Group #21444591 — subgroups

Subgroups — #21444591

Order 7808.

168 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 89 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	#16839815	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	#16839512	C₂	1	4	yes	maximal, complemented
4	C₆₁×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92418	—	2	1	no	—
6	C₄⋊#92418	—	2	1	no	—
7	C₄⋊#92418	C₄	1	2	yes	—
8	C₄⋊#92418	C₂²	1	2	yes	—
9	C₂²⋊#92418	C₄	1	2	yes	—
10	C₄×#92418	C₂²	1	2	yes	—
11	C₂×M₁₆×C₆₁	—	2	1	no	—
12	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₆₁	—	2	1	no	—
14	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	Q₈	1	1	yes	characteristic
18	M₁₆×C₆₁	—	4	2	no	—
19	C₂×C₈×C₆₁	—	2	2	no	—
20	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₆₁	D₄	1	1	yes	characteristic
22	C₄²×C₆₁	Q₈	1	1	yes	characteristic
23	C₄²×C₆₁	C₂³	1	1	yes	characteristic
24	C₂×#92418	—	2	2	no	—
25	C₂×#92418	—	4	2	no	—
26	C₂×#92418	C₂×C₄	1	4	yes	—
27	C₂³×C₆₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₆₁	—	2	2	no	—
29	C₂×C₄×C₆₁	—	4	1	no	—
30	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
34	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
35	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₆₁	—	4	2	no	—
39	#92418	—	2	4	no	—
40	#92418	—	4	2	no	—
41	C₂²×C₆₁	—	2	1	no	—
42	C₂²×C₆₁	—	2	1	no	—
43	C₂²×C₆₁	#106	1	2	yes	—
44	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₆₁	—	2	1	no	—
46	C₄×C₆₁	—	2	1	no	—
47	C₄×C₆₁	—	2	1	no	—
48	C₄×C₆₁	#106	1	2	yes	—
49	C₄×C₆₁	#131	1	2	yes	—
50	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
52	#1570	—	61	1	no	maximal, Sylow
53	C₂×C₆₁	—	2	2	no	—
54	C₂×C₆₁	#424	1	1	yes	characteristic
55	C₂×C₆₁	C₂×#106	1	1	yes	characteristic
56	C₂×C₆₁	#430	1	1	yes	characteristic
57	#431	—	61	1	no	—
58	#423	D₆₁	1	1	yes	characteristic
59	C₂×C₄⋊C₈	—	61	1	no	—
60	#334	—	61	4	no	—
61	C₆₁	#1570	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	2	1	no	—
63	C₂²×C₈	—	122	1	no	—
64	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	61	2	no	—
67	C₄⋊C₈	—	61	2	no	—
68	C₄⋊C₈	—	122	1	no	—
69	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
70	C₂²⋊C₈	—	61	2	no	—
71	C₄×C₈	—	61	2	no	—
72	C₂²×C₄	Dic₁₂₂	1	1	yes	characteristic
73	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
75	M₁₆	—	4	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	61	4	no	—
78	C₂×C₈	—	122	2	no	—
79	C₂×C₈	—	244	2	no	—
80	C₄²	Dic₁₂₂	1	1	yes	characteristic
81	C₄²	C₄×D₆₁	1	1	yes	characteristic
82	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
83	C₄²	C₂²×D₆₁	1	1	yes	characteristic
84	C₂³	#11758196	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	#11758196	1	1	yes	characteristic
88	C₂×C₄	#11758196	1	1	yes	characteristic
89	C₂×C₄	#11758196	1	1	yes	characteristic
90	C₂×C₄	#11758200	1	2	yes	—
91	C₂×C₄	#11758201	1	2	yes	—
92	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	122	4	no	—
97	C₈	—	244	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#14519708	1	2	yes	—
101	C₂²	C₂×#11758196	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#14519708	1	2	yes	—
106	C₄	#14519785	1	2	yes	—
107	C₄	C₂×#11758201	1	1	yes	characteristic
108	C₄	C₂×#11758200	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#16839790	1	1	yes	characteristic
111	C₂	C₂×#14519708	1	1	yes	characteristic
112	C₂	#16839810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839815

C₂

yes

maximal, characteristic

C₂×C₄⋊#92418

C₂

yes

maximal, characteristic

#16839512

C₂

yes

maximal, complemented

C₆₁×#423

C₂

yes

maximal, characteristic

C₂²×#92418

—

C₄⋊#92418

—

C₄⋊#92418

C₄

yes

—

C₄⋊#92418

C₂²

yes

—

C₂²⋊#92418

C₄

yes

—

C₄×#92418

C₂²

yes

—

C₂×M₁₆×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

Q₈

yes

characteristic

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

Q₈

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

—

C₂×C₄

yes

—

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

SD₁₆

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₈×C₆₁

—

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#106

yes

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#106

yes

—

C₄×C₆₁

#131

yes

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

#1570

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

yes

characteristic

C₂×C₄⋊C₈

—

#334

—

C₆₁

#1570

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄⋊C₈

C₂×D₆₁

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₂₂

yes

characteristic

C₂²×C₄

C₄×D₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₄²

Dic₁₂₂

yes

characteristic

C₄²

C₄×D₆₁

yes

characteristic

C₄²

C₂²⋊D₆₁

yes

characteristic

C₄²

C₂²×D₆₁

yes

characteristic

C₂³

#11758196

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#11758201

yes

—

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

C₈

—

C₈

—

122

—

C₈

—

244

—

C₂²

—

C₂²

—

100

C₂²

#14519708

yes

—

101

C₂²

C₂×#11758196

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#14519708

yes

—

106

C₄

#14519785

yes

—

107

C₄

C₂×#11758201

yes

characteristic

108

C₄

C₂×#11758200

yes

characteristic

109

C₂

—

110

C₂

#16839790

yes

characteristic

111

C₂

C₂×#14519708

yes

characteristic

112

C₂

#16839810

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#423	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	#423	D₆₁
hypocenter	C₆₁	#1570
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#423

C₂

socle

C₂²×C₆₁

C₂×C₄⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #16839512	C₂	semidirect
61. C₆₁	#1570	semidirect

H ≅ G/N

type

3. #16839512

C₂

semidirect

61. C₆₁

#1570

semidirect

Subgroups — #21444591

Conjugacy classes of subgroups

Complements