Finite Group Databasebeta

← #27296884

Subgroups — #27296884

Order 13568.

400 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275882C211yesmaximal, characteristic
2#20273671C212yesmaximal
3#20272736C212yesmaximal
4#20272645C211yesmaximal, characteristic
5C53×#2607C211yesmaximal, characteristic
6#16811249C2211yescharacteristic
7#16810779—22no—
8#16810486—21no—
9#16810390C2211yescharacteristic
10#16810389—21no—
11#16810359—22no—
12#16810347C2212yes—
13C53×#399—21no—
14C53×#393C2212yes—
15C53×#391—22no—
16C53×#387—22no—
17C53×#381C2211yescharacteristic
18C4×C22⋊C4×C53—21no—
19C2×#11719925—42no—
20#13133196—22no—
21#13133194—42no—
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53—21no—
24#13133141—42no—
25C22⋊#88322—22no—
26#13133124—42no—
27C22×Q8×C53D412yes—
28C23×C4×C53D411yescharacteristic
29C2×C4⋊C4×C53—22no—
30C2×C4⋊C4×C53—21no—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53D411yescharacteristic
33C2×C22⋊C4×C53C2311yescharacteristic
34C2×C42×C53—21no—
35C2×C42×C53—21no—
36#95×C53—22no—
37#95×C53—21no—
38#95×C53—22no—
39#95×C53—22no—
40#95×C53D412yes—
41C24×C53C2×D411yescharacteristic
42C2×Q8×C53—44no—
43C2×Q8×C53—42no—
44C22×C4×C53—21no—
45C22×C4×C53—22no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—22no—
49C22×C4×C53—41no—
50C22×C4×C53—41no—
51C22×C4×C53Q1612yescomplemented
52C22×C4×C53C2×D412yes—
53C22×Dic53—22no—
54C22×Dic53—41no—
55C2×Dic106—22no—
56C4⋊C4×C53—44no—
57C4⋊C4×C53—81no—
58C22⋊C4×C53—41no—
59C22⋊C4×C53—44no—
60C22⋊C4×C53—41no—
61C42×C53—81no—
62C42×C53—81no—
63C22⋊Dic53—41no—
64C22⋊Dic53—41no—
65C22⋊Dic53—41no—
66C22⋊Dic53—41no—
67#11719925—84no—
68C4⋊Dic53—22no—
69#11719919—42no—
70C2×#88322—42no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—42no—
74C23×C53#12011yescharacteristic
75C23×C53C2×Q1611yescharacteristic
76C23×C53#13711yescharacteristic
77Q8×C53—44no—
78Q8×C53—82no—
79C2×C4×C53—24no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—42no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—81no—
94C2×Dic53—42no—
95C2×Dic53—82no—
96C2×Dic53—81no—
97C2×Dic53—82no—
98Dic106—82no—
99#88322—82no—
100#12215—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—42no—
107C22×C53—81no—
108C22×C53#45111yescharacteristic
109C22×C53#45511yescharacteristic
110C22×C53B(4,𝔽2)11yescharacteristic
111C4×C53—42no—
112C4×C53—42no—
113C4×C53—44no—
114C4×C53—82no—
115C4×C53—81no—
116Dic53—82no—
117Dic53—161no—
118#2607D5311yescharacteristic
119#2030—531no—
120#1602—532no—
121#1351—532no—
122#1317—531no—
123C2×C53—21no—
124C2×C53—21no—
125C2×C53—41no—
126C2×C53—41no—
127C2×C53#160211yescharacteristic
128C2×C53#219712yes—
129#543—531no—
130#451—1062no—
131#399—21no—
132#393C2×D5312yes—
133#391—22no—
134#387—22no—
135#381—1061no—
136#381C2×D5311yescharacteristic
137C4×C22⋊C4—21no—
138#342—531no—
139#342—1061no—
140#328—1062no—
141#323—532no—
142C53#1221511yescharacteristic, complemented, Sylow
143C22×Q8C22⋊D5312yes—
144C23×C4C22⋊D5311yescharacteristic
145C2×Q16—2122no—
146C22⋊Q8—1062no—
147C22⋊Q8—2122no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—1061no—
152C2×C22⋊C4—1061no—
153C2×C22⋊C4C22⋊D5311yescharacteristic
154C2×C22⋊C4C22×D5311yescharacteristic
155C2×C42—21no—
156C2×C42—21no—
157#103—2122no—
158C22⋊C8—1062no—
159#95—22no—
160#95—21no—
161#95—22no—
162#95—22no—
163#95—2122no—
164#95C22⋊D5312yes—
165C24C2×C22⋊D5311yescharacteristic
166C2×Q8—44no—
167C2×Q8—42no—
168C2×Q8—1062no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—1062no—
177C22×C4—2121no—
178C22×C4#1171992512yescomplemented
179C22×C4C2×C22⋊D5312yes—
180Q16—4244no—
181C2×C8—2122no—
182C4⋊C4—44no—
183C4⋊C4—81no—
184C4⋊C4—1062no—
185C4⋊C4—2122no—
186C22⋊C4—41no—
187C22⋊C4—44no—
188C22⋊C4—41no—
189C22⋊C4—2121no—
190C22⋊C4—2121no—
191C22⋊C4—2121no—
192C22⋊C4—2121no—
193C42—81no—
194C42—81no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23#1313318611yescharacteristic
199C23C2×#1171992511yescharacteristic
200C23#1313324811yescharacteristic
201Q8—44no—
202Q8—82no—
203Q8—4242no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—4241no—
220C2×C4—4242no—
221C2×C4—4242no—
222C2×C4—2122no—
223C8—4242no—
224C22—22no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—81no—
231C22#1681077911yescharacteristic
232C22#1681080111yescharacteristic
233C22#1681081011yescharacteristic
234C4—42no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4—81no—
239C4—8481no—
240C4—4242no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2#2027367211yescharacteristic
246C2#2027670512yes—
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2607C2
socleC22×C53B(4,𝔽2)
prime radicalC24×C53C2×D4
hypercenter#2607D53
hypocenterC53#12215
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
51. C22×C4×C53Q16semidirect
142. C53#12215semidirect
178. C22×C4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202758822
#202736712
#202727362
#202726452
C53×#26072
#1221553