Group #28505084 — subgroups

Subgroups — #28505084

Order 15104.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#21425695	C₂	1	1	yes	maximal, characteristic
2	#21424219	C₂	1	2	yes	maximal
3	#21422200	C₂	1	1	yes	maximal, characteristic
4	#21422191	C₂	1	2	yes	maximal
5	C₅₉×#1384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14273663	C₂²	1	2	yes	—
8	#16837357	C₂²	1	1	yes	characteristic
9	#16837177	C₄	1	2	yes	complemented
10	C₁₆×Dic₅₉	C₄	1	2	yes	complemented
11	#16837172	—	2	2	no	—
12	#16837158	—	2	2	no	—
13	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#427	C₂²	1	2	yes	—
15	C₂×#11745664	—	2	2	no	—
16	C₂×#11745664	D₄	1	1	yes	characteristic
17	#14273686	D₄	1	1	yes	characteristic
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	#14273663	—	2	2	no	—
22	#14273663	—	2	2	no	—
23	#14273637	Q₈	1	2	yes	—
24	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
26	#117×C₅₉	—	2	2	no	—
27	M₃₂×C₅₉	—	2	2	no	—
28	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
29	#107×C₅₉	—	2	2	no	—
30	#106×C₅₉	—	2	2	no	—
31	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₅₉	#41	1	2	yes	—
35	C₂×C₈×C₅₉	#41	1	2	yes	—
36	C₄⋊C₄×C₅₉	—	4	2	no	—
37	C₂²⋊C₄×C₅₉	—	4	2	no	—
38	C₄²×C₅₉	—	4	2	no	—
39	C₂²⋊Dic₅₉	—	2	1	no	—
40	C₄⋊Dic₅₉	—	2	1	no	—
41	C₄×Dic₅₉	C₄⋊C₄	1	2	yes	—
42	#11745664	—	2	2	no	—
43	#11745664	—	4	2	no	—
44	#11745664	—	4	2	no	—
45	C₂×C₅₉⋊C₈	—	4	2	no	—
46	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
47	C₁₆×C₅₉	—	2	2	no	—
48	C₅₉⋊C₁₆	—	2	1	no	—
49	C₅₉⋊C₁₆	—	2	1	no	—
50	C₂³×C₅₉	#135	1	1	yes	characteristic
51	C₂×C₄×C₅₉	—	8	2	no	—
52	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₅₉	#124	1	2	yes	—
54	C₂×C₄×C₅₉	#135	1	2	yes	—
55	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
56	C₂×Dic₅₉	—	2	2	no	—
57	C₈×C₅₉	#117	1	1	yes	characteristic
58	C₈×C₅₉	#117	1	1	yes	characteristic
59	C₈×C₅₉	#125	1	2	yes	—
60	C₅₉⋊C₈	—	2	2	no	—
61	C₅₉⋊C₈	—	8	2	no	—
62	#12893	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	—	4	1	no	—
64	C₂²×C₅₉	#427	1	1	yes	characteristic
65	C₂²×C₅₉	#488	1	2	yes	—
66	C₄×C₅₉	—	16	2	no	—
67	C₄×C₅₉	#382	1	1	yes	characteristic
68	C₄×C₅₉	#427	1	1	yes	characteristic
69	C₄×C₅₉	#488	1	2	yes	—
70	Dic₅₉	—	4	2	no	—
71	#2108	—	59	1	no	—
72	#1833	—	59	2	no	—
73	#1385	—	59	1	no	—
74	#1384	D₅₉	1	1	yes	characteristic
75	#1382	—	59	2	no	—
76	C₂×C₅₉	—	2	1	no	—
77	C₂×C₅₉	—	2	2	no	—
78	C₂×C₅₉	#1835	1	1	yes	characteristic
79	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
80	C₂²×C₁₆	—	59	1	no	—
81	C₂×#108	—	59	2	no	—
82	#427	C₂×D₅₉	1	2	yes	—
83	#405	—	59	1	no	—
84	#346	—	59	2	no	—
85	C₄×C₁₆	—	59	2	no	—
86	#344	—	118	2	no	—
87	H(3,ℤ/4)	—	118	2	no	—
88	C₅₉	#12893	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₁₆	—	59	1	no	—
90	C₂×M₁₆	—	118	2	no	—
91	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
92	#117	—	2	2	no	—
93	#117	—	59	1	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	59	1	no	—
96	C₂×C₁₆	—	59	1	no	—
97	C₂×C₁₆	—	118	2	no	—
98	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
99	#108	—	118	2	no	—
100	#108	—	118	2	no	—
101	#107	—	2	2	no	—
102	#106	—	2	2	no	—
103	#97	—	59	2	no	—
104	C₄×C₈	—	59	2	no	—
105	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
106	M₁₆	—	118	2	no	—
107	M₁₆	—	236	2	no	—
108	M₁₆	—	236	2	no	—
109	C₂×C₈	—	59	2	no	—
110	C₂×C₈	—	236	2	no	—
111	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
112	C₂×C₈	#11745685	1	2	yes	—
113	C₂×C₈	#11745687	1	2	yes	—
114	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	118	1	no	—
117	C₂²⋊C₄	—	4	2	no	—
118	C₂²⋊C₄	—	118	1	no	—
119	C₄²	—	4	2	no	—
120	C₄²	—	59	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	118	1	no	—
123	C₁₆	—	118	1	no	—
124	C₂³	#14273746	1	1	yes	characteristic
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	—	118	2	no	—
127	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
128	C₂×C₄	#14273713	1	2	yes	—
129	C₂×C₄	#14273748	1	2	yes	—
130	C₂×C₄	#14273750	1	1	yes	characteristic
131	C₈	—	118	2	no	—
132	C₈	—	472	2	no	—
133	C₈	#14273683	1	1	yes	characteristic
134	C₈	#14273684	1	1	yes	characteristic
135	C₈	#14273717	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#16837439	1	1	yes	characteristic
138	C₂²	#16837755	1	2	yes	—
139	C₄	—	16	2	no	—
140	C₄	—	236	2	no	—
141	C₄	#16837266	1	1	yes	characteristic
142	C₄	#16837437	1	1	yes	characteristic
143	C₄	#16837758	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	2	1	no	—
146	C₂	#21424225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21425695

C₂

yes

maximal, characteristic

#21424219

C₂

yes

maximal

#21422200

C₂

yes

maximal, characteristic

#21422191

C₂

yes

maximal

C₅₉×#1384

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₂×#14273663

C₂²

yes

—

#16837357

C₂²

yes

characteristic

#16837177

C₄

yes

complemented

C₁₆×Dic₅₉

C₄

yes

complemented

#16837172

—

#16837158

—

C₂×M₃₂×C₅₉

C₂²

yes

characteristic

C₅₉×#427

C₂²

yes

—

C₂×#11745664

—

C₂×#11745664

D₄

yes

characteristic

#14273686

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

C₂×C₄

yes

—

—

Q₈

yes

—

C₈×Dic₅₉

C₂×C₄

yes

—

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

—

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

#107×C₅₉

—

#106×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₉

#41

yes

—

C₂×C₈×C₅₉

#41

yes

—

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

C₄⋊C₄

yes

—

#11745664

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

#135

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

#124

yes

—

C₂×C₄×C₅₉

#135

yes

—

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

#12893

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#427

yes

characteristic

C₂²×C₅₉

#488

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

#1382

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1835

yes

characteristic

C₂×M₃₂

C₂×D₅₉

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

#427

C₂×D₅₉

yes

—

#405

—

#346

—

C₄×C₁₆

—

#344

—

118

—

H(3,ℤ/4)

—

118

—

C₅₉

#12893

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

118

—

C₂²×C₈

C₂²×D₅₉

yes

characteristic

#117

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₄×D₅₉

yes

—

#108

—

118

—

100

#108

—

118

—

101

#107

—

102

#106

—

103

#97

—

104

C₄×C₈

—

105

C₂²×C₄

D₄×D₅₉

yes

characteristic

106

M₁₆

—

118

—

107

M₁₆

—

236

—

108

M₁₆

—

236

—

109

C₂×C₈

—

110

C₂×C₈

—

236

—

111

C₂×C₈

C₂×C₄×D₅₉

yes

characteristic

112

C₂×C₈

#11745685

yes

—

113

C₂×C₈

#11745687

yes

—

114

C₂×C₈

Q₈×D₅₉

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

118

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

118

—

119

C₄²

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

118

—

123

C₁₆

—

118

—

124

C₂³

#14273746

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

118

—

127

C₂×C₄

C₄⋊C₄×D₅₉

yes

characteristic

128

C₂×C₄

#14273713

yes

—

129

C₂×C₄

#14273748

yes

—

130

C₂×C₄

#14273750

yes

characteristic

131

C₈

—

118

—

132

C₈

—

472

—

133

C₈

#14273683

yes

characteristic

134

C₈

#14273684

yes

characteristic

135

C₈

#14273717

yes

—

136

C₂²

—

137

C₂²

#16837439

yes

characteristic

138

C₂²

#16837755

yes

—

139

C₄

—

140

C₄

—

236

—

141

C₄

#16837266

yes

characteristic

142

C₄

#16837437

yes

characteristic

143

C₄

#16837758

yes

—

144

C₂

—

145

C₂

—

146

C₂

#21424225

yes

characteristic

	subgroup	G/H
center	C₄	#16837437
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1384	C₂
socle	C₂×C₅₉	#1835
prime radical	C₂³×C₅₉	#135
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1384	D₅₉
hypocenter	C₅₉	#12893
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#16837437

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#1384

C₂

C₂×C₅₉

C₂³×C₅₉

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
9. #16837177	C₄	semidirect
10. C₁₆×Dic₅₉	C₄	semidirect
88. C₅₉	#12893	semidirect

H ≅ G/N

type

9. #16837177

C₄

semidirect

10. C₁₆×Dic₅₉

C₄

semidirect

88. C₅₉

#12893

semidirect

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

subgroup	index
#21425695	2
#21424219	2
#21422200	2
#21422191	2
C₅₉×#1384	2
#12893	59

subgroup

index

C₅₉×#1384

Subgroups — #28505084

Conjugacy classes of subgroups

Complements

Maximal subgroups