Finite Group Databasebeta

← #27275527

Subgroups — #27275527

Order 13568.

812 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20273106C214yesmaximal, complemented
2#20273029C212yesmaximal, complemented
3C2×C53×#540C211yesmaximal, characteristic
4#16810676C2214yes—
5D4⋊#88322—28no—
6#16810349—22no—
7#16810347—22no—
8C53×#540—28no—
9C2×#127×C53C411yescharacteristic
10C2×#125×C53C411yescharacteristic
11C2×#123×C53—22no—
12C2×#121×C53—22no—
13C2×#121×C53C412yes—
14C2×C4×D4×C53C2212yes—
15C2×#117×C53C2211yescharacteristic
16C4⋊#88322—24no—
17C22⋊#88322—24no—
18C4×#88322—24no—
19C22×D4×C53—22no—
20C22×D4×C53—22no—
21C22×D4×C53C2×C412yes—
22C23×C4×C53—22no—
23C23×C4×C53C2×C411yescharacteristic
24#127×C53—24no—
25#125×C53—24no—
26#123×C53—216no—
27#121×C53—216no—
28#121×C53—24no—
29#121×C53—24no—
30C4×D4×C53—24no—
31C4×D4×C53D418yes—
32#117×C53—24no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53C2×C412yes—
35C2×C4⋊C4×C53D412yes—
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53D412yes—
39C2×C42×C53C2×C411yescharacteristic
40C2×C42×C53C2311yescharacteristic
41C24×C53—22no—
42C24×C53C22⋊C411yescharacteristic
43C2×D4×C53—28no—
44C2×D4×C53—28no—
45C2×D4×C53—22no—
46C2×D4×C53—28no—
47C2×D4×C53—84no—
48C2×D4×C53—84no—
49C2×D4×C53C22⋊C414yes—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—24no—
53C22×C4×C53—28no—
54C22×C4×C53—84no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C22×C411yescharacteristic
58C22×C4×C53C2×D412yes—
59C4⋊C4×C53—28no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—24no—
62C4⋊C4×C53C22⋊C414yes—
63C22⋊C4×C53—28no—
64C22⋊C4×C53—28no—
65C22⋊C4×C53—24no—
66C42×C53—22no—
67C42×C53C2×D414yes—
68C2×#88322—44no—
69C23×C53—21no—
70C23×C53—24no—
71C23×C53—22no—
72C23×C53—44no—
73C23×C53—44no—
74C23×C53—42no—
75C23×C53—44no—
76C23×C53C2×C22⋊C411yescharacteristic
77D4×C53—28no—
78D4×C53—44no—
79D4×C53—88no—
80D4×C53—88no—
81C2×C4×C53—24no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—42no—
86C2×C4×C53—44no—
87C2×C4×C53—44no—
88C2×C4×C53—42no—
89C2×C4×C53—88no—
90C2×C4×C53C2×C22⋊C412yes—
91C2×C4×C53#12014yes—
92C2×C4×C53#13114yes—
93#88322—84no—
94#8326—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53—44no—
97C22×C53—42no—
98C22×C53—44no—
99C22×C53—41no—
100C22×C53—44no—
101C22×C53—82no—
102C22×C53—88no—
103C22×C53#37911yescharacteristic
104C22×C53#40812yes—
105C4×C53—24no—
106C4×C53—44no—
107C4×C53—42no—
108C4×C53—41no—
109C4×C53#40814yes—
110C2×#540D5311yescharacteristic
111#1478—534no—
112#1458—532no—
113C2×C53—22no—
114C2×C53—81no—
115C2×C53—84no—
116C2×C53#178011yescharacteristic
117C2×C53#179512yes—
118#540—28no—
119C2×#127Dic5311yescharacteristic
120C2×#125Dic5311yescharacteristic
121C2×#123—22no—
122C2×#121—22no—
123C2×#121Dic5312yes—
124C2×C4×D4C2×D5312yes—
125C2×#117C2×D5311yescharacteristic
126#431—534no—
127D4⋊C8—1068no—
128#324—1062no—
129#323—1062no—
130C53#832611yescharacteristic, complemented, Sylow
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C2×Dic5312yes—
134C23×C4—22no—
135C23×C4C2×Dic5311yescharacteristic
136#127—24no—
137#125—24no—
138#123—216no—
139#121—216no—
140#121—24no—
141#121—24no—
142C4×D4—24no—
143C4×D4C22⋊D5318yes—
144#117—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C2×Dic5312yes—
147C2×C4⋊C4C22⋊D5312yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4C22⋊D5312yes—
151C2×C42C2×Dic5311yescharacteristic
152C2×C42C22×D5311yescharacteristic
153C4⋊C8—1064no—
154C22⋊C8—1064no—
155C4×C8—1064no—
156C24—22no—
157C24C22⋊Dic5311yescharacteristic
158C2×D4—28no—
159C2×D4—28no—
160C2×D4—22no—
161C2×D4—28no—
162C2×D4—84no—
163C2×D4—84no—
164C2×D4C22⋊Dic5314yes—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—84no—
170C22×C4C22⋊Dic5311yescharacteristic
171C22×C4C22⋊Dic5312yes—
172C22×C4C22×Dic5311yescharacteristic
173C22×C4C2×C22⋊D5312yes—
174C2×C8—2124no—
175C4⋊C4—28no—
176C4⋊C4—22no—
177C4⋊C4—24no—
178C4⋊C4C22⋊Dic5314yes—
179C22⋊C4—28no—
180C22⋊C4—28no—
181C22⋊C4—24no—
182C42—22no—
183C42C2×C22⋊D5314yes—
184C23—21no—
185C23—24no—
186C23—22no—
187C23—44no—
188C23—44no—
189C23—42no—
190C23—44no—
191C23C2×C22⋊Dic5311yescharacteristic
192D4—28no—
193D4—44no—
194D4—88no—
195D4—88no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—88no—
205C2×C4C2×C22⋊Dic5312yes—
206C2×C4#1313318614yes—
207C2×C4#1313322414yes—
208C8—4244no—
209C22—22no—
210C22—41no—
211C22—44no—
212C22—44no—
213C22—44no—
214C22—42no—
215C22—82no—
216C22—88no—
217C22#1681047911yescharacteristic
218C22#1681059012yes—
219C4—24no—
220C4—41no—
221C4—44no—
222C4—42no—
223C4#1681059014yes—
224C2—22no—
225C2—84no—
226C2—81no—
227C2#2027462711yescharacteristic
228C2#2027468512yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#540C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenterC2×#540D53
hypocenterC53#8326
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#540

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20273106C2semidirect
2. #20273029C2semidirect
130. C53#8326semidirect

Maximal subgroups

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

subgroupindex
#202731062
#202730292
C2×C53×#5402
#832653