Group #28505086 — subgroups

Subgroups — #28505086

Order 15104.

234 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 111 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	#21424220	C₂	1	2	yes	maximal
2	#21422200	C₂	1	1	yes	maximal, characteristic
3	#21422198	C₂	1	1	yes	maximal, characteristic
4	#21422192	C₂	1	2	yes	maximal
5	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273663	C₂²	1	2	yes	—
7	#16837440	C₂²	1	2	yes	—
8	#16837173	—	2	2	no	—
9	#16837158	—	2	2	no	—
10	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
12	C₅₉×#405	C₂²	1	1	yes	characteristic
13	C₅₉×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₅₉	C₄	1	2	yes	complemented
15	C₂×#11745664	—	2	2	no	—
16	#14273686	—	2	2	no	—
17	#14273663	—	2	2	no	—
18	#14273663	—	2	2	no	—
19	#14273661	—	2	2	no	—
20	#14273659	—	2	2	no	—
21	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
23	#117×C₅₉	D₄	1	1	yes	characteristic
24	M₃₂×C₅₉	—	2	2	no	—
25	C₂×C₁₆×C₅₉	—	2	2	no	—
26	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
29	#97×C₅₉	Q₈	1	2	yes	—
30	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₅₉	—	2	2	no	—
33	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₅₉	#41	1	2	yes	—
37	C₂×C₈×C₅₉	#41	1	2	yes	—
38	C₄⋊C₄×C₅₉	—	2	1	no	—
39	C₂²⋊C₄×C₅₉	—	2	1	no	—
40	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
41	C₂²⋊Dic₅₉	—	4	2	no	—
42	C₄⋊Dic₅₉	—	4	2	no	—
43	C₄×Dic₅₉	—	4	2	no	—
44	#11745664	—	4	2	no	—
45	#11745664	—	4	2	no	—
46	C₂×C₅₉⋊C₈	—	4	2	no	—
47	C₁₆×C₅₉	—	2	1	no	—
48	C₁₆×C₅₉	—	2	1	no	—
49	C₁₆×C₅₉	—	2	2	no	—
50	C₂³×C₅₉	#135	1	1	yes	characteristic
51	C₂×C₄×C₅₉	—	2	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	—	8	2	no	—
57	C₈×C₅₉	—	2	2	no	—
58	C₈×C₅₉	#117	1	1	yes	characteristic
59	C₈×C₅₉	#117	1	1	yes	characteristic
60	C₈×C₅₉	#125	1	2	yes	—
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	—	4	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	—	16	2	no	—
71	#2108	D₅₉	1	1	yes	characteristic
72	#1833	—	59	2	no	—
73	#1385	—	59	1	no	—
74	#1384	—	59	1	no	—
75	#1382	—	59	2	no	—
76	C₂×C₅₉	—	2	2	no	—
77	C₂×C₅₉	—	2	1	no	—
78	C₂×C₅₉	#1835	1	1	yes	characteristic
79	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
80	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
81	C₂×#108	—	59	2	no	—
82	#427	—	59	2	no	—
83	#405	C₂×D₅₉	1	1	yes	characteristic
84	#346	Dic₅₉	1	2	yes	complemented
85	C₄×C₁₆	Dic₅₉	1	2	yes	complemented
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	—	118	2	no	—
90	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
92	#117	—	118	2	no	—
93	#117	D₂₃₆	1	1	yes	characteristic
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
98	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
99	#108	—	118	2	no	—
100	#108	—	118	2	no	—
101	#107	—	118	2	no	—
102	#106	—	118	2	no	—
103	#97	Dic₁₁₈	1	2	yes	—
104	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
105	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	236	2	no	—
108	M₁₆	—	236	2	no	—
109	C₂×C₈	—	236	2	no	—
110	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
111	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
112	C₂×C₈	#11745685	1	2	yes	—
113	C₂×C₈	#11745685	1	2	yes	—
114	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	236	2	no	—
117	C₂²⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	—	236	2	no	—
119	C₄²	—	236	2	no	—
120	C₄²	C₄⋊Dic₅₉	1	2	yes	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	2	no	—
124	C₂³	#14273749	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	472	2	no	—
127	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
128	C₂×C₄	#14273715	1	2	yes	—
129	C₂×C₄	#14273749	1	2	yes	—
130	C₂×C₄	#14273749	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	472	2	no	—
133	C₈	#14273686	1	1	yes	characteristic
134	C₈	#14273686	1	1	yes	characteristic
135	C₈	#14273719	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#16837440	1	1	yes	characteristic
138	C₂²	#16837757	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	944	2	no	—
141	C₄	#16837268	1	1	yes	characteristic
142	C₄	#16837440	1	1	yes	characteristic
143	C₄	#16837757	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#21424227	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424220

C₂

yes

maximal

#21422200

C₂

yes

maximal, characteristic

#21422198

C₂

yes

maximal, characteristic

#21422192

C₂

yes

maximal

C₅₉×#2108

C₂

yes

maximal, characteristic

C₂×#14273663

C₂²

yes

—

#16837440

C₂²

yes

—

#16837173

—

#16837158

—

C₂×M₃₂×C₅₉

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#405

C₂²

yes

characteristic

C₅₉×#346

C₄

yes

complemented

C₄×C₁₆×C₅₉

C₄

yes

—

—

—

—

—

—

C₂×M₁₆×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

#117×C₅₉

D₄

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

#97×C₅₉

Q₈

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

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

C₄⋊C₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

—

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

—

C₈×C₅₉

yes

C₈×C₅₉

yes

C₈×C₅₉

yes

—

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

—

#2108

D₅₉

yes

—

—

—

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#1835

yes

characteristic

C₂×M₃₂

C₂×D₅₉

yes

characteristic

C₂²×C₁₆

C₂×D₅₉

yes

—

—

C₂×D₅₉

yes

characteristic

#346

Dic₅₉

yes

complemented

C₄×C₁₆

Dic₅₉

yes

complemented

#344

—

118

—

H(3,ℤ/4)

—

118

—

C₅₉

#12893

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂×M₁₆

D₂₃₆

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

#117

—

118

—

#117

D₂₃₆

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

C₂×Dic₅₉

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

#108

—

118

—

100

#108

—

118

—

101

#107

—

118

—

102

#106

—

118

—

103

#97

Dic₁₁₈

yes

—

104

C₄×C₈

C₂×Dic₅₉

yes

—

105

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

106

M₁₆

—

107

M₁₆

—

236

—

108

M₁₆

—

236

—

109

C₂×C₈

—

236

—

110

C₂×C₈

C₄⋊Dic₅₉

yes

—

111

C₂×C₈

C₂×Dic₁₁₈

yes

characteristic

112

C₂×C₈

#11745685

yes

—

113

C₂×C₈

#11745685

yes

—

114

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

236

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

236

—

119

C₄²

—

236

—

120

C₄²

C₄⋊Dic₅₉

yes

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#14273749

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

472

—

127

C₂×C₄

C₂×C₄⋊Dic₅₉

yes

characteristic

128

C₂×C₄

#14273715

yes

—

129

C₂×C₄

#14273749

yes

—

130

C₂×C₄

#14273749

yes

characteristic

131

C₈

—

132

C₈

—

472

—

133

C₈

#14273686

yes

characteristic

134

C₈

#14273686

yes

characteristic

135

C₈

#14273719

yes

—

136

C₂²

—

137

C₂²

#16837440

yes

characteristic

138

C₂²

#16837757

yes

—

139

C₄

—

140

C₄

—

944

—

141

C₄

#16837268

yes

characteristic

142

C₄

#16837440

yes

characteristic

143

C₄

#16837757

yes

—

144

C₂

—

145

C₂

—

146

C₂

#21424227

yes

characteristic

	subgroup	G/H
center	C₄	#16837440
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2108	C₂
socle	C₂×C₅₉	#1835
prime radical	C₂³×C₅₉	#135
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#12893
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

subgroup

G/H

center

C₄

#16837440

commutator

C₂×C₈×C₅₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₉

Fitting

C₅₉×#2108

C₂

C₂×C₅₉

C₂³×C₅₉

C₂²×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

N	H ≅ G/N	type
13. C₅₉×#346	C₄	semidirect
14. C₄×C₁₆×C₅₉	C₄	semidirect
84. #346	Dic₅₉	semidirect
85. C₄×C₁₆	Dic₅₉	semidirect
88. C₅₉	#12893	semidirect

H ≅ G/N

type

13. C₅₉×#346

C₄

semidirect

14. C₄×C₁₆×C₅₉

C₄

semidirect

84. #346

Dic₅₉

semidirect

85. C₄×C₁₆

Dic₅₉

semidirect

88. C₅₉

#12893

semidirect

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

subgroup	index
#21424220	2
#21422200	2
#21422198	2
#21422192	2
C₅₉×#2108	2
#12893	59

subgroup

index

C₅₉×#2108

Subgroups — #28505086

Conjugacy classes of subgroups

Complements

Maximal subgroups