Group #29645577 — subgroups

Subgroups — C₆₁⋊#17309 #29645577

Order 15616.

1466 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 283 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	#21444090	C₂	1	2	yes	maximal
2	#21444089	C₂	1	12	yes	maximal
3	C₆₁×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₂₂	C₂²	1	4	yes	—
5	C₂×#14519684	C₂²	1	12	yes	—
6	C₂×#14519684	C₂²	1	12	yes	—
7	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×#95×C₆₁	C₂²	1	6	yes	—
9	C₂²×Dic₁₂₂	—	4	12	no	—
10	C₂²×Dic₁₂₂	—	4	12	no	—
11	C₂²×Dic₁₂₂	Q₈	1	16	yes	complemented
12	C₂³×Dic₆₁	C₂³	1	8	yes	—
13	#14519684	—	4	12	no	—
14	#14519684	—	4	12	no	—
15	C₂²×Q₈×C₆₁	—	4	6	no	—
16	C₂²×Q₈×C₆₁	Q₈	1	4	yes	complemented
17	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
18	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
19	#95×C₆₁	—	4	6	no	—
20	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
21	C₂×Q₈×C₆₁	—	4	12	no	—
22	C₂×Q₈×C₆₁	—	4	16	no	—
23	C₂²×C₄×C₆₁	—	2	12	no	—
24	C₂²×C₄×C₆₁	—	2	12	no	—
25	C₂²×C₄×C₆₁	—	2	6	no	—
26	C₂²×C₄×C₆₁	—	2	12	no	—
27	C₂²×C₄×C₆₁	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₆₁	C₂×Q₈	1	8	yes	—
29	C₂²×Dic₆₁	—	2	24	no	—
30	C₂²×Dic₆₁	—	2	24	no	—
31	C₂²×Dic₆₁	C₂×Q₈	1	16	yes	—
32	C₂×Dic₁₂₂	—	4	48	no	—
33	C₂×Dic₁₂₂	—	4	48	no	—
34	C₂×Dic₁₂₂	—	4	16	no	—
35	C₂³×C₆₁	C₂²×Q₈	1	4	yes	—
36	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
37	C₂³×C₆₁	H(5,𝔽₂)	1	6	yes	—
38	C₂³×C₆₁	H(5,𝔽₂)	1	4	yes	—
39	Q₈×C₆₁	—	4	16	no	—
40	C₂×C₄×C₆₁	—	2	24	no	—
41	C₂×C₄×C₆₁	—	2	6	no	—
42	C₂×C₄×C₆₁	—	2	12	no	—
43	C₂×C₄×C₆₁	—	4	12	no	—
44	C₂×C₄×C₆₁	—	4	4	no	—
45	C₂×C₄×C₆₁	—	4	12	no	—
46	C₂×Dic₆₁	—	2	48	no	—
47	C₂×Dic₆₁	—	4	24	no	—
48	C₂×Dic₆₁	—	4	8	no	—
49	Dic₁₂₂	—	4	64	no	—
50	#17309	—	61	1	no	maximal, Sylow
51	C₂²×C₆₁	#543	1	12	yes	—
52	C₂²×C₆₁	#543	1	3	yes	—
53	C₂²×C₆₁	Q₈²	1	6	yes	—
54	C₂²×C₆₁	Q₈²	1	4	yes	—
55	C₂²×C₆₁	H(3,𝔽₄)	1	6	yes	—
56	C₂²×C₆₁	H(3,𝔽₄)	1	4	yes	—
57	C₄×C₆₁	—	4	8	no	—
58	C₄×C₆₁	—	4	6	no	—
59	Dic₆₁	—	4	16	no	—
60	#1446	—	61	12	no	—
61	#1446	—	61	2	no	—
62	#1446	D₆₁	1	1	yes	characteristic
63	C₂×C₆₁	#2811	1	4	yes	—
64	C₂×C₆₁	#2811	1	6	yes	—
65	C₂×C₆₁	#2840	1	4	yes	—
66	C₂×C₆₁	#2840	1	1	yes	characteristic
67	C₂³×Q₈	—	61	4	no	—
68	C₂³×Q₈	C₂×D₆₁	1	1	yes	characteristic
69	C₂×#95	—	61	12	no	—
70	C₂×#95	—	61	12	no	—
71	C₂×#95	C₂×D₆₁	1	6	yes	—
72	C₆₁	#17309	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	4	6	no	—
74	C₂²×Q₈	—	61	16	no	—
75	C₂²×Q₈	—	244	12	no	—
76	C₂²×Q₈	—	244	12	no	—
77	C₂²×Q₈	Dic₁₂₂	1	4	yes	complemented
78	C₂³×C₄	—	61	8	no	—
79	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
80	C₂³×C₄	C₂²×D₆₁	1	4	yes	—
81	#95	—	4	6	no	—
82	#95	—	244	12	no	—
83	#95	—	244	12	no	—
84	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
85	C₂×Q₈	—	4	12	no	—
86	C₂×Q₈	—	4	16	no	—
87	C₂×Q₈	—	244	48	no	—
88	C₂×Q₈	—	244	48	no	—
89	C₂×Q₈	—	244	16	no	—
90	C₂²×C₄	—	2	6	no	—
91	C₂²×C₄	—	2	12	no	—
92	C₂²×C₄	—	2	12	no	—
93	C₂²×C₄	—	2	12	no	—
94	C₂²×C₄	—	61	16	no	—
95	C₂²×C₄	—	122	24	no	—
96	C₂²×C₄	—	122	24	no	—
97	C₂²×C₄	C₂×Dic₁₂₂	1	6	yes	—
98	C₂²×C₄	Q₈×D₆₁	1	8	yes	—
99	C₂³	C₂×Q₈×D₆₁	1	4	yes	—
100	C₂³	C₂²×Dic₁₂₂	1	1	yes	characteristic
101	C₂³	#14519822	1	6	yes	—
102	C₂³	#14519823	1	4	yes	—
103	Q₈	—	4	16	no	—
104	Q₈	—	244	64	no	—
105	C₂×C₄	—	2	24	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	4	12	no	—
109	C₂×C₄	—	4	12	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	122	48	no	—
112	C₂×C₄	—	244	24	no	—
113	C₂×C₄	—	244	8	no	—
114	C₂²	#16840387	1	12	yes	—
115	C₂²	#16840388	1	3	yes	—
116	C₂²	#16840492	1	6	yes	—
117	C₂²	Q₈×Dic₁₂₂	1	4	yes	—
118	C₂²	#16840504	1	6	yes	—
119	C₂²	#16840505	1	4	yes	—
120	C₄	—	4	8	no	—
121	C₄	—	4	6	no	—
122	C₄	—	244	16	no	—
123	C₂	#21453811	1	4	yes	—
124	C₂	#21453812	1	6	yes	—
125	C₂	#21454071	1	4	yes	—
126	C₂	#21454072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21444090

C₂

yes

maximal

#21444089

C₂

yes

maximal

C₆₁×#1446

C₂

yes

maximal, characteristic

C₂³×Dic₁₂₂

C₂²

yes

—

C₂×#14519684

C₂²

yes

—

C₂×#14519684

C₂²

yes

—

C₂³×Q₈×C₆₁

C₂²

yes

characteristic

C₂×#95×C₆₁

C₂²

yes

—

C₂²×Dic₁₂₂

—

C₂²×Dic₁₂₂

—

C₂²×Dic₁₂₂

Q₈

yes

complemented

C₂³×Dic₆₁

C₂³

yes

—

#14519684

—

#14519684

—

C₂²×Q₈×C₆₁

—

C₂²×Q₈×C₆₁

Q₈

yes

complemented

C₂³×C₄×C₆₁

C₂³

yes

—

C₂³×C₄×C₆₁

C₂³

yes

—

#95×C₆₁

—

C₂⁴×C₆₁

C₂⁴

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×Q₈

yes

—

C₂²×C₄×C₆₁

C₂×Q₈

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂³×C₆₁

C₂²×Q₈

yes

—

C₂³×C₆₁

C₂²×Q₈

yes

characteristic

C₂³×C₆₁

H(5,𝔽₂)

yes

—

C₂³×C₆₁

H(5,𝔽₂)

yes

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

#17309

—

maximal, Sylow

C₂²×C₆₁

#543

yes

—

C₂²×C₆₁

#543

yes

—

C₂²×C₆₁

Q₈²

yes

—

C₂²×C₆₁

Q₈²

yes

—

C₂²×C₆₁

H(3,𝔽₄)

yes

—

C₂²×C₆₁

H(3,𝔽₄)

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

#1446

—

#1446

—

#1446

D₆₁

yes

characteristic

C₂×C₆₁

#2811

yes

—

C₂×C₆₁

#2811

yes

—

C₂×C₆₁

#2840

yes

—

C₂×C₆₁

#2840

yes

characteristic

C₂³×Q₈

—

C₂³×Q₈

C₂×D₆₁

yes

characteristic

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₆₁

yes

—

C₆₁

#17309

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

244

—

C₂²×Q₈

—

244

—

C₂²×Q₈

Dic₁₂₂

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂²×D₆₁

yes

—

C₂³×C₄

C₂²×D₆₁

yes

—

#95

—

#95

—

244

—

#95

—

244

—

C₂⁴

C₂³×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂×Dic₁₂₂

yes

—

C₂²×C₄

Q₈×D₆₁

yes

—

C₂³

C₂×Q₈×D₆₁

yes

—

100

C₂³

C₂²×Dic₁₂₂

yes

characteristic

101

C₂³

#14519822

yes

—

102

C₂³

#14519823

yes

—

103

Q₈

—

104

Q₈

—

244

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

122

—

112

C₂×C₄

—

244

—

113

C₂×C₄

—

244

—

114

C₂²

#16840387

yes

—

115

C₂²

#16840388

yes

—

116

C₂²

#16840492

yes

—

117

C₂²

Q₈×Dic₁₂₂

yes

—

118

C₂²

#16840504

yes

—

119

C₂²

#16840505

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

244

—

123

C₂

#21453811

yes

—

124

C₂

#21453812

yes

—

125

C₂

#21454071

yes

—

126

C₂

#21454072

yes

characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₆₁
commutator, socle, prime radical, cosocle	C₂⁴×C₆₁	C₂⁴
Fitting	C₆₁×#1446	C₂
hypercenter	#1446	D₆₁
hypocenter	C₆₁	#17309
LCS	G ▷ C₂⁴×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ #1446

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₆₁

commutator, socle, prime radical, cosocle

C₂⁴×C₆₁

C₂⁴

Fitting

C₆₁×#1446

C₂

D₆₁

C₆₁

G ▷ C₂⁴×C₆₁ ▷ C₆₁

C₁ ◁ C₂⁴ ◁ #1446

N	H ≅ G/N	type
11. C₂²×Dic₁₂₂	Q₈	semidirect
16. C₂²×Q₈×C₆₁	Q₈	semidirect
72. C₆₁	#17309	semidirect
77. C₂²×Q₈	Dic₁₂₂	semidirect

H ≅ G/N

type

11. C₂²×Dic₁₂₂

Q₈

semidirect

16. C₂²×Q₈×C₆₁

Q₈

semidirect

72. C₆₁

#17309

semidirect

77. C₂²×Q₈

Dic₁₂₂

semidirect

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

subgroup	index
#21444090	2
#21444089	2
C₆₁×#1446	2
#17309	61

subgroup

index

#21444090

#21444089

C₆₁×#1446

#17309

Subgroups — C₆₁⋊#17309 #29645577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C61⋊#17309 #29645577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁⋊#17309 #29645577