Group #28500529 — subgroups

Subgroups — #28500529

Order 15104.

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

normal subgroups only

This group has 212 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	C₁₆×Dic₁₁₈	C₂	1	2	yes	maximal
2	C₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21422160	C₂	1	1	yes	maximal, characteristic
4	#21422076	C₂	1	2	yes	maximal
5	C₅₉×#1312	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₁₈	C₂²	1	2	yes	—
7	C₄×Dic₂₃₆	C₄	1	1	yes	characteristic
8	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
9	Dic₅₉⋊C₁₆	—	2	2	no	—
10	C₁₆×Dic₅₉	—	2	2	no	—
11	#16837155	C₄	1	1	yes	characteristic
12	#16837134	C₄	1	2	yes	—
13	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₈²×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₃₆	C₈	1	1	yes	characteristic
17	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	#14273661	C₈	1	1	yes	characteristic
20	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
21	Dic₅₉⋊C₈	—	2	2	no	—
22	#14273652	C₈	1	2	yes	—
23	C₈×Dic₅₉	—	2	2	no	—
24	C₂×C₁₆×C₅₉	—	2	1	no	—
25	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
26	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₁₈	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
35	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
37	Dic₅₉⋊C₄	—	2	2	no	—
38	C₄×Dic₅₉	—	2	2	no	—
39	C₂×C₅₉⋊C₈	—	2	2	no	—
40	Dic₂₃₆	C₁₆	1	4	yes	complemented
41	C₁₆×C₅₉	—	2	2	no	—
42	C₁₆×C₅₉	—	4	1	no	—
43	C₅₉⋊C₁₆	—	4	2	no	—
44	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
47	C₂×Dic₅₉	—	2	2	no	—
48	C₂×Dic₅₉	—	2	2	no	—
49	Dic₁₁₈	—	2	2	no	—
50	Dic₁₁₈	C₂×C₁₆	1	4	yes	—
51	C₈×C₅₉	—	2	1	no	—
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	—	4	1	no	—
54	C₈×C₅₉	C₂×C₁₆	1	2	yes	—
55	C₈×C₅₉	#137	1	2	yes	—
56	C₅₉⋊C₈	—	4	2	no	—
57	#12059	—	59	1	no	maximal, Sylow
58	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
59	C₄×C₅₉	—	2	1	no	—
60	C₄×C₅₉	#441	1	1	yes	characteristic
61	C₄×C₅₉	#444	1	1	yes	characteristic
62	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
63	C₄×C₅₉	#504	1	1	yes	characteristic
64	Dic₅₉	—	2	4	no	—
65	Dic₅₉	—	4	2	no	—
66	Q₈×C₁₆	—	59	2	no	—
67	C₈×Q₁₆	—	59	1	no	—
68	#1367	—	59	1	no	—
69	Q₈⋊C₁₆	—	59	2	no	—
70	#1312	D₅₉	1	1	yes	characteristic
71	C₂×C₅₉	#1579	1	1	yes	characteristic
72	C₂×C₅₉	D₄×C₁₆	1	1	yes	characteristic
73	C₂×C₅₉	#2171	1	1	yes	characteristic
74	C₈×Q₈	—	59	2	no	—
75	C₄×Q₁₆	—	59	1	no	—
76	C₄⋊C₁₆	—	59	2	no	—
77	C₄⋊C₁₆	—	118	2	no	—
78	C₄×C₁₆	—	118	2	no	—
79	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
80	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
81	#335	—	59	1	no	—
82	Q₈⋊C₈	—	59	2	no	—
83	C₈²	C₂×D₅₉	1	1	yes	characteristic
84	C₅₉	#12059	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	59	1	no	—
86	C₄×Q₈	—	59	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	118	2	no	—
89	C₂×C₁₆	D₂₃₆	1	2	yes	—
90	#107	—	59	1	no	—
91	C₄⋊C₈	—	59	2	no	—
92	C₄⋊C₈	—	118	2	no	—
93	#103	—	59	2	no	—
94	C₄×C₈	—	118	2	no	—
95	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
96	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
98	C₂×Q₈	—	59	2	no	—
99	Q₁₆	—	59	4	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	118	2	no	—
102	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
103	C₂×C₈	C₈×D₅₉	1	1	yes	characteristic
104	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
105	C₂×C₈	#11745685	1	1	yes	characteristic
106	C₄⋊C₄	—	59	2	no	—
107	C₄⋊C₄	—	118	2	no	—
108	C₄²	—	118	2	no	—
109	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	236	2	no	—
113	Q₈	—	59	4	no	—
114	Q₈	—	118	2	no	—
115	C₂×C₄	—	118	2	no	—
116	C₂×C₄	—	118	2	no	—
117	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
119	C₂×C₄	#14273733	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	236	2	no	—
124	C₈	C₁₆×D₅₉	1	2	yes	—
125	C₈	#14273760	1	2	yes	—
126	C₂²	C₈×D₂₃₆	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	118	4	no	—
129	C₄	—	236	2	no	—
130	C₄	#16837510	1	1	yes	characteristic
131	C₄	#16837525	1	1	yes	characteristic
132	C₄	C₂×C₁₆×D₅₉	1	1	yes	characteristic
133	C₄	#16837818	1	1	yes	characteristic
134	C₂	#21422916	1	1	yes	characteristic
135	C₂	C₁₆×D₂₃₆	1	1	yes	characteristic
136	C₂	#21425939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₁₈

C₂

yes

maximal

C₈×Dic₂₃₆

C₂

yes

maximal, characteristic

#21422160

C₂

yes

maximal, characteristic

#21422076

C₂

yes

maximal

C₅₉×#1312

C₂

yes

maximal, characteristic

C₈×Dic₁₁₈

C₂²

yes

—

C₄×Dic₂₃₆

C₄

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

—

Dic₅₉⋊C₁₆

—

C₁₆×Dic₅₉

—

#16837155

C₄

yes

characteristic

#16837134

C₄

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

C₈

yes

characteristic

C₄×Dic₁₁₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₁₆

—

#14273661

C₈

yes

characteristic

C₄⋊(C₅₉⋊C₈)

C₂×C₄

yes

—

Dic₅₉⋊C₈

—

#14273652

C₈

yes

—

C₈×Dic₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

C₂×C₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₄⋊Dic₅₉

C₂×C₈

yes

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

C₁₆

yes

complemented

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

#131

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

C₂×C₁₆

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₁₆

yes

—

C₈×C₅₉

#137

yes

—

C₅₉⋊C₈

—

#12059

—

maximal, Sylow

C₂²×C₅₉

C₈×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

#504

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Q₈×C₁₆

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

D₅₉

yes

characteristic

C₂×C₅₉

#1579

yes

characteristic

C₂×C₅₉

D₄×C₁₆

yes

characteristic

C₂×C₅₉

#2171

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₄×C₁₆

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12059

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

D₂₃₆

yes

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

#103

—

C₄×C₈

—

118

—

C₄×C₈

C₄×D₅₉

yes

characteristic

C₄×C₈

C₄×D₅₉

yes

characteristic

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

118

—

102

C₂×C₈

C₈×D₅₉

yes

characteristic

103

C₂×C₈

C₈×D₅₉

yes

characteristic

104

C₂×C₈

C₂×D₂₃₆

yes

characteristic

105

C₂×C₈

#11745685

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

118

—

108

C₄²

—

118

—

109

C₄²

C₂×C₄×D₅₉

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

236

—

113

Q₈

—

114

Q₈

—

118

—

115

C₂×C₄

—

118

—

116

C₂×C₄

—

118

—

117

C₂×C₄

C₄×D₂₃₆

yes

characteristic

118

C₂×C₄

C₂×C₈×D₅₉

yes

characteristic

119

C₂×C₄

#14273733

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

236

—

124

C₈

C₁₆×D₅₉

yes

—

125

C₈

#14273760

yes

—

126

C₂²

C₈×D₂₃₆

yes

characteristic

127

C₄

—

128

C₄

—

118

—

129

C₄

—

236

—

130

C₄

#16837510

yes

characteristic

131

C₄

#16837525

yes

characteristic

132

C₄

C₂×C₁₆×D₅₉

yes

characteristic

133

C₄

#16837818

yes

characteristic

134

C₂

#21422916

yes

characteristic

135

C₂

C₁₆×D₂₃₆

yes

characteristic

136

C₂

#21425939

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1312	C₂
socle, prime radical	C₂²×C₅₉	C₈×D₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#1312	D₅₉
hypocenter	C₅₉	#12059
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₂₃₆

commutator

C₄×C₅₉

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1312

C₂

socle, prime radical

C₂²×C₅₉

C₈×D₄

cosocle

C₄×C₈×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
40. Dic₂₃₆	C₁₆	semidirect
84. C₅₉	#12059	semidirect

H ≅ G/N

type

40. Dic₂₃₆

C₁₆

semidirect

84. C₅₉

#12059

semidirect

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

subgroup	index
C₁₆×Dic₁₁₈	2
C₈×Dic₂₃₆	2
#21422160	2
#21422076	2
C₅₉×#1312	2
#12059	59

subgroup

index

C₁₆×Dic₁₁₈

C₈×Dic₂₃₆

#21422160

#21422076

C₅₉×#1312

#12059

Subgroups — #28500529

Conjugacy classes of subgroups

Complements

Maximal subgroups