Finite Group Databasebeta

← #27294375

Subgroups — #27294375

Order 13568.

592 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 107 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#20280520C211yesmaximal, characteristic
2C2×#16810347C211yesmaximal, characteristic
3#20272738C212yesmaximal, complemented
4#20272735C212yesmaximal, complemented
5C2×C53×#342C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic, complemented
7C2×C22⋊#88322—21no—
8#16810559C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810546—21no—
11#16810546—21no—
12#16810535C411yescharacteristic, complemented
13#16810535C2211yescharacteristic
14#16810496—21no—
15#16810483C411yescharacteristic, complemented
16#16810479—21no—
17#16810347—21no—
18#16810347C2212yes—
19C22×C22⋊C4×C53—21no—
20C22×C22⋊C4×C53C2211yescharacteristic
21C53×#342—21no—
22C53×#342C2212yes—
23C22×Dic106C2×C411yescharacteristic, complemented
24C22×#88322—41no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic, complemented
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C22⋊#88322—22no—
39C22⋊#88322—41no—
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—81no—
62C24×C53—81no—
63C24×C53C22⋊C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
65C24×C53SD1612yes—
66C24×C53Q1612yes—
67C24×C53C2×D411yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—44no—
73C22×C4×C53—42no—
74C22×C4×C53—81no—
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—
86C2×Dic106—41no—
87C2×Dic106—81no—
88C22⋊C4×C53—24no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—44no—
94C22⋊C4×C53—42no—
95C22⋊Dic53—41no—
96C22⋊Dic53—41no—
97C22⋊Dic53—42no—
98C22⋊Dic53—81no—
99C22⋊Dic53—81no—
100C22⋊Dic53—81no—
101C22⋊Dic53—81no—
102C4⋊Dic53—41no—
103C4⋊Dic53—81no—
104#11719919—42no—
105#11719919—81no—
106#11719919—81no—
107C2×#88322—42no—
108C2×#88322—82no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—81no—
126C23×C53—81no—
127C23×C53—82no—
128C23×C53—82no—
129C23×C53—84no—
130C23×C53#9911yescharacteristic
131C23×C53#9911yescharacteristic
132C23×C53#10312yes—
133C23×C53#10312yes—
134C23×C53#10412yes—
135C23×C53C2×C22⋊C411yescharacteristic
136C23×C53C2×SD1611yescharacteristic
137C23×C53C2×Q1611yescharacteristic
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—44no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53—88no—
152C2×C4×C53—82no—
153C2×C4×C53—82no—
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—
165C2×Dic53—41no—
166Dic106—82no—
167#88322—82no—
168#11716—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—22no—
177C22×C53—21no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—44no—
184C22×C53—44no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—82no—
192C22×C53—82no—
193C22×C53#32812yes—
194C22×C53#32812yes—
195C22×C53C2×#9911yescharacteristic
196C22×C53C2×#10311yescharacteristic
197C22×C53C2×#10411yescharacteristic
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—84no—
203Dic53—81no—
204Dic53—81no—
205Dic53—81no—
206Dic53—81no—
207#2626—531no—
208C2×#342D5311yescharacteristic
209C2×#323—531no—
210#1352—532no—
211#1351—532no—
212C2×C53—21no—
213C2×C53—22no—
214C2×C53—21no—
215C2×C53—22no—
216C2×C53—41no—
217C2×C53—42no—
218C2×C53—41no—
219C2×C53C2×#32811yescharacteristic
220C2×C53#212111yescharacteristic
221C2×C53#212711yescharacteristic
222C2×C22⋊Q8—531no—
223C22×C22⋊C4—21no—
224C22×C22⋊C4C2×D5311yescharacteristic
225C2×C22⋊C8—1061no—
226#399—531no—
227#399—531no—
228#396—1061no—
229#396—1061no—
230#393—531no—
231#393—531no—
232#385—1061no—
233#380—531no—
234#379—1061no—
235#342—21no—
236#342C2×D5312yes—
237#323—532no—
238#323—1061no—
239C53#1171611yescharacteristic, complemented, Sylow
240C25C22⋊D5311yescharacteristic
241C22×Q8—531no—
242C23×C4—41no—
243C23×C4C4×D5311yescharacteristic
244C23×C4D21211yescharacteristic
245C22×C8—2121no—
246C22⋊Q8—2121no—
247C22⋊Q8—2121no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1061no—
261C2×C22⋊C4—1061no—
262C2×C22⋊C4—1061no—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4C4×D5311yescharacteristic
266C2×C22⋊C4D21211yescharacteristic
267C2×C22⋊C4C22⋊D5311yescharacteristic
268C2×C22⋊C4C22×D5311yescharacteristic
269C22⋊C8—1062no—
270C22⋊C8—2121no—
271#95—531no—
272#95—531no—
273#95—1061no—
274#95—1061no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24#1171992111yescharacteristic
281C24#1171992111yescharacteristic
282C24#1171992312yes—
283C24#1171992512yes—
284C24C2×C22⋊D5311yescharacteristic
285C2×Q8—4241no—
286C2×Q8—2121no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—1061no—
300C22×C4—1061no—
301C22×C4#1171992111yescharacteristic
302C22×C4#1171992111yescharacteristic
303C22×C4C2×C4×D5311yescharacteristic
304C22×C4C2×D21211yescharacteristic
305C2×C8—4242no—
306C2×C8—2122no—
307C4⋊C4—4241no—
308C4⋊C4—4241no—
309C4⋊C4—4241no—
310C4⋊C4—2121no—
311C4⋊C4—2122no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—4241no—
320C22⋊C4—4241no—
321C22⋊C4—4241no—
322C22⋊C4—4241no—
323C22⋊C4—2121no—
324C22⋊C4—2121no—
325C22⋊C4—2122no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#1313313111yescharacteristic
348C23#1313313111yescharacteristic
349C23#1313314012yes—
350C23#1313314012yes—
351C23#1313314512yes—
352C23C2×#1171992111yescharacteristic
353C23C2×#1171992311yescharacteristic
354C23C2×#1171992511yescharacteristic
355Q8—4242no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—88no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—2121no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2121no—
384C8—4242no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#1681035912yes—
410C22#1681035912yes—
411C22C2×#1313313111yescharacteristic
412C22C2×#1313314011yescharacteristic
413C22C2×#1313314511yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—84no—
419C4—4241no—
420C4—4241no—
421C4—4241no—
422C4—4241no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×#1681035911yescharacteristic
431C2#2027635911yescharacteristic
432C2#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#11716
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. #20272735C2semidirect
6. C2×#13133196C4semidirect
8. #16810559C4semidirect
12. #16810535C4semidirect
15. #16810483C4semidirect
23. C22×Dic106C2×C4semidirect
28. C2×C4⋊Dic53C2×C4semidirect
42. #13133124C2×C4semidirect
43. #13133124C2×C4semidirect
239. C53#11716semidirect

Maximal subgroups

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

subgroupindex
#202805202
C2×#168103472
#202727382
#202727352
C2×C53×#3422
#1171653