Finite Group Databasebeta

← #27296915

Subgroups — #27296915

Order 13568.

392 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 59 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#20282661C211yesmaximal, characteristic
2#20273757C212yesmaximal, complemented
3#20272741C212yesmaximal, complemented
4#20272647C211yesmaximal, characteristic
5C53×#2027C211yesmaximal, characteristic
6#16810956—22no—
7#16810562C2212yes—
8#16810535—22no—
9#16810529—21no—
10#16810503—21no—
11#16810498—22no—
12#16810360—22no—
13#16810349C2212yes—
14C53×#544C2211yescharacteristic
15C53×#418—21no—
16C53×#390C2211yescharacteristic
17C53×#343C2211yescharacteristic
18C53×#339—21no—
19C22×Dic106—21no—
20C23×Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53D412yes—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—21no—
25#13133150—42no—
26#13133141—42no—
27C22⋊#88322—22no—
28#13133124—22no—
29#13133124—22no—
30#13133124—22no—
31#13133124D412yes—
32C22×D4×C53D411yescharacteristic
33C2×M16×C53—21no—
34C4⋊Q8×C53—41no—
35#125×C53—41no—
36C22⋊Q8×C53—22no—
37#117×C53—21no—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C42×C53D411yescharacteristic
41#102×C53—41no—
42#102×C53—41no—
43C24×C53—21no—
44C2×Q8×C53—22no—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×D4×C53—41no—
48C2×D4×C53Q1612yes—
49C22×C4×C53—21no—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C22×Dic53—82no—
55C22×Dic53—81no—
56C2×Dic106—42no—
57C2×Dic106—81no—
58C2×Dic106—81no—
59M16×C53—41no—
60M16×C53—41no—
61C2×C8×C53—41no—
62C4⋊C4×C53—21no—
63C4⋊C4×C53—21no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—42no—
67C22⋊C4×C53—44no—
68C22⋊C4×C53—41no—
69C42×C53—41no—
70C42×C53—41no—
71C22⋊Dic53—82no—
72C22⋊Dic53—81no—
73C4⋊Dic53—42no—
74C4⋊Dic53—81no—
75C2×#88322—42no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53—41no—
80C23×C53—41no—
81C23×C53#12011yescharacteristic
82Q8×C53—82no—
83D4×C53—42no—
84D4×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53C2×Q1611yescharacteristic
95C2×C4×C53#13711yescharacteristic
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—84no—
103Dic106—84no—
104C8×C53—81no—
105#88322—82no—
106#12221—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—44no—
110C22×C53—42no—
111C22×C53—82no—
112C22×C53—81no—
113C22×C53—81no—
114C22×C53#45111yescharacteristic
115C22×C53#45511yescharacteristic
116C22×C53B(4,𝔽2)11yescharacteristic
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—42no—
120C4×C53—42no—
121C4×C53—81no—
122C4×C53—81no—
123Dic53—82no—
124Dic53—82no—
125#2804—531no—
126#2027D5311yescharacteristic
127#1617—532no—
128#1353—532no—
129#1318—531no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53—82no—
133C2×C53#160211yescharacteristic
134C2×C53#219811yescharacteristic
135C2×C53#220211yescharacteristic
136#544C2×D5311yescharacteristic
137#484—1062no—
138#418—21no—
139#400—532no—
140#393—1062no—
141#391—1061no—
142#390C2×D5311yescharacteristic
143#386—1061no—
144#385—1062no—
145#343C2×D5311yescharacteristic
146#339—21no—
147#328—1062no—
148#324—532no—
149C53#1222111yescharacteristic, complemented, Sylow
150C22×Q8—1061no—
151C22×D4C22⋊D5311yescharacteristic
152C23×C4—1061no—
153C2×M16—21no—
154C4⋊Q8—41no—
155#125—41no—
156C22⋊Q8—22no—
157#117—21no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4C22×D5311yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1061no—
164C2×C42C22⋊D5311yescharacteristic
165#107—2122no—
166#103—2122no—
167#102—41no—
168#102—41no—
169C22⋊C8—1062no—
170#95—532no—
171#95—1062no—
172#95—1062no—
173#95—1062no—
174C24—21no—
175C2×Q8—22no—
176C2×Q8—4241no—
177C2×Q8—4241no—
178C2×Q8—2122no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1171992512yes—
183C22×C4—21no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4—4242no—
187C22×C4—4241no—
188C22×C4D4×D5312yes—
189C22×C4C2×C22⋊D5311yescharacteristic
190M16—41no—
191M16—41no—
192C2×C8—41no—
193C2×C8—2122no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—4241no—
200C4⋊C4—2122no—
201C22⋊C4—44no—
202C22⋊C4—41no—
203C22⋊C4—4242no—
204C22⋊C4—4241no—
205C42—41no—
206C42—41no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23#1313318511yescharacteristic
213Q8—82no—
214Q8—4244no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—4244no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4C2×#1171992511yescharacteristic
234C2×C4#1313324711yescharacteristic
235C8—81no—
236C8—4242no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1681077811yescharacteristic
245C22#1681080011yescharacteristic
246C22#1681080911yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—81no—
252C4—81no—
253C4—4242no—
254C4—4242no—
255C2—21no—
256C2—21no—
257C2—82no—
258C2#2027367111yescharacteristic
259C2#2027671011yescharacteristic
260C2#2027673111yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2027C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
hypercenter#2027D53
hypocenterC53#12221
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2027

Complements

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

NH ≅ G/Ntype
2. #20273757C2semidirect
3. #20272741C2semidirect
149. C53#12221semidirect

Maximal subgroups

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

subgroupindex
#202826612
#202737572
#202727412
#202726472
C53×#20272
#1222153