Group #21447987 — subgroups

Subgroups — #21447987

Order 7808.

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	#16839880	C₂	1	1	yes	maximal, characteristic
2	#16839584	C₂	1	2	yes	maximal, complemented
3	#16839583	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839571	C₂	1	2	yes	maximal
5	C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	#14519799	C₂²	1	1	yes	characteristic
7	#14519783	—	2	1	no	—
8	#14519717	C₂²	1	2	yes	—
9	#14519713	C₂²	1	1	yes	characteristic
10	#14519705	—	2	1	no	—
11	C₄⋊Q₈×C₆₁	—	2	1	no	—
12	#106×C₆₁	—	2	1	no	—
13	#106×C₆₁	C₂²	1	2	yes	—
14	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₆₁	—	4	1	no	—
16	#11758222	—	2	1	no	—
17	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
18	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₆₁	—	2	2	no	—
20	C₄⋊C₄×C₆₁	—	4	1	no	—
21	C₄²×C₆₁	D₄	1	1	yes	characteristic
22	#11758194	—	2	1	no	—
23	Dic₂₄₄	Q₈	1	1	yes	characteristic, complemented
24	D₄₈₈	Q₈	1	1	yes	characteristic, complemented
25	#11758190	—	2	1	no	—
26	#11758189	—	4	1	no	—
27	C₈×D₆₁	—	4	1	no	—
28	#11758185	—	2	2	no	—
29	Q₈×C₆₁	—	8	2	no	—
30	C₂×C₄×C₆₁	—	2	1	no	—
31	C₂×C₄×C₆₁	—	4	2	no	—
32	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂²⋊D₆₁	—	4	1	no	—
34	D₂₄₄	—	2	1	no	—
35	C₄×D₆₁	—	4	1	no	—
36	Dic₁₂₂	—	2	1	no	—
37	C₈×C₆₁	SD₁₆	1	1	yes	characteristic
38	C₈×C₆₁	SD₁₆	1	1	yes	characteristic
39	C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₆₁	#41	1	1	yes	characteristic
41	#92418	—	4	1	no	—
42	C₂²×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₂×D₆₁	—	4	1	no	—
44	C₄×C₆₁	—	2	2	no	—
45	C₄×C₆₁	—	8	2	no	—
46	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₆₁	—	4	1	no	—
49	#2230	—	61	1	no	maximal, Sylow
50	C₂×C₆₁	—	2	1	no	—
51	C₂×C₆₁	#476	1	1	yes	characteristic
52	D₆₁	—	8	1	no	—
53	#498	D₆₁	1	1	yes	characteristic, complemented
54	#443	—	61	1	no	—
55	#365	—	61	2	no	—
56	#364	—	61	1	no	—
57	#360	—	61	2	no	—
58	C₆₁	#2230	1	1	yes	characteristic, complemented, Sylow
59	#135	—	61	1	no	—
60	#131	—	122	1	no	—
61	C₄⋊Q₈	—	2	1	no	—
62	M₃₂	—	61	2	no	—
63	#108	—	61	1	no	—
64	#106	—	2	1	no	—
65	#106	C₂×D₆₁	1	2	yes	—
66	#104	—	122	1	no	—
67	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
68	#41	—	122	1	no	—
69	C₂×Q₈	—	4	1	no	—
70	Q₁₆	—	61	1	no	—
71	SD₁₆	—	122	1	no	—
72	D₈	—	61	1	no	—
73	M₁₆	—	122	1	no	—
74	M₁₆	—	244	1	no	—
75	C₂×C₈	—	244	1	no	—
76	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
81	C₁₆	—	122	2	no	—
82	Q₈	—	8	2	no	—
83	Q₈	—	122	1	no	—
84	D₄	—	122	1	no	—
85	D₄	—	244	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	244	1	no	—
89	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
90	C₈	—	244	1	no	—
91	C₈	#11758201	1	1	yes	characteristic
92	C₈	#11758201	1	1	yes	characteristic
93	C₈	Q₈×D₆₁	1	1	yes	characteristic
94	C₈	#11758226	1	1	yes	characteristic
95	C₂²	—	244	1	no	—
96	C₂²	#14519803	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	8	2	no	—
99	C₄	—	244	1	no	—
100	C₄	#14519753	1	1	yes	characteristic
101	C₄	C₂×#11758201	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	488	1	no	—
104	C₂	#16840049	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839880

C₂

yes

maximal, characteristic

#16839584

C₂

yes

maximal, complemented

#16839583

C₂

yes

maximal, characteristic, complemented

#16839571

C₂

yes

maximal

C₆₁×#498

C₂

yes

maximal, characteristic, complemented

#14519799

C₂²

yes

characteristic

#14519783

—

#14519717

C₂²

yes

—

#14519713

C₂²

yes

characteristic

#14519705

—

C₄⋊Q₈×C₆₁

—

#106×C₆₁

—

#106×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

#11758222

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₄

yes

characteristic

#11758194

—

Dic₂₄₄

Q₈

yes

characteristic, complemented

D₄₈₈

Q₈

yes

characteristic, complemented

#11758190

—

#11758189

—

C₈×D₆₁

—

#11758185

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

SD₁₆

yes

characteristic

C₈×C₆₁

SD₁₆

yes

characteristic

C₈×C₆₁

C₂×Q₈

yes

C₈×C₆₁

yes

—

C₂²×C₆₁

AGL(1,ℤ/8)

yes

characteristic

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

—

#2230

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#476

yes

characteristic

D₆₁

—

#498

D₆₁

yes

characteristic, complemented

—

—

—

—

C₆₁

#2230

yes

characteristic, complemented, Sylow

#135

—

#131

—

122

—

C₄⋊Q₈

—

M₃₂

—

#108

—

#106

—

#106

C₂×D₆₁

yes

—

#104

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

#41

—

122

—

C₂×Q₈

—

Q₁₆

—

SD₁₆

—

122

—

D₈

—

M₁₆

—

122

—

M₁₆

—

244

—

C₂×C₈

—

244

—

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₆₁

yes

characteristic

C₁₆

—

122

—

Q₈

—

Q₈

—

122

—

D₄

—

122

—

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

244

—

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

C₈

—

244

—

C₈

yes

C₈

yes

C₈

Q₈×D₆₁

yes

characteristic

C₈

#11758226

yes

characteristic

C₂²

—

244

—

C₂²

#14519803

yes

characteristic

C₄

—

C₄

—

C₄

—

244

—

100

C₄

#14519753

yes

characteristic

101

C₄

C₂×#11758201

yes

characteristic

102

C₂

—

103

C₂

—

488

—

104

C₂

#16840049

yes

characteristic

	subgroup	G/H
center	C₂	#16840049
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×#498	C₂
socle	C₂×C₆₁	#476
hypercenter	#498	D₆₁
hypocenter	C₆₁	#2230
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

subgroup

G/H

center

C₂

#16840049

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×#498

C₂

C₂×C₆₁

D₆₁

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

N	H ≅ G/N	type
2. #16839584	C₂	semidirect
3. #16839583	C₂	semidirect
5. C₆₁×#498	C₂	semidirect
23. Dic₂₄₄	Q₈	semidirect
24. D₄₈₈	Q₈	semidirect
53. #498	D₆₁	semidirect
58. C₆₁	#2230	semidirect

H ≅ G/N

type

2. #16839584

C₂

semidirect

3. #16839583

C₂

semidirect

5. C₆₁×#498

C₂

semidirect

23. Dic₂₄₄

Q₈

semidirect

24. D₄₈₈

Q₈

semidirect

53. #498

D₆₁

semidirect

58. C₆₁

#2230

semidirect

Subgroups — #21447987

Conjugacy classes of subgroups

Complements