Group #21461361 — subgroups

Subgroups — D₆₁⋊#334 #21461361

Order 7808.

200 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 94 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	D₆₁⋊(C₄⋊C₈)	C₂	1	2	yes	maximal
2	#16840624	C₂	1	2	yes	maximal, complemented
3	#16840623	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×D₆₁⋊C₈	—	2	2	no	—
6	#14519844	—	2	2	no	—
7	C₄⋊#92420	C₂²	1	4	yes	—
8	C₂×C₈×D₆₁	C₄	1	1	yes	characteristic
9	C₂×C₈×D₆₁	C₄	1	1	yes	characteristic
10	C₄²×D₆₁	C₂²	1	1	yes	characteristic
11	C₄×#92418	C₄	1	1	yes	characteristic
12	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
15	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
17	C₂×C₄×D₆₁	Q₈	1	1	yes	characteristic
18	C₂×#92420	—	2	4	no	—
19	D₆₁⋊C₈	—	4	4	no	—
20	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₆₁	D₄	1	1	yes	characteristic
24	C₄×Dic₆₁	Q₈	1	1	yes	characteristic
25	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
26	C₂×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂×#92418	C₂×C₄	1	1	yes	characteristic
28	C₈×D₆₁	—	2	2	no	—
29	C₈×D₆₁	C₈	1	4	yes	complemented
30	C₂²×D₆₁	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	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
37	C₄×D₆₁	—	2	2	no	—
38	C₄×D₆₁	C₂×C₈	1	2	yes	—
39	C₄×D₆₁	M₁₆	1	2	yes	—
40	C₄×D₆₁	SD₁₆	1	4	yes	—
41	#92420	—	4	4	no	—
42	C₈×C₆₁	—	2	1	no	—
43	C₈×C₆₁	C₂×C₈	1	2	yes	complemented
44	#92418	—	2	1	no	—
45	#92418	C₂×C₈	1	2	yes	complemented
46	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×D₆₁	C₄⋊C₈	1	2	yes	—
48	C₂×D₆₁	#106	1	2	yes	—
49	C₂×D₆₁	#108	1	2	yes	—
50	C₄×C₆₁	—	2	1	no	—
51	C₄×C₆₁	#106	1	2	yes	—
52	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
53	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
54	Dic₆₁	—	2	1	no	—
55	Dic₆₁	C₄⋊C₈	1	2	yes	—
56	Dic₆₁	C₂×SD₁₆	1	2	yes	—
57	C₂×#334	—	61	1	no	maximal, Sylow
58	C₂×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₆₁	C₂×#106	1	1	yes	characteristic
60	C₂×C₆₁	C₂×#108	1	1	yes	characteristic
61	D₆₁	#334	1	4	yes	complemented
62	C₂×C₄⋊C₈	—	61	2	no	—
63	C₂×C₄×C₈	—	61	1	no	—
64	#334	—	61	2	no	—
65	#334	—	61	2	no	—
66	C₆₁	C₂×#334	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	—	61	1	no	—
68	C₂²×C₈	—	61	1	no	—
69	C₂²×C₈	—	122	2	no	—
70	C₂×C₄²	—	61	1	no	—
71	C₄⋊C₈	—	61	4	no	—
72	C₄⋊C₈	—	122	2	no	—
73	C₄×C₈	—	61	1	no	—
74	C₄×C₈	—	61	1	no	—
75	C₄×C₈	—	61	1	no	—
76	C₄×C₈	#6923	1	1	yes	characteristic
77	C₂²×C₄	—	61	1	no	—
78	C₂²×C₄	—	61	1	no	—
79	C₂²×C₄	—	61	1	no	—
80	C₂×C₈	—	61	1	no	—
81	C₂×C₈	—	61	4	no	—
82	C₂×C₈	—	61	1	no	—
83	C₂×C₈	—	122	2	no	—
84	C₂×C₈	—	122	4	no	—
85	C₂×C₈	—	244	4	no	—
86	C₂×C₈	C₂×#6923	1	1	yes	characteristic
87	C₂×C₈	C₂×#6923	1	1	yes	characteristic
88	C₄²	—	61	1	no	—
89	C₄²	—	61	1	no	—
90	C₄²	—	61	1	no	—
91	C₄²	C₂×#6923	1	1	yes	characteristic
92	C₂³	—	61	1	no	—
93	C₂×C₄	—	61	1	no	—
94	C₂×C₄	—	61	1	no	—
95	C₂×C₄	—	61	4	no	—
96	C₂×C₄	—	61	2	no	—
97	C₂×C₄	—	61	2	no	—
98	C₂×C₄	—	61	1	no	—
99	C₂×C₄	—	122	2	no	—
100	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
101	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
102	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	61	2	no	—
105	C₈	—	122	1	no	—
106	C₈	—	244	4	no	—
107	C₈	D₆₁⋊C₈	1	2	yes	complemented
108	C₂²	—	61	2	no	—
109	C₂²	—	61	2	no	—
110	C₂²	—	61	2	no	—
111	C₂²	C₂×C₄⋊#6923	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	61	2	no	—
114	C₄	—	61	2	no	—
115	C₄	—	122	1	no	—
116	C₄	#14519845	1	2	yes	—
117	C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
118	C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
119	C₂	—	61	4	no	—
120	C₂	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
121	C₂	C₂×#14519845	1	1	yes	characteristic
122	C₂	D₆₁⋊#108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₆₁⋊(C₄⋊C₈)

C₂

yes

maximal

#16840624

C₂

yes

maximal, complemented

#16840623

C₂

yes

maximal, complemented

C₄×C₈×D₆₁

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₈

—

#14519844

—

C₄⋊#92420

C₂²

yes

—

C₂×C₈×D₆₁

C₄

yes

characteristic

C₂×C₈×D₆₁

C₄

yes

characteristic

C₄²×D₆₁

C₂²

yes

characteristic

C₄×#92418

C₄

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

Q₈

yes

characteristic

C₂×#92420

—

D₆₁⋊C₈

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

Q₈

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

C₈

yes

complemented

C₂²×D₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×Q₈

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₂×C₈

yes

—

C₄×D₆₁

M₁₆

yes

—

C₄×D₆₁

SD₁₆

yes

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₈

yes

complemented

#92418

—

#92418

C₂×C₈

yes

complemented

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×D₆₁

C₄⋊C₈

yes

—

C₂×D₆₁

#106

yes

—

C₂×D₆₁

#108

yes

—

C₄×C₆₁

—

C₄×C₆₁

#106

yes

—

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

C₂×M₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

C₄⋊C₈

yes

—

Dic₆₁

C₂×SD₁₆

yes

—

C₂×#334

—

maximal, Sylow

C₂×C₆₁

C₂×C₄⋊C₈

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#334

—

#334

—

C₆₁

C₂×#334

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6923

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

100

C₂×C₄

C₄⋊#6923

yes

characteristic

101

C₂×C₄

C₄⋊#6923

yes

characteristic

102

C₂×C₄

C₂²×#6923

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

122

—

106

C₈

—

244

—

107

C₈

D₆₁⋊C₈

yes

complemented

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄⋊#6923

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

122

—

116

C₄

#14519845

yes

—

117

C₄

C₂×D₆₁⋊C₈

yes

characteristic

118

C₄

D₆₁⋊M₁₆

yes

characteristic

119

C₂

—

120

C₂

D₆₁⋊(C₄⋊C₈)

yes

characteristic

121

C₂

C₂×#14519845

yes

characteristic

122

C₂

D₆₁⋊#108

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#6923
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄×C₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6923
hypocenter	C₆₁	C₂×#334
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄⋊#6923

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄×C₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₄⋊C₄

C₄×C₈

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. #16840624	C₂	semidirect
3. #16840623	C₂	semidirect
29. C₈×D₆₁	C₈	semidirect
43. C₈×C₆₁	C₂×C₈	semidirect
45. #92418	C₂×C₈	semidirect
61. D₆₁	#334	semidirect
66. C₆₁	C₂×#334	semidirect
107. C₈	D₆₁⋊C₈	semidirect

H ≅ G/N

type

2. #16840624

C₂

semidirect

3. #16840623

C₂

semidirect

29. C₈×D₆₁

C₈

semidirect

43. C₈×C₆₁

C₂×C₈

semidirect

45. #92418

C₂×C₈

semidirect

61. D₆₁

#334

semidirect

66. C₆₁

C₂×#334

semidirect

107. C₈

D₆₁⋊C₈

semidirect

Subgroups — D₆₁⋊#334 #21461361

Conjugacy classes of subgroups

Complements

Subgroups — D61⋊#334 #21461361

Conjugacy classes of subgroups

Complements

Subgroups — D₆₁⋊#334 #21461361