Finite Group Databasebeta

← #27293656

Subgroups — #27293656

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#20279196C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3#20272658C212yesmaximal, complemented
4#20272648C212yesmaximal, complemented
5C2×C53×#324C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7#16810587—21no—
8#16810561—21no—
9#16810561—21no—
10#16810528C411yescharacteristic
11#16810528C2211yescharacteristic
12#16810488C411yescharacteristic
13#16810488C2211yescharacteristic
14#16810486—21no—
15#16810486—21no—
16C4×C22⋊Dic53C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—41no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C22⋊#88322—22no—
34C22⋊#88322—41no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C22×C8×C53—41no—
48C2×C4⋊C4×C53—41no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53C2311yescharacteristic
54C22⋊C8×C53—22no—
55C22⋊C8×C53—41no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×Dic106—22no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×C8×C53—42no—
81C2×C8×C53—82no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C22⋊Dic53—42no—
87C22⋊Dic53—41no—
88C22⋊Dic53—41no—
89C4⋊Dic53—41no—
90#11719919—42no—
91C4×Dic53—22no—
92C4×Dic53—41no—
93#11719917—82no—
94C2×#88322—42no—
95C23×C53—22no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—41no—
99C23×C53#9911yescharacteristic
100C23×C53#9911yescharacteristic
101C23×C53C2×C22⋊C411yescharacteristic
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53#13112yes—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—42no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133Dic106—82no—
134C8×C53—82no—
135#88322—82no—
136#11551—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53#35412yes—
147C22×C53#40811yescharacteristic
148C22×C53C2×#9911yescharacteristic
149C22×C53#41311yescharacteristic
150C4×C53—22no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—41no—
156Dic53—81no—
157Dic53—81no—
158Dic53—81no—
159Dic53—81no—
160#2503—531no—
161#1463—531no—
162C2×#324D5311yescharacteristic
163#1324—532no—
164#1319—532no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53—22no—
168C2×C53—41no—
169C2×C53#146811yescharacteristic
170C2×C53C2×#35411yescharacteristic
171C2×C53#213311yescharacteristic
172C2×C22⋊Q8—531no—
173C22×C4⋊C4C2×D5311yescharacteristic
174#407—1061no—
175C2×C22⋊C8—21no—
176#400—1061no—
177#400—1061no—
178#391—531no—
179#391—531no—
180#382—531no—
181#382—531no—
182#381—1061no—
183#381—1061no—
184C4×C22⋊C4—531no—
185#324—21no—
186#324—532no—
187#324—1061no—
188#324C2×D5312yes—
189C53#1155111yescharacteristic, complemented, Sylow
190C22×Q8—531no—
191C23×C4C4×D5311yescharacteristic
192C23×C4D21211yescharacteristic
193C23×C4C22⋊D5311yescharacteristic
194C2×M16—2121no—
195C22×C8—41no—
196C22⋊Q8—2121no—
197C22⋊Q8—2121no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4C4×D5311yescharacteristic
203C2×C4⋊C4D21211yescharacteristic
204C2×C4⋊C4C22⋊D5311yescharacteristic
205C2×C4⋊C4C22×D5311yescharacteristic
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C42—531no—
209C2×C42—531no—
210C22⋊C8—22no—
211C22⋊C8—41no—
212C22⋊C8—1062no—
213C22⋊C8—2121no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217#95—1061no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223C24#1171992111yescharacteristic
224C2×Q8—1062no—
225C2×Q8—2121no—
226C2×Q8—2121no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4C2×C4×D5311yescharacteristic
245C22×C4C2×D21211yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—4242no—
248C2×C8—42no—
249C2×C8—82no—
250C2×C8—2122no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2121no—
256C4⋊C4—2122no—
257C22⋊C4—2122no—
258C22⋊C4—2121no—
259C22⋊C4—2121no—
260C42—1062no—
261C42—2121no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—41no—
266C23#1313313111yescharacteristic
267C23#1313313111yescharacteristic
268C23C2×#1171992111yescharacteristic
269Q8—4242no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2122no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4#1313322312yes—
301C8—82no—
302C8—4242no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22#1681041712yes—
313C22#1681058911yescharacteristic
314C22C2×#1313313111yescharacteristic
315C22#1681060511yescharacteristic
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—4241no—
323C4—4241no—
324C4—4241no—
325C4—4241no—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—41no—
330C2#2027306111yescharacteristic
331C2C2×#1681041711yescharacteristic
332C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#324C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#324D53
hypocenterC53#11551
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #20272658C2semidirect
4. #20272648C2semidirect
189. C53#11551semidirect

Maximal subgroups

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

subgroupindex
#202791962
#202730452
#202726582
#202726482
C2×C53×#3242
#1155153