Group #22562385 — subgroups

Subgroups — #22562385

Order 8576.

120 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16863220	C₂	1	2	yes	maximal, complemented
3	#16863217	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863206	C₂	1	2	yes	maximal
5	#443×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524035	—	2	1	no	—
7	#14523983	C₂²	1	2	yes	—
8	#14523977	—	2	1	no	—
9	#14523975	C₂²	1	2	yes	—
10	#135×C₆₇	C₂²	1	1	yes	characteristic
11	#131×C₆₇	—	2	1	no	—
12	#108×C₆₇	C₂²	1	1	yes	characteristic
13	#104×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	#41×C₆₇	—	2	1	no	—
19	Q₁₆×C₆₇	Q₈	1	1	yes	characteristic
20	SD₁₆×C₆₇	—	2	1	no	—
21	D₈×C₆₇	Q₈	1	1	yes	characteristic
22	M₁₆×C₆₇	—	2	1	no	—
23	M₁₆×C₆₇	—	4	1	no	—
24	C₂×C₈×C₆₇	—	4	1	no	—
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₄²×C₆₇	D₄	1	1	yes	characteristic
28	Q₈×C₆₇	—	2	1	no	—
29	D₄×C₆₇	—	2	1	no	—
30	D₄×C₆₇	—	4	1	no	—
31	C₂×C₄×C₆₇	—	2	1	no	—
32	C₂×C₄×C₆₇	—	4	1	no	—
33	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₆₇	—	4	2	no	—
35	C₈×C₆₇	—	4	1	no	—
36	C₈×C₆₇	SD₁₆	1	1	yes	characteristic
37	C₈×C₆₇	SD₁₆	1	1	yes	characteristic
38	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₆₇	#41	1	1	yes	characteristic
40	C₂²×C₆₇	—	4	1	no	—
41	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
42	C₄×C₆₇	—	4	1	no	—
43	C₄×C₆₇	—	4	1	no	—
44	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
45	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
46	Dic₆₇	—	8	2	no	—
47	C₂×C₆₇	—	2	1	no	—
48	C₂×C₆₇	—	8	1	no	—
49	C₂×C₆₇	#476	1	1	yes	characteristic
50	#2231	—	67	1	no	maximal, Sylow
51	C₆₇	#2231	1	1	yes	characteristic, complemented, Sylow
52	#499	—	67	1	no	—
53	#443	D₆₇	1	1	yes	characteristic
54	#365	—	67	2	no	—
55	#364	—	67	1	no	—
56	#360	—	67	2	no	—
57	#135	C₂×D₆₇	1	1	yes	characteristic
58	#131	—	2	1	no	—
59	#125	—	134	1	no	—
60	M₃₂	—	67	2	no	—
61	#108	C₂×D₆₇	1	1	yes	characteristic
62	#107	—	134	1	no	—
63	#106	—	67	2	no	—
64	#104	—	2	1	no	—
65	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
66	#41	—	2	1	no	—
67	Q₁₆	Dic₁₃₄	1	1	yes	characteristic
68	SD₁₆	—	2	1	no	—
69	D₈	Dic₁₃₄	1	1	yes	characteristic
70	M₁₆	—	2	1	no	—
71	M₁₆	—	4	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
75	C₄⋊C₄	—	268	2	no	—
76	C₄⋊C₄	—	134	2	no	—
77	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
78	C₁₆	—	134	2	no	—
79	Q₈	—	2	1	no	—
80	D₄	—	2	1	no	—
81	D₄	—	4	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	268	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
86	C₈	—	4	1	no	—
87	C₈	#11761616	1	1	yes	characteristic
88	C₈	#11761616	1	1	yes	characteristic
89	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
90	C₈	#11761628	1	1	yes	characteristic
91	C₂²	—	4	1	no	—
92	C₂²	#14524068	1	1	yes	characteristic
93	C₄	—	4	1	no	—
94	C₄	—	4	1	no	—
95	C₄	—	536	2	no	—
96	C₄	#14524022	1	1	yes	characteristic
97	C₄	C₂×#11761616	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	8	1	no	—
100	C₂	#16863686	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863798

C₂

yes

maximal, characteristic, complemented

#16863220

C₂

yes

maximal, complemented

#16863217

C₂

yes

maximal, characteristic, complemented

#16863206

C₂

yes

maximal

#443×C₆₇

C₂

yes

maximal, characteristic

#14524035

—

#14523983

C₂²

yes

—

#14523977

—

#14523975

C₂²

yes

—

#135×C₆₇

C₂²

yes

characteristic

#131×C₆₇

—

#108×C₆₇

C₂²

yes

characteristic

#104×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

Q₁₆×C₆₇

Q₈

yes

characteristic

SD₁₆×C₆₇

—

D₈×C₆₇

Q₈

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

SD₁₆

yes

characteristic

C₈×C₆₇

SD₁₆

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#499

—

#443

D₆₇

yes

—

—

—

C₂×D₆₇

yes

characteristic

#131

—

#125

—

134

—

M₃₂

—

#108

C₂×D₆₇

yes

characteristic

#107

—

134

—

#106

—

#104

—

C₄×C₈

C₂×D₆₇

yes

characteristic

#41

—

Q₁₆

Dic₁₃₄

yes

characteristic

SD₁₆

—

D₈

Dic₁₃₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄²

C₂²⋊D₆₇

yes

characteristic

C₁₆

—

134

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

C₂×Dic₁₃₄

yes

characteristic

C₈

#11761628

yes

characteristic

C₂²

—

C₂²

#14524068

yes

characteristic

C₄

—

C₄

—

C₄

—

536

—

C₄

yes

C₄

yes

C₂

—

C₂

—

100

C₂

#16863686

yes

characteristic

	subgroup	G/H
center	C₂	#16863686
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#443×C₆₇	C₂
socle	C₂×C₆₇	#476
hypercenter	#443	D₆₇
hypocenter	C₆₇	#2231
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#16863686

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#443×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
1. #16863798	C₂	semidirect
2. #16863220	C₂	semidirect
3. #16863217	C₂	semidirect
51. C₆₇	#2231	semidirect

H ≅ G/N

type

1. #16863798

C₂

semidirect

2. #16863220

C₂

semidirect

3. #16863217

C₂

semidirect

51. C₆₇

#2231

semidirect

Subgroups — #22562385

Conjugacy classes of subgroups

Complements