Group #28554253 — subgroups

Subgroups — C₅₉⋊#17309 #28554253

Order 15104.

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	#21422367	C₂	1	2	yes	maximal
2	#21422366	C₂	1	12	yes	maximal
3	C₅₉×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₁₈	C₂²	1	4	yes	—
5	C₂×C₅₉⋊#95	C₂²	1	12	yes	—
6	C₂×C₅₉⋊#95	C₂²	1	12	yes	—
7	C₂³×Q₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₂×#95×C₅₉	C₂²	1	6	yes	—
9	C₂²×Dic₁₁₈	—	4	12	no	—
10	C₂²×Dic₁₁₈	—	4	12	no	—
11	C₂²×Dic₁₁₈	Q₈	1	16	yes	complemented
12	C₂³×Dic₅₉	C₂³	1	8	yes	—
13	C₅₉⋊#95	—	4	12	no	—
14	C₅₉⋊#95	—	4	12	no	—
15	C₂²×Q₈×C₅₉	—	4	6	no	—
16	C₂²×Q₈×C₅₉	Q₈	1	4	yes	complemented
17	C₂³×C₄×C₅₉	C₂³	1	4	yes	—
18	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
19	#95×C₅₉	—	4	6	no	—
20	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
21	C₂×Q₈×C₅₉	—	4	12	no	—
22	C₂×Q₈×C₅₉	—	4	16	no	—
23	C₂²×C₄×C₅₉	—	2	6	no	—
24	C₂²×C₄×C₅₉	—	2	12	no	—
25	C₂²×C₄×C₅₉	—	2	12	no	—
26	C₂²×C₄×C₅₉	—	2	12	no	—
27	C₂²×C₄×C₅₉	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₅₉	C₂×Q₈	1	8	yes	—
29	C₂²×Dic₅₉	—	2	24	no	—
30	C₂²×Dic₅₉	—	2	24	no	—
31	C₂²×Dic₅₉	C₂×Q₈	1	16	yes	—
32	C₂×Dic₁₁₈	—	4	48	no	—
33	C₂×Dic₁₁₈	—	4	48	no	—
34	C₂×Dic₁₁₈	—	4	16	no	—
35	C₂³×C₅₉	C₂²×Q₈	1	4	yes	—
36	C₂³×C₅₉	C₂²×Q₈	1	1	yes	characteristic
37	C₂³×C₅₉	H(5,𝔽₂)	1	6	yes	—
38	C₂³×C₅₉	H(5,𝔽₂)	1	4	yes	—
39	Q₈×C₅₉	—	4	16	no	—
40	C₂×C₄×C₅₉	—	2	24	no	—
41	C₂×C₄×C₅₉	—	2	6	no	—
42	C₂×C₄×C₅₉	—	2	12	no	—
43	C₂×C₄×C₅₉	—	4	12	no	—
44	C₂×C₄×C₅₉	—	4	4	no	—
45	C₂×C₄×C₅₉	—	4	12	no	—
46	C₂×Dic₅₉	—	2	48	no	—
47	C₂×Dic₅₉	—	4	24	no	—
48	C₂×Dic₅₉	—	4	8	no	—
49	Dic₁₁₈	—	4	64	no	—
50	#17309	—	59	1	no	maximal, Sylow
51	C₂²×C₅₉	#543	1	12	yes	—
52	C₂²×C₅₉	#543	1	3	yes	—
53	C₂²×C₅₉	Q₈²	1	6	yes	—
54	C₂²×C₅₉	Q₈²	1	4	yes	—
55	C₂²×C₅₉	H(3,𝔽₄)	1	6	yes	—
56	C₂²×C₅₉	H(3,𝔽₄)	1	4	yes	—
57	C₄×C₅₉	—	4	8	no	—
58	C₄×C₅₉	—	4	6	no	—
59	Dic₅₉	—	4	16	no	—
60	#1446	—	59	12	no	—
61	#1446	—	59	2	no	—
62	#1446	D₅₉	1	1	yes	characteristic
63	C₂×C₅₉	#2811	1	4	yes	—
64	C₂×C₅₉	#2811	1	6	yes	—
65	C₂×C₅₉	#2840	1	4	yes	—
66	C₂×C₅₉	#2840	1	1	yes	characteristic
67	C₂³×Q₈	—	59	4	no	—
68	C₂³×Q₈	C₂×D₅₉	1	1	yes	characteristic
69	C₂×#95	—	59	12	no	—
70	C₂×#95	—	59	12	no	—
71	C₂×#95	C₂×D₅₉	1	6	yes	—
72	C₅₉	#17309	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	4	6	no	—
74	C₂²×Q₈	—	59	16	no	—
75	C₂²×Q₈	—	236	12	no	—
76	C₂²×Q₈	—	236	12	no	—
77	C₂²×Q₈	Dic₁₁₈	1	4	yes	complemented
78	C₂³×C₄	—	59	8	no	—
79	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
80	C₂³×C₄	C₂²×D₅₉	1	4	yes	—
81	#95	—	4	6	no	—
82	#95	—	236	12	no	—
83	#95	—	236	12	no	—
84	C₂⁴	C₂³×D₅₉	1	1	yes	characteristic
85	C₂×Q₈	—	4	12	no	—
86	C₂×Q₈	—	4	16	no	—
87	C₂×Q₈	—	236	48	no	—
88	C₂×Q₈	—	236	48	no	—
89	C₂×Q₈	—	236	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	—	59	16	no	—
95	C₂²×C₄	—	118	24	no	—
96	C₂²×C₄	—	118	24	no	—
97	C₂²×C₄	C₂×Dic₁₁₈	1	6	yes	—
98	C₂²×C₄	Q₈×D₅₉	1	8	yes	—
99	C₂³	C₂×Q₈×D₅₉	1	4	yes	—
100	C₂³	C₂²×Dic₁₁₈	1	1	yes	characteristic
101	C₂³	#14273772	1	6	yes	—
102	C₂³	#14273773	1	4	yes	—
103	Q₈	—	4	16	no	—
104	Q₈	—	236	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	—	118	48	no	—
112	C₂×C₄	—	236	24	no	—
113	C₂×C₄	—	236	8	no	—
114	C₂²	#16838027	1	12	yes	—
115	C₂²	#16838028	1	3	yes	—
116	C₂²	#16838132	1	6	yes	—
117	C₂²	Q₈×Dic₁₁₈	1	4	yes	—
118	C₂²	#16838144	1	6	yes	—
119	C₂²	#16838145	1	4	yes	—
120	C₄	—	4	8	no	—
121	C₄	—	4	6	no	—
122	C₄	—	236	16	no	—
123	C₂	#21432088	1	4	yes	—
124	C₂	#21432089	1	6	yes	—
125	C₂	#21432348	1	4	yes	—
126	C₂	#21432349	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21422367

C₂

yes

maximal

#21422366

C₂

yes

maximal

C₅₉×#1446

C₂

yes

maximal, characteristic

C₂³×Dic₁₁₈

C₂²

yes

—

C₂×C₅₉⋊#95

C₂²

yes

—

C₂×C₅₉⋊#95

C₂²

yes

—

C₂³×Q₈×C₅₉

C₂²

yes

characteristic

C₂×#95×C₅₉

C₂²

yes

—

C₂²×Dic₁₁₈

—

C₂²×Dic₁₁₈

—

C₂²×Dic₁₁₈

Q₈

yes

complemented

C₂³×Dic₅₉

C₂³

yes

—

C₅₉⋊#95

—

C₅₉⋊#95

—

C₂²×Q₈×C₅₉

—

C₂²×Q₈×C₅₉

Q₈

yes

complemented

C₂³×C₄×C₅₉

C₂³

yes

—

C₂³×C₄×C₅₉

C₂³

yes

—

#95×C₅₉

—

C₂⁴×C₅₉

C₂⁴

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×Q₈

yes

—

C₂²×C₄×C₅₉

C₂×Q₈

yes

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

—

C₂²×Dic₅₉

C₂×Q₈

yes

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂³×C₅₉

C₂²×Q₈

yes

—

C₂³×C₅₉

C₂²×Q₈

yes

characteristic

C₂³×C₅₉

H(5,𝔽₂)

yes

—

C₂³×C₅₉

H(5,𝔽₂)

yes

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

#17309

—

maximal, Sylow

C₂²×C₅₉

#543

yes

—

C₂²×C₅₉

#543

yes

—

C₂²×C₅₉

Q₈²

yes

—

C₂²×C₅₉

Q₈²

yes

—

C₂²×C₅₉

H(3,𝔽₄)

yes

—

C₂²×C₅₉

H(3,𝔽₄)

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

#1446

—

#1446

—

#1446

D₅₉

yes

characteristic

C₂×C₅₉

#2811

yes

—

C₂×C₅₉

#2811

yes

—

C₂×C₅₉

#2840

yes

—

C₂×C₅₉

#2840

yes

characteristic

C₂³×Q₈

—

C₂³×Q₈

C₂×D₅₉

yes

characteristic

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₅₉

yes

—

C₅₉

#17309

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

236

—

C₂²×Q₈

—

236

—

C₂²×Q₈

Dic₁₁₈

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂²×D₅₉

yes

—

C₂³×C₄

C₂²×D₅₉

yes

—

#95

—

#95

—

236

—

#95

—

236

—

C₂⁴

C₂³×D₅₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

236

—

C₂×Q₈

—

236

—

C₂×Q₈

—

236

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

118

—

C₂²×C₄

—

118

—

C₂²×C₄

C₂×Dic₁₁₈

yes

—

C₂²×C₄

Q₈×D₅₉

yes

—

C₂³

C₂×Q₈×D₅₉

yes

—

100

C₂³

C₂²×Dic₁₁₈

yes

characteristic

101

C₂³

#14273772

yes

—

102

C₂³

#14273773

yes

—

103

Q₈

—

104

Q₈

—

236

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

118

—

112

C₂×C₄

—

236

—

113

C₂×C₄

—

236

—

114

C₂²

#16838027

yes

—

115

C₂²

#16838028

yes

—

116

C₂²

#16838132

yes

—

117

C₂²

Q₈×Dic₁₁₈

yes

—

118

C₂²

#16838144

yes

—

119

C₂²

#16838145

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

236

—

123

C₂

#21432088

yes

—

124

C₂

#21432089

yes

—

125

C₂

#21432348

yes

—

126

C₂

#21432349

yes

characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₉
commutator, socle, prime radical, cosocle	C₂⁴×C₅₉	C₂⁴
Fitting	C₅₉×#1446	C₂
hypercenter	#1446	D₅₉
hypocenter	C₅₉	#17309
LCS	G ▷ C₂⁴×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ #1446

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₅₉

commutator, socle, prime radical, cosocle

C₂⁴×C₅₉

C₂⁴

Fitting

C₅₉×#1446

C₂

D₅₉

C₅₉

G ▷ C₂⁴×C₅₉ ▷ C₅₉

C₁ ◁ C₂⁴ ◁ #1446

N	H ≅ G/N	type
11. C₂²×Dic₁₁₈	Q₈	semidirect
16. C₂²×Q₈×C₅₉	Q₈	semidirect
72. C₅₉	#17309	semidirect
77. C₂²×Q₈	Dic₁₁₈	semidirect

H ≅ G/N

type

11. C₂²×Dic₁₁₈

Q₈

semidirect

16. C₂²×Q₈×C₅₉

Q₈

semidirect

72. C₅₉

#17309

semidirect

77. C₂²×Q₈

Dic₁₁₈

semidirect

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

subgroup	index
#21422367	2
#21422366	2
C₅₉×#1446	2
#17309	59

subgroup

index

#21422367

#21422366

C₅₉×#1446

#17309

Subgroups — C₅₉⋊#17309 #28554253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C59⋊#17309 #28554253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉⋊#17309 #28554253