Group #28500528 — subgroups

Subgroups — #28500528

Order 15104.

212 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 117 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	Q₈×C₅₉⋊C₁₆	C₂	1	2	yes	maximal
2	#21422160	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₅₉⋊C₁₆)	C₂	1	2	yes	maximal
4	#21422014	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
7	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
8	C₄×C₅₉⋊C₁₆	—	2	2	no	—
9	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₅₉	C₂²	1	2	yes	—
12	C₄×Q₁₆×C₅₉	C₄	1	1	yes	characteristic
13	C₅₉×#335	C₄	1	1	yes	characteristic
14	C₅₉×Q₈⋊C₈	C₄	1	2	yes	—
15	C₈²×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₅₉⋊C₁₆	—	2	2	no	—
17	C₂×C₅₉⋊C₁₆	—	2	1	no	—
18	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
19	C₂×Q₁₆×C₅₉	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
21	#107×C₅₉	C₈	1	1	yes	characteristic
22	C₄⋊C₈×C₅₉	—	2	2	no	—
23	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
24	#103×C₅₉	C₈	1	2	yes	—
25	C₄×C₈×C₅₉	—	2	2	no	—
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×Q₈×C₅₉	C₂×C₈	1	2	yes	—
30	Q₁₆×C₅₉	C₁₆	1	4	yes	complemented
31	C₂×C₈×C₅₉	—	2	2	no	—
32	C₂×C₈×C₅₉	—	2	1	no	—
33	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₅₉	—	2	2	no	—
38	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
39	C₄²×C₅₉	—	2	2	no	—
40	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
41	C₅₉⋊C₁₆	—	2	2	no	—
42	C₅₉⋊C₁₆	—	4	1	no	—
43	C₅₉⋊C₁₆	—	4	2	no	—
44	Q₈×C₅₉	—	2	2	no	—
45	Q₈×C₅₉	C₂×C₁₆	1	4	yes	—
46	C₂×C₄×C₅₉	—	2	2	no	—
47	C₂×C₄×C₅₉	—	2	2	no	—
48	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
51	C₈×C₅₉	—	2	1	no	—
52	C₈×C₅₉	—	2	1	no	—
53	C₈×C₅₉	—	4	1	no	—
54	C₈×C₅₉	—	4	2	no	—
55	C₈×C₅₉	C₂×C₁₆	1	2	yes	—
56	C₈×C₅₉	#137	1	2	yes	—
57	#12059	—	59	1	no	maximal, Sylow
58	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
59	C₄×C₅₉	—	2	4	no	—
60	C₄×C₅₉	—	2	1	no	—
61	C₄×C₅₉	—	4	2	no	—
62	C₄×C₅₉	#441	1	1	yes	characteristic
63	C₄×C₅₉	#444	1	1	yes	characteristic
64	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
65	C₄×C₅₉	#504	1	1	yes	characteristic
66	Q₈×C₁₆	—	59	2	no	—
67	C₈×Q₁₆	D₅₉	1	1	yes	characteristic
68	#1367	—	59	1	no	—
69	Q₈⋊C₁₆	—	59	2	no	—
70	#1312	—	59	1	no	—
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	C₂×D₅₉	1	2	yes	—
75	C₄×Q₁₆	Dic₅₉	1	1	yes	characteristic
76	C₄⋊C₁₆	—	59	2	no	—
77	C₄⋊C₁₆	—	118	2	no	—
78	C₄×C₁₆	—	59	1	no	—
79	C₄×C₁₆	—	59	1	no	—
80	C₄×C₁₆	—	118	2	no	—
81	#335	Dic₅₉	1	1	yes	characteristic
82	Q₈⋊C₈	Dic₅₉	1	2	yes	—
83	C₈²	C₂×D₅₉	1	1	yes	characteristic
84	C₅₉	#12059	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	C₅₉⋊C₈	1	1	yes	characteristic
86	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
87	C₂×C₁₆	—	59	2	no	—
88	C₂×C₁₆	—	118	2	no	—
89	C₂×C₁₆	—	118	1	no	—
90	#107	C₅₉⋊C₈	1	1	yes	characteristic
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
93	#103	C₅₉⋊C₈	1	2	yes	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
98	C₂×Q₈	C₂×C₅₉⋊C₈	1	2	yes	—
99	Q₁₆	C₅₉⋊C₁₆	1	4	yes	complemented
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
103	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
104	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
105	C₂×C₈	#11745687	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×C₅₉⋊C₈	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
110	C₁₆	—	118	2	no	—
111	C₁₆	—	236	1	no	—
112	C₁₆	—	236	2	no	—
113	Q₈	—	2	2	no	—
114	Q₈	C₂×C₅₉⋊C₁₆	1	4	yes	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
119	C₂×C₄	#14273734	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	4	2	no	—
124	C₈	C₂×C₅₉⋊C₁₆	1	2	yes	—
125	C₈	#14273759	1	2	yes	—
126	C₂²	D₄×C₅₉⋊C₈	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	4	no	—
129	C₄	—	4	2	no	—
130	C₄	#16837509	1	1	yes	characteristic
131	C₄	#16837524	1	1	yes	characteristic
132	C₄	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
133	C₄	#16837819	1	1	yes	characteristic
134	C₂	#21422915	1	1	yes	characteristic
135	C₂	D₄×C₅₉⋊C₁₆	1	1	yes	characteristic
136	C₂	#21425938	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₅₉⋊C₁₆

C₂

yes

maximal

#21422160

C₂

yes

maximal, characteristic

Q₈⋊(C₅₉⋊C₁₆)

C₂

yes

maximal

#21422014

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₅₉

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₁₆)

—

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

—

C₄×C₅₉⋊C₁₆

—

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₄×C₅₉⋊C₁₆

C₂²

yes

characteristic

C₈×Q₈×C₅₉

C₂²

yes

—

C₄×Q₁₆×C₅₉

C₄

yes

characteristic

C₅₉×#335

C₄

yes

characteristic

C₅₉×Q₈⋊C₈

C₄

yes

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

D₄

yes

—

C₂×Q₁₆×C₅₉

C₈

yes

characteristic

C₄×Q₈×C₅₉

C₂×C₄

yes

—

#107×C₅₉

C₈

yes

characteristic

C₄⋊C₈×C₅₉

—

C₄⋊C₈×C₅₉

C₂×C₄

yes

—

#103×C₅₉

C₈

yes

—

C₄×C₈×C₅₉

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

C₂×C₈

yes

—

Q₁₆×C₅₉

C₁₆

yes

complemented

C₂×C₈×C₅₉

—

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

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

C₂×C₈

yes

—

C₄²×C₅₉

—

C₄²×C₅₉

C₂²×C₄

yes

characteristic

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

Q₈×C₅₉

C₂×C₁₆

yes

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₉

#131

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₂×C₁₆

yes

—

C₈×C₅₉

#137

yes

—

#12059

—

maximal, Sylow

C₂²×C₅₉

C₈×D₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

#504

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

D₅₉

yes

characteristic

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₂×C₅₉

#1579

yes

characteristic

C₂×C₅₉

D₄×C₁₆

yes

characteristic

C₂×C₅₉

#2171

yes

characteristic

C₈×Q₈

C₂×D₅₉

yes

—

C₄×Q₁₆

Dic₅₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

118

—

#335

Dic₅₉

yes

characteristic

Q₈⋊C₈

Dic₅₉

yes

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12059

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₅₉⋊C₈

yes

characteristic

C₄×Q₈

C₂×Dic₅₉

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

—

118

—

#107

C₅₉⋊C₈

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×Dic₅₉

yes

—

#103

C₅₉⋊C₈

yes

—

C₄×C₈

—

C₄×C₈

C₂×Dic₅₉

yes

characteristic

C₄×C₈

C₂×Dic₅₉

yes

characteristic

C₄×C₈

C₂²×D₅₉

yes

characteristic

C₂×Q₈

C₂×C₅₉⋊C₈

yes

—

Q₁₆

C₅₉⋊C₁₆

yes

complemented

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₅₉⋊C₈

yes

characteristic

103

C₂×C₈

C₂×C₅₉⋊C₈

yes

characteristic

104

C₂×C₈

D₄×D₅₉

yes

characteristic

105

C₂×C₈

#11745687

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×C₅₉⋊C₈

yes

—

108

C₄²

—

109

C₄²

C₂²×Dic₅₉

yes

characteristic

110

C₁₆

—

118

—

111

C₁₆

—

236

—

112

C₁₆

—

236

—

113

Q₈

—

114

Q₈

C₂×C₅₉⋊C₁₆

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₄×Dic₅₉

yes

characteristic

118

C₂×C₄

C₂²×C₅₉⋊C₈

yes

characteristic

119

C₂×C₄

#14273734

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×C₅₉⋊C₁₆

yes

—

125

C₈

#14273759

yes

—

126

C₂²

D₄×C₅₉⋊C₈

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#16837509

yes

characteristic

131

C₄

#16837524

yes

characteristic

132

C₄

C₂²×C₅₉⋊C₁₆

yes

characteristic

133

C₄

#16837819

yes

characteristic

134

C₂

#21422915

yes

characteristic

135

C₂

D₄×C₅₉⋊C₁₆

yes

characteristic

136

C₂

#21425938

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₄×C₅₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₈×Q₁₆×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	C₈×D₄
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₈×Q₁₆	D₅₉
hypocenter	C₅₉	#12059
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₅₉

commutator

C₄×C₅₉

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₈×Q₁₆×C₅₉

C₂

socle, prime radical

C₂²×C₅₉

C₈×D₄

cosocle

C₄×C₈×C₅₉

C₂³

hypercenter

C₈×Q₁₆

D₅₉

hypocenter

C₅₉

#12059

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
30. Q₁₆×C₅₉	C₁₆	semidirect
84. C₅₉	#12059	semidirect
99. Q₁₆	C₅₉⋊C₁₆	semidirect

H ≅ G/N

type

30. Q₁₆×C₅₉

C₁₆

semidirect

84. C₅₉

#12059

semidirect

99. Q₁₆

C₅₉⋊C₁₆

semidirect

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

subgroup	index
Q₈×C₅₉⋊C₁₆	2
#21422160	2
Q₈⋊(C₅₉⋊C₁₆)	2
#21422014	2
C₈×Q₁₆×C₅₉	2
#12059	59

subgroup

index

Q₈×C₅₉⋊C₁₆

#21422160

Q₈⋊(C₅₉⋊C₁₆)

#21422014

C₈×Q₁₆×C₅₉

#12059

Subgroups — #28500528

Conjugacy classes of subgroups

Complements

Maximal subgroups