Finite Group Databasebeta

← #27295234

Subgroups — #27295234

Order 13568.

452 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 95 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#20279891C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20272740C212yesmaximal, complemented
4#20272737C212yesmaximal, complemented
5C53×#1460C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic, complemented
7C2×#13133132—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810543C411yescharacteristic, complemented
11#16810535C2211yescharacteristic
12#16810507—21no—
13#16810503—21no—
14#16810496—21no—
15#16810483—21no—
16#16810481—21no—
17C2×#13133124C411yescharacteristic, complemented
18#16810390C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#431—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C23×Dic53—21no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53—41no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36#13133132—42no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—41no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×D4×C53D411yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C4×D4×C53—41no—
49C4×D4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2311yescharacteristic
53C2×C42×C53D411yescharacteristic
54C4⋊C8×C53—41no—
55C22⋊C8×C53—22no—
56C4×C8×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53SD1612yes—
62C2×D4×C53Q1612yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—81no—
68C22×C4×C53—81no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—42no—
79C22×Dic53—42no—
80C22×Dic53—42no—
81C22×Dic53—81no—
82C2×Dic106—41no—
83C2×Dic106—81no—
84C2×C8×C53—42no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C42×C53—21no—
89C42×C53—21no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C22⋊Dic53—81no—
95C4⋊Dic53—41no—
96C4⋊Dic53—82no—
97#11719919—42no—
98#11719919—41no—
99#11719919—41no—
100#11719919—81no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53#10312yes—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53C2×SD1611yescharacteristic
112C23×C53C2×Q1611yescharacteristic
113D4×C53—42no—
114D4×C53—42no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#9911yescharacteristic
130C2×C4×C53#10312yes—
131C2×Dic53—22no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—84no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146C2×Dic53—82no—
147Dic106—82no—
148C8×C53—82no—
149#11902—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—21no—
155C22×C53—42no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—42no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53C2×#9911yescharacteristic
165C22×C53C2×#10311yescharacteristic
166C22×C53#42111yescharacteristic
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—22no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173Dic53—42no—
174Dic53—81no—
175Dic53—81no—
176Dic53—81no—
177Dic53—162no—
178#2564—531no—
179#1799—531no—
180#1460D5311yescharacteristic
181#1352—532no—
182#1351—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—41no—
186C2×C53—42no—
187C2×C53—41no—
188C2×C53#152011yescharacteristic
189C2×C53#212111yescharacteristic
190C2×C53#212611yescharacteristic
191C2×C22⋊Q8—531no—
192C2×C4×D4C2×D5311yescharacteristic
193#431—21no—
194C2×#99—1061no—
195#400—531no—
196#399—531no—
197#395—531no—
198#393—531no—
199#387—1061no—
200#386—1061no—
201#385—1061no—
202#380—1061no—
203#380—1061no—
204C2×#95—531no—
205#342—532no—
206#323—21no—
207#323C2×D5312yes—
208C53#1190211yescharacteristic, complemented, Sylow
209C22×Q8—531no—
210C22×D4D21211yescharacteristic
211C23×C4—1061no—
212C23×C4C4×D5311yescharacteristic
213C23×C4D21211yescharacteristic
214C22⋊Q8—2121no—
215C22⋊Q8—2121no—
216C4×D4—41no—
217C4×D4—41no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—2121no—
223C2×C4⋊C4C4×D5311yescharacteristic
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4C22⋊D5311yescharacteristic
229C2×C22⋊C4C22×D5311yescharacteristic
230C2×C42C22⋊D5311yescharacteristic
231C4⋊C8—41no—
232#99—2122no—
233C22⋊C8—22no—
234C4×C8—41no—
235#95—531no—
236#95—531no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242#95—2121no—
243C24#1171992111yescharacteristic
244C24C2×D21211yescharacteristic
245C2×Q8—4241no—
246C2×Q8—2121no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4#1171991312yes—
250C2×D4Dic21212yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—4241no—
262C22×C4—2122no—
263C22×C4—2122no—
264C22×C4—2122no—
265C22×C4#1171992111yescharacteristic
266C22×C4#1171992111yescharacteristic
267C22×C4#1171992111yescharacteristic
268C22×C4C2×C4×D5311yescharacteristic
269C22×C4C2×C22⋊D5311yescharacteristic
270C2×C8—42no—
271C4⋊C4—41no—
272C4⋊C4—4242no—
273C4⋊C4—4241no—
274C4⋊C4—2121no—
275C4⋊C4—2122no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—4241no—
281C22⋊C4—2122no—
282C22⋊C4—2122no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C42—21no—
286C42—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23#1313314212yes—
296C23C2×#1171992111yescharacteristic
297C23C2×#1171991311yescharacteristic
298C23C2×Dic21211yescharacteristic
299Q8—4242no—
300D4—42no—
301D4—42no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—1062no—
316C2×C4—4242no—
317C2×C4—4242no—
318C2×C4—4242no—
319C2×C4—4244no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4#1313313111yescharacteristic
332C2×C4#1313313111yescharacteristic
333C2×C4#1313314212yes—
334C8—82no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—81no—
349C22C2×#1313313111yescharacteristic
350C22C2×#1313314211yescharacteristic
351C22#1681063911yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—8482no—
359C4—4241no—
360C4—4241no—
361C4—4241no—
362C4—2122no—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—42no—
367C2—41no—
368C2#2027328811yescharacteristic
369C2#2027636211yescharacteristic
370C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1460C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1460D53
hypocenterC53#11902
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #20272740C2semidirect
4. #20272737C2semidirect
6. C2×#13133196C4semidirect
8. #16810562C4semidirect
10. #16810543C4semidirect
17. C2×#13133124C4semidirect
208. C53#11902semidirect

Maximal subgroups

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

subgroupindex
#202798912
#202747032
#202727402
#202727372
C53×#14602
#1190253