Finite Group Databasebeta

← #27296916

Subgroups — #27296916

Order 13568.

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

normal subgroups only

This group has 392 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#20275871C211yesmaximal, characteristic
2#20273756C212yesmaximal, complemented
3#20272742C212yesmaximal, complemented
4#20272647C211yesmaximal, characteristic
5C53×#2804C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810958—22no—
8#16810628—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C53×#400C2212yes—
14C53×#393—22no—
15C53×#391—21no—
16C53×#390C2211yescharacteristic
17C53×#386—21no—
18C53×#385—22no—
19C2×#11719917—21no—
20#13133216—41no—
21#13133208—41no—
22#13133196—22no—
23#13133176—21no—
24#13133150—42no—
25#13133140—42no—
26#13133138—41no—
27#13133138—41no—
28C22⋊#88322—22no—
29C22×Q8×C53—21no—
30C22×D4×C53D411yescharacteristic
31C23×C4×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53D412yes—
34C2×C4⋊C4×C53C2311yescharacteristic
35C2×C22⋊C4×C53—22no—
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—22no—
38C2×C42×C53D411yescharacteristic
39#95×C53—22no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53D412yes—
43C24×C53—21no—
44C2×Q8×C53—42no—
45C2×Q8×C53—81no—
46C2×Q8×C53—81no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53Q1612yes—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—82no—
55C22×C4×C53—81no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C2×Dic106—22no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—81no—
62C22⋊C4×C53—44no—
63C22⋊C4×C53—82no—
64C22⋊C4×C53—81no—
65C42×C53—41no—
66C22⋊Dic53—41no—
67C4⋊Dic53—21no—
68C4⋊Dic53—21no—
69C4⋊Dic53—22no—
70#11719919—42no—
71C4×Dic53—41no—
72#11719917—41no—
73#11719917—41no—
74C2×#88322—42no—
75C2×#88322—41no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53—41no—
80C23×C53—41no—
81C23×C53#12011yescharacteristic
82Q8×C53—84no—
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—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—84no—
99C2×C4×C53C2×Q1611yescharacteristic
100C2×C4×C53#13711yescharacteristic
101C2×Dic53—41no—
102C2×Dic53—41no—
103Dic106—82no—
104#88322—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—82no—
122C4×C53—82no—
123Dic53—81no—
124Dic53—81no—
125#2804D5311yescharacteristic
126#2027—531no—
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#544—531no—
137#484—1062no—
138#418—1061no—
139#400C2×D5312yes—
140#393—22no—
141#391—21no—
142#390C2×D5311yescharacteristic
143#386—21no—
144#385—22no—
145#343—531no—
146#339—1061no—
147#328—1062no—
148#324—532no—
149C53#1222111yescharacteristic, complemented, Sylow
150C22×Q8—21no—
151C22×D4C22⋊D5311yescharacteristic
152C23×C4—21no—
153C2×M16—1061no—
154C4⋊Q8—2121no—
155#125—2121no—
156C22⋊Q8—1062no—
157#117—1061no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C22⋊D5312yes—
160C2×C4⋊C4C22×D5311yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C42C22⋊D5311yescharacteristic
165#107—2122no—
166#103—2122no—
167#102—2121no—
168#102—2121no—
169C22⋊C8—1062no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95C22⋊D5312yes—
174C24—21no—
175C2×Q8—42no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×Q8—1062no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1171992512yes—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—82no—
187C22×C4—81no—
188C22×C4C2×C22⋊D5311yescharacteristic
189C22×C4C2×C22⋊D5312yes—
190M16—2121no—
191M16—2121no—
192C2×C8—2122no—
193C2×C8—2121no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—1061no—
198C4⋊C4—1061no—
199C4⋊C4—1062no—
200C4⋊C4—2122no—
201C22⋊C4—44no—
202C22⋊C4—82no—
203C22⋊C4—81no—
204C22⋊C4—2121no—
205C42—41no—
206C42—2121no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23#1313318611yescharacteristic
213Q8—84no—
214Q8—4242no—
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—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—2121no—
232C2×C4—2121no—
233C2×C4C2×#1171992511yescharacteristic
234C2×C4#1313324811yescharacteristic
235C8—4241no—
236C8—4242no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#1681077911yescharacteristic
245C22#1681080111yescharacteristic
246C22#1681081011yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—82no—
252C4—82no—
253C4—4241no—
254C4—4241no—
255C2—21no—
256C2—21no—
257C2—82no—
258C2#2027367211yescharacteristic
259C2#2027671111yescharacteristic
260C2#2027673211yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2804C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
hypercenter#2804D53
hypocenterC53#12221
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2804

Complements

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

NH ≅ G/Ntype
2. #20273756C2semidirect
3. #20272742C2semidirect
149. C53#12221semidirect

Maximal subgroups

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

subgroupindex
#202758712
#202737562
#202727422
#202726472
C53×#28042
#1222153