Group #22562377 — subgroups

Subgroups — #22562377

Order 8576.

126 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16863514	C₂	1	1	yes	maximal, characteristic
2	#16863218	C₂	1	2	yes	maximal, complemented
3	#16863217	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863205	C₂	1	2	yes	maximal
5	#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14524065	C₂²	1	1	yes	characteristic
7	#14524049	—	2	1	no	—
8	#14523983	C₂²	1	2	yes	—
9	#14523979	C₂²	1	1	yes	characteristic
10	#14523971	—	2	1	no	—
11	C₄⋊Q₈×C₆₇	—	2	1	no	—
12	#106×C₆₇	—	2	1	no	—
13	#106×C₆₇	C₂²	1	2	yes	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#11761628	—	2	1	no	—
16	Dic₂₆₈	Q₈	1	1	yes	characteristic, complemented
17	#11761617	—	2	1	no	—
18	D₅₃₆	Q₈	1	1	yes	characteristic, complemented
19	#11761612	—	2	1	no	—
20	#11761611	—	4	1	no	—
21	C₈×D₆₇	—	4	1	no	—
22	C₆₇⋊C₁₆	—	2	2	no	—
23	C₂×Q₈×C₆₇	—	4	1	no	—
24	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₆₇	—	2	2	no	—
27	C₄⋊C₄×C₆₇	—	4	1	no	—
28	C₄²×C₆₇	D₄	1	1	yes	characteristic
29	Q₈×C₆₇	—	8	2	no	—
30	C₂×C₄×C₆₇	—	2	1	no	—
31	C₂×C₄×C₆₇	—	4	2	no	—
32	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
33	C₂²⋊D₆₇	—	4	1	no	—
34	D₂₆₈	—	2	1	no	—
35	C₄×D₆₇	—	4	1	no	—
36	Dic₁₃₄	—	2	1	no	—
37	C₈×C₆₇	SD₁₆	1	1	yes	characteristic
38	C₈×C₆₇	SD₁₆	1	1	yes	characteristic
39	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₆₇	#41	1	1	yes	characteristic
41	C₆₇⋊C₈	—	4	1	no	—
42	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₂×D₆₇	—	4	1	no	—
44	C₄×C₆₇	—	2	2	no	—
45	C₄×C₆₇	—	8	2	no	—
46	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₆₇	—	4	1	no	—
49	C₂×C₆₇	—	2	1	no	—
50	C₂×C₆₇	#476	1	1	yes	characteristic
51	D₆₇	—	8	1	no	—
52	#2230	—	67	1	no	maximal, Sylow
53	C₆₇	#2230	1	1	yes	characteristic, complemented, Sylow
54	#498	D₆₇	1	1	yes	characteristic, complemented
55	#443	—	67	1	no	—
56	#365	—	67	2	no	—
57	#364	—	67	1	no	—
58	#360	—	67	2	no	—
59	#135	—	67	1	no	—
60	#131	—	134	1	no	—
61	C₄⋊Q₈	—	2	1	no	—
62	M₃₂	—	67	2	no	—
63	#108	—	67	1	no	—
64	#106	—	2	1	no	—
65	#106	C₂×D₆₇	1	2	yes	—
66	#104	—	134	1	no	—
67	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
68	#41	—	134	1	no	—
69	C₂×Q₈	—	4	1	no	—
70	Q₁₆	—	67	1	no	—
71	SD₁₆	—	134	1	no	—
72	D₈	—	67	1	no	—
73	M₁₆	—	268	1	no	—
74	M₁₆	—	134	1	no	—
75	C₂×C₈	—	268	1	no	—
76	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
81	C₁₆	—	134	2	no	—
82	Q₈	—	8	2	no	—
83	Q₈	—	134	1	no	—
84	D₄	—	268	1	no	—
85	D₄	—	134	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	268	1	no	—
89	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
90	C₈	—	268	1	no	—
91	C₈	#11761615	1	1	yes	characteristic
92	C₈	#11761615	1	1	yes	characteristic
93	C₈	Q₈×D₆₇	1	1	yes	characteristic
94	C₈	#11761629	1	1	yes	characteristic
95	C₂²	—	268	1	no	—
96	C₂²	#14524069	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	8	2	no	—
99	C₄	—	268	1	no	—
100	C₄	#14524019	1	1	yes	characteristic
101	C₄	C₂×#11761615	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	536	1	no	—
104	C₂	#16863683	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863514

C₂

yes

maximal, characteristic

#16863218

C₂

yes

maximal, complemented

#16863217

C₂

yes

maximal, characteristic, complemented

#16863205

C₂

yes

maximal

#498×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524065

C₂²

yes

characteristic

#14524049

—

#14523983

C₂²

yes

—

#14523979

C₂²

yes

characteristic

#14523971

—

C₄⋊Q₈×C₆₇

—

#106×C₆₇

—

#106×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

#11761628

—

Dic₂₆₈

Q₈

yes

characteristic, complemented

#11761617

—

D₅₃₆

Q₈

yes

characteristic, complemented

#11761612

—

#11761611

—

C₈×D₆₇

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²⋊D₆₇

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

SD₁₆

yes

characteristic

C₈×C₆₇

SD₁₆

yes

characteristic

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#476

yes

characteristic

D₆₇

—

#2230

—

maximal, Sylow

C₆₇

#2230

yes

characteristic, complemented, Sylow

#498

D₆₇

yes

characteristic, complemented

—

—

—

—

—

—

134

—

C₄⋊Q₈

—

M₃₂

—

#108

—

#106

—

#106

C₂×D₆₇

yes

—

#104

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

#41

—

134

—

C₂×Q₈

—

Q₁₆

—

SD₁₆

—

134

—

D₈

—

M₁₆

—

268

—

M₁₆

—

134

—

C₂×C₈

—

268

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₆₇

yes

characteristic

C₁₆

—

134

—

Q₈

—

Q₈

—

134

—

D₄

—

268

—

D₄

—

134

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

C₈

—

268

—

C₈

yes

C₈

yes

C₈

Q₈×D₆₇

yes

characteristic

C₈

#11761629

yes

characteristic

C₂²

—

268

—

C₂²

#14524069

yes

characteristic

C₄

—

C₄

—

C₄

—

268

—

100

C₄

#14524019

yes

characteristic

101

C₄

C₂×#11761615

yes

characteristic

102

C₂

—

103

C₂

—

536

—

104

C₂

#16863683

yes

characteristic

	subgroup	G/H
center	C₂	#16863683
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#498×C₆₇	C₂
socle	C₂×C₆₇	#476
hypercenter	#498	D₆₇
hypocenter	C₆₇	#2230
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

subgroup

G/H

center

C₂

#16863683

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#498×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

N	H ≅ G/N	type
2. #16863218	C₂	semidirect
3. #16863217	C₂	semidirect
5. #498×C₆₇	C₂	semidirect
16. Dic₂₆₈	Q₈	semidirect
18. D₅₃₆	Q₈	semidirect
53. C₆₇	#2230	semidirect
54. #498	D₆₇	semidirect

H ≅ G/N

type

2. #16863218

C₂

semidirect

3. #16863217

C₂

semidirect

5. #498×C₆₇

C₂

semidirect

16. Dic₂₆₈

Q₈

semidirect

18. D₅₃₆

Q₈

semidirect

53. C₆₇

#2230

semidirect

54. #498

D₆₇

semidirect

Subgroups — #22562377

Conjugacy classes of subgroups

Complements