Group #28505109 — subgroups

Subgroups — #28505109

Order 15104.

342 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 203 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×#16837223	C₂	1	4	yes	maximal
2	#21425852	C₂	1	2	yes	maximal
3	C₅₉×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273667	C₂²	1	4	yes	—
5	#16837223	—	2	8	no	—
6	C₄⋊(C₅₉⋊C₁₆)	—	2	4	no	—
7	C₂×#97×C₅₉	C₄	1	2	yes	—
8	C₂×#97×C₅₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#376	C₄	1	1	yes	characteristic
12	#14273667	—	2	8	no	—
13	C₂×C₅₉⋊C₁₆	—	2	4	no	—
14	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
19	#97×C₅₉	—	2	4	no	—
20	#97×C₅₉	C₈	1	4	yes	—
21	C₄×C₈×C₅₉	D₄	1	2	yes	—
22	C₄×C₈×C₅₉	D₄	1	4	yes	—
23	C₄×C₈×C₅₉	Q₈	1	4	yes	—
24	C₄×C₈×C₅₉	Q₈	1	2	yes	—
25	#95×C₅₉	—	2	2	no	—
26	C₂²×C₄×C₅₉	—	2	2	no	—
27	C₂²×C₄×C₅₉	M₁₆	1	2	yes	—
28	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₉	—	2	2	no	—
30	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
32	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
34	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
35	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
37	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₅₉	—	2	2	no	—
39	C₄²×C₅₉	C₄⋊C₄	1	4	yes	—
40	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
41	C₄²×C₅₉	C₂×C₈	1	2	yes	—
42	C₅₉⋊C₁₆	—	4	4	no	—
43	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₉	—	2	4	no	—
45	C₂×C₄×C₅₉	—	4	2	no	—
46	C₂×C₄×C₅₉	—	4	4	no	—
47	C₂×C₄×C₅₉	C₄⋊C₈	1	4	yes	—
48	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
52	C₈×C₅₉	—	2	2	no	—
53	C₈×C₅₉	C₄⋊C₈	1	8	yes	—
54	C₈×C₅₉	#127	1	1	yes	characteristic
55	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
56	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
57	#12901	—	59	1	no	maximal, Sylow
58	C₂²×C₅₉	—	2	2	no	—
59	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂²×C₅₉	#423	1	2	yes	—
61	C₄×C₅₉	—	2	4	no	—
62	C₄×C₅₉	—	4	4	no	—
63	C₄×C₅₉	#384	1	1	yes	characteristic
64	C₄×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
65	C₄×C₅₉	#423	1	1	yes	characteristic
66	C₂×#364	—	59	4	no	—
67	#2150	—	59	2	no	—
68	#1726	D₅₉	1	1	yes	characteristic
69	C₂×C₅₉	—	2	2	no	—
70	C₂×C₅₉	#1842	1	1	yes	characteristic
71	C₂×C₅₉	#2153	1	2	yes	—
72	C₂×M₃₂	—	59	4	no	—
73	C₂×#97	Dic₅₉	1	2	yes	—
74	C₂×#97	Dic₅₉	1	1	yes	characteristic
75	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
76	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
77	#376	Dic₅₉	1	1	yes	characteristic
78	#364	—	118	8	no	—
79	C₄⋊C₁₆	—	118	4	no	—
80	C₅₉	#12901	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
83	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
84	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
85	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
86	M₃₂	—	118	8	no	—
87	C₂×C₁₆	—	118	4	no	—
88	#97	—	2	4	no	—
89	#97	C₅₉⋊C₈	1	4	yes	—
90	C₄×C₈	Dic₁₁₈	1	4	yes	—
91	C₄×C₈	Dic₁₁₈	1	2	yes	—
92	C₄×C₈	D₂₃₆	1	4	yes	—
93	C₄×C₈	D₂₃₆	1	2	yes	—
94	#95	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#11745664	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₂×C₅₉⋊C₈	1	4	yes	—
100	C₂×C₈	#11745664	1	4	yes	—
101	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
102	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
103	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
105	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
106	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
107	C₄²	—	2	2	no	—
108	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
109	C₄²	C₄⋊Dic₅₉	1	4	yes	—
110	C₄²	C₄⋊Dic₅₉	1	2	yes	—
111	C₁₆	—	236	4	no	—
112	C₂³	C₂×#11745664	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
120	C₂×C₄	C₂×#11745664	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	C₄⋊(C₅₉⋊C₈)	1	8	yes	—
123	C₈	#14273724	1	1	yes	characteristic
124	C₈	#14273727	1	2	yes	—
125	C₈	#14273727	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
128	C₂²	#16837426	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#16837272	1	1	yes	characteristic
132	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	2	yes	—
133	C₄	#16837426	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#21424248	1	1	yes	characteristic
136	C₂	#21425862	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837223

C₂

yes

maximal

#21425852

C₂

yes

maximal

C₅₉×#1726

C₂

yes

maximal, characteristic

C₂×#14273667

C₂²

yes

—

#16837223

—

C₄⋊(C₅₉⋊C₁₆)

—

C₂×#97×C₅₉

C₄

yes

—

C₂×#97×C₅₉

C₄

yes

characteristic

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#376

C₄

yes

characteristic

#14273667

—

C₂×C₅₉⋊C₁₆

—

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

characteristic

#97×C₅₉

—

#97×C₅₉

C₈

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

Q₈

yes

—

#95×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

M₁₆

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₄⋊C₄

yes

—

C₄²×C₅₉

C₂×C₈

yes

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×M₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₄⋊C₈

yes

—

C₈×C₅₉

#127

yes

characteristic

C₈×C₅₉

C₄⋊Q₈

yes

—

C₈×C₅₉

C₄⋊Q₈

yes

characteristic

#12901

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₉

#423

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#384

yes

characteristic

C₄×C₅₉

C₂×C₄⋊C₈

yes

—

C₄×C₅₉

yes

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

#1842

yes

characteristic

C₂×C₅₉

#2153

yes

—

C₂×M₃₂

—

C₂×#97

Dic₅₉

yes

—

C₂×#97

Dic₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

#376

Dic₅₉

yes

characteristic

#364

—

118

—

C₄⋊C₁₆

—

118

—

C₅₉

#12901

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

—

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

characteristic

M₃₂

—

118

—

C₂×C₁₆

—

118

—

#97

—

#97

C₅₉⋊C₈

yes

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

Dic₁₁₈

yes

—

C₄×C₈

D₂₃₆

yes

—

C₄×C₈

D₂₃₆

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11745664

yes

—

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×C₅₉⋊C₈

yes

—

100

C₂×C₈

#11745664

yes

—

101

C₂×C₈

C₄⋊Dic₅₉

yes

—

102

C₂×C₈

C₄⋊Dic₅₉

yes

—

103

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

104

C₂×C₈

C₂×Dic₁₁₈

yes

—

105

C₂×C₈

C₂×D₂₃₆

yes

—

106

C₂×C₈

C₂×D₂₃₆

yes

characteristic

107

C₄²

—

108

C₄²

C₂×C₅₉⋊C₈

yes

—

109

C₄²

C₄⋊Dic₅₉

yes

—

110

C₄²

C₄⋊Dic₅₉

yes

—

111

C₁₆

—

236

—

112

C₂³

C₂×#11745664

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄⋊(C₅₉⋊C₈)

yes

—

117

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

—

118

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

119

C₂×C₄

C₂²×C₅₉⋊C₈

yes

characteristic

120

C₂×C₄

C₂×#11745664

yes

—

121

C₈

—

122

C₈

C₄⋊(C₅₉⋊C₈)

yes

—

123

C₈

#14273724

yes

characteristic

124

C₈

#14273727

yes

—

125

C₈

#14273727

yes

characteristic

126

C₂²

—

127

C₂²

C₂×C₄⋊(C₅₉⋊C₈)

yes

characteristic

128

C₂²

#16837426

yes

—

129

C₄

—

130

C₄

—

131

C₄

#16837272

yes

characteristic

132

C₄

C₂×C₄⋊(C₅₉⋊C₈)

yes

—

133

C₄

#16837426

yes

characteristic

134

C₂

—

135

C₂

#21424248

yes

characteristic

136

C₂

#21425862

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1726	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₈
prime radical	C₂³×C₅₉	C₂×M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1726	D₅₉
hypocenter	C₅₉	#12901
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1726

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₈

prime radical

C₂³×C₅₉

C₂×M₁₆

cosocle

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
80. C₅₉	#12901	semidirect

H ≅ G/N

type

80. C₅₉

#12901

semidirect

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

subgroup	index
C₂×#16837223	2
#21425852	2
C₅₉×#1726	2
#12901	59

subgroup

index

C₂×#16837223

#21425852

C₅₉×#1726

#12901

Subgroups — #28505109

Conjugacy classes of subgroups

Complements

Maximal subgroups