Group #28503050 — subgroups

Subgroups — #28503050

Order 15104.

192 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 69 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	1	yes	maximal, characteristic
2	#21423497	C₂	1	2	yes	maximal
3	#21422168	C₂	1	1	yes	maximal, characteristic
4	#21422154	C₂	1	2	yes	maximal
5	C₅₉×#2252	C₂	1	1	yes	maximal, characteristic
6	#16837694	—	2	2	no	—
7	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
8	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₅₉⋊C₁₆	—	2	1	no	—
10	C₁₆×Dic₅₉	—	2	1	no	—
11	#16837155	C₂²	1	2	yes	—
12	#16837151	—	2	2	no	—
13	C₅₉×#500	C₂²	1	2	yes	—
14	C₅₉×#366	—	2	1	no	—
15	C₅₉×#366	—	2	1	no	—
16	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
18	C₂×C₅₉⋊C₁₆	—	2	1	no	—
19	#14273660	—	4	2	no	—
20	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₅₉⋊C₈	—	2	1	no	—
23	#14273650	—	4	2	no	—
24	C₈×Dic₅₉	—	2	1	no	—
25	C₄⋊Q₈×C₅₉	—	2	2	no	—
26	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
28	#107×C₅₉	—	2	2	no	—
29	#107×C₅₉	—	2	2	no	—
30	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₅₉	—	4	2	no	—
32	C₂×Dic₁₁₈	D₈	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	—	4	2	no	—
36	C₄⋊C₄×C₅₉	—	4	2	no	—
37	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₅₉	D₈	1	1	yes	characteristic
39	Dic₅₉⋊C₄	—	2	1	no	—
40	C₄×Dic₅₉	—	2	1	no	—
41	C₂×C₅₉⋊C₈	—	2	1	no	—
42	C₂×C₅₉⋊C₈	—	4	2	no	—
43	C₁₆×C₅₉	—	2	1	no	—
44	C₁₆×C₅₉	Q₁₆	1	2	yes	—
45	C₅₉⋊C₁₆	—	4	1	no	—
46	Q₈×C₅₉	—	16	2	no	—
47	C₂×C₄×C₅₉	—	8	2	no	—
48	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
51	C₂×Dic₅₉	—	2	1	no	—
52	C₂×Dic₅₉	—	2	1	no	—
53	Dic₁₁₈	—	2	1	no	—
54	Dic₁₁₈	Q₃₂	1	2	yes	—
55	C₈×C₅₉	—	2	1	no	—
56	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₅₉	#137	1	1	yes	characteristic
58	C₅₉⋊C₈	—	4	1	no	—
59	C₅₉⋊C₈	—	8	2	no	—
60	#12458	—	59	1	no	maximal, Sylow
61	C₂²×C₅₉	#466	1	1	yes	characteristic
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	—	16	2	no	—
64	C₄×C₅₉	#462	1	1	yes	characteristic
65	C₄×C₅₉	#473	1	1	yes	characteristic
66	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
67	C₄×C₅₉	#508	1	1	yes	characteristic
68	Dic₅₉	—	2	2	no	—
69	Dic₅₉	—	4	1	no	—
70	#2252	D₅₉	1	1	yes	characteristic
71	Q₈×C₁₆	—	59	1	no	—
72	#1676	—	59	2	no	—
73	#1371	—	59	1	no	—
74	#1365	—	59	2	no	—
75	C₂×C₅₉	#1674	1	1	yes	characteristic
76	C₂×C₅₉	#2216	1	1	yes	characteristic
77	C₂×C₅₉	#2221	1	1	yes	characteristic
78	#500	C₂×D₅₉	1	2	yes	—
79	#477	—	118	2	no	—
80	C₈×Q₈	—	59	1	no	—
81	#366	—	2	1	no	—
82	#366	—	2	1	no	—
83	C₄⋊C₁₆	—	59	1	no	—
84	C₄⋊C₁₆	—	118	1	no	—
85	C₄×C₁₆	—	118	1	no	—
86	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
87	#335	—	59	2	no	—
88	#333	—	118	2	no	—
89	C₅₉	#12458	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	59	1	no	—
92	C₂×C₁₆	—	118	1	no	—
93	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
94	C₂×C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	236	2	no	—
98	C₄⋊C₈	—	59	1	no	—
99	C₄⋊C₈	—	118	1	no	—
100	C₄⋊C₈	—	118	2	no	—
101	#103	—	236	2	no	—
102	C₄×C₈	—	118	1	no	—
103	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	59	1	no	—
106	C₂×C₈	—	118	1	no	—
107	C₂×C₈	—	236	2	no	—
108	C₂×C₈	#11745687	1	1	yes	characteristic
109	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	59	1	no	—
113	C₄⋊C₄	—	118	1	no	—
114	C₄²	—	118	1	no	—
115	C₄²	D₄×D₅₉	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	236	1	no	—
118	C₁₆	#11745672	1	2	yes	—
119	Q₈	—	16	2	no	—
120	Q₈	—	59	2	no	—
121	Q₈	—	118	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	118	1	no	—
124	C₂×C₄	—	118	1	no	—
125	C₂×C₄	#14273698	1	1	yes	characteristic
126	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
127	C₂×C₄	#14273746	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	472	2	no	—
130	C₈	—	236	1	no	—
131	C₈	C₂×#11745672	1	1	yes	characteristic
132	C₈	#14273758	1	1	yes	characteristic
133	C₂²	#16837635	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	16	2	no	—
136	C₄	—	118	2	no	—
137	C₄	—	236	1	no	—
138	C₄	#16837609	1	1	yes	characteristic
139	C₄	#16837674	1	1	yes	characteristic
140	C₄	Q₃₂×D₅₉	1	1	yes	characteristic
141	C₄	#16837835	1	1	yes	characteristic
142	C₂	#21423487	1	1	yes	characteristic
143	C₂	#21426184	1	1	yes	characteristic
144	C₂	#21426213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₁₈

C₂

yes

maximal, characteristic

#21423497

C₂

yes

maximal

#21422168

C₂

yes

maximal, characteristic

#21422154

C₂

yes

maximal

C₅₉×#2252

C₂

yes

maximal, characteristic

#16837694

—

C₈×Dic₁₁₈

C₂²

yes

characteristic

C₄⋊(C₅₉⋊C₁₆)

C₂²

yes

characteristic

Dic₅₉⋊C₁₆

—

C₁₆×Dic₅₉

—

#16837155

C₂²

yes

—

#16837151

—

C₅₉×#500

C₂²

yes

—

C₅₉×#366

—

C₅₉×#366

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₄×Dic₁₁₈

D₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

#14273660

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

D₄

yes

characteristic

Dic₅₉⋊C₈

—

#14273650

—

C₈×Dic₅₉

—

C₄⋊Q₈×C₅₉

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

characteristic

#107×C₅₉

—

#107×C₅₉

—

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×Dic₁₁₈

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

D₈

yes

characteristic

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₁₆×C₅₉

Q₁₆

yes

—

C₅₉⋊C₁₆

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

#135

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

Q₃₂

yes

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₈×C₅₉

#137

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

C₄×C₅₉

#508

yes

characteristic

Dic₅₉

—

Dic₅₉

—

#2252

D₅₉

yes

characteristic

Q₈×C₁₆

—

#1676

—

#1371

—

#1365

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×D₅₉

yes

—

#477

—

118

—

C₈×Q₈

—

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

118

—

C₄×C₁₆

—

118

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

#335

—

#333

—

118

—

C₅₉

#12458

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₂²⋊D₅₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₅₉

yes

—

—

—

236

—

C₄⋊C₈

—

C₄⋊C₈

—

118

—

100

C₄⋊C₈

—

118

—

101

#103

—

236

—

102

C₄×C₈

—

118

—

103

C₄×C₈

C₂²×D₅₉

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×C₈

—

118

—

107

C₂×C₈

—

236

—

108

C₂×C₈

#11745687

yes

characteristic

109

C₂×C₈

C₂×C₂²⋊D₅₉

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

118

—

114

C₄²

—

118

—

115

C₄²

D₄×D₅₉

yes

characteristic

116

C₁₆

—

117

C₁₆

—

236

—

118

C₁₆

#11745672

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

118

—

122

C₂×C₄

—

123

C₂×C₄

—

118

—

124

C₂×C₄

—

118

—

125

C₂×C₄

#14273698

yes

characteristic

126

C₂×C₄

D₈×D₅₉

yes

characteristic

127

C₂×C₄

#14273746

yes

characteristic

128

C₈

—

129

C₈

—

472

—

130

C₈

—

236

—

131

C₈

C₂×#11745672

yes

characteristic

132

C₈

#14273758

yes

characteristic

133

C₂²

#16837635

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

118

—

137

C₄

—

236

—

138

C₄

#16837609

yes

characteristic

139

C₄

#16837674

yes

characteristic

140

C₄

Q₃₂×D₅₉

yes

characteristic

141

C₄

#16837835

yes

characteristic

142

C₂

#21423487

yes

characteristic

143

C₂

#21426184

yes

characteristic

144

C₂

#21426213

yes

characteristic

	subgroup	G/H
center	C₂²	#16837635
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2252	C₂
socle, prime radical	C₂²×C₅₉	#466
hypercenter	#2252	D₅₉
hypocenter	C₅₉	#12458
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16837635

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#2252

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
89. C₅₉	#12458	semidirect

H ≅ G/N

type

89. C₅₉

#12458

semidirect

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

subgroup	index
C₁₆×Dic₁₁₈	2
#21423497	2
#21422168	2
#21422154	2
C₅₉×#2252	2
#12458	59

subgroup

index

C₁₆×Dic₁₁₈

C₅₉×#2252

Subgroups — #28503050

Conjugacy classes of subgroups

Complements

Maximal subgroups