Finite Group Databasebeta

← #27272640

Subgroups — #27272640

Order 13568.

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

normal subgroups only

This group has 470 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#20283053C211yesmaximal, characteristic
2#20274694C211yesmaximal, characteristic
3#20274476C211yesmaximal, characteristic
4#20272540C211yesmaximal, characteristic, complemented
5#20272540C211yesmaximal, characteristic, complemented
6#20272538C211yesmaximal, characteristic, complemented
7C53×#1271C211yesmaximal, characteristic, complemented
8C2×#13133197C411yescharacteristic
9C2×C42×D53C2211yescharacteristic
10C2×#13133146—21no—
11C2×#13133129—21no—
12C2×#13133129—21no—
13C2×#13133127—21no—
14#16810529C2211yescharacteristic
15#16810528C411yescharacteristic
16#16810519C411yescharacteristic
17#16810510—21no—
18#16810509—21no—
19#16810494—21no—
20#16810485—21no—
21#16810480C411yescharacteristic
22#16810378C2211yescharacteristic
23#16810378C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×#88322—21no—
32C22×#88322—21no—
33#13133197—42no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C42×D53—42no—
43#13133146—42no—
44#13133129—24no—
45#13133129—42no—
46#13133127—42no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59#95×C53—21no—
60C23×D53C22⋊C411yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C2×C4×D53—22no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—22no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×Dic106—21no—
84C2×Dic106—21no—
85C2×Dic106—41no—
86C2×Dic106—41no—
87C2×C8×C53—22no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C42×C53—41no—
99#11719921—42no—
100#11719921—42no—
101C4⋊Dic53—42no—
102C4⋊Dic53—41no—
103#11719919—42no—
104C4×Dic53—22no—
105C4×Dic53—22no—
106C4×Dic53—41no—
107C2×#88322—22no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113C23×C53C2×C22⋊C411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53#10412yes—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53#13111yescharacteristic
134C2×C4×C53#13111yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53#13711yescharacteristic
137C2×C4×C53#13711yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53#9912yes—
154C4×D53—42no—
155C4×D53—42no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—42no—
159C4×D53—42no—
160Dic106—81no—
161Dic106—81no—
162C8×C53—42no—
163C8×C53—81no—
164#88322—42no—
165#88322—81no—
166#7701—531nomaximal, Sylow
167C22×C53#40811yescharacteristic
168C22×C53C2×#9911yescharacteristic
169C22×C53#41311yescharacteristic
170C22×C53#41711yescharacteristic
171C22×C53#41911yescharacteristic
172C22×C53C2×#10411yescharacteristic
173C22×C53#42111yescharacteristic
174C2×D53—22no—
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—81no—
188C4×C53—81no—
189Dic53—22no—
190Dic53—41no—
191Dic53—41no—
192Dic53—81no—
193Dic53—81no—
194#2855—531no—
195#1797—531no—
196#1749—531no—
197#1272—531no—
198#1272—531no—
199#1271—531no—
200#1271D5311yescharacteristic, complemented
201C2×C53#146811yescharacteristic
202C2×C53#148311yescharacteristic
203C2×C53#149711yescharacteristic
204C2×C53#150811yescharacteristic
205C2×C53#151311yescharacteristic
206C2×C53#153211yescharacteristic
207C2×C53#155911yescharacteristic
208D53—42no—
209C2×C22⋊Q8—531no—
210C22×C42—531no—
211C2×C4⋊C8—1061no—
212C2×C22⋊C8—1061no—
213C2×C22⋊C8—1061no—
214C2×#97—1061no—
215#391—531no—
216#391—531no—
217#389—531no—
218#389C2×D5311yescharacteristic
219#387—1061no—
220#387—1061no—
221#385—1061no—
222#381—1061no—
223#380—531no—
224#336—531no—
225#336—531no—
226#336C2×D5311yescharacteristic
227#336C2×D5311yescharacteristic
228C53#770111yescharacteristic, complemented, Sylow
229C22×Q8—531no—
230C23×C4—531no—
231C23×C4—531no—
232C23×C4—531no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—1061no—
236C22×C8—1061no—
237C22⋊Q8—2122no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4C4×D5311yescharacteristic
243C2×C4⋊C4C4×D5311yescharacteristic
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—2122no—
250C2×C42C22×D5311yescharacteristic
251C4⋊C8—2122no—
252C22⋊C8—1064no—
253C22⋊C8—2122no—
254#97—2122no—
255#95—21no—
256#95—1061no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262C24—531no—
263C2×Q8—1061no—
264C2×Q8—1061no—
265C2×Q8—2121no—
266C2×Q8—2121no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—1062no—
273C22×C4—1062no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—2121no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4C2×C4×D5311yescharacteristic
287C22×C4D4×D5311yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—1062no—
296C2×C8—2121no—
297C2×C8—2121no—
298C2×C8—2121no—
299C2×C8—2121no—
300C2×C8—2121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—2121no—
307C4⋊C4—2122no—
308C4⋊C4—2122no—
309C22⋊C4—2122no—
310C22⋊C4—2122no—
311C42—41no—
312C42—1062no—
313C42—1062no—
314C42—2121no—
315C23—532no—
316C23—1061no—
317C23—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—1061no—
322C23C22⋊C4×D5311yescharacteristic
323Q8—4241no—
324Q8—4241no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—532no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4—2122no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4#1313322311yescharacteristic
359C2×C4#1313322511yescharacteristic
360C2×C4#1313324111yescharacteristic
361C2×C4#1313324611yescharacteristic
362C2×C4#1313324611yescharacteristic
363C2×C4#1313324911yescharacteristic
364C8—42no—
365C8—81no—
366C8—4241no—
367C8—2122no—
368C22—1062no—
369C22—1062no—
370C22—1062no—
371C22—2121no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22#1681058811yescharacteristic
376C22#99×D5311yescharacteristic
377C22#1681060611yescharacteristic
378C22#1681062111yescharacteristic
379C22#1681063011yescharacteristic
380C22#104×D5311yescharacteristic
381C22#1681063811yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—1062no—
391C4—4241no—
392C4—4241no—
393C4—2121no—
394C4—2121no—
395C2—2122no—
396C2#2027306211yescharacteristic
397C2#2027312511yescharacteristic
398C2#2027318611yescharacteristic
399C2#2027323311yescharacteristic
400C2#2027325511yescharacteristic
401C2#2027333411yescharacteristic
402C2#2027344411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1271D53
hypocenterC53#7701
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #20272540C2semidirect
5. #20272540C2semidirect
6. #20272538C2semidirect
7. C53×#1271C2semidirect
200. #1271D53semidirect
228. C53#7701semidirect

Maximal subgroups

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

subgroupindex
#202830532
#202746942
#202744762
#202725402
#202725382
C53×#12712
#770153