Finite Group Databasebeta

← #28317680

Subgroups — #28317680

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291647C211yesmaximal, characteristic, complemented
2#20291641C211yesmaximal, characteristic, complemented
3#20290358C212yesmaximal, complemented
4#20290328C212yesmaximal, complemented
5#20290316C212yesmaximal, complemented
6SD16⋊#88324C212yesmaximal, complemented
7#20290272C211yesmaximal, characteristic, complemented
8#20290050C211yesmaximal, characteristic, complemented
9#20290050C211yesmaximal, characteristic, complemented
10#20289924C211yesmaximal, characteristic, complemented
11#135×Dic53C211yesmaximal, characteristic
12#16811674C2212yes—
13Q8×#88324C2212yes—
14#16811658—22no—
15#16811658C2212yes—
16#16811654—22no—
17D4×#88324C2212yes—
18#16811643—21no—
19#16811640—21no—
20#16811640C2211yescharacteristic
21#16811626—21no—
22#16811626C2211yescharacteristic
23#16811626C2211yescharacteristic
24C2×#13133284—21no—
25C4×#11719940C2211yescharacteristic
26#16811487C2212yes—
27#16811486C2212yes—
28Q8⋊#88324C2212yes—
29Q8⋊#88324C2212yes—
30D4⋊#88324C2212yes—
31D4⋊#88324C2212yes—
32#88324⋊C8C2212yes—
33#16811464C2212yes—
34#41×Dic53C2211yescharacteristic
35#41×Dic53C2211yescharacteristic
36Q16×Dic53C411yescharacteristic
37Q16×Dic53C2211yescharacteristic
38SD16×Dic53C411yescharacteristic
39SD16×Dic53C411yescharacteristic
40SD16×Dic53C2211yescharacteristic
41SD16×Dic53C2211yescharacteristic
42D8×Dic53C411yescharacteristic
43D8×Dic53C2211yescharacteristic
44#16810661C411yescharacteristic
45#16810619C411yescharacteristic
46#16810619C411yescharacteristic
47C2×C8×Dic53C2211yescharacteristic
48C2×#135×C53C411yescharacteristic
49C2×#11719940—21no—
50C2×#11719940—21no—
51C2×#11719940—21no—
52C2×#11719940—22no—
53C22×#88324—22no—
54#13133284—21no—
55#13133284—21no—
56#13133284—22no—
57#13133284D411yescharacteristic
58#13133284D411yescharacteristic
59C4⋊#88324—22no—
60C4⋊#88324C2312yes—
61C4⋊#88324C2312yes—
62C22⋊#88324—22no—
63C22⋊#88324—22no—
64C22⋊#88324D412yes—
65#13133272—21no—
66#13133272—22no—
67C4×#88324—22no—
68C4×#88324C2312yes—
69C4×#88324C2312yes—
70C22×#88322C2×C411yescharacteristic
71Q8×Dic53C2×C411yescharacteristic
72Q8×Dic53C2×C411yescharacteristic
73Q8×Dic53C2311yescharacteristic
74Q8×Dic53C2311yescharacteristic
75D4×Dic53—21no—
76D4×Dic53—21no—
77D4×Dic53—21no—
78D4×Dic53—21no—
79D4×Dic53C2×C411yescharacteristic
80D4×Dic53C2×C411yescharacteristic
81D4×Dic53C2311yescharacteristic
82D4×Dic53C2311yescharacteristic
83#13133176—21no—
84#13133176—21no—
85#13133176C2×C411yescharacteristic
86#13133176C2×C411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90#13133151C2×C411yescharacteristic
91#13133151C2×C411yescharacteristic
92#13133149C2×C411yescharacteristic
93#13133149C2×C411yescharacteristic
94#13133141C2×C411yescharacteristic
95#13133141C2×C411yescharacteristic
96#13133141C2×C411yescharacteristic
97#13133141C2×C411yescharacteristic
98#13133138C2×C411yescharacteristic
99#13133138C2×C411yescharacteristic
100#13133138C2×C411yescharacteristic
101#13133138C2×C411yescharacteristic
102C8×Dic53C2×C411yescharacteristic
103C8×Dic53C2×C411yescharacteristic
104C8×Dic53C2311yescharacteristic
105C8×Dic53C2311yescharacteristic
106C2×#41×C53C2×C411yescharacteristic
107C2×#41×C53C2×C411yescharacteristic
108#135×C53—24no—
109C2×Q16×C53C2×C411yescharacteristic
110C2×SD16×C53C2×C411yescharacteristic
111C2×SD16×C53C2×C411yescharacteristic
112C2×D8×C53C2×C411yescharacteristic
113C22×C8×C53C2×C411yescharacteristic
114#41×C53—22no—
115#41×C53—22no—
116#41×C53—41no—
117#41×C53—41no—
118C2×Q8×C53C22×C411yescharacteristic
119C2×Q8×C53C22×C411yescharacteristic
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53C22×C411yescharacteristic
123C2×D4×C53C22×C411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C22×C411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53#4111yescharacteristic
133#11719940—42no—
134#11719940—82no—
135#11719940—81no—
136#11719940—81no—
137C2×#88324—22no—
138C2×#88324—22no—
139C2×#88324—22no—
140C2×#88324—82no—
141C2×#88324C2×D412yes—
142C2×#88324C2×D412yes—
143Q16×C53—22no—
144SD16×C53—22no—
145SD16×C53M1614yescomplemented
146D8×C53—22no—
147C2×C8×C53—22no—
148C2×C8×C53C22×C411yescharacteristic
149C2×C8×C53C22×C411yescharacteristic
150C22⋊Dic53—21no—
151C22⋊Dic53—21no—
152C22⋊Dic53—21no—
153C22⋊Dic53—21no—
154C22⋊Dic53—21no—
155C22⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53C22×C411yescharacteristic
161C4⋊Dic53C22×C411yescharacteristic
162C4⋊Dic53C22×C411yescharacteristic
163C4⋊Dic53C22×C411yescharacteristic
164C4×Dic53—21no—
165C4×Dic53—21no—
166C4×Dic53—21no—
167C4×Dic53—21no—
168C4×Dic53—21no—
169C4×Dic53—21no—
170C4×Dic53C22×C411yescharacteristic
171C4×Dic53C2×D411yescharacteristic
172C4×Dic53#4111yescharacteristic
173C4×Dic53C2411yescharacteristic
174C2×#88322—41no—
175C2×#88322C22×C411yescharacteristic
176C2×#88322C22×C411yescharacteristic
177C23×C53—21no—
178C23×C53—21no—
179C23×C53C4×D411yescharacteristic
180Q8×C53—21no—
181Q8×C53—21no—
182Q8×C53—21no—
183Q8×C53C2×M1612yes—
184D4×C53—21no—
185D4×C53—21no—
186D4×C53—21no—
187D4×C53—42no—
188D4×C53—42no—
189D4×C53C2×M1612yes—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53C4×D411yescharacteristic
197C2×C4×C53C23×C411yescharacteristic
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—22no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53C4×D411yescharacteristic
212C2×Dic53C4×D411yescharacteristic
213C2×Dic53C22×D411yescharacteristic
214C2×Dic53C2×#4111yescharacteristic
215#88324—22no—
216#88324—22no—
217#88324—42no—
218#88324—42no—
219#88324—42no—
220C8×C53—21no—
221C8×C53C2×M1612yes—
222#88322—21no—
223#88322—21no—
224#18061—531nomaximal, Sylow
225C22×C53—21no—
226C22×C53—22no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—41no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—41no—
233C22×C53—41no—
234C22×C53C2×C4×D411yescharacteristic
235C4×C53—21no—
236C4×C53—22no—
237C4×C53—41no—
238C4×C53C22×M1611yescharacteristic
239C4×C53#56811yescharacteristic
240Dic53—21no—
241Dic53—21no—
242Dic53—41no—
243Dic53—41no—
244Dic53—41no—
245Dic53—41no—
246Dic53#44412yes—
247Dic53#57611yescharacteristic
248Dic53#57811yescharacteristic
249#2991—531no—
250#2990—531no—
251C4×#135—531no—
252#1592—532no—
253#1584—532no—
254#1581—532no—
255SD16⋊C8—532no—
256#1567—531no—
257#1489—531no—
258#1489—531no—
259#1451—531no—
260C2×C53—41no—
261C2×C53—42no—
262C2×C53—81no—
263C2×C53D4×M1611yescharacteristic
264C2×C53#295111yescharacteristic
265C2×C53C2×#44411yescharacteristic
266C2×#135#615711yescharacteristic
267C4×#41—531no—
268C4×#41—531no—
269#446—532no—
270C8×Q8—532no—
271C4×Q16—531no—
272C4×Q16—531no—
273C4×SD16—531no—
274C4×SD16—531no—
275C4×SD16—531no—
276C4×SD16—531no—
277C4×D8—531no—
278C4×D8—531no—
279#436—532no—
280#436—1062no—
281#435—1062no—
282C8×D4—532no—
283#432—1061no—
284#431—531no—
285#431—1061no—
286#427—531no—
287#423—531no—
288#423—531no—
289#423—1061no—
290C2×C4⋊C8—1061no—
291#416—531no—
292#416—531no—
293C4×M16—531no—
294C2×C4×C8—531no—
295#335—532no—
296#335—532no—
297Q8⋊C8—532no—
298Q8⋊C8—532no—
299D4⋊C8—532no—
300D4⋊C8—532no—
301#322—532no—
302#322—532no—
303C53#1806111yescharacteristic, complemented, Sylow
304C2×#41C2×#615711yescharacteristic
305C2×#41C2×#615711yescharacteristic
306#135—24no—
307C2×Q16C2×#615711yescharacteristic
308C2×SD16C2×#615711yescharacteristic
309C2×SD16C2×#615711yescharacteristic
310C2×D8C2×#615711yescharacteristic
311C2×M16—1062no—
312C2×M16—1061no—
313C2×M16—1061no—
314C2×M16—1061no—
315C22×C8—531no—
316C22×C8—1062no—
317C22×C8C2×#615711yescharacteristic
318C4×Q8—531no—
319C4×Q8—531no—
320C4×Q8—531no—
321C4×Q8—531no—
322C4×D4—531no—
323C4×D4—531no—
324C4×D4—531no—
325C4×D4—531no—
326C4×D4—1061no—
327C4×D4—1061no—
328C4×D4—1061no—
329C4×D4—1061no—
330#117—531no—
331#117—531no—
332#117—1061no—
333#117—1061no—
334C2×C42—531no—
335C2×C42—1061no—
336C2×C42—1061no—
337#107—531no—
338#107—531no—
339#106—531no—
340#106—531no—
341C4⋊C8—531no—
342C4⋊C8—531no—
343C4⋊C8—532no—
344C4⋊C8—532no—
345C4⋊C8—1061no—
346C4⋊C8—1061no—
347C4⋊C8—1062no—
348C4⋊C8—1062no—
349#103—531no—
350#103—531no—
351#103—531no—
352#103—531no—
353#102—531no—
354#102—531no—
355#102—531no—
356#102—531no—
357C22⋊C8—532no—
358C22⋊C8—1062no—
359C22⋊C8—1062no—
360#97—1061no—
361#97—1062no—
362C4×C8—532no—
363C4×C8—531no—
364C4×C8—531no—
365C4×C8—531no—
366C4×C8—531no—
367C4×C8—532no—
368C4×C8—1062no—
369#41—22no—
370#41—22no—
371#41—41no—
372#41—41no—
373C2×Q8C22×#615711yescharacteristic
374C2×Q8C22×#615711yescharacteristic
375C2×D4—21no—
376C2×D4—21no—
377C2×D4C22×#615711yescharacteristic
378C2×D4C22×#615711yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—531no—
382C22×C4—531no—
383C22×C4—1061no—
384C22×C4—1061no—
385C22×C4—1061no—
386C22×C4—1061no—
387C22×C4C22×#615711yescharacteristic
388Q16—22no—
389SD16—22no—
390SD16#1171994014yescomplemented
391D8—22no—
392M16—4242no—
393M16—4241no—
394M16—4241no—
395M16—2122no—
396C2×C8—22no—
397C2×C8—532no—
398C2×C8—532no—
399C2×C8—531no—
400C2×C8—531no—
401C2×C8—1062no—
402C2×C8—1062no—
403C2×C8—1062no—
404C2×C8—4242no—
405C2×C8—2121no—
406C2×C8C22×#615711yescharacteristic
407C2×C8C22×#615711yescharacteristic
408C4⋊C4—531no—
409C4⋊C4—531no—
410C4⋊C4—531no—
411C4⋊C4—531no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1061no—
415C4⋊C4—1061no—
416C22⋊C4—1061no—
417C22⋊C4—1061no—
418C22⋊C4—1061no—
419C22⋊C4—1061no—
420C22⋊C4—1061no—
421C22⋊C4—1061no—
422C42—531no—
423C42—531no—
424C42—531no—
425C42—531no—
426C42—1061no—
427C42—1061no—
428C42—1061no—
429C42—1061no—
430C42—1061no—
431C42—1061no—
432C23—21no—
433C23—21no—
434C23D4×#615711yescharacteristic
435Q8—21no—
436Q8—21no—
437Q8—21no—
438Q8C2×#1171994012yes—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—42no—
444D4C2×#1171994012yes—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—531no—
452C2×C4—531no—
453C2×C4—531no—
454C2×C4—531no—
455C2×C4—1061no—
456C2×C4—1062no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—2121no—
463C2×C4—2121no—
464C2×C4—2121no—
465C2×C4—2121no—
466C2×C4—2122no—
467C2×C4—2122no—
468C2×C4D4×#615711yescharacteristic
469C2×C4C23×#615711yescharacteristic
470C8—21no—
471C8—1062no—
472C8—1062no—
473C8—1061no—
474C8—1061no—
475C8—2122no—
476C8—2122no—
477C8—2122no—
478C8C2×#1171994012yes—
479C22—21no—
480C22—22no—
481C22—21no—
482C22—21no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22C2×D4×#615711yescharacteristic
489C4—21no—
490C4—22no—
491C4—41no—
492C4—531no—
493C4—531no—
494C4—532no—
495C4—1061no—
496C4—1061no—
497C4—2121no—
498C4—2121no—
499C4—2121no—
500C4—2121no—
501C4C22×#1171994011yescharacteristic
502C4#1681170111yescharacteristic
503C2—41no—
504C2—42no—
505C2—81no—
506C2D4×#1171994011yescharacteristic
507C2#2029147711yescharacteristic
508C2D53⋊#44411yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
Fitting, prime radicalC2×#135×C53C4
socleC22×C53C2×C4×D4
cosocleC4×Dic53C24
hypercenterC2×#135#6157
hypocenterC53#18061
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #20291647C2semidirect
2. #20291641C2semidirect
3. #20290358C2semidirect
4. #20290328C2semidirect
5. #20290316C2semidirect
6. SD16⋊#88324C2semidirect
7. #20290272C2semidirect
8. #20290050C2semidirect
9. #20290050C2semidirect
10. #20289924C2semidirect
145. SD16×C53M16semidirect
303. C53#18061semidirect
390. SD16#11719940semidirect

Maximal subgroups

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

subgroupindex
#202916472
#202916412
#202903582
#202903282
#202903162
SD16⋊#883242
#202902722
#202900502
#202899242
#135×Dic532
#1806153