Finite Group Databasebeta

← #27272933

Subgroups — #27272933

Order 13568.

424 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 83 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#20282703C211yesmaximal, characteristic
2#20276045C212yesmaximal
3#20272541C211yesmaximal, characteristic
4#20272537C212yesmaximal
5C53×#2016C211yesmaximal, characteristic
6C2×#13133148—22no—
7#16810562—21no—
8#16810555—22no—
9#16810530—21no—
10#16810521C2212yes—
11#16810510—22no—
12#16810385—22no—
13#16810378C2212yes—
14C22×Q16×C53—21no—
15C2×C4⋊Q8×C53C2211yescharacteristic
16C2×#103×C53—21no—
17C53×#391C2211yescharacteristic
18C53×#336C2211yescharacteristic
19C22×#88322—22no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53D412yes—
23C2×C4⋊Dic53D412yes—
24C2×#11719919—22no—
25C2×C4×Dic53—21no—
26#13133148—44no—
27#13133124—21no—
28#13133124—22no—
29#13133124—22no—
30#13133124—22no—
31C22×Q8×C53—21no—
32C22×Q8×C53D411yescharacteristic
33C2×Q16×C53—42no—
34C2×Q16×C53—42no—
35C2×Q16×C53—42no—
36C22×C8×C53—21no—
37C4⋊Q8×C53—42no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C42×C53C2311yescharacteristic
41#103×C53—42no—
42#95×C53—22no—
43C2×Q8×C53—21no—
44C2×Q8×C53—22no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53SD1612yes—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C22×Dic53—22no—
59C22×Dic53—22no—
60Q16×C53—48no—
61C2×C8×C53—22no—
62C2×C8×C53—41no—
63C2×C8×C53—41no—
64C4⋊C4×C53—24no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—41no—
67C42×C53—41no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C4⋊Dic53—42no—
71C4⋊Dic53—81no—
72C4⋊Dic53—81no—
73#11719919—42no—
74#11719919—42no—
75C4×Dic53—81no—
76C2×#88322—24no—
77C2×#88322—42no—
78C2×#88322—42no—
79C23×C53#12011yescharacteristic
80Q8×C53—42no—
81Q8×C53—42no—
82Q8×C53—42no—
83Q8×C53—81no—
84Q8×C53—81no—
85Q8×C53—81no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—81no—
97C2×C4×C53—81no—
98C2×C4×C53C2×SD1611yescharacteristic
99C2×C4×C53#13512yes—
100C2×C4×C53#13711yescharacteristic
101C2×C4×C53#13712yes—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C8×C53—42no—
109#88322—44no—
110#7760—531nomaximal, Sylow
111C22×C53#44811yescharacteristic
112C22×C53#45212yes—
113C22×C53#45311yescharacteristic
114C22×C53#45612yes—
115C22×C53B(4,𝔽2)11yescharacteristic
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—42no—
121C4×C53—82no—
122C4×C53—81no—
123Dic53—82no—
124Dic53—82no—
125#2809—531no—
126#2058—532no—
127#2016D5311yescharacteristic
128#1272—531no—
129#1271—532no—
130C2×C53#159611yescharacteristic
131C2×C53#160312yes—
132C2×C53#163411yescharacteristic
133C2×C53#165312yes—
134C2×C53#166011yescharacteristic
135C22×Q16—21no—
136C2×C4⋊Q8C2×D5311yescharacteristic
137C2×#106—1062no—
138C2×#103—21no—
139#400—1061no—
140#398—1062no—
141#391—1061no—
142#391C2×D5311yescharacteristic
143#389—532no—
144#387—1062no—
145#340—1062no—
146#336—532no—
147#336C2×D5311yescharacteristic
148C53#776011yescharacteristic, complemented, Sylow
149C22×Q8—21no—
150C22×Q8C22⋊D5311yescharacteristic
151C2×Q16—42no—
152C2×Q16—42no—
153C2×Q16—42no—
154C22×C8—21no—
155C22×C8—1062no—
156C4⋊Q8—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4—1061no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4C22⋊D5311yescharacteristic
164C2×C42—1061no—
165C2×C42C22×D5311yescharacteristic
166#106—2124no—
167#103—42no—
168#95—22no—
169#95—1061no—
170#95—1062no—
171#95—1062no—
172#95—1062no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8#1171992312yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—1062no—
187C22×C4—1062no—
188C22×C4D4×D5312yes—
189C22×C4C2×C22⋊D5311yescharacteristic
190Q16—48no—
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—1064no—
195C2×C8—2122no—
196C2×C8—2122no—
197C4⋊C4—24no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—1062no—
201C4⋊C4—1062no—
202C4⋊C4—4241no—
203C4⋊C4—4241no—
204C4⋊C4—2122no—
205C4⋊C4—2122no—
206C4⋊C4—2122no—
207C42—41no—
208C42—4241no—
209C23#1313318511yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—81no—
214Q8—81no—
215Q8—81no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4C2×#1171992311yescharacteristic
235C2×C4#1313324012yes—
236C2×C4#1313324711yescharacteristic
237C2×C4#1313324912yes—
238C8—42no—
239C8—2124no—
240C22#1681076111yescharacteristic
241C22#1681078412yes—
242C22#1681079011yescharacteristic
243C22#1681080412yes—
244C22#1681080911yescharacteristic
245C4—22no—
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—81no—
251C4—82no—
252C4—4242no—
253C4—4242no—
254C2#2027363711yescharacteristic
255C2#2027367712yes—
256C2#2027385811yescharacteristic
257C2#2027399012yes—
258C2#2027402411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2016C2
socle, prime radicalC23×C53#120
hypercenter#2016D53
hypocenterC53#7760
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2016

Complements

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

NH ≅ G/Ntype
148. C53#7760semidirect

Maximal subgroups

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

subgroupindex
#202827032
#202760452
#202725412
#202725372
C53×#20162
#776053