Group #21444644 — subgroups

Subgroups — #21444644

Order 7808.

192 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic
2	C₈×D₂₄₄	C₂	1	2	yes	maximal, complemented
3	#16839513	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839491	C₂	1	2	yes	maximal, complemented
5	C₆₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₈₈	C₄	1	1	yes	characteristic
7	C₂×C₈×D₆₁	—	2	2	no	—
8	C₄×D₂₄₄	C₂²	1	2	yes	—
9	#14519711	C₄	1	1	yes	characteristic
10	C₄⋊#92418	C₂²	1	2	yes	—
11	#14519699	C₄	1	2	yes	—
12	#14519689	—	2	2	no	—
13	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₆₁	—	2	2	no	—
18	C₂×C₈×C₆₁	—	2	1	no	—
19	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	D₄	1	2	yes	—
22	C₄²×C₆₁	C₂³	1	1	yes	characteristic
23	#11758198	—	2	2	no	—
24	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
25	C₂×#92418	—	2	2	no	—
26	D₄₈₈	C₈	1	4	yes	complemented
27	C₈×D₆₁	—	4	4	no	—
28	C₂²×D₆₁	—	2	2	no	—
29	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
32	C₂×Dic₆₁	—	2	2	no	—
33	D₂₄₄	—	2	2	no	—
34	D₂₄₄	C₂×C₈	1	4	yes	complemented
35	C₄×D₆₁	—	4	2	no	—
36	C₄×D₆₁	—	4	2	no	—
37	C₈×C₆₁	—	2	1	no	—
38	C₈×C₆₁	—	2	2	no	—
39	C₈×C₆₁	—	4	1	no	—
40	C₈×C₆₁	C₂×C₈	1	2	yes	complemented
41	#92418	—	4	2	no	—
42	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
43	C₂×D₆₁	—	2	4	no	—
44	C₂×D₆₁	—	4	2	no	—
45	C₂×D₆₁	—	4	2	no	—
46	C₄×C₆₁	—	2	1	no	—
47	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₆₁	#131	1	1	yes	characteristic
49	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
50	Dic₆₁	—	4	2	no	—
51	#1580	—	61	1	no	maximal, Sylow
52	C₂×C₆₁	C₈×D₄	1	1	yes	characteristic
53	C₂×C₆₁	#442	1	1	yes	characteristic
54	C₂×C₆₁	#444	1	1	yes	characteristic
55	D₆₁	—	4	4	no	—
56	C₄×D₈	—	61	1	no	—
57	C₈×D₄	—	61	2	no	—
58	#334	—	61	1	no	—
59	D₄⋊C₈	—	61	2	no	—
60	#322	D₆₁	1	1	yes	characteristic, complemented
61	C₆₁	#1580	1	1	yes	characteristic, complemented, Sylow
62	C₂×D₈	—	61	1	no	—
63	C₂²×C₈	—	122	2	no	—
64	C₄×D₄	—	61	2	no	—
65	#107	—	61	1	no	—
66	C₄⋊C₈	—	61	2	no	—
67	#102	—	61	2	no	—
68	C₂²⋊C₈	—	122	2	no	—
69	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
70	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
71	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
72	C₂×D₄	—	61	2	no	—
73	C₂²×C₄	—	122	2	no	—
74	D₈	—	61	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	122	2	no	—
77	C₂×C₈	—	244	4	no	—
78	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
79	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
80	C₂×C₈	D₂₄₄	1	2	yes	—
81	C₄⋊C₄	—	61	2	no	—
82	C₂²⋊C₄	—	122	2	no	—
83	C₄²	C₂²×D₆₁	1	1	yes	characteristic
84	C₂³	—	122	2	no	—
85	D₄	—	61	4	no	—
86	D₄	—	122	2	no	—
87	C₂×C₄	—	122	2	no	—
88	C₂×C₄	—	244	2	no	—
89	C₂×C₄	—	244	2	no	—
90	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
91	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
92	C₂×C₄	#11758222	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	2	2	no	—
95	C₈	—	4	1	no	—
96	C₈	—	244	2	no	—
97	C₈	C₈×D₆₁	1	2	yes	complemented
98	C₂²	—	122	4	no	—
99	C₂²	—	244	2	no	—
100	C₂²	—	244	2	no	—
101	C₂²	C₄×D₂₄₄	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	244	2	no	—
104	C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
105	C₄	#14519783	1	1	yes	characteristic
106	C₄	#14519805	1	2	yes	—
107	C₂	—	244	4	no	—
108	C₂	C₈×D₂₄₄	1	1	yes	characteristic
109	C₂	#16839875	1	1	yes	characteristic
110	C₂	#16839885	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄₈₈

C₂

yes

maximal, characteristic

C₈×D₂₄₄

C₂

yes

maximal, complemented

#16839513

C₂

yes

maximal, characteristic, complemented

#16839491

C₂

yes

maximal, complemented

C₆₁×#322

C₂

yes

maximal, characteristic, complemented

C₂×D₄₈₈

C₄

yes

characteristic

C₂×C₈×D₆₁

—

C₄×D₂₄₄

C₂²

yes

—

#14519711

C₄

yes

characteristic

C₄⋊#92418

C₂²

yes

—

#14519699

C₄

yes

—

#14519689

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂×#92418

—

D₄₈₈

C₈

yes

complemented

C₈×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

C₂×C₈

yes

complemented

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₈

yes

complemented

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²×C₈

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

Dic₆₁

—

#1580

—

maximal, Sylow

C₂×C₆₁

C₈×D₄

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₄×D₈

—

C₈×D₄

—

#334

—

D₄⋊C₈

—

#322

D₆₁

yes

characteristic, complemented

C₆₁

#1580

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

122

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

122

—

D₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₄×D₆₁

yes

characteristic

C₂×C₈

C₄×D₆₁

yes

characteristic

C₂×C₈

D₂₄₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

122

—

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

—

122

—

D₄

—

D₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

#11758222

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

244

—

C₈

C₈×D₆₁

yes

complemented

C₂²

—

122

—

C₂²

—

244

—

100

C₂²

—

244

—

101

C₂²

C₄×D₂₄₄

yes

characteristic

102

C₄

—

103

C₄

—

244

—

104

C₄

C₂×C₈×D₆₁

yes

characteristic

105

C₄

#14519783

yes

characteristic

106

C₄

#14519805

yes

—

107

C₂

—

244

—

108

C₂

C₈×D₂₄₄

yes

characteristic

109

C₂

#16839875

yes

characteristic

110

C₂

#16839885

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#322	C₂
socle	C₂²×C₆₁	C₄×D₄
hypercenter	#322	D₆₁
hypocenter	C₆₁	#1580
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₂₄₄

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#322

C₂

socle

C₂²×C₆₁

C₄×D₄

D₆₁

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
2. C₈×D₂₄₄	C₂	semidirect
3. #16839513	C₂	semidirect
4. #16839491	C₂	semidirect
5. C₆₁×#322	C₂	semidirect
26. D₄₈₈	C₈	semidirect
34. D₂₄₄	C₂×C₈	semidirect
40. C₈×C₆₁	C₂×C₈	semidirect
60. #322	D₆₁	semidirect
61. C₆₁	#1580	semidirect
97. C₈	C₈×D₆₁	semidirect

H ≅ G/N

type

2. C₈×D₂₄₄

C₂

semidirect

3. #16839513

C₂

semidirect

4. #16839491

C₂

semidirect

5. C₆₁×#322

C₂

semidirect

26. D₄₈₈

C₈

semidirect

34. D₂₄₄

C₂×C₈

semidirect

40. C₈×C₆₁

C₂×C₈

semidirect

60. #322

D₆₁

semidirect

61. C₆₁

#1580

semidirect

97. C₈

C₈×D₆₁

semidirect

Subgroups — #21444644

Conjugacy classes of subgroups

Complements