Finite Group Databasebeta

← #27294656

Subgroups — #27294656

Order 13568.

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

normal subgroups only

This group has 532 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#20279671C211yesmaximal, characteristic
2#20273056C211yesmaximal, characteristic, complemented
3#20272714C212yesmaximal, complemented
4#20272711C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic, complemented
6C2×#13133191C411yescharacteristic, complemented
7C2×#13133189—21no—
8#16810680—21no—
9#16810553C411yescharacteristic, complemented
10#16810548C2211yescharacteristic
11#16810540C411yescharacteristic, complemented
12#16810532C2211yescharacteristic
13#16810514—21no—
14#16810502—21no—
15#16810500—21no—
16C2×#13133124C411yescharacteristic, complemented
17#16810347—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C4×C22⋊C4×C53—21no—
22C53×#342C2212yes—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25C22×C22⋊D53—21no—
26C23×Dic53—21no—
27#13133191—41no—
28#13133191—41no—
29#13133189—44no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C4⋊#88322—41no—
38C22⋊#88322—22no—
39#13133128—41no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
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×C53—22no—
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×C53—41no—
55C2×C42×C53D411yescharacteristic
56#99×C53—24no—
57#95×C53—41no—
58C24×C53C22⋊C411yescharacteristic
59C24×C53C2×D411yescharacteristic
60C23×D53—21no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—41no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—42no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C2×C22⋊D53—42no—
77C2×C22⋊D53—42no—
78C2×C22⋊D53—42no—
79C2×C22⋊D53—42no—
80C2×C22⋊D53—42no—
81C2×C22⋊D53—81no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—42no—
86C22×Dic53—42no—
87C22×Dic53—81no—
88C2×D212—41no—
89C2×D212—81no—
90C4⋊C4×C53—41no—
91C22⋊C4×C53—24no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—42no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—82no—
98C22⋊Dic53—42no—
99#11719921—42no—
100#11719921—42no—
101#11719921—41no—
102#11719921—41no—
103#11719921—81no—
104C4⋊Dic53—41no—
105#11719919—42no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53#10412yes—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53AGL(1,ℤ/8)12yes—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125D4×C53—44no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—44no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—82no—
142C2×C4×C53—82no—
143C2×C4×C53#9911yescharacteristic
144C2×C4×C53#9911yescharacteristic
145C22⋊D53—84no—
146C22⋊D53—84no—
147C2×Dic53—22no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—84no—
157D212—82no—
158#88322—82no—
159#11775—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—42no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—81no—
172C22×C53C2×#9911yescharacteristic
173C22×C53#41811yescharacteristic
174C22×C53C2×#10411yescharacteristic
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C2×D53—81no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—84no—
189Dic53—42no—
190Dic53—81no—
191Dic53—81no—
192#2543—531no—
193C4×#99D5311yescharacteristic, complemented
194#1466—531no—
195#1344—532no—
196#1343—532no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—42no—
200C2×C53—42no—
201C2×C53#150711yescharacteristic
202C2×C53#212111yescharacteristic
203C2×C53#212611yescharacteristic
204D53—161no—
205C2×#121—531no—
206C2×#121—1061no—
207C2×C4×D4C2×D5311yescharacteristic
208#432—1061no—
209C2×#99—21no—
210#398—531no—
211#397—531no—
212#394—531no—
213#392—531no—
214#388—1061no—
215#386—1061no—
216#386—1061no—
217C4×C22⋊C4—21no—
218C2×#95—531no—
219#342C2×D5312yes—
220#323—532no—
221#323—1061no—
222C53#1177511yescharacteristic, complemented, Sylow
223C22×D4—531no—
224C22×D4—1061no—
225C22×D4—1061no—
226C22×D4C22⋊D5311yescharacteristic
227C23×C4—1061no—
228C23×C4C4×D5311yescharacteristic
229C23×C4C22⋊D5311yescharacteristic
230#121—2124no—
231#121—2121no—
232#121—2121no—
233C4×D4—41no—
234C4×D4—41no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4C4×D5311yescharacteristic
238C2×C22⋊C4—22no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4D21211yescharacteristic
245C2×C22⋊C4C22×D5311yescharacteristic
246C2×C42—41no—
247C2×C42D21211yescharacteristic
248C4⋊C8—2121no—
249#99—24no—
250C22⋊C8—1062no—
251#97—2121no—
252#95—41no—
253#95—531no—
254#95—531no—
255#95—1061no—
256#95—1061no—
257#95—1061no—
258C24—1061no—
259C24#1171992111yescharacteristic
260C24C2×C22⋊D5311yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—4241no—
265C2×D4—4241no—
266C2×D4—2121no—
267C2×D4—2122no—
268C2×D4—2122no—
269C2×D4—2122no—
270C2×D4—2122no—
271C2×D4—2122no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—4241no—
283C22×C4—2122no—
284C22×C4—2122no—
285C22×C4#1171992111yescharacteristic
286C22×C4#1171992111yescharacteristic
287C22×C4#1171992111yescharacteristic
288C22×C4C2×C4×D5311yescharacteristic
289C22×C4C2×D21211yescharacteristic
290C2×C8—2122no—
291C4⋊C4—41no—
292C4⋊C4—2121no—
293C4⋊C4—2122no—
294C22⋊C4—24no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—4241no—
299C22⋊C4—2122no—
300C22⋊C4—2122no—
301C22⋊C4—2122no—
302C22⋊C4—2121no—
303C22⋊C4—2121no—
304C42—21no—
305C42—21no—
306C42—82no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—2121no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23—2121no—
320C23—2121no—
321C23—2121no—
322C23#1313314512yes—
323C23C2×#1171992111yescharacteristic
324C23#1313324212yes—
325D4—44no—
326D4—4242no—
327D4—4244no—
328D4—4244no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—44no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—1062no—
347C2×C4—4244no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4#1313313111yescharacteristic
357C2×C4#1313313111yescharacteristic
358C8—4242no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—22no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22—4241no—
375C22—4241no—
376C22—4241no—
377C22—4241no—
378C22C2×#1313313111yescharacteristic
379C22#1681062711yescharacteristic
380C22C2×#1313314511yescharacteristic
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—84no—
388C4—4241no—
389C4—4241no—
390C4—2122no—
391C2—21no—
392C2—21no—
393C2—42no—
394C2—42no—
395C2—8481no—
396C2#2027323011yescharacteristic
397C2#2027635911yescharacteristic
398C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radical#20279671C2
cosocleC2×C22⋊C4×C53C23
hypercenterC4×#99D53
hypocenterC53#11775
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #20273056C2semidirect
3. #20272714C2semidirect
4. #20272711C2semidirect
5. C4×#99×C53C2semidirect
6. C2×#13133191C4semidirect
9. #16810553C4semidirect
11. #16810540C4semidirect
16. C2×#13133124C4semidirect
193. C4×#99D53semidirect
222. C53#11775semidirect

Maximal subgroups

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

subgroupindex
#202796712
#202730562
#202727142
#202727112
C4×#99×C532
#1177553