Group #21446801 — subgroups

Subgroups — #21446801

Order 7808.

244 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	#16839608	C₂	1	1	yes	maximal, characteristic
3	#16839521	C₂	1	4	yes	maximal
4	C₆₁×#533	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758194	C₂²	1	2	yes	—
6	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
7	C₄⋊#92418	—	2	2	no	—
8	#14519684	—	2	2	no	—
9	C₄⋊Q₈×C₆₁	C₄	1	2	yes	complemented
10	#125×C₆₁	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₆₁	—	2	2	no	—
12	C₄×Q₈×C₆₁	—	2	2	no	—
13	#117×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₆₁	—	2	2	no	—
16	C₂²×C₄×C₆₁	D₄	1	2	yes	—
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	—	2	2	no	—
19	C₄⋊C₄×C₆₁	—	2	4	no	—
20	C₄⋊C₄×C₆₁	—	2	2	no	—
21	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
22	C₂²⋊C₄×C₆₁	—	2	2	no	—
23	C₄²×C₆₁	—	2	2	no	—
24	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₆₁	D₄	1	2	yes	—
27	C₄×Dic₆₁	—	2	2	no	—
28	C₄×Dic₆₁	Q₈	1	4	yes	complemented
29	#11758194	—	2	4	no	—
30	C₂×#92418	—	2	2	no	—
31	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₆₁	—	4	4	no	—
33	C₂×C₄×C₆₁	—	2	4	no	—
34	C₂×C₄×C₆₁	—	4	2	no	—
35	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₆₁	#41	1	2	yes	—
41	C₂×Dic₆₁	—	2	4	no	—
42	C₂×Dic₆₁	—	4	4	no	—
43	#92418	—	4	2	no	—
44	C₂²×C₆₁	—	2	2	no	—
45	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₆₁	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₆₁	—	2	2	no	—
48	C₄×C₆₁	—	2	2	no	—
49	C₄×C₆₁	—	4	4	no	—
50	C₄×C₆₁	C₄×Q₈	1	2	yes	—
51	C₄×C₆₁	#124	1	1	yes	characteristic
52	C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₆₁	—	4	4	no	—
54	#1995	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	—	2	2	no	—
56	C₂×C₆₁	#391	1	1	yes	characteristic
57	C₂×C₆₁	#421	1	2	yes	—
58	#533	D₆₁	1	1	yes	characteristic
59	#423	—	61	1	no	—
60	#376	—	61	1	no	—
61	#339	—	61	4	no	—
62	C₆₁	#1995	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₁₆	—	61	2	no	—
64	C₄⋊Q₈	Dic₆₁	1	2	yes	complemented
65	#125	Dic₆₁	1	2	yes	complemented
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₆₁	1	2	yes	—
69	C₂×C₄²	—	61	2	no	—
70	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
71	C₄⋊C₈	—	122	2	no	—
72	#95	—	122	2	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	—	122	2	no	—
75	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
76	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
77	M₁₆	—	122	4	no	—
78	C₂×C₈	—	122	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	61	4	no	—
85	C₄²	—	122	2	no	—
86	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
87	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
88	C₄²	C₂²⋊D₆₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	122	4	no	—
94	C₂×C₄	—	244	4	no	—
95	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
96	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
97	C₂×C₄	Q₈×D₆₁	1	2	yes	—
98	C₂×C₄	#11758226	1	2	yes	—
99	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
101	C₈	—	244	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
104	C₂²	#14519753	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	244	4	no	—
109	C₄	Q₈×Dic₆₁	1	2	yes	—
110	C₄	#14519766	1	1	yes	characteristic
111	C₄	#14519775	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#16839669	1	1	yes	characteristic
114	C₂	#16839779	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16839786

C₂

yes

maximal, characteristic

#16839608

C₂

yes

maximal, characteristic

#16839521

C₂

yes

maximal

C₆₁×#533

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₄⋊#92418

—

#14519684

—

C₄⋊Q₈×C₆₁

C₄

yes

complemented

#125×C₆₁

C₄

yes

complemented

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

Q₈

yes

complemented

#11758194

—

C₂×#92418

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

#124

yes

characteristic

C₄×C₆₁

C₄⋊Q₈

yes

characteristic

Dic₆₁

—

#1995

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#391

yes

characteristic

C₂×C₆₁

#421

yes

—

#533

D₆₁

yes

—

—

—

C₆₁

#1995

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

Dic₆₁

yes

complemented

#125

Dic₆₁

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

122

—

C₄²

C₂×Dic₆₁

yes

characteristic

C₄²

C₂×Dic₆₁

yes

characteristic

C₄²

C₂²⋊D₆₁

yes

—

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂²⋊Dic₆₁

yes

—

C₂×C₄

C₂²⋊Dic₆₁

yes

—

C₂×C₄

Q₈×D₆₁

yes

—

C₂×C₄

#11758226

yes

—

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

101

C₈

—

244

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

104

C₂²

#14519753

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

244

—

109

C₄

Q₈×Dic₆₁

yes

—

110

C₄

#14519766

yes

characteristic

111

C₄

#14519775

yes

characteristic

112

C₂

—

113

C₂

#16839669

yes

characteristic

114

C₂

#16839779

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#533	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	#533	D₆₁
hypocenter	C₆₁	#1995
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#533

C₂

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₆₁	C₄	semidirect
10. #125×C₆₁	C₄	semidirect
28. C₄×Dic₆₁	Q₈	semidirect
62. C₆₁	#1995	semidirect
64. C₄⋊Q₈	Dic₆₁	semidirect
65. #125	Dic₆₁	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₆₁

C₄

semidirect

10. #125×C₆₁

C₄

semidirect

28. C₄×Dic₆₁

Q₈

semidirect

62. C₆₁

#1995

semidirect

64. C₄⋊Q₈

Dic₆₁

semidirect

65. #125

Dic₆₁

semidirect

Subgroups — #21446801

Conjugacy classes of subgroups

Complements