Group #21445189 — subgroups

Subgroups — #21445189

Order 7808.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16840142	C₂	1	1	yes	maximal, characteristic
2	#16840047	C₂	1	2	yes	maximal
3	#16839515	C₂	1	1	yes	maximal, characteristic
4	#16839509	C₂	1	2	yes	maximal
5	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₄₄	—	2	1	no	—
7	C₂×Dic₂₄₄	—	2	1	no	—
8	#14519777	C₂²	1	2	yes	—
9	#14519708	—	2	2	no	—
10	C₄⋊#92418	C₂²	1	2	yes	—
11	#14519701	—	2	2	no	—
12	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₆₁	—	2	1	no	—
14	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₁	—	2	1	no	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
18	C₂×Dic₁₂₂	—	2	2	no	—
19	C₂×C₈×C₆₁	—	2	1	no	—
20	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₆₁	—	2	1	no	—
23	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
24	C₄²×C₆₁	—	2	1	no	—
25	C₄²×C₆₁	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₆₁	—	2	2	no	—
27	C₄⋊Dic₆₁	—	2	2	no	—
28	C₂×#92418	—	2	2	no	—
29	Dic₂₄₄	—	4	2	no	—
30	Dic₂₄₄	—	4	2	no	—
31	Q₈×C₆₁	—	2	1	no	—
32	Q₈×C₆₁	Q₁₆	1	2	yes	complemented
33	C₂×C₄×C₆₁	—	2	1	no	—
34	C₂×C₄×C₆₁	—	2	1	no	—
35	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
38	C₂×Dic₆₁	—	4	2	no	—
39	Dic₁₂₂	—	4	2	no	—
40	Dic₁₂₂	—	4	2	no	—
41	C₈×C₆₁	—	2	1	no	—
42	C₈×C₆₁	—	4	1	no	—
43	C₈×C₆₁	SD₁₆	1	2	yes	—
44	#92418	—	4	2	no	—
45	C₂²×C₆₁	#121	1	1	yes	characteristic
46	C₄×C₆₁	—	2	2	no	—
47	C₄×C₆₁	—	2	1	no	—
48	C₄×C₆₁	—	4	1	no	—
49	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₆₁	#135	1	1	yes	characteristic
52	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₆₁	—	8	2	no	—
54	#1670	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	#460	1	1	yes	characteristic
56	C₂×C₆₁	#467	1	1	yes	characteristic
57	C₂×C₆₁	#473	1	1	yes	characteristic
58	#494	—	61	1	no	—
59	#475	—	61	2	no	—
60	C₈×Q₈	D₆₁	1	1	yes	characteristic
61	#335	—	61	1	no	—
62	#332	—	61	2	no	—
63	C₆₁	#1670	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	122	1	no	—
65	C₂×Q₁₆	—	122	1	no	—
66	C₄⋊Q₈	—	61	2	no	—
67	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
68	#106	—	122	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	61	2	no	—
71	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
72	#103	—	122	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
75	C₂×Q₈	—	122	2	no	—
76	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
77	Q₁₆	—	244	2	no	—
78	Q₁₆	—	244	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	122	2	no	—
81	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	122	2	no	—
85	C₄⋊C₄	—	122	2	no	—
86	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₆₁	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	244	2	no	—
91	Q₈	—	244	2	no	—
92	Q₈	Dic₂₄₄	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	244	2	no	—
96	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
97	C₂×C₄	#11758222	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	244	2	no	—
102	C₈	#11758201	1	2	yes	—
103	C₂²	#14519751	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	488	2	no	—
108	C₄	C₂×#11758201	1	1	yes	characteristic
109	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
110	C₄	#14519799	1	1	yes	characteristic
111	C₄	#14519802	1	1	yes	characteristic
112	C₂	#16839962	1	1	yes	characteristic
113	C₂	#16840003	1	1	yes	characteristic
114	C₂	#16840037	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840142

C₂

yes

maximal, characteristic

#16840047

C₂

yes

maximal

#16839515

C₂

yes

maximal, characteristic

#16839509

C₂

yes

maximal

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

C₂²

yes

—

#14519708

—

C₄⋊#92418

C₂²

yes

—

#14519701

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

—

Q₈×C₆₁

—

Q₈×C₆₁

Q₁₆

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

SD₁₆

yes

—

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

C₈×Q₈

D₆₁

yes

characteristic

#335

—

#332

—

C₆₁

#1670

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

#106

—

122

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

122

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

C₂×Q₈

D₂₄₄

yes

characteristic

Q₁₆

—

244

—

Q₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

D₂₄₄

yes

characteristic

C₄²

—

C₄²

C₂²×D₆₁

yes

characteristic

Q₈

—

Q₈

—

244

—

Q₈

—

244

—

Q₈

Dic₂₄₄

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

244

—

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

#11758222

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

244

—

102

C₈

#11758201

yes

—

103

C₂²

#14519751

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

488

—

108

C₄

C₂×#11758201

yes

characteristic

109

C₄

C₂×Dic₂₄₄

yes

characteristic

110

C₄

#14519799

yes

characteristic

111

C₄

#14519802

yes

characteristic

112

C₂

#16839962

yes

characteristic

113

C₂

#16840003

yes

characteristic

114

C₂

#16840037

yes

characteristic

	subgroup	G/H
center	C₂²	#14519751
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	#121
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#1670
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14519751

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

#121

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

#1670

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
32. Q₈×C₆₁	Q₁₆	semidirect
63. C₆₁	#1670	semidirect
92. Q₈	Dic₂₄₄	semidirect

H ≅ G/N

type

32. Q₈×C₆₁

Q₁₆

semidirect

63. C₆₁

#1670

semidirect

92. Q₈

Dic₂₄₄

semidirect

Subgroups — #21445189

Conjugacy classes of subgroups

Complements