Group #28500708 — subgroups

Subgroups — #28500708

Order 15104.

198 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 69 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	#21425968	C₂	1	1	yes	maximal, characteristic
2	#21423004	C₂	1	2	yes	maximal
3	#21422065	C₂	1	2	yes	maximal
4	#21422016	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#1368	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₇₂	C₄	1	1	yes	characteristic
7	#16837529	—	2	2	no	—
8	C₄×Dic₂₃₆	C₂²	1	2	yes	—
9	#16837230	C₄	1	1	yes	characteristic
10	#16837207	C₄	1	2	yes	—
11	#16837178	C₂²	1	1	yes	characteristic
12	#16837132	—	2	2	no	—
13	#16837123	C₂²	1	1	yes	characteristic
14	C₅₉×#346	C₂²	1	1	yes	characteristic
15	C₅₉×#334	C₂²	1	2	yes	—
16	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₁₈	—	2	2	no	—
18	Dic₄₇₂	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	—	2	1	no	—
20	#14273661	C₂×C₄	1	2	yes	—
21	Dic₅₉⋊C₈	—	4	2	no	—
22	#14273652	—	2	2	no	—
23	#14273637	—	4	2	no	—
24	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₅₉	—	2	2	no	—
28	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₁₈	—	2	2	no	—
30	C₂×C₈×C₅₉	—	4	2	no	—
31	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
33	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₄⋊Dic₅₉	—	2	2	no	—
35	Dic₅₉⋊C₄	—	4	2	no	—
36	C₄×Dic₅₉	—	4	2	no	—
37	C₂×C₅₉⋊C₈	—	2	2	no	—
38	Dic₂₃₆	—	2	2	no	—
39	Dic₂₃₆	—	2	2	no	—
40	C₁₆×C₅₉	—	2	1	no	—
41	C₁₆×C₅₉	M₁₆	1	2	yes	—
42	C₅₉⋊C₁₆	—	4	1	no	—
43	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×Dic₅₉	—	4	2	no	—
46	C₂×Dic₅₉	—	4	2	no	—
47	Dic₁₁₈	—	4	2	no	—
48	Dic₁₁₈	—	4	2	no	—
49	C₈×C₅₉	—	2	1	no	—
50	C₈×C₅₉	—	8	2	no	—
51	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
52	C₈×C₅₉	#131	1	1	yes	characteristic
53	C₅₉⋊C₈	—	4	2	no	—
54	#12088	—	59	1	no	maximal, Sylow
55	C₂²×C₅₉	#442	1	1	yes	characteristic
56	C₄×C₅₉	—	2	2	no	—
57	C₄×C₅₉	#436	1	1	yes	characteristic
58	C₄×C₅₉	#444	1	1	yes	characteristic
59	Dic₅₉	—	8	2	no	—
60	Dic₅₉	—	8	2	no	—
61	#2176	—	59	1	no	—
62	#1593	—	59	2	no	—
63	#1368	D₅₉	1	1	yes	characteristic
64	#1334	—	59	2	no	—
65	#1313	—	59	1	no	—
66	C₂×C₅₉	#1594	1	1	yes	characteristic
67	C₂×C₅₉	#2180	1	2	yes	—
68	C₂×Q₃₂	—	59	1	no	—
69	#446	—	118	2	no	—
70	C₄×Q₁₆	—	59	2	no	—
71	#367	—	59	1	no	—
72	#358	—	59	2	no	—
73	#346	—	59	1	no	—
74	#346	C₂×D₅₉	1	1	yes	characteristic
75	#334	C₂×D₅₉	1	2	yes	—
76	Q₈⋊C₈	—	118	2	no	—
77	#322	—	59	1	no	—
78	C₅₉	#12088	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	59	2	no	—
80	C₄×Q₈	—	118	2	no	—
81	Q₃₂	—	118	2	no	—
82	C₂×C₁₆	—	118	1	no	—
83	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
84	C₂×C₁₆	C₄×D₅₉	1	1	yes	characteristic
85	#107	—	59	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	236	2	no	—
88	#103	—	118	2	no	—
89	#97	—	236	2	no	—
90	C₄×C₈	—	59	2	no	—
91	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
92	C₂×Q₈	—	118	2	no	—
93	Q₁₆	—	118	2	no	—
94	Q₁₆	—	118	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	118	2	no	—
97	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
98	C₂×C₈	#11745689	1	1	yes	characteristic
99	C₄⋊C₄	—	118	2	no	—
100	C₄⋊C₄	—	236	2	no	—
101	C₄²	—	236	2	no	—
102	C₄²	D₄×D₅₉	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	236	1	no	—
105	C₁₆	#11745659	1	2	yes	—
106	Q₈	—	236	2	no	—
107	Q₈	—	236	2	no	—
108	C₂×C₄	—	236	2	no	—
109	C₂×C₄	—	236	2	no	—
110	C₂×C₄	#14273689	1	1	yes	characteristic
111	C₂×C₄	#14273754	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	8	2	no	—
114	C₈	—	236	2	no	—
115	C₈	C₂×#11745659	1	1	yes	characteristic
116	C₈	#14273735	1	1	yes	characteristic
117	C₂²	#16837513	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	472	2	no	—
120	C₄	—	472	2	no	—
121	C₄	#16837479	1	1	yes	characteristic
122	C₄	#16837523	1	1	yes	characteristic
123	C₂	#21423009	1	1	yes	characteristic
124	C₂	#21425986	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21425968

C₂

yes

maximal, characteristic

#21423004

C₂

yes

maximal

#21422065

C₂

yes

maximal

#21422016

C₂

yes

maximal, characteristic

C₅₉×#1368

C₂

yes

maximal, characteristic

C₂×Dic₄₇₂

C₄

yes

characteristic

#16837529

—

C₄×Dic₂₃₆

C₂²

yes

—

#16837230

C₄

yes

characteristic

#16837207

C₄

yes

—

#16837178

C₂²

yes

characteristic

#16837132

—

#16837123

C₂²

yes

characteristic

C₅₉×#346

C₂²

yes

characteristic

C₅₉×#334

C₂²

yes

—

C₂×Dic₂₃₆

C₂×C₄

yes

—

C₄×Dic₁₁₈

—

Dic₄₇₂

—

C₂×C₅₉⋊C₁₆

—

#14273661

C₂×C₄

yes

—

Dic₅₉⋊C₈

—

#14273652

—

#14273637

—

C₄×C₅₉⋊C₈

D₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

C₁₆×C₅₉

—

C₁₆×C₅₉

M₁₆

yes

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×M₁₆

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

Dic₅₉

—

Dic₅₉

—

#2176

—

#1593

—

#1368

D₅₉

yes

characteristic

#1334

—

#1313

—

C₂×C₅₉

#1594

yes

characteristic

C₂×C₅₉

#2180

yes

—

C₂×Q₃₂

—

#446

—

118

—

C₄×Q₁₆

—

—

—

—

C₂×D₅₉

yes

characteristic

#334

C₂×D₅₉

yes

—

Q₈⋊C₈

—

118

—

#322

—

C₅₉

#12088

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

118

—

Q₃₂

—

118

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₄×D₅₉

yes

characteristic

C₂×C₁₆

C₄×D₅₉

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

236

—

#103

—

118

—

#97

—

236

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

118

—

Q₁₆

—

118

—

Q₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

C₂×C₈

#11745689

yes

characteristic

C₄⋊C₄

—

118

—

100

C₄⋊C₄

—

236

—

101

C₄²

—

236

—

102

C₄²

D₄×D₅₉

yes

characteristic

103

C₁₆

—

104

C₁₆

—

236

—

105

C₁₆

#11745659

yes

—

106

Q₈

—

236

—

107

Q₈

—

236

—

108

C₂×C₄

—

236

—

109

C₂×C₄

—

236

—

110

C₂×C₄

#14273689

yes

characteristic

111

C₂×C₄

#14273754

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

236

—

115

C₈

C₂×#11745659

yes

characteristic

116

C₈

#14273735

yes

characteristic

117

C₂²

#16837513

yes

characteristic

118

C₄

—

119

C₄

—

472

—

120

C₄

—

472

—

121

C₄

#16837479

yes

characteristic

122

C₄

#16837523

yes

characteristic

123

C₂

#21423009

yes

characteristic

124

C₂

#21425986

yes

—

	subgroup	G/H
center	C₂²	#16837513
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1368	C₂
socle, prime radical	C₂²×C₅₉	#442
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1368	D₅₉
hypocenter	C₅₉	#12088
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#16837513

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1368

C₂

socle, prime radical

C₂²×C₅₉

#442

cosocle

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
78. C₅₉	#12088	semidirect

H ≅ G/N

type

78. C₅₉

#12088

semidirect

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

subgroup	index
#21425968	2
#21423004	2
#21422065	2
#21422016	2
C₅₉×#1368	2
#12088	59

subgroup

index

C₅₉×#1368

Subgroups — #28500708

Conjugacy classes of subgroups

Complements

Maximal subgroups