Group #21448740 — subgroups

Subgroups — #21448740

Order 7808.

552 conjugacy classes of subgroups; only the 313 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 313 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×Q₈×Dic₆₁	C₂	1	2	yes	maximal
2	#16839661	C₂	1	6	yes	maximal
3	#16839628	C₂	1	3	yes	maximal
4	C₄×C₄⋊Dic₆₁	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₆₁	—	2	8	no	—
7	C₂×C₄⋊Dic₆₁	C₂²	1	6	yes	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	6	yes	—
9	C₂×C₄×Dic₆₁	C₂²	1	6	yes	—
10	#14519684	C₂²	1	1	yes	characteristic
11	#14519684	C₂²	1	3	yes	—
12	#14519684	C₂²	1	6	yes	—
13	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₆₁	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	3	yes	—
17	C₂×Q₈×C₆₁	C₂×C₄	1	4	yes	—
18	C₂×Q₈×C₆₁	D₄	1	4	yes	—
19	C₂×Q₈×C₆₁	Q₈	1	4	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
22	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
23	C₂²×Dic₆₁	C₂³	1	6	yes	—
24	C₂²×Dic₆₁	C₂³	1	2	yes	—
25	C₄⋊C₄×C₆₁	C₂×C₄	1	12	yes	—
26	C₄²×C₆₁	C₂×C₄	1	12	yes	—
27	C₄⋊Dic₆₁	—	2	12	no	—
28	C₄⋊Dic₆₁	—	2	12	no	—
29	C₄×Dic₆₁	—	2	12	no	—
30	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
31	Q₈×C₆₁	C₄⋊C₄	1	16	yes	complemented
32	C₂×C₄×C₆₁	—	2	6	no	—
33	C₂×C₄×C₆₁	—	2	2	no	—
34	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₆₁	C₂²×C₄	1	6	yes	—
36	C₂×C₄×C₆₁	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₆₁	C₂×D₄	1	6	yes	—
38	C₂×C₄×C₆₁	C₂×Q₈	1	6	yes	—
39	C₂×Dic₆₁	—	2	2	no	—
40	C₂×Dic₆₁	—	2	2	no	—
41	C₂×Dic₆₁	—	2	6	no	—
42	C₂×Dic₆₁	—	2	6	no	—
43	C₂×Dic₆₁	—	2	6	no	—
44	C₂×Dic₆₁	#41	1	4	yes	—
45	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
46	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
48	C₂²×C₆₁	C₂×#41	1	2	yes	—
49	C₂²×C₆₁	2₋¹⁺⁴	1	2	yes	—
50	C₄×C₆₁	—	2	6	no	—
51	C₄×C₆₁	—	2	2	no	—
52	C₄×C₆₁	C₂×C₄⋊C₄	1	12	yes	—
53	Dic₆₁	—	2	4	no	—
54	Dic₆₁	—	4	6	no	—
55	#2351	—	61	1	no	maximal, Sylow
56	C₂×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₆₁	C₄×#41	1	1	yes	characteristic
58	C₂×C₆₁	#519	1	1	yes	characteristic
59	C₂×C₆₁	#548	1	2	yes	—
60	C₂×C₆₁	#557	1	2	yes	—
61	C₂×C₄×Q₈	—	61	2	no	—
62	C₂×C₄×Q₈	D₆₁	1	1	yes	characteristic
63	#389	—	61	6	no	—
64	#382	—	61	3	no	—
65	C₄×C₄⋊C₄	—	61	3	no	—
66	C₆₁	#2351	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic
68	C₄×Q₈	—	122	8	no	—
69	C₄×Q₈	Dic₆₁	1	8	yes	complemented
70	C₂×C₄⋊C₄	—	61	6	no	—
71	C₂×C₄⋊C₄	—	61	6	no	—
72	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
73	C₂×C₄²	—	61	6	no	—
74	C₂×C₄²	C₂×D₆₁	1	3	yes	—
75	#95	—	61	1	no	—
76	#95	—	61	3	no	—
77	#95	—	61	6	no	—
78	C₂×Q₈	Dic₁₂₂	1	4	yes	—
79	C₂×Q₈	D₂₄₄	1	4	yes	—
80	C₂×Q₈	C₂×Dic₆₁	1	4	yes	—
81	C₂²×C₄	—	61	2	no	—
82	C₂²×C₄	—	61	6	no	—
83	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
85	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
86	C₄⋊C₄	—	122	12	no	—
87	C₄⋊C₄	—	122	12	no	—
88	C₄⋊C₄	C₂×Dic₆₁	1	12	yes	—
89	C₄²	—	122	12	no	—
90	C₄²	C₂×Dic₆₁	1	12	yes	—
91	C₂³	C₂³×D₆₁	1	1	yes	characteristic
92	Q₈	C₄⋊Dic₆₁	1	16	yes	complemented
93	C₂×C₄	—	2	6	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	61	4	no	—
96	C₂×C₄	—	122	2	no	—
97	C₂×C₄	—	122	2	no	—
98	C₂×C₄	—	122	6	no	—
99	C₂×C₄	—	122	6	no	—
100	C₂×C₄	—	122	6	no	—
101	C₂×C₄	C₂×Dic₁₂₂	1	6	yes	—
102	C₂×C₄	C₂×D₂₄₄	1	6	yes	—
103	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
104	C₂×C₄	C₂²×Dic₆₁	1	6	yes	—
105	C₂×C₄	C₂²×Dic₆₁	1	6	yes	—
106	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
107	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
108	C₂²	C₂²×Dic₁₂₂	1	1	yes	characteristic
109	C₂²	#41×D₆₁	1	2	yes	—
110	C₂²	#14519825	1	2	yes	—
111	C₄	—	2	6	no	—
112	C₄	—	2	2	no	—
113	C₄	—	122	4	no	—
114	C₄	—	244	6	no	—
115	C₄	C₂×C₄⋊Dic₆₁	1	12	yes	—
116	C₂	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
117	C₂	#41×Dic₆₁	1	1	yes	characteristic
118	C₂	#16840240	1	1	yes	characteristic
119	C₂	#16840424	1	2	yes	—
120	C₂	#16840488	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₆₁

C₂

yes

maximal

#16839661

C₂

yes

maximal

#16839628

C₂

yes

maximal

C₄×C₄⋊Dic₆₁

C₂

yes

maximal

C₂×C₄×Q₈×C₆₁

C₂

yes

maximal, characteristic

Q₈×Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

#14519684

C₂²

yes

characteristic

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

C₂²×Q₈×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₂³×C₆₁

C₂⁴

yes

characteristic

Q₈×C₆₁

C₄⋊C₄

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#41

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

C₂×#41

yes

—

C₂²×C₆₁

2₋¹⁺⁴

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

—

Dic₆₁

—

Dic₆₁

—

#2351

—

maximal, Sylow

C₂×C₆₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₆₁

C₄×#41

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#557

yes

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₆₁

yes

characteristic

#389

—

#382

—

C₄×C₄⋊C₄

—

C₆₁

#2351

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

122

—

C₄×Q₈

Dic₆₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

—

#95

—

#95

—

#95

—

C₂×Q₈

Dic₁₂₂

yes

—

C₂×Q₈

D₂₄₄

yes

—

C₂×Q₈

C₂×Dic₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₄²

—

122

—

C₄²

C₂×Dic₆₁

yes

—

C₂³

C₂³×D₆₁

yes

characteristic

Q₈

C₄⋊Dic₆₁

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

100

C₂×C₄

—

122

—

101

C₂×C₄

C₂×Dic₁₂₂

yes

—

102

C₂×C₄

C₂×D₂₄₄

yes

—

103

C₂×C₄

C₂²×Dic₆₁

yes

—

104

C₂×C₄

C₂²×Dic₆₁

yes

—

105

C₂×C₄

C₂²×Dic₆₁

yes

—

106

C₂²

C₂³×Dic₆₁

yes

characteristic

107

C₂²

C₂²×D₂₄₄

yes

characteristic

108

C₂²

C₂²×Dic₁₂₂

yes

characteristic

109

C₂²

#41×D₆₁

yes

—

110

C₂²

#14519825

yes

—

111

C₄

—

112

C₄

—

113

C₄

—

122

—

114

C₄

—

244

—

115

C₄

C₂×C₄⋊Dic₆₁

yes

—

116

C₂

C₂²×C₄⋊Dic₆₁

yes

characteristic

117

C₂

#41×Dic₆₁

yes

characteristic

118

C₂

#16840240

yes

characteristic

119

C₂

#16840424

yes

—

120

C₂

#16840488

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator	C₂²×C₆₁	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂⁴
hypercenter	C₂×C₄×Q₈	D₆₁
hypocenter	C₆₁	#2351
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator

C₂²×C₆₁

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂⁴

hypercenter

C₂×C₄×Q₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
14. C₄×Q₈×C₆₁	C₄	semidirect
31. Q₈×C₆₁	C₄⋊C₄	semidirect
66. C₆₁	#2351	semidirect
69. C₄×Q₈	Dic₆₁	semidirect
92. Q₈	C₄⋊Dic₆₁	semidirect

H ≅ G/N

type

14. C₄×Q₈×C₆₁

C₄

semidirect

31. Q₈×C₆₁

C₄⋊C₄

semidirect

66. C₆₁

#2351

semidirect

69. C₄×Q₈

Dic₆₁

semidirect

92. Q₈

C₄⋊Dic₆₁

semidirect

Subgroups — #21448740

Conjugacy classes of subgroups

Complements