Finite Group Databasebeta

← #27537006

Subgroups — #27537006

Order 13568.

1124 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811436C211yesmaximal, characteristic
2C2×#16811039C211yesmaximal, characteristic, complemented
3#20283311C211yesmaximal, characteristic, complemented
4#20276435C211yesmaximal, characteristic, complemented
5#20276433C212yesmaximal, complemented
6#20276426C212yesmaximal, complemented
7#20276425C211yesmaximal, characteristic, complemented
8#20276423C211yesmaximal, characteristic, complemented
9#20276421C212yesmaximal, complemented
10#358×D53C212yesmaximal, complemented
11C2×C53×#358C211yesmaximal, characteristic, complemented
12C2×#13133265—21no—
13#16811436—24no—
14#16811436C418yescomplemented
15C2×#13133249—22no—
16C2×#13133248—21no—
17C2×#13133240C2212yes—
18C2×#13133239C2211yescharacteristic
19C2×Q16×D53C2212yes—
20C2×#13133223C2211yescharacteristic
21#16811104—21no—
22#16811039—24no—
23C2×#13133157C2211yescharacteristic
24C2×#13133156C2212yes—
25C2×C16×D53C2212yes—
26#16810664—22no—
27#16810663—21no—
28#16810661C2211yescharacteristic
29#16810660C2212yes—
30#107×D53C2212yes—
31#16810428C2211yescharacteristic
32#16810428C2211yescharacteristic
33#16810427C2212yes—
34#16810427C2212yes—
35#16810426C2212yes—
36#16810425C2211yescharacteristic
37#16810425C2211yescharacteristic
38#16810423C2212yes—
39C22×Q16×C53C2211yescharacteristic
40C22×C16×C53C2211yescharacteristic
41C2×#107×C53C2211yescharacteristic
42C53×#358C2212yes—
43C53×#358C2212yes—
44#13133265—28no—
45#13133265—44no—
46C2×#11719949—22no—
47C2×#11719949—41no—
48C2×#11719945—21no—
49C2×#11719945—42no—
50C2×#11719945D411yescharacteristic
51C2×Q8×D53—22no—
52C2×Q8×D53—41no—
53#13133249—216no—
54#13133248—28no—
55#13133240—24no—
56#13133240C2×C418yescomplemented
57#13133239—22no—
58#13133239C2×C414yescomplemented
59C2×Dic212C2311yescharacteristic
60C2×#11719925—22no—
61Q16×D53—24no—
62Q16×D53C2×C418yescomplemented
63C2×#11719913—21no—
64C2×#11719924—22no—
65C2×D424C2311yescharacteristic
66#13133223C2×C414yes—
67#13133223D414yes—
68C2×#11719917D411yescharacteristic
69C2×#11719912D412yes—
70C2×C8×D53C2312yes—
71#13133182—41no—
72#13133179—41no—
73C4×C22⋊D53—42no—
74#13133176—21no—
75#13133175—22no—
76C4⋊C4×D53—22no—
77#13133157—22no—
78#13133156—24no—
79C2×#11719908C2312yes—
80C16×D53—24no—
81#13133151C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133150—22no—
84#13133149—21no—
85#13133148—22no—
86C22×Q8×C53—21no—
87C2×Q16×C53—21no—
88C2×Q16×C53—22no—
89C2×Q16×C53C2×C414yescomplemented
90C2×Q16×C53C2312yes—
91C22×C8×C53C2311yescharacteristic
92C2×C4⋊C4×C53—21no—
93C2×C16×C53—22no—
94C2×C16×C53C2311yescharacteristic
95C2×C16×C53C2311yescharacteristic
96#107×C53—21no—
97#107×C53C2312yes—
98C2×Q8×C53—24no—
99C2×Q8×C53—22no—
100C2×Q8×C53—41no—
101C2×Q8×C53—42no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C2×C22⋊D53—41no—
106C2×C22⋊D53D812yes—
107#11719949—28no—
108#11719949—48no—
109#11719949—44no—
110Q8×D53—28no—
111Q8×D53—44no—
112Q8×D53—48no—
113#11719945—24no—
114#11719945—42no—
115#11719945—416no—
116#11719945C22⋊C414yes—
117#11719945SD1614yes—
118C2×D212—21no—
119C2×D212—42no—
120C2×D212D811yescharacteristic
121C2×C4×D53—42no—
122C2×C4×D53—41no—
123C2×C4×D53—42no—
124C2×C4×D53—42no—
125C2×C4×D53C2×D412yes—
126C2×Dic106—21no—
127C2×Dic106—42no—
128C2×Dic106D811yescharacteristic
129Q16×C53—24no—
130Q16×C53—22no—
131Q16×C53C22×C414yes—
132C2×C8×C53C22×C412yes—
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D412yes—
135C2×C8×C53C2411yescharacteristic
136C4⋊C4×C53—22no—
137C4⋊C4×C53—41no—
138C22⋊Dic53—41no—
139#11719925—28no—
140#11719924—28no—
141#11719921—42no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144#11719919—42no—
145C4×Dic53—41no—
146C4×Dic53—42no—
147#11719917C22⋊C414yes—
148C2×#88322C2×D412yes—
149Dic212—21no—
150Dic212C22×C412yes—
151D424—21no—
152D424C22×C412yes—
153#11719913—24no—
154#11719912C22⋊C418yes—
155C8×D53C22×C414yes—
156C8×D53C2×D414yes—
157C16×C53—21no—
158C16×C53—21no—
159#11719908—22no—
160C23×C53C2×D811yescharacteristic
161C22×D53—41no—
162C22×D53C2×D812yes—
163Q8×C53—24no—
164Q8×C53—42no—
165Q8×C53—44no—
166C2×C4×C53—42no—
167C2×C4×C53—44no—
168C2×C4×C53—42no—
169C2×C4×C53—161no—
170C2×C4×C53C2×C22⋊C412yes—
171C2×C4×C53C2×D811yescharacteristic
172C2×C4×C53C2×SD1612yes—
173C2×C4×C53C22×D411yescharacteristic
174C22⋊D53—44no—
175C22⋊D53#10218yes—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53C2×D812yes—
180D212—22no—
181D212—48no—
182D212—41no—
183D212#10214yes—
184C4×D53—48no—
185C4×D53—44no—
186C4×D53—48no—
187C4×D53—162no—
188C4×D53C2×C22⋊C414yes—
189C4×D53C2×SD1614yes—
190Dic106—22no—
191Dic106—41no—
192Dic106—48no—
193Dic106#10214yes—
194C8×C53C2×C22⋊C412yes—
195C8×C53C23×C411yescharacteristic
196C8×C53C22×D411yescharacteristic
197#88322C2×C22⋊C414yes—
198#33382—531nomaximal, Sylow
199C22×C53—21no—
200C22×C53—21no—
201C22×C53C2×#10212yes—
202C22×C53C22×D811yescharacteristic
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—42no—
206C2×D53—81no—
207C2×D53—81no—
208C2×D53C2×#10214yes—
209C4×C53—44no—
210C4×C53—82no—
211C4×C53C2×#10212yes—
212C4×C53C22×C22⋊C411yescharacteristic
213C4×C53C22×SD1611yescharacteristic
214Dic53—42no—
215Dic53—81no—
216Dic53—81no—
217Dic53C2×#10214yes—
218C2×#578—531no—
219C2×#504—531no—
220#2913—531no—
221#2141—532no—
222#2141—531no—
223#2139—531no—
224#2139—532no—
225#2138—531no—
226#2138—532no—
227C2×#358—532no—
228C2×#358D5311yescharacteristic, complemented
229C2×C53—22no—
230C2×C53C22×#10211yescharacteristic
231C2×C53#341611yescharacteristic
232C2×C53#341711yescharacteristic
233D53—24no—
234D53—82no—
235C2×2−1+4—1061no—
236#578—538no—
237#578—1064no—
238C2×#137—1062no—
239C2×#137—1061no—
240C2×#135—532no—
241C2×#135—531no—
242C22×Q16—532no—
243C22×Q16C2×D5311yescharacteristic
244C2×#131—531no—
245#520—1061no—
246#504—1064no—
247C2×M32—531no—
248C2×M32—532no—
249C22×C16—532no—
250C22×C16C2×D5311yescharacteristic
251#428—1062no—
252#428—1061no—
253#427—531no—
254#427—532no—
255C2×#107—532no—
256C2×#107C2×D5311yescharacteristic
257#358—532no—
258#358—532no—
259#358—531no—
260#358—531no—
261#358—532no—
262#358C2×D5312yes—
263#358C2×D5312yes—
264#357—531no—
265#357—531no—
266#357—532no—
267C53#3338211yescharacteristic, complemented, Sylow
2682−1+4—1068no—
2692−1+4—2124no—
270C2×#41—531no—
271C2×#41—1062no—
272C2×#41—1061no—
273C2×#41—2122no—
274C2×#41—2121no—
275C22×Q8—21no—
276C22×Q8—1062no—
277C22×Q8—2121no—
278#137—10616no—
279#137—1068no—
280#135—538no—
281#135—534no—
282#135—1064no—
283#135—1062no—
284C2×Q16—21no—
285C2×Q16—22no—
286C2×Q16—538no—
287C2×Q16—531no—
288C2×Q16—1064no—
289C2×Q16—1062no—
290C2×Q16C4×D5314yescomplemented
291C2×Q16C22×D5312yes—
292C2×SD16—1062no—
293C2×SD16—1061no—
294C2×D8—531no—
295#131—534no—
296#131—534no—
297C2×M16—531no—
298C2×M16—532no—
299C22×C8—532no—
300C22×C8C22×D5311yescharacteristic
301C4×Q8—2121no—
302C4×D4—2121no—
303C4×D4—2122no—
304#117—1062no—
305#117—1061no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—1062no—
308M32—1064no—
309M32—1062no—
310C2×C16—22no—
311C2×C16—532no—
312C2×C16—1064no—
313C2×C16C22×D5311yescharacteristic
314C2×C16C22×D5311yescharacteristic
315#107—21no—
316#107—531no—
317#107—531no—
318#107—1062no—
319#107C22×D5312yes—
320#106—1062no—
321#106—1061no—
322#41—534no—
323#41—534no—
324#41—1068no—
325#41—1064no—
326#41—21216no—
327#41—2124no—
328#41—2128no—
329#41—2122no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—531no—
335C2×Q8—1068no—
336C2×Q8—1061no—
337C2×Q8—2122no—
338C2×Q8—2124no—
339C2×Q8—2128no—
340C2×D4—532no—
341C2×D4—531no—
342C2×D4—1061no—
343C2×D4—2122no—
344C2×D4—2121no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—532no—
348C22×C4—2122no—
349C22×C4—2122no—
350C22×C4—2122no—
351C22×C4—2121no—
352C22×C4D4×D5311yescharacteristic
353Q16—24no—
354Q16—22no—
355Q16—532no—
356Q16—1068no—
357Q16—1061no—
358Q16C2×C4×D5314yes—
359SD16—1068no—
360SD16—1064no—
361D8—532no—
362D8—1061no—
363M16—534no—
364M16—538no—
365C2×C8—532no—
366C2×C8—534no—
367C2×C8—534no—
368C2×C8C2×C4×D5312yes—
369C2×C8D4×D5311yescharacteristic
370C2×C8D4×D5312yes—
371C2×C8C23×D5311yescharacteristic
372C4⋊C4—22no—
373C4⋊C4—41no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4—2122no—
377C22⋊C4—2122no—
378C22⋊C4—2121no—
379C42—2122no—
380C42—2121no—
381C16—21no—
382C16—21no—
383C16—1062no—
384C23—532no—
385C23—2121no—
386C23D8×D5311yescharacteristic
387Q8—24no—
388Q8—44no—
389Q8—42no—
390Q8—534no—
391Q8—1062no—
392Q8—2128no—
393Q8—2121no—
394D4—538no—
395D4—534no—
396D4—1062no—
397D4—2128no—
398D4—2124no—
399D4—2121no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—161no—
404C2×C4—532no—
405C2×C4—534no—
406C2×C4—534no—
407C2×C4—8482no—
408C2×C4—2128no—
409C2×C4—2128no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2124no—
414C2×C4C22⋊C4×D5312yes—
415C2×C4D8×D5311yescharacteristic
416C2×C4SD16×D5312yes—
417C2×C4C2×D4×D5311yescharacteristic
418C8—534no—
419C8C22⋊C4×D5312yes—
420C8C22×C4×D5311yescharacteristic
421C8C2×D4×D5311yescharacteristic
422C22—21no—
423C22—21no—
424C22—534no—
425C22—1062no—
426C22—1062no—
427C22—4241no—
428C22—4241no—
429C22—2122no—
430C22#102×D5312yes—
431C22C2×D8×D5311yescharacteristic
432C4—44no—
433C4—82no—
434C4—534no—
435C4—4241no—
436C4—4241no—
437C4—2122no—
438C4#102×D5312yes—
439C4C2×C22⋊C4×D5311yescharacteristic
440C4C2×SD16×D5311yescharacteristic
441C2—22no—
442C2—1064no—
443C2—4242no—
444C2C2×#102×D5311yescharacteristic
445C2#2028785711yescharacteristic
446C2#2028786411yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC2×C53×#358C2
socleC22×C53C22×D8
prime radicalC2×#16811436C2
cosocleC2×C8×C53C24
hypercenterC2×#358D53
hypocenterC53#33382
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×#16811039C2semidirect
3. #20283311C2semidirect
4. #20276435C2semidirect
5. #20276433C2semidirect
6. #20276426C2semidirect
7. #20276425C2semidirect
8. #20276423C2semidirect
9. #20276421C2semidirect
10. #358×D53C2semidirect
11. C2×C53×#358C2semidirect
14. #16811436C4semidirect
56. #13133240C2×C4semidirect
58. #13133239C2×C4semidirect
62. Q16×D53C2×C4semidirect
89. C2×Q16×C53C2×C4semidirect
228. C2×#358D53semidirect
267. C53#33382semidirect
290. C2×Q16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168114362
C2×#168110392
#202833112
#202764352
#202764332
#202764262
#202764252
#202764232
#202764212
#358×D532
C2×C53×#3582
#3338253