Group #28505141 — subgroups

Subgroups — #28505141

Order 15104.

306 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 179 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×#16837232	C₂	1	4	yes	maximal
2	#21424268	C₂	1	2	yes	maximal
3	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273663	—	2	2	no	—
5	C₂×#14273663	C₂²	1	4	yes	—
6	#16837426	—	2	2	no	—
7	#16837232	—	2	8	no	—
8	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
10	C₅₉×#346	C₄	1	4	yes	—
11	C₂×#11745664	—	2	4	no	—
12	#14273663	—	2	8	no	—
13	#14273663	—	4	4	no	—
14	C₄⋊(C₅₉⋊C₈)	—	4	4	no	—
15	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₅₉	D₄	1	4	yes	—
20	C₂×C₁₆×C₅₉	Q₈	1	4	yes	—
21	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₉	Q₈	1	2	yes	—
23	C₂²×C₄×C₅₉	Q₁₆	1	2	yes	—
24	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₉	D₈	1	2	yes	—
27	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
28	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
30	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
31	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
33	C₄²×C₅₉	D₈	1	2	yes	—
34	#11745664	—	4	4	no	—
35	#11745664	—	4	4	no	—
36	C₂×C₅₉⋊C₈	—	4	4	no	—
37	C₁₆×C₅₉	C₄⋊C₄	1	8	yes	—
38	C₂³×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₉	—	4	2	no	—
40	C₂×C₄×C₅₉	#107	1	4	yes	—
41	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
44	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₈×C₅₉	#107	1	4	yes	—
48	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
49	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
50	C₅₉⋊C₈	—	8	4	no	—
51	#12910	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
54	C₂²×C₅₉	#494	1	1	yes	characteristic
55	C₂²×C₅₉	#500	1	1	yes	characteristic
56	C₄×C₅₉	—	2	4	no	—
57	C₄×C₅₉	#384	1	1	yes	characteristic
58	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
59	C₄×C₅₉	#493	1	1	yes	characteristic
60	C₄×C₅₉	#500	1	1	yes	characteristic
61	C₂×#368	—	59	4	no	—
62	C₂×#346	D₅₉	1	1	yes	characteristic
63	#1848	—	59	2	no	—
64	C₂×C₅₉	—	2	1	no	—
65	C₂×C₅₉	—	2	1	no	—
66	C₂×C₅₉	#1845	1	1	yes	characteristic
67	C₂×C₅₉	#2161	1	2	yes	—
68	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
69	C₂×#108	—	59	4	no	—
70	C₂×#108	—	118	2	no	—
71	#423	—	118	2	no	—
72	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
73	#368	—	118	8	no	—
74	#346	Dic₅₉	1	4	yes	—
75	C₅₉	#12910	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	118	4	no	—
77	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
79	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
80	C₂×C₁₆	Dic₁₁₈	1	4	yes	—
81	C₂×C₁₆	D₂₃₆	1	4	yes	—
82	C₂×C₁₆	C₂×Dic₅₉	1	4	yes	—
83	#108	—	118	8	no	—
84	#108	—	236	4	no	—
85	C₄⋊C₈	—	236	4	no	—
86	C₄×C₈	Dic₁₁₈	1	2	yes	—
87	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
88	C₂²×C₄	Dic₂₃₆	1	2	yes	—
89	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
90	M₁₆	—	236	4	no	—
91	M₁₆	—	236	4	no	—
92	C₂×C₈	—	236	4	no	—
93	C₂×C₈	D₄₇₂	1	2	yes	—
94	C₂×C₈	Dic₂₃₆	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
96	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
97	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
98	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
99	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
100	C₄²	D₄₇₂	1	2	yes	—
101	C₄²	C₄⋊Dic₅₉	1	2	yes	—
102	C₁₆	C₄⋊Dic₅₉	1	8	yes	—
103	C₂³	C₂×Dic₂₃₆	1	1	yes	characteristic
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	#14273661	1	4	yes	—
106	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
107	C₂×C₄	#14273724	1	1	yes	characteristic
108	C₂×C₄	#14273727	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
110	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
112	C₈	—	472	4	no	—
113	C₈	#14273661	1	4	yes	—
114	C₈	C₂×C₄⋊Dic₅₉	1	2	yes	—
115	C₈	#14273727	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#14273661	1	1	yes	characteristic
118	C₂²	#16837782	1	1	yes	characteristic
119	C₂²	#16837807	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	#16837272	1	1	yes	characteristic
122	C₄	C₂×#14273661	1	1	yes	characteristic
123	C₄	#16837779	1	1	yes	characteristic
124	C₄	#16837807	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	#21424256	1	1	yes	characteristic
128	C₂	#21425891	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837232

C₂

yes

maximal

#21424268

C₂

yes

maximal

C₂×C₅₉×#346

C₂

yes

maximal, characteristic

C₂×#14273663

—

C₂×#14273663

C₂²

yes

—

#16837426

—

#16837232

—

C₂²×C₁₆×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#346

C₄

yes

—

C₂×#11745664

—

#14273663

—

#14273663

—

C₄⋊(C₅₉⋊C₈)

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

Q₁₆

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

D₈

yes

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

D₈

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

C₄⋊C₄

yes

—

C₂³×C₅₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#107

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

characteristic

C₈×C₅₉

#107

yes

—

C₈×C₅₉

C₂×C₄⋊C₄

yes

—

C₈×C₅₉

C₄⋊Q₈

yes

—

C₅₉⋊C₈

—

#12910

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

D₅₉

yes

characteristic

#1848

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1845

yes

characteristic

C₂×C₅₉

#2161

yes

—

C₂²×C₁₆

C₂×D₅₉

yes

—

C₂×#108

—

C₂×#108

—

118

—

#423

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#368

—

118

—

#346

Dic₅₉

yes

—

C₅₉

#12910

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

D₂₃₆

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

D₂₃₆

yes

characteristic

C₂×C₁₆

Dic₁₁₈

yes

—

C₂×C₁₆

D₂₃₆

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

#108

—

118

—

#108

—

236

—

C₄⋊C₈

—

236

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₂²×C₄

Dic₂₃₆

yes

—

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

M₁₆

—

236

—

M₁₆

—

236

—

C₂×C₈

—

236

—

C₂×C₈

D₄₇₂

yes

—

C₂×C₈

Dic₂₃₆

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₂×Dic₁₁₈

yes

—

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

C₂×C₈

C₂×D₂₃₆

yes

—

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

100

C₄²

D₄₇₂

yes

—

101

C₄²

C₄⋊Dic₅₉

yes

—

102

C₁₆

C₄⋊Dic₅₉

yes

—

103

C₂³

C₂×Dic₂₃₆

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#14273661

yes

—

106

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

107

C₂×C₄

#14273724

yes

characteristic

108

C₂×C₄

#14273727

yes

characteristic

109

C₂×C₄

C₂×D₄₇₂

yes

characteristic

110

C₂×C₄

C₂×D₄₇₂

yes

characteristic

111

C₂×C₄

C₂×Dic₂₃₆

yes

characteristic

112

C₈

—

472

—

113

C₈

#14273661

yes

—

114

C₈

C₂×C₄⋊Dic₅₉

yes

—

115

C₈

#14273727

yes

—

116

C₂²

—

117

C₂²

C₂×#14273661

yes

characteristic

118

C₂²

#16837782

yes

characteristic

119

C₂²

#16837807

yes

characteristic

120

C₄

—

121

C₄

#16837272

yes

characteristic

122

C₄

C₂×#14273661

yes

characteristic

123

C₄

#16837779

yes

characteristic

124

C₄

#16837807

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

#21424256

yes

characteristic

128

C₂

#21425891

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂×#107
prime radical	C₂³×C₅₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#12910
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₇₂

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₂×C₅₉×#346

C₂

socle

C₂²×C₅₉

C₂×#107

prime radical

C₂³×C₅₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
75. C₅₉	#12910	semidirect

H ≅ G/N

type

75. C₅₉

#12910

semidirect

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

subgroup	index
C₂×#16837232	2
#21424268	2
C₂×C₅₉×#346	2
#12910	59

subgroup

index

C₂×#16837232

#21424268

C₂×C₅₉×#346

#12910

Subgroups — #28505141

Conjugacy classes of subgroups

Complements

Maximal subgroups