Group #29510701 — subgroups

Subgroups — C₆₁×#7268 #29510701

Order 15616.

238 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 118 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×C₆₁×#364	C₂	1	2	yes	maximal
2	C₆₁×#1844	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₆₁	—	2	2	no	—
4	C₂×#106×C₆₁	—	2	2	no	—
5	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
6	C₆₁×#384	C₄	1	2	yes	—
7	C₆₁×#364	—	2	2	no	—
8	C₆₁×#364	C₄	1	4	yes	complemented
9	C₂²×C₈×C₆₁	D₄	1	2	yes	—
10	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₆₁	—	4	2	no	—
12	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
13	M₃₂×C₆₁	—	2	4	no	—
14	M₃₂×C₆₁	—	4	2	no	—
15	C₂×C₁₆×C₆₁	—	4	2	no	—
16	#106×C₆₁	—	4	2	no	—
17	#106×C₆₁	—	4	2	no	—
18	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
20	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	—	2	1	no	—
22	C₂²×C₄×C₆₁	—	4	2	no	—
23	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₆₁	D₈	1	2	yes	—
27	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
28	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
29	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
30	C₄⋊C₄×C₆₁	—	4	4	no	—
31	C₄⋊C₄×C₆₁	—	8	4	no	—
32	C₄²×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	C₄²	1	1	yes	characteristic
34	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₁₆×C₆₁	—	4	4	no	—
36	C₂³×C₆₁	#100	1	1	yes	characteristic
37	C₂×C₄×C₆₁	—	2	4	no	—
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	—	8	4	no	—
41	C₂×C₄×C₆₁	—	8	2	no	—
42	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
43	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
44	C₂×C₄×C₆₁	#102	1	2	yes	—
45	C₂×C₄×C₆₁	#103	1	2	yes	—
46	C₈×C₆₁	#102	1	2	yes	—
47	C₈×C₆₁	#103	1	2	yes	—
48	C₈×C₆₁	#106	1	2	yes	—
49	C₈×C₆₁	#107	1	2	yes	—
50	#7268	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₆₁	—	2	2	no	—
52	C₂²×C₆₁	#331	1	1	yes	characteristic
53	C₂²×C₆₁	#333	1	1	yes	characteristic
54	C₂²×C₆₁	#343	1	1	yes	characteristic
55	C₄×C₆₁	—	2	2	no	—
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	—	16	2	no	—
58	C₄×C₆₁	#332	1	2	yes	—
59	C₄×C₆₁	#340	1	2	yes	—
60	C₂×#364	C₂×C₆₁	1	2	yes	—
61	#1844	C₂×C₆₁	1	1	yes	characteristic
62	C₂×C₆₁	—	2	1	no	—
63	C₂×C₆₁	—	2	1	no	—
64	C₂×C₆₁	#1296	1	1	yes	characteristic
65	C₂×C₆₁	#1359	1	2	yes	—
66	C₂×M₃₂	—	2	2	no	—
67	C₂×#106	—	2	2	no	—
68	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
69	#384	C₄×C₆₁	1	2	yes	—
70	#364	—	2	2	no	—
71	#364	C₄×C₆₁	1	4	yes	complemented
72	C₆₁	#7268	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²×C₈	D₄×C₆₁	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄⋊C₄	—	4	2	no	—
76	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
77	M₃₂	—	2	4	no	—
78	M₃₂	—	4	2	no	—
79	C₂×C₁₆	—	4	2	no	—
80	#106	—	4	2	no	—
81	#106	—	4	2	no	—
82	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
83	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
84	C₄×C₈	Q₈×C₆₁	1	1	yes	characteristic
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊C₄×C₆₁	1	2	yes	—
89	C₂×C₈	C₄⋊C₄×C₆₁	1	2	yes	—
90	C₂×C₈	D₈×C₆₁	1	2	yes	—
91	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
92	C₂×C₈	SD₁₆×C₆₁	1	2	yes	—
93	C₂×C₈	Q₁₆×C₆₁	1	2	yes	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	C₄²×C₆₁	1	1	yes	characteristic
98	C₄²	C₄⋊C₄×C₆₁	1	1	yes	characteristic
99	C₁₆	—	4	4	no	—
100	C₂³	#100×C₆₁	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	#95×C₆₁	1	1	yes	characteristic
107	C₂×C₄	#101×C₆₁	1	1	yes	characteristic
108	C₂×C₄	#102×C₆₁	1	2	yes	—
109	C₂×C₄	#103×C₆₁	1	2	yes	—
110	C₈	#102×C₆₁	1	2	yes	—
111	C₈	#103×C₆₁	1	2	yes	—
112	C₈	#106×C₆₁	1	2	yes	—
113	C₈	#107×C₆₁	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	C₆₁×#331	1	1	yes	characteristic
116	C₂²	C₆₁×#333	1	1	yes	characteristic
117	C₂²	C₆₁×#343	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	C₆₁×#332	1	2	yes	—
122	C₄	C₆₁×#340	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	C₆₁×#1296	1	1	yes	characteristic
126	C₂	C₆₁×#1359	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₆₁×#364

C₂

yes

maximal

C₆₁×#1844

C₂

yes

maximal, characteristic

C₂×M₃₂×C₆₁

—

C₂×#106×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#384

C₄

yes

—

C₆₁×#364

—

C₆₁×#364

C₄

yes

complemented

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

M₃₂×C₆₁

—

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

#106×C₆₁

—

#106×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

Q₈

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

Q₁₆

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₁₆×C₆₁

—

C₂³×C₆₁

#100

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₈×C₆₁

#102

yes

—

C₈×C₆₁

#103

yes

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

#107

yes

—

#7268

C₆₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#332

yes

—

C₄×C₆₁

#340

yes

—

C₂×#364

C₂×C₆₁

yes

—

#1844

C₂×C₆₁

yes

characteristic

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1296

yes

characteristic

C₂×C₆₁

#1359

yes

—

C₂×M₃₂

—

C₂×#106

—

C₂×C₄×C₈

C₂²×C₆₁

yes

characteristic

#384

C₄×C₆₁

yes

—

#364

—

#364

C₄×C₆₁

yes

complemented

C₆₁

#7268

yes

characteristic, direct product factor, Sylow

C₂²×C₈

D₄×C₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄×C₆₁

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#106

—

#106

—

C₄×C₈

C₂×C₄×C₆₁

yes

—

C₄×C₈

D₄×C₆₁

yes

characteristic

C₄×C₈

Q₈×C₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₆₁

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₆₁

yes

—

C₂×C₈

C₄⋊C₄×C₆₁

yes

—

C₂×C₈

D₈×C₆₁

yes

—

C₂×C₈

SD₁₆×C₆₁

yes

—

C₂×C₈

SD₁₆×C₆₁

yes

—

C₂×C₈

Q₁₆×C₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄²×C₆₁

yes

characteristic

C₄²

C₄⋊C₄×C₆₁

yes

characteristic

C₁₆

—

100

C₂³

#100×C₆₁

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#95×C₆₁

yes

characteristic

107

C₂×C₄

#101×C₆₁

yes

characteristic

108

C₂×C₄

#102×C₆₁

yes

—

109

C₂×C₄

#103×C₆₁

yes

—

110

C₈

#102×C₆₁

yes

—

111

C₈

#103×C₆₁

yes

—

112

C₈

#106×C₆₁

yes

—

113

C₈

#107×C₆₁

yes

—

114

C₂²

—

115

C₂²

C₆₁×#331

yes

characteristic

116

C₂²

C₆₁×#333

yes

characteristic

117

C₂²

C₆₁×#343

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₆₁×#332

yes

—

122

C₄

C₆₁×#340

yes

—

123

C₂

—

124

C₂

—

125

C₂

C₆₁×#1296

yes

characteristic

126

C₂

C₆₁×#1359

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₁	#331
commutator	C₄²	C₄²×C₆₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₆₁
prime radical	C₂³×C₆₁	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₁

#331

commutator

C₄²

C₄²×C₆₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₆₁

prime radical

C₂³×C₆₁

#100

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

N	H ≅ G/N	type
8. C₆₁×#364	C₄	semidirect
50. #7268	C₆₁	direct (N × H)
71. #364	C₄×C₆₁	semidirect
72. C₆₁	#7268	direct (N × H)

H ≅ G/N

type

8. C₆₁×#364

C₄

semidirect

50. #7268

C₆₁

direct (N × H)

71. #364

C₄×C₆₁

semidirect

72. C₆₁

#7268

direct (N × H)

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

subgroup	index
C₂×C₆₁×#364	2
C₆₁×#1844	2
#7268	61

subgroup

index

C₂×C₆₁×#364

C₆₁×#1844

#7268

Subgroups — C₆₁×#7268 #29510701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61×#7268 #29510701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7268 #29510701