Finite Group Databasebeta

← #27294482

Subgroups — #27294482

Order 13568.

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

normal subgroups only

This group has 400 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#20280575C211yesmaximal, characteristic
2C4×#13133136C211yesmaximal, characteristic, complemented
3#20272746C212yesmaximal, complemented
4#20272719C212yesmaximal
5C53×#1461C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810558—21no—
10#16810543C411yescharacteristic
11#16810543C2211yescharacteristic
12#16810540C411yescharacteristic
13#16810540C2211yescharacteristic
14#16810534—21no—
15#16810514—21no—
16#16810514—21no—
17#16810491C411yescharacteristic
18#16810392C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#431—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×D212C2×C411yescharacteristic
25C2×#11719917—22no—
26#13133205—41no—
27#13133205—41no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719921—21no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36#13133136—24no—
37#13133128—41no—
38C4×#88322—41no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×Q8×C53D411yescharacteristic
46C4×Q8×C53—41no—
47C4×Q8×C53—41no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C4⋊C8×C53—41no—
55C22⋊C8×C53—22no—
56C4×C8×C53—41no—
57C23×D53—21no—
58C2×Q8×C53—22no—
59C2×Q8×C53—22no—
60C2×Q8×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C2×D212—41no—
72C2×D212—81no—
73C2×Dic106—41no—
74C2×Dic106—81no—
75C2×C8×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85#11719921—42no—
86#11719921—41no—
87#11719921—41no—
88#11719921—42no—
89#11719921—42no—
90#11719921—81no—
91#11719921—81no—
92#11719919—81no—
93#11719919—81no—
94#11719917—24no—
95#11719917—42no—
96C2×#88322—42no—
97C23×C53C2×C22⋊C411yescharacteristic
98C22×D53—41no—
99C22×D53—41no—
100C22×D53—41no—
101C22×D53—41no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—41no—
105Q8×C53—44no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#10412yes—
123C2×C4×C53#13512yes—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133D212—82no—
134Dic106—82no—
135C8×C53—82no—
136#88322—44no—
137#11738—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53C2×#9911yescharacteristic
141C22×C53#41611yescharacteristic
142C22×C53C2×#10411yescharacteristic
143C2×D53—81no—
144C2×D53—81no—
145C2×D53—81no—
146C2×D53—81no—
147C2×D53—81no—
148C2×D53—81no—
149C2×D53—81no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158Dic53—81no—
159Dic53—81no—
160Dic53—81no—
161#2632—531no—
162C4×#101—531no—
163#1461D5311yescharacteristic, complemented
164#1354—532no—
165#1345—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53#150111yescharacteristic
169C2×C53#212211yescharacteristic
170C2×C53#213311yescharacteristic
171D53—161no—
172C2×#124—531no—
173C2×C4×Q8C2×D5311yescharacteristic
174#431—21no—
175C2×#101—1061no—
176C4×M16—1061no—
177#399—1061no—
178#395—531no—
179#395—531no—
180#394—531no—
181#394—531no—
182#393—1061no—
183#388—1061no—
184#388—1061no—
185#383—531no—
186#343—532no—
187#324—21no—
188#324C2×D5312yes—
189C53#1173811yescharacteristic, complemented, Sylow
190C22×Q8—531no—
191C22×Q8D21211yescharacteristic
192C22×D4—531no—
193C2×M16—1062no—
194#124—2121no—
195#124—2121no—
196C4×Q8—41no—
197C4×Q8—41no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4C4×D5311yescharacteristic
201C2×C4⋊C4C22⋊D5311yescharacteristic
202C2×C4⋊C4C22×D5311yescharacteristic
203C2×C22⋊C4—1061no—
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1061no—
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C42C4×D5311yescharacteristic
210C2×C42D21211yescharacteristic
211C2×C42C22⋊D5311yescharacteristic
212C4⋊C8—41no—
213#101—1064no—
214C22⋊C8—22no—
215#97—2121no—
216C4×C8—41no—
217C4×C8—2121no—
218#95—531no—
219#95—531no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224C24—1061no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—4241no—
229C2×Q8—2121no—
230C2×D4—4241no—
231C2×D4—2121no—
232C22×C4—1061no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4#1171992111yescharacteristic
236C22×C4#1171992111yescharacteristic
237C22×C4#1171992111yescharacteristic
238C22×C4#1171992111yescharacteristic
239C22×C4C2×C4×D5311yescharacteristic
240C22×C4C2×D21211yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—1064no—
243M16—2122no—
244C2×C8—42no—
245C2×C8—2122no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—4241no—
250C4⋊C4—4241no—
251C22⋊C4—4241no—
252C22⋊C4—4241no—
253C22⋊C4—2122no—
254C22⋊C4—2121no—
255C22⋊C4—2121no—
256C22⋊C4—2122no—
257C22⋊C4—2122no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—2121no—
265C23—2121no—
266C23—2121no—
267C23—2121no—
268C23—2121no—
269C23—2121no—
270C23—2121no—
271C23C2×#1171992111yescharacteristic
272Q8—44no—
273Q8—4242no—
274D4—4242no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4#1313313111yescharacteristic
299C2×C4#1313313111yescharacteristic
300C2×C4#1313314312yes—
301C2×C4#1313323912yes—
302C8—82no—
303C8—2124no—
304C22—21no—
305C22—21no—
306C22—4241no—
307C22—4241no—
308C22—4241no—
309C22—4241no—
310C22—4241no—
311C22—4241no—
312C22—4241no—
313C22C2×#1313313111yescharacteristic
314C22#1681062011yescharacteristic
315C22C2×#1313314311yescharacteristic
316C4—22no—
317C4—22no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—4241no—
325C4—4241no—
326C4—4241no—
327C2—21no—
328C2—21no—
329C2—8481no—
330C2#2027320711yescharacteristic
331C2#2027636411yescharacteristic
332C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1461C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1461D53
hypocenterC53#11738
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. C4×#13133136C2semidirect
3. #20272746C2semidirect
5. C53×#1461C2semidirect
163. #1461D53semidirect
189. C53#11738semidirect

Maximal subgroups

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

subgroupindex
#202805752
C4×#131331362
#202727462
#202727192
C53×#14612
#1173853