Group #21441675 — subgroups

Subgroups — C₆₁×#1634 #21441675

Order 7808.

218 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 68 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	C₆₁×#555	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#476	C₂	1	2	yes	maximal, complemented
4	C₆₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×Q₁₆×C₆₁	—	2	2	no	—
7	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
9	#124×C₆₁	—	2	2	no	—
10	#121×C₆₁	—	2	2	no	—
11	C₄×D₄×C₆₁	C₂²	1	2	yes	—
12	#117×C₆₁	—	2	2	no	—
13	#106×C₆₁	—	2	2	no	—
14	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	#102×C₆₁	—	2	2	no	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₁	—	2	1	no	—
18	C₂×Q₈×C₆₁	D₄	1	2	yes	—
19	C₂×D₄×C₆₁	—	2	2	no	—
20	C₂×D₄×C₆₁	D₄	1	2	yes	—
21	C₂²×C₄×C₆₁	—	2	2	no	—
22	Q₁₆×C₆₁	—	4	4	no	—
23	C₂×C₈×C₆₁	—	2	1	no	—
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₄⋊C₄×C₆₁	—	2	2	no	—
27	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
28	C₂²⋊C₄×C₆₁	—	2	4	no	—
29	C₂²⋊C₄×C₆₁	—	2	2	no	—
30	C₄²×C₆₁	—	2	2	no	—
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₂³×C₆₁	—	2	2	no	—
33	Q₈×C₆₁	—	4	2	no	—
34	Q₈×C₆₁	—	4	2	no	—
35	D₄×C₆₁	—	2	2	no	—
36	D₄×C₆₁	—	2	2	no	—
37	D₄×C₆₁	—	8	2	no	—
38	C₂×C₄×C₆₁	—	2	2	no	—
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	4	2	no	—
41	C₂×C₄×C₆₁	—	4	2	no	—
42	C₂×C₄×C₆₁	—	4	1	no	—
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
45	C₈×C₆₁	—	2	2	no	—
46	C₈×C₆₁	—	4	2	no	—
47	C₂²×C₆₁	—	4	2	no	—
48	C₂²×C₆₁	—	4	2	no	—
49	C₂²×C₆₁	—	4	2	no	—
50	C₂²×C₆₁	#120	1	1	yes	characteristic
51	C₄×C₆₁	—	2	1	no	—
52	C₄×C₆₁	—	4	2	no	—
53	C₄×C₆₁	—	4	2	no	—
54	C₄×C₆₁	—	8	1	no	—
55	C₄×C₆₁	#135	1	2	yes	—
56	C₄×C₆₁	#137	1	2	yes	—
57	#1634	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×C₆₁	—	8	2	no	—
59	C₂×C₆₁	#452	1	2	yes	—
60	C₂×C₆₁	#453	1	1	yes	characteristic
61	#555	C₂×C₆₁	1	1	yes	characteristic
62	#494	C₂×C₆₁	1	1	yes	characteristic, complemented
63	#476	C₂×C₆₁	1	2	yes	complemented
64	#333	C₂×C₆₁	1	1	yes	characteristic, complemented
65	D₄⋊C₈	C₂×C₆₁	1	2	yes	complemented
66	C₆₁	#1634	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
69	C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂²×C₆₁	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	2	2	no	—
75	C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
76	#102	—	2	2	no	—
77	C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	D₄×C₆₁	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	D₄×C₆₁	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	D₄×C₆₁	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂³×C₆₁	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
105	C₂×C₄	C₂×D₄×C₆₁	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	4	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#120×C₆₁	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	8	1	no	—
116	C₄	#135×C₆₁	1	2	yes	—
117	C₄	#137×C₆₁	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	C₆₁×#452	1	2	yes	—
120	C₂	C₆₁×#453	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₁×#555

C₂

yes

maximal, characteristic

C₆₁×#494

C₂

yes

maximal, characteristic, complemented

C₆₁×#476

C₂

yes

maximal, complemented

C₆₁×#333

C₂

yes

maximal, characteristic, complemented

C₆₁×D₄⋊C₈

C₂

yes

maximal, complemented

C₂×Q₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#124×C₆₁

—

#121×C₆₁

—

C₄×D₄×C₆₁

C₂²

yes

—

#117×C₆₁

—

#106×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

—

#102×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₂²⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

C₂³×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#120

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

#137

yes

—

#1634

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₁

—

C₂×C₆₁

#452

yes

—

C₂×C₆₁

#453

yes

characteristic

#555

C₂×C₆₁

yes

characteristic

#494

C₂×C₆₁

yes

characteristic, complemented

#476

C₂×C₆₁

yes

complemented

#333

C₂×C₆₁

yes

characteristic, complemented

D₄⋊C₈

C₂×C₆₁

yes

complemented

C₆₁

#1634

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₆₁

yes

characteristic

C₄⋊Q₈

C₂²×C₆₁

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂²×C₆₁

yes

—

#117

—

#106

—

C₄⋊C₈

C₂²×C₆₁

yes

—

#102

—

C₄×C₈

C₂²×C₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₆₁

yes

—

C₂×D₄

—

C₂×D₄

D₄×C₆₁

yes

—

C₂²×C₄

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₆₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³×C₆₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₄×C₆₁

yes

characteristic

105

C₂×C₄

C₂×D₄×C₆₁

yes

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#120×C₆₁

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#135×C₆₁

yes

—

117

C₄

#137×C₆₁

yes

—

118

C₂

—

119

C₂

C₆₁×#452

yes

—

120

C₂

C₆₁×#453

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₆₁
prime radical	C₆₁×#555	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₆₁

prime radical

C₆₁×#555

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

N	H ≅ G/N	type
2. C₆₁×#494	C₂	semidirect
3. C₆₁×#476	C₂	semidirect
4. C₆₁×#333	C₂	semidirect
5. C₆₁×D₄⋊C₈	C₂	semidirect
57. #1634	C₆₁	direct (N × H)
62. #494	C₂×C₆₁	semidirect
63. #476	C₂×C₆₁	semidirect
64. #333	C₂×C₆₁	semidirect
65. D₄⋊C₈	C₂×C₆₁	semidirect
66. C₆₁	#1634	direct (N × H)

H ≅ G/N

type

2. C₆₁×#494

C₂

semidirect

3. C₆₁×#476

C₂

semidirect

4. C₆₁×#333

C₂

semidirect

5. C₆₁×D₄⋊C₈

C₂

semidirect

57. #1634

C₆₁

direct (N × H)

62. #494

C₂×C₆₁

semidirect

63. #476

C₂×C₆₁

semidirect

64. #333

C₂×C₆₁

semidirect

65. D₄⋊C₈

C₂×C₆₁

semidirect

66. C₆₁

#1634

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₁×#555	2
C₆₁×#494	2
C₆₁×#476	2
C₆₁×#333	2
C₆₁×D₄⋊C₈	2
#1634	61

subgroup

index

C₆₁×#555

C₆₁×#494

C₆₁×#476

C₆₁×#333

C₆₁×D₄⋊C₈

#1634

Subgroups — C₆₁×#1634 #21441675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#1634 #21441675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#1634 #21441675