Finite Group Databasebeta

← #27277620

Subgroups — #27277620

Order 13568.

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

normal subgroups only

This group has 398 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#20277316C211yesmaximal, characteristic
2#20274793C211yesmaximal, characteristic
3#20274793C211yesmaximal, characteristic
4#20272624C211yesmaximal, characteristic
5#20272624C211yesmaximal, characteristic
6#20272623C211yesmaximal, characteristic
7C53×#1306C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810587C2211yescharacteristic
11#16810508—22no—
12#16810506—22no—
13#16810486C2211yescharacteristic
14#16810481C412yes—
15C22⋊C4×Dic53C412yes—
16#16810391—22no—
17#16810391—22no—
18C53×#407C2211yescharacteristic
19C53×#407C2211yescharacteristic
20C53×#381C2211yescharacteristic
21C23×Dic53C2×C411yescharacteristic
22C23×Dic53C2×C411yescharacteristic
23C2×#11719917—21no—
24C2×#11719917—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53D412yes—
30C2×C4⋊Dic53D412yes—
31C2×#11719919—22no—
32C2×C22⋊Dic53C2×C412yes—
33C2×C4×Dic53—22no—
34C22⋊#88322—21no—
35C22⋊#88322—21no—
36C22⋊#88322—21no—
37C22⋊#88322—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—22no—
41#13133124—22no—
42C23×C4×C53C2311yescharacteristic
43C2×M16×C53—21no—
44C2×M16×C53—21no—
45C2×C22⋊C4×C53C2×C412yes—
46C22⋊C8×C53—21no—
47C22⋊C8×C53—21no—
48C22⋊C8×C53—21no—
49C22⋊C8×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52C24×C53C22×C411yescharacteristic
53C22×C4×C53—22no—
54C22×C4×C53—42no—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53#4111yescharacteristic
58C22×C4×C53#4111yescharacteristic
59C22×C4×C53#4111yescharacteristic
60C22×C4×C53#4111yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53—22no—
66C22×Dic53—81no—
67C22×Dic53—81no—
68C22×Dic53C22⋊C412yes—
69C22×Dic53C22⋊C412yes—
70M16×C53—42no—
71M16×C53—42no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C22⋊C4×C53—42no—
75C22⋊Dic53—42no—
76C4⋊Dic53—22no—
77C4⋊Dic53—22no—
78C4⋊Dic53—22no—
79C4⋊Dic53—22no—
80#11719919—82no—
81C4×Dic53—82no—
82#11719917—42no—
83#11719917—42no—
84C2×#88322—41no—
85C2×#88322—41no—
86C23×C53—22no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53#11711yescharacteristic
91C23×C53#11711yescharacteristic
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53#12311yescharacteristic
102C2×C4×C53#12311yescharacteristic
103C2×C4×C53#12311yescharacteristic
104C2×C4×C53#12311yescharacteristic
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—81no—
113C2×Dic53—81no—
114C2×Dic53—81no—
115C2×Dic53—81no—
116C8×C53—81no—
117C8×C53—81no—
118#88322—81no—
119#88322—81no—
120#8755—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—44no—
130C22×C53#38111yescharacteristic
131C22×C53#41012yes—
132C4×C53—21no—
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—82no—
137Dic53—42no—
138Dic53—42no—
139Dic53—82no—
140#2313—531no—
141#1821—531no—
142#1821—531no—
143#1306—531no—
144#1306—531no—
145#1306—531no—
146#1306D5311yescharacteristic
147C2×C53—22no—
148C2×C53—41no—
149C2×C53—41no—
150C2×C53#182011yescharacteristic
151C2×C53#182911yescharacteristic
152C2×C53#183011yescharacteristic
153C22×C4⋊C4—531no—
154#407—531no—
155#407—531no—
156#407C2×D5311yescharacteristic
157#407C2×D5311yescharacteristic
158#387—1062no—
159#387—1062no—
160#381—531no—
161#381C2×D5311yescharacteristic
162#380—532no—
163C4×C22⋊C4—532no—
164#343—1062no—
165#343—1062no—
166C53#875511yescharacteristic, complemented, Sylow
167C23×C4—531no—
168C23×C4—531no—
169C23×C4C22×D5311yescharacteristic
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—1061no—
173C2×M16—1061no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4—532no—
176C2×C4⋊C4—1061no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4—1062no—
181C2×C22⋊C4—532no—
182C2×C22⋊C4C4×D5312yes—
183C2×C42—1062no—
184C22⋊C8—21no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—1061no—
189C22⋊C8—1061no—
190C22⋊C8—1061no—
191C22⋊C8—1061no—
192#95—21no—
193#95—21no—
194#95—1062no—
195#95—1062no—
196#95—1061no—
197#95—1061no—
198C24C2×C4×D5311yescharacteristic
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—532no—
202C22×C4—532no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4—1062no—
208C22×C4—4241no—
209C22×C4—4241no—
210C22×C4#1171994511yescharacteristic
211C22×C4#1171994511yescharacteristic
212C22×C4D4×D5311yescharacteristic
213C22×C4D4×D5311yescharacteristic
214C22×C4#1171994711yescharacteristic
215C22×C4#1171994911yescharacteristic
216M16—42no—
217M16—42no—
218M16—2122no—
219M16—2122no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—2121no—
223C2×C8—2121no—
224C4⋊C4—1062no—
225C4⋊C4—1062no—
226C4⋊C4—1062no—
227C4⋊C4—1062no—
228C4⋊C4—4242no—
229C22⋊C4—42no—
230C22⋊C4—2122no—
231C42—4242no—
232C23—22no—
233C23—41no—
234C23—41no—
235C23C22⋊C4×D5311yescharacteristic
236C23#1313317411yescharacteristic
237C23#1313317511yescharacteristic
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—1062no—
248C2×C4—1062no—
249C2×C4—4241no—
250C2×C4—4241no—
251C2×C4—4241no—
252C2×C4—4241no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4—2122no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4#1313319911yescharacteristic
259C2×C4#1313319911yescharacteristic
260C2×C4#1313320211yescharacteristic
261C2×C4#1313320211yescharacteristic
262C8—81no—
263C8—81no—
264C8—4241no—
265C8—4241no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22#1681048511yescharacteristic
276C22#1681059512yes—
277C4—21no—
278C4—21no—
279C4—21no—
280C4—21no—
281C4—82no—
282C4—4242no—
283C4—2122no—
284C4—2122no—
285C2—22no—
286C2—41no—
287C2—41no—
288C2#2027479111yescharacteristic
289C2#2027482511yescharacteristic
290C2#2027482811yescharacteristic
subgroupG/H
centerC22#16810485
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1306C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenter#1306D53
hypocenterC53#8755
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
166. C53#8755semidirect

Maximal subgroups

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

subgroupindex
#202773162
#202747932
#202726242
#202726232
C53×#13062
#875553