇	Q₈	1	2	yes	—
16	#11761612	—	2	2	no	—
17	#11761612	C₂×C₄	1	4	yes	—
18	C₂×C₆₇⋊C₈	—	2	2	no	—
19	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₆₇	D₄	1	2	yes	—
23	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₆₇	—	2	2	no	—
25	M₁₆×C₆₇	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₆₇	—	2	2	no	—
27	C₂×C₈×C₆₇	—	2	2	no	—
28	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	#41	1	2	yes	—
34	C₂×Dic₆₇	—	4	2	no	—
35	C₂×Dic₆₇	C₄⋊C₄	1	4	yes	—
36	C₈×C₆₇	—	2	2	no	—
37	C₈×C₆₇	—	2	4	no	—
38	C₆₇⋊C₈	—	2	4	no	—
39	C₆₇⋊C₈	C₄²	1	4	yes	—
40	C₂²×C₆₇	—	2	2	no	—
41	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₆₇	C₄×Q₈	1	2	yes	—
43	C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₆₇	#117	1	1	yes	characteristic
45	C₄×C₆₇	C₄×D₄	1	2	yes	—
46	Dic₆₇	—	2	4	no	—
47	C₂×C₆₇	—	2	2	no	—
48	C₂×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₆₇	#430	1	2	yes	—
50	#1778	—	67	1	no	maximal, Sylow
51	C₆₇	#1778	1	1	yes	characteristic, complemented, Sylow
52	C₂×#108	—	67	2	no	—
53	C₄×M₁₆	—	67	2	no	—
54	C₂×#97	—	67	1	no	—
55	#341	—	67	1	no	—
56	#341	D₆₇	1	1	yes	characteristic
57	C₂×M₁₆	—	67	2	no	—
58	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
59	C₂²×C₈	—	67	1	no	—
60	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
61	C₂×C₄²	—	67	1	no	—
62	#108	—	67	8	no	—
63	#97	—	67	4	no	—
64	#97	—	134	2	no	—
65	C₄×C₈	—	134	2	no	—
66	C₂²×C₄	—	67	2	no	—
67	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	67	4	no	—
70	M₁₆	—	134	2	no	—
71	M₁₆	C₄×D₆₇	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	67	2	no	—
75	C₂×C₈	—	67	4	no	—
76	C₂×C₈	—	134	2	no	—
77	C₂×C₈	C₄×D₆₇	1	2	yes	—
78	C₄²	—	67	2	no	—
79	C₄²	—	67	2	no	—
80	C₂³	Q₈×D₆₇	1	1	yes	characteristic
81	C₂×C₄	—	268	2	no	—
82	C₂×C₄	—	67	4	no	—
83	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
85	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
86	C₂×C₄	#11761627	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	67	4	no	—
90	C₈	—	134	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₆₇	1	1	yes	characteristic
93	C₂²	#14524008	1	2	yes	—
94	C₄	—	134	4	no	—
95	C₄	C₄²×D₆₇	1	1	yes	characteristic
96	C₄	#14523999	1	1	yes	characteristic
97	C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#16863247	1	1	yes	characteristic
100	C₂	#16863443	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523978

C₂

yes

maximal

M₁₆×Dic₆₇

C₂

yes

maximal

C₂×#14523953

C₂

yes

maximal, characteristic

#16863161

C₂

yes

maximal, characteristic

#341×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523978

C₄

yes

complemented

#14523953

—

#14523953

C₄

yes

—

C₈×Dic₆₇

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

Q₈

yes

—

#11761612

—

#11761612

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₄⋊C₄

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₄²

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

C₂×C₄²

yes

characteristic

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₄×D₄

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₆₇

#430

yes

—

#1778

—

maximal, Sylow

C₆₇

#1778

yes

characteristic, complemented, Sylow

C₂×#108

—

C₄×M₁₆

—

C₂×#97

—

#341

—

#341

D₆₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

#97

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

134

—

M₁₆

C₄×D₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

—

C₄²

—

C₄²

—

C₂³

Q₈×D₆₇

yes

characteristic

C₂×C₄

—

268

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂×C₄

C₂×C₄×D₆₇

yes

—

C₂×C₄

D₄×D₆₇

yes

characteristic

C₂×C₄

#11761627

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

134

—

C₂²

—

C₂²

C₄⋊C₄×D₆₇

yes

characteristic

C₂²

#14524008

yes

—

C₄

—

134

—

C₄

C₄²×D₆₇

yes

characteristic

C₄

#14523999

yes

characteristic

C₄

(C₂²⋊D₆₇)⋊C₄

yes

—

C₂

—

C₂

#16863247

yes

characteristic

100

C₂

#16863443

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₆₇
commutator	C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#341×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₄
hypercenter	#341	D₆₇
hypocenter	C₆₇	#1778
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₆₇

commutator

C₄×C₆₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#341×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₄⋊C₄

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #14523978	C₄	semidirect
51. C₆₇	#1778	semidirect

H ≅ G/N

type

9. #14523978

C₄

semidirect

51. C₆₇

#1778

semidirect

Subgroups — #22560114

Conjugacy classes of subgroups

Complements