Group #21461319 — subgroups

Subgroups — D₆₁⋊#333 #21461319

Order 7808.

256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 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	#16840622	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×D₆₁⋊C₈	—	2	2	no	—
5	#14519844	C₂²	1	4	yes	—
6	C₄⋊#92420	—	2	2	no	—
7	C₂×Q₈×D₆₁	C₄	1	2	yes	—
8	#14519777	C₄	1	1	yes	characteristic
9	#14519776	—	2	2	no	—
10	#14519775	C₂²	1	2	yes	—
11	C₄⋊C₄×D₆₁	—	2	2	no	—
12	C₄²×D₆₁	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
15	Q₈×D₆₁	—	4	4	no	—
16	C₂×C₄×D₆₁	—	2	2	no	—
17	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₆₁	D₄	1	2	yes	—
19	C₂×Dic₁₂₂	—	2	2	no	—
20	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
21	C₂×#92420	—	2	4	no	—
22	D₆₁⋊C₈	—	4	4	no	—
23	C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄⋊Dic₆₁	—	2	2	no	—
26	#11758196	—	2	4	no	—
27	C₄×Dic₆₁	D₄	1	2	yes	—
28	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
29	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
30	Q₈×C₆₁	—	4	2	no	—
31	C₂×C₄×C₆₁	—	2	2	no	—
32	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
33	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₆₁	—	2	2	no	—
35	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
37	C₄×D₆₁	—	2	2	no	—
38	C₄×D₆₁	—	4	4	no	—
39	C₄×D₆₁	SD₁₆	1	4	yes	—
40	C₄×D₆₁	Q₁₆	1	4	yes	—
41	Dic₁₂₂	—	4	4	no	—
42	Dic₁₂₂	—	4	2	no	—
43	#92420	—	4	4	no	—
44	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×D₆₁	#100	1	2	yes	—
46	C₂×D₆₁	#103	1	4	yes	—
47	C₄×C₆₁	—	2	1	no	—
48	C₄×C₆₁	—	4	2	no	—
49	C₄×C₆₁	#103	1	4	yes	—
50	Dic₆₁	—	2	1	no	—
51	Dic₆₁	—	4	2	no	—
52	Dic₆₁	C₂×SD₁₆	1	2	yes	—
53	Dic₆₁	C₂×Q₁₆	1	2	yes	—
54	C₂×#333	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	C₂×#100	1	1	yes	characteristic
56	C₂×C₆₁	C₂×#103	1	2	yes	—
57	D₆₁	#333	1	4	yes	complemented
58	C₂×C₄⋊Q₈	—	61	1	no	—
59	C₂×C₄⋊C₈	—	61	2	no	—
60	#333	—	61	4	no	—
61	C₆₁	C₂×#333	1	1	yes	characteristic, complemented, Sylow
62	C₂²×Q₈	—	61	2	no	—
63	C₂²×C₈	—	122	2	no	—
64	C₄⋊Q₈	—	61	2	no	—
65	C₄⋊Q₈	—	61	1	no	—
66	C₄⋊Q₈	—	122	2	no	—
67	C₄⋊Q₈	#6923	1	1	yes	characteristic
68	C₂×C₄⋊C₄	—	122	2	no	—
69	C₂×C₄²	—	61	1	no	—
70	C₄⋊C₈	—	61	4	no	—
71	C₄⋊C₈	—	122	2	no	—
72	C₂×Q₈	—	61	2	no	—
73	C₂×Q₈	—	122	2	no	—
74	C₂×Q₈	—	244	4	no	—
75	C₂×Q₈	C₂×#6923	1	2	yes	—
76	C₂²×C₄	—	61	2	no	—
77	C₂²×C₄	—	61	1	no	—
78	C₂²×C₄	—	122	2	no	—
79	C₂×C₈	—	122	4	no	—
80	C₂×C₈	—	244	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	122	4	no	—
83	C₄⋊C₄	—	122	2	no	—
84	C₄²	—	61	2	no	—
85	C₄²	—	61	1	no	—
86	C₄²	C₂×#6923	1	1	yes	characteristic
87	C₂³	—	61	1	no	—
88	Q₈	—	4	2	no	—
89	Q₈	—	244	2	no	—
90	Q₈	—	244	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	61	2	no	—
93	C₂×C₄	—	61	4	no	—
94	C₂×C₄	—	61	4	no	—
95	C₂×C₄	—	61	1	no	—
96	C₂×C₄	—	122	2	no	—
97	C₂×C₄	—	122	2	no	—
98	C₂×C₄	—	244	4	no	—
99	C₂×C₄	C₂²⋊#6923	1	2	yes	—
100	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
101	C₈	—	244	4	no	—
102	C₂²	—	61	2	no	—
103	C₂²	—	61	4	no	—
104	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	4	2	no	—
107	C₄	—	61	2	no	—
108	C₄	—	61	2	no	—
109	C₄	—	122	1	no	—
110	C₄	—	244	2	no	—
111	C₄	#14519840	1	4	yes	—
112	C₂	—	61	4	no	—
113	C₂	D₆₁⋊#100	1	1	yes	characteristic
114	C₂	C₂×#14519840	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₆₁⋊(C₄⋊C₈)

C₂

yes

maximal

#16840622

C₂

yes

maximal, complemented

C₄⋊Q₈×D₆₁

C₂

yes

maximal, characteristic

C₂×D₆₁⋊C₈

—

#14519844

C₂²

yes

—

C₄⋊#92420

—

C₂×Q₈×D₆₁

C₄

yes

—

#14519777

C₄

yes

characteristic

#14519776

—

#14519775

C₂²

yes

—

C₄⋊C₄×D₆₁

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

—

D₆₁⋊C₈

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂²×D₆₁

C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

SD₁₆

yes

—

C₄×D₆₁

Q₁₆

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

#100

yes

—

C₂×D₆₁

#103

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#103

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×SD₁₆

yes

—

Dic₆₁

C₂×Q₁₆

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

#333

yes

complemented

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

#333

—

C₆₁

C₂×#333

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

#6923

yes

characteristic

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂×Q₈

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

C₄²

—

C₄²

C₂×#6923

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

244

—

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂²⋊#6923

yes

—

100

C₂×C₄

C₂²×#6923

yes

characteristic

101

C₈

—

244

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊#6923

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

122

—

110

C₄

—

244

—

111

C₄

#14519840

yes

—

112

C₂

—

113

C₂

D₆₁⋊#100

yes

characteristic

114

C₂

C₂×#14519840

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	C₂×#333
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄⋊Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #16840622	C₂	semidirect
57. D₆₁	#333	semidirect
61. C₆₁	C₂×#333	semidirect

H ≅ G/N

type

2. #16840622

C₂

semidirect

57. D₆₁

#333

semidirect

61. C₆₁

C₂×#333

semidirect

Subgroups — D₆₁⋊#333 #21461319

Conjugacy classes of subgroups

Complements

Subgroups — D61⋊#333 #21461319

Conjugacy classes of subgroups

Complements

Subgroups — D₆₁⋊#333 #21461319