Finite Group Databasebeta

← #27277630

Subgroups — #27277630

Order 13568.

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

normal subgroups only

This group has 394 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#20277317C211yesmaximal, characteristic
2#20273044C211yesmaximal, characteristic
3#20273044C211yesmaximal, characteristic
4#20272622C211yesmaximal, characteristic
5#20272621C211yesmaximal, characteristic
6#20272620C211yesmaximal, characteristic
7C53×#1305C211yesmaximal, characteristic
8C22×#11719919C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810587C2211yescharacteristic
11#16810510—22no—
12#16810504—22no—
13#16810484C412yes—
14#16810483C2211yescharacteristic
15C22⋊C4×Dic53C412yes—
16#16810348—22no—
17#16810348—22no—
18C53×#407C2211yescharacteristic
19C53×#407C2211yescharacteristic
20C53×#380C2211yescharacteristic
21C23×Dic53C2×C411yescharacteristic
22C23×Dic53C2×C411yescharacteristic
23C2×#11719917—21no—
24C2×#11719917—21no—
25C2×C4⋊Dic53—21no—
26C2×#11719919—21no—
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919D412yes—
31C2×#11719919D412yes—
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—22no—
40#13133124—24no—
41C23×C4×C53C2311yescharacteristic
42C2×M16×C53—21no—
43C2×M16×C53—21no—
44C2×C4⋊C4×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—
51C24×C53C22×C411yescharacteristic
52C22×C4×C53—22no—
53C22×C4×C53—81no—
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53#4111yescharacteristic
57C22×C4×C53#4111yescharacteristic
58C22×C4×C53#4111yescharacteristic
59C22×C4×C53#4111yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—22no—
65C22×Dic53—41no—
66C22×Dic53—41no—
67C22×Dic53—81no—
68C22×Dic53C22⋊C412yes—
69C22×Dic53C22⋊C412yes—
70M16×C53—81no—
71M16×C53—81no—
72C2×C8×C53—41no—
73C2×C8×C53—41no—
74C4⋊C4×C53—81no—
75C22⋊C4×C53—42no—
76C22⋊Dic53—42no—
77C4⋊Dic53—81no—
78#11719919—22no—
79#11719919—22no—
80#11719919—22no—
81#11719919—22no—
82C4×Dic53—82no—
83#11719917—81no—
84#11719917—81no—
85C2×#88322—41no—
86C2×#88322—41no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53C2×C22⋊C411yescharacteristic
92C23×C53#11711yescharacteristic
93C23×C53#11711yescharacteristic
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53#12311yescharacteristic
104C2×C4×C53#12311yescharacteristic
105C2×C4×C53#12311yescharacteristic
106C2×C4×C53#12311yescharacteristic
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—81no—
118C2×Dic53—81no—
119C8×C53—81no—
120C8×C53—81no—
121#88322—81no—
122#88322—81no—
123#8758—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53#38111yescharacteristic
135C22×C53#41311yescharacteristic
136C22×C53#41311yescharacteristic
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—21no—
141C4×C53—82no—
142Dic53—42no—
143Dic53—42no—
144Dic53—82no—
145#2313—531no—
146#1463—531no—
147#1463—531no—
148#1305—531no—
149#1305—531no—
150#1305—531no—
151#1305D5311yescharacteristic
152C2×C53—21no—
153C2×C53—21no—
154C2×C53—41no—
155C2×C53—41no—
156C2×C53#182211yescharacteristic
157C2×C53#182811yescharacteristic
158C2×C53#183011yescharacteristic
159C22×C4⋊C4—531no—
160#407—531no—
161#407—531no—
162#407C2×D5311yescharacteristic
163#407C2×D5311yescharacteristic
164#387—1062no—
165#387—1062no—
166#381—532no—
167#380—531no—
168#380C2×D5311yescharacteristic
169C4×C22⋊C4—532no—
170#324—1062no—
171#324—1062no—
172C53#875811yescharacteristic, complemented, Sylow
173C23×C4—531no—
174C23×C4—531no—
175C23×C4C22×D5311yescharacteristic
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—1061no—
179C2×M16—1061no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—532no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C22⋊C4—532no—
189C2×C22⋊C4C4×D5312yes—
190C2×C42—1062no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—1061no—
196C22⋊C8—1061no—
197C22⋊C8—1061no—
198C22⋊C8—1061no—
199#95—21no—
200#95—1062no—
201#95—1064no—
202#95—1061no—
203C24C2×C4×D5311yescharacteristic
204C22×C4—22no—
205C22×C4—81no—
206C22×C4—532no—
207C22×C4—532no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—1061no—
212C22×C4—1062no—
213C22×C4—4241no—
214C22×C4—2121no—
215C22×C4—2121no—
216C22×C4#1171994511yescharacteristic
217C22×C4#1171994511yescharacteristic
218C22×C4D4×D5311yescharacteristic
219C22×C4D4×D5311yescharacteristic
220C22×C4#1171994711yescharacteristic
221C22×C4#1171994911yescharacteristic
222M16—81no—
223M16—81no—
224M16—4241no—
225M16—4241no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—2121no—
229C2×C8—2121no—
230C4⋊C4—81no—
231C4⋊C4—1062no—
232C4⋊C4—1062no—
233C4⋊C4—1062no—
234C4⋊C4—1062no—
235C4⋊C4—4241no—
236C22⋊C4—42no—
237C22⋊C4—2122no—
238C42—4242no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23C22⋊C4×D5311yescharacteristic
244C23#1313317411yescharacteristic
245C23#1313317511yescharacteristic
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—4241no—
260C2×C4—4241no—
261C2×C4—2122no—
262C2×C4—2122no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4#1313319911yescharacteristic
268C2×C4#1313319911yescharacteristic
269C2×C4#1313320211yescharacteristic
270C2×C4#1313320211yescharacteristic
271C8—81no—
272C8—81no—
273C8—4241no—
274C8—4241no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—42no—
284C22—42no—
285C22#1681048511yescharacteristic
286C22#1681060411yescharacteristic
287C22#1681060611yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—82no—
293C4—4242no—
294C4—2122no—
295C4—2122no—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2#2027479711yescharacteristic
301C2#2027482211yescharacteristic
302C2#2027482811yescharacteristic
subgroupG/H
centerC22#16810485
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1305C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenter#1305D53
hypocenterC53#8758
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
172. C53#8758semidirect

Maximal subgroups

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

subgroupindex
#202773172
#202730442
#202726222
#202726212
#202726202
C53×#13052
#875853