Finite Group Databasebeta

← #27295122

Subgroups — #27295122

Order 13568.

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

normal subgroups only

This group has 522 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#20279671C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic, complemented
3#20272716C212yesmaximal, complemented
4#20272713C212yesmaximal, complemented
5C53×#1460C211yesmaximal, characteristic, complemented
6C2×#13133191C411yescharacteristic, complemented
7C2×#13133189—21no—
8C2×#13133132—21no—
9#16810553C411yescharacteristic, complemented
10#16810548C2211yescharacteristic
11#16810540C411yescharacteristic, complemented
12#16810532C2211yescharacteristic
13#16810514—21no—
14#16810502—21no—
15#16810500—21no—
16#16810483—21no—
17C2×#13133124C411yescharacteristic, complemented
18#16810390C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#431—21no—
21C53×#323—21no—
22C53×#323C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C22⋊D53—21no—
26C23×Dic53—21no—
27#13133191—41no—
28#13133191—41no—
29#13133189—42no—
30#13133189—42no—
31C2×C4⋊Dic53—41no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40#13133132—42no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—41no—
45#13133124C2×C411yescharacteristic
46#13133124C2×C411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C23×C4×C53C2×C411yescharacteristic
49C23×C4×C53D411yescharacteristic
50C4×D4×C53—41no—
51C4×D4×C53—41no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53C2311yescharacteristic
55C2×C42×C53D411yescharacteristic
56C4⋊C8×C53—41no—
57C22⋊C8×C53—22no—
58C4×C8×C53—41no—
59C24×C53C22⋊C411yescharacteristic
60C24×C53C2×D411yescharacteristic
61C23×D53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53D812yes—
65C2×D4×C53SD1612yes—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—81no—
71C22×C4×C53—81no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C2×C22⋊D53—42no—
78C2×C22⋊D53—42no—
79C2×C22⋊D53—42no—
80C2×C22⋊D53—42no—
81C2×C22⋊D53—41no—
82C2×C22⋊D53—41no—
83C2×C22⋊D53—81no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—42no—
88C22×Dic53—42no—
89C22×Dic53—42no—
90C22×Dic53—81no—
91C2×D212—41no—
92C2×D212—81no—
93C2×C8×C53—42no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C42×C53—21no—
98C42×C53—21no—
99C22⋊Dic53—41no—
100C22⋊Dic53—41no—
101C22⋊Dic53—42no—
102C22⋊Dic53—42no—
103#11719921—42no—
104#11719921—41no—
105#11719921—41no—
106#11719921—42no—
107#11719921—81no—
108C4⋊Dic53—41no—
109C4⋊Dic53—82no—
110#11719919—42no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53#10212yes—
120C23×C53C2×C22⋊C411yescharacteristic
121C23×C53C2×D811yescharacteristic
122C23×C53C2×SD1611yescharacteristic
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130D4×C53—42no—
131D4×C53—42no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53#9911yescharacteristic
146C2×C4×C53#9911yescharacteristic
147C2×C4×C53#10212yes—
148C22⋊D53—82no—
149C22⋊D53—82no—
150C22⋊D53—84no—
151C2×Dic53—22no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—84no—
161C2×Dic53—82no—
162C2×Dic53—82no—
163C2×Dic53—82no—
164D212—82no—
165C8×C53—82no—
166#11876—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—22no—
171C22×C53—21no—
172C22×C53—42no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—42no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—81no—
181C22×C53C2×#9911yescharacteristic
182C22×C53C2×#10211yescharacteristic
183C22×C53#42111yescharacteristic
184C2×D53—81no—
185C2×D53—81no—
186C2×D53—81no—
187C2×D53—81no—
188C2×D53—81no—
189C2×D53—81no—
190C2×D53—81no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—22no—
194C4×C53—41no—
195C4×C53—41no—
196C4×C53—41no—
197Dic53—42no—
198Dic53—81no—
199Dic53—81no—
200Dic53—162no—
201#2543—531no—
202#1799—531no—
203#1460D5311yescharacteristic, complemented
204#1344—532no—
205#1343—532no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—41no—
209C2×C53—42no—
210C2×C53—41no—
211C2×C53#151911yescharacteristic
212C2×C53#212111yescharacteristic
213C2×C53#212611yescharacteristic
214D53—161no—
215C2×#121—531no—
216C2×#121—1061no—
217C2×C4×D4C2×D5311yescharacteristic
218#431—21no—
219C2×#99—1061no—
220#398—531no—
221#397—531no—
222#394—531no—
223#392—531no—
224#388—1061no—
225#386—1061no—
226#386—1061no—
227#380—1061no—
228C2×#95—531no—
229#342—532no—
230#323—21no—
231#323C2×D5312yes—
232C53#1187611yescharacteristic, complemented, Sylow
233C22×D4—531no—
234C22×D4—1061no—
235C22×D4—1061no—
236C22×D4D21211yescharacteristic
237C23×C4—1061no—
238C23×C4C4×D5311yescharacteristic
239C23×C4D21211yescharacteristic
240#121—2122no—
241#121—2122no—
242#121—2121no—
243#121—2121no—
244C4×D4—41no—
245C4×D4—41no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—2121no—
249C2×C4⋊C4C4×D5311yescharacteristic
250C2×C22⋊C4—1061no—
251C2×C22⋊C4—1061no—
252C2×C22⋊C4—1061no—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1062no—
256C2×C22⋊C4C22⋊D5311yescharacteristic
257C2×C22⋊C4C22×D5311yescharacteristic
258C2×C42C22⋊D5311yescharacteristic
259C4⋊C8—41no—
260#99—2122no—
261C22⋊C8—22no—
262C4×C8—41no—
263#95—531no—
264#95—531no—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268#95—2121no—
269C24—1061no—
270C24#1171992111yescharacteristic
271C24C2×D21211yescharacteristic
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—4241no—
275C2×D4—4241no—
276C2×D4—2121no—
277C2×D4—2122no—
278C2×D4—2122no—
279C2×D4—2122no—
280C2×D4—2122no—
281C2×D4—2121no—
282C2×D4—2121no—
283C2×D4#1171991312yes—
284C2×D4D42412yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—4241no—
295C22×C4—2122no—
296C22×C4—2122no—
297C22×C4—2122no—
298C22×C4#1171992111yescharacteristic
299C22×C4#1171992111yescharacteristic
300C22×C4#1171992111yescharacteristic
301C22×C4C2×C4×D5311yescharacteristic
302C22×C4C2×C22⋊D5311yescharacteristic
303C2×C8—42no—
304C4⋊C4—41no—
305C4⋊C4—4242no—
306C4⋊C4—2121no—
307C4⋊C4—2122no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—4241no—
311C22⋊C4—2122no—
312C22⋊C4—2122no—
313C22⋊C4—2122no—
314C22⋊C4—2121no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C22⋊C4—2122no—
319C42—21no—
320C42—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—2121no—
330C23—2121no—
331C23—2121no—
332C23—2121no—
333C23—2121no—
334C23—2121no—
335C23—2121no—
336C23#1313313912yes—
337C23C2×#1171992111yescharacteristic
338C23C2×D42411yescharacteristic
339C23C2×#1171991311yescharacteristic
340D4—42no—
341D4—42no—
342D4—4242no—
343D4—4244no—
344D4—4242no—
345D4—4242no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—1062no—
360C2×C4—4242no—
361C2×C4—4242no—
362C2×C4—4242no—
363C2×C4—4244no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4#1313313111yescharacteristic
373C2×C4#1313313111yescharacteristic
374C2×C4#1313313912yes—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—4241no—
391C22—4241no—
392C22—4241no—
393C22—4241no—
394C22—4241no—
395C22—4241no—
396C22—4241no—
397C22C2×#1313313111yescharacteristic
398C22C2×#1313313911yescharacteristic
399C22#1681063911yescharacteristic
400C4—22no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—8482no—
407C4—4241no—
408C4—4241no—
409C4—2122no—
410C2—21no—
411C2—21no—
412C2—41no—
413C2—42no—
414C2—41no—
415C2—8481no—
416C2#2027328411yescharacteristic
417C2#2027636211yescharacteristic
418C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1460C2
socleC22×C53C2×#99
prime radical#20279671C2
cosocleC2×C22⋊C4×C53C23
hypercenter#1460D53
hypocenterC53#11876
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
2. #20274703C2semidirect
3. #20272716C2semidirect
4. #20272713C2semidirect
5. C53×#1460C2semidirect
6. C2×#13133191C4semidirect
9. #16810553C4semidirect
11. #16810540C4semidirect
17. C2×#13133124C4semidirect
203. #1460D53semidirect
232. C53#11876semidirect

Maximal subgroups

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

subgroupindex
#202796712
#202747032
#202727162
#202727132
C53×#14602
#1187653