Finite Group Databasebeta

← #27294271

Subgroups — #27294271

Order 13568.

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

normal subgroups only

This group has 670 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#20280449C211yesmaximal, characteristic
2C2×#16810347C211yesmaximal, characteristic, complemented
3#20272714C212yesmaximal, complemented
4#20272711C212yesmaximal, complemented
5C2×C53×#342C211yesmaximal, characteristic, complemented
6C2×#13133191C411yescharacteristic, complemented
7C2×#13133186—21no—
8C2×C22⋊#88322—21no—
9#16810548C411yescharacteristic, complemented
10#16810548C2211yescharacteristic
11#16810539—21no—
12#16810539—21no—
13#16810532C411yescharacteristic, complemented
14#16810532C2211yescharacteristic
15#16810502—21no—
16#16810483C411yescharacteristic, complemented
17#16810347—21no—
18#16810347C2212yes—
19C22×C22⋊C4×C53—21no—
20C22×C22⋊C4×C53C2211yescharacteristic
21C53×#342—21no—
22C53×#342C2212yes—
23C22×D212C2×C411yescharacteristic, complemented
24C22×C22⋊D53—21no—
25C22×C22⋊D53—21no—
26C22×C22⋊D53—21no—
27C22×#88322—41no—
28#13133191—41no—
29#13133191—41no—
30#13133186—42no—
31#13133186—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C411yescharacteristic, complemented
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C22⋊#88322—22no—
41C22⋊#88322—41no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic, complemented
45#13133124C2×C411yescharacteristic, complemented
46C25×C53D411yescharacteristic
47C23×C4×C53—41no—
48C23×C4×C53C2×C411yescharacteristic
49C23×C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—24no—
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53—41no—
53C2×C22⋊C4×C53—41no—
54C2×C22⋊C4×C53—41no—
55C2×C22⋊C4×C53—42no—
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C22⋊C4×C53D411yescharacteristic
59C2×C22⋊C4×C53C2311yescharacteristic
60C24×C53—21no—
61C24×C53—21no—
62C24×C53—22no—
63C24×C53—81no—
64C24×C53—81no—
65C24×C53C22⋊C411yescharacteristic
66C24×C53C22⋊C411yescharacteristic
67C24×C53D812yes—
68C24×C53SD1612yes—
69C24×C53C2×D411yescharacteristic
70C23×D53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—44no—
76C22×C4×C53—42no—
77C22×C4×C53—81no—
78C22×C4×C53—81no—
79C22×C4×C53—82no—
80C22×C4×C53—82no—
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C2×C22⋊D53—41no—
86C2×C22⋊D53—41no—
87C2×C22⋊D53—42no—
88C2×C22⋊D53—42no—
89C2×C22⋊D53—42no—
90C2×C22⋊D53—41no—
91C2×C22⋊D53—41no—
92C2×C22⋊D53—81no—
93C2×C22⋊D53—81no—
94C2×C22⋊D53—81no—
95C2×C22⋊D53—81no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—21no—
99C2×D212—41no—
100C2×D212—81no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C22⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—44no—
107C22⋊C4×C53—42no—
108C22⋊Dic53—42no—
109C22⋊Dic53—42no—
110C22⋊Dic53—41no—
111C22⋊Dic53—41no—
112#11719921—42no—
113#11719921—81no—
114#11719921—81no—
115C4⋊Dic53—41no—
116C4⋊Dic53—81no—
117C2×#88322—42no—
118C2×#88322—82no—
119C23×C53—22no—
120C23×C53—22no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—22no—
124C23×C53—22no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—81no—
136C23×C53—81no—
137C23×C53—82no—
138C23×C53—82no—
139C23×C53—84no—
140C23×C53#9911yescharacteristic
141C23×C53#9911yescharacteristic
142C23×C53#10212yes—
143C23×C53#10212yes—
144C23×C53#10412yes—
145C23×C53C2×C22⋊C411yescharacteristic
146C23×C53C2×D811yescharacteristic
147C23×C53C2×SD1611yescharacteristic
148C22×D53—41no—
149C22×D53—41no—
150C22×D53—41no—
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—41no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—44no—
165C2×C4×C53—81no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53—88no—
169C2×C4×C53—82no—
170C2×C4×C53—82no—
171C22⋊D53—84no—
172C22⋊D53—82no—
173C22⋊D53—82no—
174C22⋊D53—82no—
175C22⋊D53—82no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185D212—82no—
186#88322—82no—
187#11692—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—22no—
196C22×C53—21no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—42no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—44no—
203C22×C53—44no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—81no—
209C22×C53—81no—
210C22×C53—82no—
211C22×C53—82no—
212C22×C53#32712yes—
213C22×C53#32712yes—
214C22×C53C2×#9911yescharacteristic
215C22×C53C2×#10211yescharacteristic
216C22×C53C2×#10411yescharacteristic
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C2×D53—81no—
224C4×C53—41no—
225C4×C53—41no—
226C4×C53—41no—
227C4×C53—41no—
228C4×C53—84no—
229Dic53—81no—
230Dic53—81no—
231Dic53—81no—
232#2617—531no—
233C2×#342D5311yescharacteristic, complemented
234C2×#323—531no—
235#1344—532no—
236#1343—532no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—22no—
240C2×C53—22no—
241C2×C53—41no—
242C2×C53—42no—
243C2×C53—41no—
244C2×C53C2×#32711yescharacteristic
245C2×C53#212111yescharacteristic
246C2×C53#212711yescharacteristic
247D53—161no—
248C2×#121—531no—
249C2×#120—1061no—
250C22×C22⋊C4—21no—
251C22×C22⋊C4C2×D5311yescharacteristic
252C2×C22⋊C8—1061no—
253#397—531no—
254#397—531no—
255#394—1061no—
256#394—1061no—
257#392—531no—
258#392—531no—
259#386—1061no—
260#380—531no—
261#342—21no—
262#342C2×D5312yes—
263#323—532no—
264#323—1061no—
265C53#1169211yescharacteristic, complemented, Sylow
266C25C22⋊D5311yescharacteristic
267C22×D4—531no—
268C22×D4—1061no—
269C22×D4—1061no—
270C22×D4—1061no—
271C23×C4—41no—
272C23×C4C4×D5311yescharacteristic
273C23×C4D21211yescharacteristic
274C22×C8—2121no—
275#121—2121no—
276#121—2121no—
277#120—2122no—
278#120—2122no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4—1061no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—1061no—
288C2×C22⋊C4—1061no—
289C2×C22⋊C4—1061no—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4—1061no—
292C2×C22⋊C4—1061no—
293C2×C22⋊C4C4×D5311yescharacteristic
294C2×C22⋊C4D21211yescharacteristic
295C2×C22⋊C4C22⋊D5311yescharacteristic
296C2×C22⋊C4C22×D5311yescharacteristic
297C22⋊C8—1062no—
298C22⋊C8—2121no—
299#95—531no—
300#95—531no—
301#95—1061no—
302#95—1061no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—81no—
307C24—81no—
308C24—1061no—
309C24#1171992111yescharacteristic
310C24#1171992111yescharacteristic
311C24#1171992212yes—
312C24#1171992412yes—
313C24C2×C22⋊D5311yescharacteristic
314C2×D4—4241no—
315C2×D4—4241no—
316C2×D4—4241no—
317C2×D4—4241no—
318C2×D4—4241no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2122no—
323C2×D4—2122no—
324C2×D4—2122no—
325C2×D4—2121no—
326C2×D4—2121no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—1061no—
338C22×C4—1061no—
339C22×C4—1061no—
340C22×C4#1171992111yescharacteristic
341C22×C4#1171992111yescharacteristic
342C22×C4C2×C4×D5311yescharacteristic
343C22×C4C2×D21211yescharacteristic
344C2×C8—4242no—
345C2×C8—2122no—
346C4⋊C4—4241no—
347C4⋊C4—2121no—
348C22⋊C4—24no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—44no—
354C22⋊C4—42no—
355C22⋊C4—4241no—
356C22⋊C4—4241no—
357C22⋊C4—2122no—
358C22⋊C4—2122no—
359C22⋊C4—2122no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C23—21no—
363C23—21no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—81no—
379C23—81no—
380C23—82no—
381C23—82no—
382C23—84no—
383C23—2121no—
384C23—2121no—
385C23—2121no—
386C23—2121no—
387C23—2121no—
388C23—2121no—
389C23—2121no—
390C23#1313313111yescharacteristic
391C23#1313313111yescharacteristic
392C23#1313313712yes—
393C23#1313313712yes—
394C23#1313314512yes—
395C23C2×#1171992111yescharacteristic
396C23C2×#1171992211yescharacteristic
397C23C2×#1171992411yescharacteristic
398D4—4242no—
399D4—4244no—
400D4—4242no—
401D4—4242no—
402D4—4242no—
403D4—4242no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—88no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—2121no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2121no—
429C8—4242no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—41no—
443C22—41no—
444C22—44no—
445C22—44no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—81no—
451C22—81no—
452C22—82no—
453C22—82no—
454C22—4241no—
455C22—4241no—
456C22—4241no—
457C22—4241no—
458C22—4241no—
459C22—4241no—
460C22—4241no—
461C22#1681035612yes—
462C22#1681035612yes—
463C22C2×#1313313111yescharacteristic
464C22C2×#1313313711yescharacteristic
465C22C2×#1313314511yescharacteristic
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—84no—
471C4—4241no—
472C4—4241no—
473C4—4241no—
474C2—21no—
475C2—22no—
476C2—21no—
477C2—22no—
478C2—41no—
479C2—42no—
480C2—41no—
481C2—8481no—
482C2C2×#1681035611yescharacteristic
483C2#2027635911yescharacteristic
484C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC2×C53×#342C2
socleC22×C53C2×#99
prime radical#20280449C2
cosocleC2×C22⋊C4×C53C23
hypercenterC2×#342D53
hypocenterC53#11692
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
2. C2×#16810347C2semidirect
3. #20272714C2semidirect
4. #20272711C2semidirect
5. C2×C53×#342C2semidirect
6. C2×#13133191C4semidirect
9. #16810548C4semidirect
13. #16810532C4semidirect
16. #16810483C4semidirect
23. C22×D212C2×C4semidirect
33. C2×C4⋊Dic53C2×C4semidirect
44. #13133124C2×C4semidirect
45. #13133124C2×C4semidirect
233. C2×#342D53semidirect
265. C53#11692semidirect

Maximal subgroups

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

subgroupindex
#202804492
C2×#168103472
#202727142
#202727112
C2×C53×#3422
#1169253