Group #21447870 — subgroups

Subgroups — #21447870

Order 7808.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₂×Dic₄₈₈	C₂	1	2	yes	maximal
2	#16840142	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
4	#16839567	C₂	1	2	yes	maximal
5	C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₄₄	—	2	1	no	—
7	C₂×Dic₂₄₄	C₂²	1	2	yes	—
8	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
9	#14519777	—	2	1	no	—
10	C₄×Dic₁₂₂	—	2	1	no	—
11	Dic₄₈₈	—	2	4	no	—
12	#14519711	C₂²	1	1	yes	characteristic
13	#14519702	—	2	1	no	—
14	C₂×C₁₆×C₆₁	C₂²	1	2	yes	—
15	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₂₂	—	2	1	no	—
17	C₂×Dic₁₂₂	—	2	2	no	—
18	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
20	C₄²×C₆₁	D₄	1	1	yes	characteristic
21	C₄⋊Dic₆₁	—	2	1	no	—
22	C₄⋊Dic₆₁	—	4	1	no	—
23	C₄⋊Dic₆₁	—	4	1	no	—
24	#11758196	—	4	1	no	—
25	C₄×Dic₆₁	—	4	1	no	—
26	Dic₂₄₄	—	2	2	no	—
27	Dic₂₄₄	—	2	1	no	—
28	Dic₂₄₄	—	2	2	no	—
29	Dic₂₄₄	—	4	2	no	—
30	Dic₂₄₄	D₄	1	2	yes	—
31	C₁₆×C₆₁	—	2	2	no	—
32	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
33	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₆₁	—	4	1	no	—
36	C₂×Dic₆₁	—	4	1	no	—
37	C₂×Dic₆₁	—	4	2	no	—
38	Dic₁₂₂	—	2	2	no	—
39	Dic₁₂₂	—	4	1	no	—
40	Dic₁₂₂	—	4	2	no	—
41	Dic₁₂₂	—	4	2	no	—
42	C₈×C₆₁	—	2	1	no	—
43	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₆₁	#41	1	1	yes	characteristic
45	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
46	C₄×C₆₁	—	2	1	no	—
47	C₄×C₆₁	Q₃₂	1	2	yes	complemented
48	C₄×C₆₁	#121	1	1	yes	characteristic
49	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₆₁	—	4	2	no	—
51	Dic₆₁	—	8	2	no	—
52	Dic₆₁	—	8	1	no	—
53	#2209	—	61	1	no	maximal, Sylow
54	C₂×C₆₁	#459	1	1	yes	characteristic
55	C₂×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
56	C₂×C₆₁	#510	1	1	yes	characteristic
57	C₂×Q₃₂	—	61	2	no	—
58	#494	—	61	1	no	—
59	C₄×Q₁₆	—	61	1	no	—
60	C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
61	#358	—	61	2	no	—
62	C₆₁	#2209	1	1	yes	characteristic, complemented, Sylow
63	C₂×Q₁₆	—	61	1	no	—
64	C₂×Q₁₆	—	61	2	no	—
65	C₂×Q₁₆	—	122	1	no	—
66	C₄⋊Q₈	—	122	1	no	—
67	C₄×Q₈	—	122	1	no	—
68	Q₃₂	—	122	4	no	—
69	C₂×C₁₆	C₂×D₆₁	1	2	yes	—
70	#107	—	61	1	no	—
71	#103	—	122	1	no	—
72	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
73	C₂×Q₈	—	122	1	no	—
74	C₂×Q₈	—	122	2	no	—
75	Q₁₆	—	61	2	no	—
76	Q₁₆	—	122	2	no	—
77	Q₁₆	—	122	1	no	—
78	Q₁₆	—	122	2	no	—
79	Q₁₆	—	244	2	no	—
80	C₂×C₈	D₂₄₄	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
82	C₄⋊C₄	—	122	1	no	—
83	C₄⋊C₄	—	244	1	no	—
84	C₄⋊C₄	—	244	1	no	—
85	C₄⋊C₄	—	244	1	no	—
86	C₄²	—	244	1	no	—
87	C₄²	D₂₄₄	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	122	2	no	—
90	Q₈	—	244	2	no	—
91	Q₈	—	244	2	no	—
92	Q₈	—	244	1	no	—
93	C₂×C₄	—	244	1	no	—
94	C₂×C₄	—	244	2	no	—
95	C₂×C₄	—	244	1	no	—
96	C₂×C₄	D₄₈₈	1	1	yes	characteristic
97	C₂×C₄	D₄₈₈	1	1	yes	characteristic
98	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₆₁	1	1	yes	characteristic
101	C₈	#11758226	1	1	yes	characteristic
102	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	488	1	no	—
105	C₄	—	488	2	no	—
106	C₄	—	244	2	no	—
107	C₄	Dic₄₈₈	1	2	yes	complemented
108	C₄	#14519752	1	1	yes	characteristic
109	C₄	#14519805	1	1	yes	characteristic
110	C₂	#16839958	1	1	yes	characteristic
111	C₂	C₂×Dic₄₈₈	1	1	yes	characteristic
112	C₂	#16840205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₈₈

C₂

yes

maximal

#16840142

C₂

yes

maximal, characteristic

C₄×Dic₂₄₄

C₂

yes

maximal, characteristic

#16839567

C₂

yes

maximal

C₆₁×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

C₂²

yes

—

C₂×Dic₂₄₄

C₂²

yes

characteristic

#14519777

—

C₄×Dic₁₂₂

—

Dic₄₈₈

—

#14519711

C₂²

yes

characteristic

#14519702

—

C₂×C₁₆×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

D₄

yes

—

C₁₆×C₆₁

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×D₄

yes

characteristic

C₈×C₆₁

#41

yes

characteristic

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

Q₃₂

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×Q₃₂

yes

characteristic

C₂×C₆₁

#510

yes

characteristic

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

D₆₁

yes

characteristic

#358

—

C₆₁

#2209

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

122

—

C₄×Q₈

—

122

—

Q₃₂

—

122

—

C₂×C₁₆

C₂×D₆₁

yes

—

#107

—

#103

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

Q₁₆

—

Q₁₆

—

122

—

Q₁₆

—

122

—

Q₁₆

—

122

—

Q₁₆

—

244

—

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

—

244

—

C₄²

D₂₄₄

yes

characteristic

C₁₆

—

Q₈

—

122

—

Q₈

—

244

—

Q₈

—

244

—

Q₈

—

244

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

D₄₈₈

yes

characteristic

C₂×C₄

D₄₈₈

yes

characteristic

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₈

—

100

C₈

D₄×D₆₁

yes

characteristic

101

C₈

#11758226

yes

characteristic

102

C₂²

C₂×D₄₈₈

yes

characteristic

103

C₄

—

104

C₄

—

488

—

105

C₄

—

488

—

106

C₄

—

244

—

107

C₄

Dic₄₈₈

yes

complemented

108

C₄

#14519752

yes

characteristic

109

C₄

#14519805

yes

characteristic

110

C₂

#16839958

yes

characteristic

111

C₂

C₂×Dic₄₈₈

yes

characteristic

112

C₂

#16840205

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×C₄⋊C₁₆	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	#2209
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₈₈

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×C₄⋊C₁₆

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₄⋊C₁₆

D₆₁

hypocenter

C₆₁

#2209

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₆₁	Q₃₂	semidirect
62. C₆₁	#2209	semidirect
107. C₄	Dic₄₈₈	semidirect

H ≅ G/N

type

47. C₄×C₆₁

Q₃₂

semidirect

62. C₆₁

#2209

semidirect

107. C₄

Dic₄₈₈

semidirect

Subgroups — #21447870

Conjugacy classes of subgroups

Complements