Finite Group Databasebeta

← #27294556

Subgroups — #27294556

Order 13568.

586 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 87 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#20278989C211yesmaximal, characteristic
2#20273042C211yesmaximal, characteristic
3#20272738C212yesmaximal, complemented
4#20272714C212yesmaximal, complemented
5C2×C53×#342C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic, complemented
7#16810587—21no—
8#16810559C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810548C411yescharacteristic, complemented
11#16810548C2211yescharacteristic
12#16810511—21no—
13#16810511—21no—
14#16810496—21no—
15#16810486C411yescharacteristic, complemented
16#16810479—21no—
17#16810347—21no—
18#16810347C2212yes—
19C22×C22⋊C4×C53—21no—
20C22×C22⋊C4×C53C2211yescharacteristic
21C53×#342—21no—
22C53×#342C2212yes—
23C2×#11719917—41no—
24#13133176—41no—
25#13133176—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic, complemented
27C2×C4⋊Dic53C2×C411yescharacteristic, complemented
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C4×Dic53—21no—
37C22⋊#88322—22no—
38C22⋊#88322—41no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic, complemented
43#13133124C2×C411yescharacteristic, complemented
44C25×C53D411yescharacteristic
45C23×C4×C53—41no—
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C2×C22⋊C4×C53—24no—
49C2×C22⋊C4×C53—22no—
50C2×C22⋊C4×C53—41no—
51C2×C22⋊C4×C53—41no—
52C2×C22⋊C4×C53—41no—
53C2×C22⋊C4×C53—42no—
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C22⋊C4×C53C2311yescharacteristic
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—22no—
61C24×C53—21no—
62C24×C53—21no—
63C24×C53—42no—
64C24×C53—81no—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C22⋊C411yescharacteristic
67C24×C53C2×D411yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—42no—
73C22×C4×C53—44no—
74C22×C4×C53—42no—
75C22×C4×C53—81no—
76C22×C4×C53—82no—
77C22×C4×C53—82no—
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22⋊C4×C53—24no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—44no—
92C22⋊C4×C53—42no—
93C22⋊Dic53—42no—
94C22⋊Dic53—41no—
95C22⋊Dic53—41no—
96C22⋊Dic53—42no—
97C22⋊Dic53—41no—
98C22⋊Dic53—41no—
99C22⋊Dic53—81no—
100C22⋊Dic53—81no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103#11719919—81no—
104#11719919—81no—
105C4×Dic53—42no—
106#11719917—82no—
107C2×#88322—42no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—41no—
120C23×C53—42no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—42no—
126C23×C53—81no—
127C23×C53—81no—
128C23×C53—84no—
129C23×C53—82no—
130C23×C53—82no—
131C23×C53#9911yescharacteristic
132C23×C53#9911yescharacteristic
133C23×C53#10412yes—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53AGL(1,ℤ/8)11yescharacteristic
136C23×C53#13711yescharacteristic
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—44no—
149C2×C4×C53—81no—
150C2×C4×C53—88no—
151C2×C4×C53—82no—
152C2×C4×C53—82no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165#88322—82no—
166#11755—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—22no—
176C22×C53—21no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—44no—
183C22×C53—44no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53—82no—
191C22×C53—82no—
192C22×C53#35212yes—
193C22×C53C2×#9911yescharacteristic
194C22×C53#41711yescharacteristic
195C22×C53C2×#10411yescharacteristic
196C4×C53—41no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—84no—
201Dic53—81no—
202Dic53—81no—
203Dic53—81no—
204Dic53—81no—
205#2482—531no—
206C2×#342D5311yescharacteristic
207#1462—531no—
208#1352—532no—
209#1344—532no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—22no—
213C2×C53—41no—
214C2×C53—41no—
215C2×C53—42no—
216C2×C53—41no—
217C2×C53#150411yescharacteristic
218C2×C53C2×#35211yescharacteristic
219C2×C53#212711yescharacteristic
220C2×#117—531no—
221C22×C22⋊C4—21no—
222C22×C22⋊C4C2×D5311yescharacteristic
223#407—1061no—
224#399—531no—
225#399—531no—
226#397—531no—
227#397—531no—
228#388—1061no—
229#388—1061no—
230#385—1061no—
231#381—531no—
232#379—1061no—
233#342—21no—
234#342C2×D5312yes—
235#323—532no—
236#323—1061no—
237C53#1175511yescharacteristic, complemented, Sylow
238C25C22⋊D5311yescharacteristic
239C23×C4—41no—
240C23×C4C4×D5311yescharacteristic
241C23×C4D21211yescharacteristic
242C2×M16—2121no—
243#117—2121no—
244#117—2121no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—1061no—
256C2×C22⋊C4—1061no—
257C2×C22⋊C4—1061no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1061no—
261C2×C22⋊C4C4×D5311yescharacteristic
262C2×C22⋊C4D21211yescharacteristic
263C2×C22⋊C4C22⋊D5311yescharacteristic
264C2×C22⋊C4C22×D5311yescharacteristic
265C2×C42—1061no—
266C22⋊C8—1062no—
267C22⋊C8—2121no—
268#95—531no—
269#95—531no—
270#95—1061no—
271#95—1061no—
272#95—1061no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—42no—
279C24—81no—
280C24#1171992111yescharacteristic
281C24#1171992111yescharacteristic
282C24C2×C22⋊D5311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4#1171992111yescharacteristic
298C22×C4#1171992111yescharacteristic
299C22×C4C2×C4×D5311yescharacteristic
300C22×C4C2×D21211yescharacteristic
301M16—4242no—
302C2×C8—2122no—
303C4⋊C4—4241no—
304C4⋊C4—4241no—
305C4⋊C4—2121no—
306C4⋊C4—2121no—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—4241no—
315C22⋊C4—4241no—
316C22⋊C4—2122no—
317C22⋊C4—2121no—
318C22⋊C4—2121no—
319C22⋊C4—2122no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C42—2122no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23#1313313111yescharacteristic
347C23#1313313111yescharacteristic
348C23#1313314512yes—
349C23C2×#1171992111yescharacteristic
350C23#1313324311yescharacteristic
351C23#1313324811yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—81no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C8—4242no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22#1681041212yes—
407C22C2×#1313313111yescharacteristic
408C22#1681062211yescharacteristic
409C22C2×#1313314511yescharacteristic
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—84no—
415C4—4241no—
416C4—4241no—
417C4—4241no—
418C4—4241no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—41no—
426C2#2027321711yescharacteristic
427C2C2×#1681041211yescharacteristic
428C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC2×C53×#342C2
socleC22×C53C2×#99
prime radicalC25×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC2×#342D53
hypocenterC53#11755
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. #20272738C2semidirect
4. #20272714C2semidirect
6. C2×#13133176C4semidirect
8. #16810559C4semidirect
10. #16810548C4semidirect
15. #16810486C4semidirect
26. C2×C4⋊Dic53C2×C4semidirect
27. C2×C4⋊Dic53C2×C4semidirect
42. #13133124C2×C4semidirect
43. #13133124C2×C4semidirect
237. C53#11755semidirect

Maximal subgroups

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

subgroupindex
#202789892
#202730422
#202727382
#202727142
C2×C53×#3422
#1175553