Finite Group Databasebeta

← #27283601

Subgroups — #27283601

Order 13568.

564 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810852C211yesmaximal, characteristic
2#20275700C211yesmaximal, characteristic, complemented
3#20275551C211yesmaximal, characteristic
4#20274740C211yesmaximal, characteristic, complemented
5#20273288C211yesmaximal, characteristic, complemented
6#20273284C211yesmaximal, characteristic
7C8×C22⋊C4×C53C211yesmaximal, characteristic, complemented
8C22×#11719913C411yescharacteristic, complemented
9C2×#13133196C2211yescharacteristic
10C2×#13133191C2211yescharacteristic
11#16810852—21no—
12#16810852—21no—
13#16810852—21no—
14#16810852—21no—
15C2×#13133149—21no—
16C2×#13133149C411yescharacteristic, complemented
17C2×#13133142C411yescharacteristic, complemented
18C2×#13133139C411yescharacteristic, complemented
19C2×#13133131—22no—
20#16810483C2211yescharacteristic
21#16810483C2211yescharacteristic
22#16810389—22no—
23#16810386—21no—
24#16810361—22no—
25#16810358—22no—
26C23×C8×C53C2211yescharacteristic
27C2×C22⋊C8×C53C2211yescharacteristic
28C2×C4×C8×C53—21no—
29C4×C22⋊C4×C53C2211yescharacteristic
30C53×#336—21no—
31C22×Dic106C2×C411yescharacteristic
32C22×D212C2×C411yescharacteristic
33C22×C22⋊D53—21no—
34C2×#11719913—21no—
35C2×#11719913—21no—
36C2×#11719913—42no—
37#13133196—21no—
38#13133196—21no—
39#13133196—21no—
40#13133196—21no—
41#13133191—21no—
42#13133191—21no—
43#13133191—21no—
44#13133191—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53C2×C411yescharacteristic
48C2×C4⋊Dic53C2×C411yescharacteristic
49C2×#11719919—21no—
50C2×C22⋊Dic53—21no—
51C2×C22⋊Dic53—21no—
52C2×C22⋊Dic53—21no—
53C2×#11719921—21no—
54#13133149—21no—
55#13133149—21no—
56#13133149—42no—
57#13133142—21no—
58#13133142—21no—
59#13133139—21no—
60#13133139—21no—
61#13133131—44no—
62#13133124—21no—
63#13133124—21no—
64C23×C4×C53C2311yescharacteristic
65C22×C8×C53—21no—
66C22×C8×C53—42no—
67C22×C8×C53C2×C411yescharacteristic
68C22×C8×C53C2×C411yescharacteristic
69C22×C8×C53D411yescharacteristic
70C22×C8×C53D411yescharacteristic
71C22×C8×C53D411yescharacteristic
72C22×C8×C53D411yescharacteristic
73C2×C22⋊C4×C53D412yes—
74C2×C42×C53—21no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53D412yes—
77C4×C8×C53—42no—
78#95×C53—21no—
79C24×C53C2×D411yescharacteristic
80C23×D53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C2×C22⋊D53—41no—
91C2×C22⋊D53—41no—
92C2×C22⋊D53—42no—
93C2×C22⋊D53—42no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—21no—
97C22×Dic53—41no—
98C22×Dic53—41no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212—41no—
102C2×D212—41no—
103C2×Dic106—21no—
104C2×Dic106—21no—
105C2×Dic106—41no—
106C2×Dic106—41no—
107C2×C8×C53—22no—
108C2×C8×C53—22no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C2×C8×C53C22⋊C411yescharacteristic
117C2×C8×C53C22⋊C411yescharacteristic
118C2×C8×C53C22⋊C411yescharacteristic
119C2×C8×C53C22⋊C411yescharacteristic
120C22⋊C4×C53—22no—
121C22⋊C4×C53SD1614yescomplemented
122C42×C53—41no—
123C42×C53—41no—
124C22⋊Dic53—41no—
125C22⋊Dic53—41no—
126C22⋊Dic53—42no—
127C22⋊Dic53—42no—
128#11719921—41no—
129#11719921—41no—
130C4⋊Dic53—21no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134C4⋊Dic53—41no—
135C4⋊Dic53—41no—
136C4⋊Dic53—41no—
137C4⋊Dic53—41no—
138#11719919—41no—
139#11719919—41no—
140#11719913—42no—
141#11719913—42no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53C4×D411yescharacteristic
145C23×C53#12111yescharacteristic
146C23×C53#12711yescharacteristic
147C23×C53C2×SD1612yes—
148C23×C53#13512yes—
149C22×D53—41no—
150C22×D53—42no—
151C22×D53—81no—
152C22×D53—81no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—24no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53C2×C22⋊C411yescharacteristic
166C2×C4×C53C4×D411yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12411yescharacteristic
169C22⋊D53—82no—
170C22⋊D53—81no—
171C22⋊D53—81no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—81no—
177C2×Dic53—81no—
178C2×Dic53—81no—
179C2×Dic53—81no—
180C2×Dic53—81no—
181C2×Dic53—81no—
182C2×Dic53—81no—
183C2×Dic53—81no—
184D212—41no—
185D212—41no—
186D212—81no—
187Dic106—41no—
188Dic106—41no—
189Dic106—81no—
190C8×C53—21no—
191C8×C53—21no—
192C8×C53—21no—
193C8×C53—42no—
194C8×C53#9912yescomplemented
195#9781—531nomaximal, Sylow
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53#39011yescharacteristic
205C22×C53C4×SD1612yes—
206C22×C53#46512yes—
207C22×C53#49211yescharacteristic
208C22×C53#49511yescharacteristic
209C2×D53—81no—
210C2×D53—81no—
211C2×D53—81no—
212C2×D53—162no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—44no—
217C4×C53C2×#9911yescharacteristic
218C4×C53#41011yescharacteristic
219Dic53—81no—
220Dic53—81no—
221Dic53—81no—
222Dic53—161no—
223Dic53—161no—
224C2×#465—531no—
225#1996—531no—
226#1964—531no—
227#1807—531no—
228C8×C22⋊C4D5311yescharacteristic, complemented
229#1520—531no—
230#1519—531no—
231C2×C53—21no—
232C2×C53—21no—
233C2×C53—22no—
234C2×C53—22no—
235C2×C53#196411yescharacteristic
236C2×C53#196811yescharacteristic
237C2×C53#197611yescharacteristic
238D53—161no—
239C22×SD16—531no—
240C23×C8C2×D5311yescharacteristic
241C2×C22⋊Q8—531no—
242C2×#121—531no—
243#465—1061no—
244#465—1061no—
245#465—1061no—
246#465—1061no—
247C2×#106—531no—
248C2×#106—1061no—
249C2×#103—531no—
250C2×#102—531no—
251C2×#99—1062no—
252C2×C22⋊C8C2×D5311yescharacteristic
253C2×C4×C8—21no—
254#380—531no—
255#380—531no—
256C4×C22⋊C4C2×D5311yescharacteristic
257#342—1062no—
258#340—1061no—
259#336—21no—
260#328—1062no—
261#327—1062no—
262C53#978111yescharacteristic, complemented, Sylow
263C22×Q8—531no—
264C22×D4—531no—
265C22×D4—1061no—
266C23×C4C22×D5311yescharacteristic
267C2×SD16—1061no—
268C2×SD16—1061no—
269C2×SD16—2122no—
270C22×C8—21no—
271C22×C8—42no—
272C22×C8C4×D5311yescharacteristic
273C22×C8C4×D5311yescharacteristic
274C22×C8D21211yescharacteristic
275C22×C8D21211yescharacteristic
276C22×C8C22⋊D5311yescharacteristic
277C22×C8C22⋊D5311yescharacteristic
278C22⋊Q8—1061no—
279C22⋊Q8—1061no—
280C22⋊Q8—1061no—
281C22⋊Q8—1061no—
282#121—1061no—
283#121—1061no—
284#121—1061no—
285#121—1061no—
286C2×C4⋊C4—531no—
287C2×C4⋊C4—531no—
288C2×C4⋊C4—1061no—
289C2×C4⋊C4—1061no—
290C2×C4⋊C4—1061no—
291C2×C22⋊C4—1061no—
292C2×C22⋊C4—1061no—
293C2×C22⋊C4—1061no—
294C2×C22⋊C4—1061no—
295C2×C22⋊C4D21212yes—
296C2×C42—21no—
297#106—1061no—
298#106—1061no—
299#106—2122no—
300#103—1061no—
301#103—1061no—
302#102—1061no—
303#102—1061no—
304#99—2124no—
305C22⋊C8—21no—
306C22⋊C8D21212yes—
307C4×C8—42no—
308#95—21no—
309#95—1061no—
310#95—1061no—
311C24—1061no—
312C24C2×D21211yescharacteristic
313C2×Q8—1061no—
314C2×Q8—1061no—
315C2×Q8—2121no—
316C2×Q8—2121no—
317C2×D4—1061no—
318C2×D4—1061no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2122no—
324C2×D4—2122no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—2121no—
332C22×C4—2121no—
333C22×C4C2×C4×D5311yescharacteristic
334C22×C4C2×D21211yescharacteristic
335C22×C4C2×D21211yescharacteristic
336C22×C4#1171994511yescharacteristic
337C22×C4#1171994511yescharacteristic
338C22×C4C2×C22⋊D5311yescharacteristic
339SD16—2122no—
340SD16—2122no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8#1171992111yescharacteristic
351C2×C8#1171992111yescharacteristic
352C2×C8#1171992111yescharacteristic
353C2×C8#1171992111yescharacteristic
354C4⋊C4—1061no—
355C4⋊C4—1061no—
356C4⋊C4—1061no—
357C4⋊C4—1061no—
358C4⋊C4—2121no—
359C4⋊C4—2121no—
360C4⋊C4—2121no—
361C4⋊C4—2121no—
362C4⋊C4—2121no—
363C4⋊C4—2121no—
364C22⋊C4—22no—
365C22⋊C4—2122no—
366C22⋊C4—2122no—
367C22⋊C4—2121no—
368C22⋊C4—2121no—
369C22⋊C4—2121no—
370C22⋊C4—2121no—
371C22⋊C4#1171991314yescomplemented
372C42—41no—
373C42—41no—
374C23—41no—
375C23—41no—
376C23—4241no—
377C23—4241no—
378C23—2122no—
379C23—2121no—
380C23C4×D21211yescharacteristic
381C23#1313319111yescharacteristic
382C23#1313321411yescharacteristic
383C23C2×#1171991312yes—
384C23#1313323912yes—
385Q8—4241no—
386Q8—2121no—
387Q8—2121no—
388D4—4241no—
389D4—4242no—
390D4—4241no—
391D4—4241no—
392D4—2121no—
393D4—2121no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—4241no—
407C2×C4—4241no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—4241no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4—2122no—
418C2×C4C2×#1171992111yescharacteristic
419C2×C4C4×D21211yescharacteristic
420C2×C4#1313319111yescharacteristic
421C2×C4#1313320511yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—42no—
426C8#1313313112yescomplemented
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—8482no—
436C22—4241no—
437C22—4241no—
438C22—4241no—
439C22#1681052511yescharacteristic
440C22C4×#1171991312yes—
441C22#1681085212yes—
442C22#1681099611yescharacteristic
443C22#1681100711yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—44no—
448C4—8481no—
449C4—8481no—
450C4—4241no—
451C4—4241no—
452C4—4241no—
453C4C2×#1313313111yescharacteristic
454C4#1681059611yescharacteristic
455C2—22no—
456C2—21no—
457C2—21no—
458C2—22no—
459C2—8481no—
460C2#2027555111yescharacteristic
461C2#2027556911yescharacteristic
462C2#2027561311yescharacteristic
subgroupG/H
centerC22#16810525
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC8×C22⋊C4×C53C2
socleC22×C53#390
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterC8×C22⋊C4D53
hypocenterC53#9781
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20275700C2semidirect
4. #20274740C2semidirect
5. #20273288C2semidirect
7. C8×C22⋊C4×C53C2semidirect
8. C22×#11719913C4semidirect
16. C2×#13133149C4semidirect
17. C2×#13133142C4semidirect
18. C2×#13133139C4semidirect
121. C22⋊C4×C53SD16semidirect
194. C8×C53#99semidirect
228. C8×C22⋊C4D53semidirect
262. C53#9781semidirect
371. C22⋊C4#11719913semidirect
426. C8#13133131semidirect

Maximal subgroups

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

subgroupindex
C2×#168108522
#202757002
#202755512
#202747402
#202732882
#202732842
C8×C22⋊C4×C532
#978153