Group #21444578 — subgroups

Subgroups — #21444578

Order 7808.

184 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16839786	C₂	1	2	yes	maximal
2	#16839515	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758194	—	2	2	no	—
5	C₄⋊#92418	—	2	2	no	—
6	C₄⋊#92418	C₂²	1	4	yes	—
7	C₂²×C₈×C₆₁	C₄	1	2	yes	—
8	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
11	C₄×C₈×C₆₁	C₂²	1	2	yes	—
12	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₆₁	—	2	4	no	—
16	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₆₁	D₄	1	1	yes	characteristic
20	C₄²×C₆₁	Q₈	1	1	yes	characteristic
21	C₄²×C₆₁	C₂³	1	1	yes	characteristic
22	#11758194	—	4	4	no	—
23	C₂×#92418	—	2	4	no	—
24	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₆₁	—	2	2	no	—
26	C₂×C₄×C₆₁	—	2	2	no	—
27	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	D₈	1	2	yes	—
31	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
32	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
35	C₈×C₆₁	—	2	2	no	—
36	C₈×C₆₁	M₁₆	1	4	yes	complemented
37	#92418	—	4	4	no	—
38	C₂²×C₆₁	—	2	1	no	—
39	C₂²×C₆₁	—	2	1	no	—
40	C₂²×C₆₁	#107	1	2	yes	—
41	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₆₁	—	2	1	no	—
43	C₄×C₆₁	—	2	1	no	—
44	C₄×C₆₁	#107	1	2	yes	—
45	C₄×C₆₁	#108	1	2	yes	—
46	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
47	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
48	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
49	#1567	—	61	1	no	maximal, Sylow
50	C₂×C₆₁	—	2	2	no	—
51	C₂×C₆₁	#423	1	1	yes	characteristic
52	C₂×C₆₁	C₂×#107	1	1	yes	characteristic
53	C₂×C₆₁	C₂×#108	1	1	yes	characteristic
54	#423	—	61	2	no	—
55	C₂×C₄×C₈	D₆₁	1	1	yes	characteristic
56	#335	—	61	4	no	—
57	C₆₁	#1567	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₁₆	—	122	2	no	—
59	C₂²×C₈	Dic₆₁	1	2	yes	—
60	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
61	C₄⋊C₈	—	61	4	no	—
62	C₄⋊C₈	—	122	2	no	—
63	C₄×C₈	Dic₆₁	1	1	yes	characteristic
64	C₄×C₈	Dic₆₁	1	1	yes	characteristic
65	C₄×C₈	C₂×D₆₁	1	2	yes	—
66	C₂²×C₄	Dic₁₂₂	1	1	yes	characteristic
67	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
68	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
69	M₁₆	—	244	4	no	—
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	122	4	no	—
72	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
73	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
74	C₄²	Dic₁₂₂	1	1	yes	characteristic
75	C₄²	D₂₄₄	1	1	yes	characteristic
76	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
77	C₄²	C₂²×D₆₁	1	1	yes	characteristic
78	C₂³	C₄⋊Dic₆₁	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	D₄₈₈	1	2	yes	—
82	C₂×C₄	Dic₂₄₄	1	2	yes	—
83	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
84	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
85	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
86	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
87	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
88	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	244	4	no	—
91	C₈	#11758194	1	4	yes	complemented
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	#14519711	1	2	yes	—
95	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	#14519711	1	2	yes	—
99	C₄	#14519713	1	2	yes	—
100	C₄	C₂×#11758194	1	2	yes	—
101	C₄	C₂×D₄₈₈	1	1	yes	characteristic
102	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
103	C₂	—	2	2	no	—
104	C₂	#16839786	1	1	yes	characteristic
105	C₂	C₂×#14519711	1	1	yes	characteristic
106	C₂	C₂×#14519713	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839786

C₂

yes

maximal

#16839515

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

—

C₄⋊#92418

—

C₄⋊#92418

C₂²

yes

—

C₂²×C₈×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₄

yes

characteristic

C₄×C₈×C₆₁

C₄

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

Q₈

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

Q₈

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

M₁₆

yes

complemented

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#107

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#107

yes

—

C₄×C₆₁

#108

yes

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

characteristic

#1567

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄×C₈

D₆₁

yes

characteristic

#335

—

C₆₁

#1567

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

Dic₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄×C₈

Dic₆₁

yes

characteristic

C₄×C₈

Dic₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

—

C₂²×C₄

Dic₁₂₂

yes

characteristic

C₂²×C₄

D₂₄₄

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₄²

Dic₁₂₂

yes

characteristic

C₄²

D₂₄₄

yes

characteristic

C₄²

C₂×Dic₆₁

yes

characteristic

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄₈₈

yes

—

C₂×C₄

Dic₂₄₄

yes

—

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

C₈

—

C₈

—

244

—

C₈

#11758194

yes

complemented

C₂²

—

C₂²

—

C₂²

#14519711

yes

—

C₂²

C₂×C₄⋊Dic₆₁

yes

characteristic

C₄

—

C₄

—

C₄

#14519711

yes

—

C₄

#14519713

yes

—

100

C₄

C₂×#11758194

yes

—

101

C₄

C₂×D₄₈₈

yes

characteristic

102

C₄

C₂×Dic₂₄₄

yes

characteristic

103

C₂

—

104

C₂

#16839786

yes

characteristic

105

C₂

C₂×#14519711

yes

characteristic

106

C₂

C₂×#14519713

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₂×C₄×C₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₆₁
hypocenter	C₆₁	#1567
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₂×C₄×C₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂×C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₆₁

hypocenter

C₆₁

#1567

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16839515	C₂	semidirect
36. C₈×C₆₁	M₁₆	semidirect
57. C₆₁	#1567	semidirect
91. C₈	#11758194	semidirect

H ≅ G/N

type

2. #16839515

C₂

semidirect

36. C₈×C₆₁

M₁₆

semidirect

57. C₆₁

#1567

semidirect

91. C₈

#11758194

semidirect

Subgroups — #21444578

Conjugacy classes of subgroups

Complements