Finite Group Databasebeta

← #557×#6157 #28328804

Subgroups — #557×#6157 #28328804

Order 13568.

2220 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291249C216yesmaximal
2#20291244C213yesmaximal
3#20291234C213yesmaximal
4#20291230C212yesmaximal
5Q8×C4⋊#6157C211yesmaximal, characteristic
6C4⋊Q8×#6157C213yesmaximal
7#125×#6157C216yesmaximal
8C4×Q8×#6157C212yesmaximal
9C4×Q8×#6157C213yesmaximal
10C4×Q8×#6157C211yesmaximal, characteristic
11#557×D53C211yesmaximal, characteristic
12C2×Q8×#6157C2213yes—
13C2×Q8×#6157C2212yes—
14C2×Q8×#6157C2211yescharacteristic
15C2×Q8×#6157C2213yes—
16C2×Q8×#6157C2211yescharacteristic
17(C2×#6157)⋊Q8C2216yes—
18#16811574C2216yes—
19#16811574C2216yes—
20#16811574C2216yes—
21#16811574C2216yes—
22#16811572C22112yes—
23#16811572C22112yes—
24#16811569C2213yes—
25#16811567C2216yes—
26C4⋊C4×#6157C2211yescharacteristic
27C4⋊C4×#6157C2216yes—
28C4⋊C4×#6157C2213yes—
29C4⋊C4×#6157C2216yes—
30C4⋊C4×#6157C2213yes—
31C4⋊C4×#6157C2216yes—
32C4⋊C4×#6157C2216yes—
33C4⋊C4×#6157C2213yes—
34C4⋊C4×#6157C2216yes—
35C4×C4⋊#6157C2213yes—
36C4×C4⋊#6157C2216yes—
37C4×C4⋊#6157C2213yes—
38C4×C4⋊#6157C2216yes—
39C42×#6157C2216yes—
40C42×#6157C2213yes—
41#16811352C411yescharacteristic, complemented
42#16811351C413yescomplemented
43#16811350C416yescomplemented
44#16811349C412yescomplemented
45#16811348C413yescomplemented
46C4⋊Q8×D53C2213yes—
47#125×D53C2216yes—
48C4×Q8×D53C2211yescharacteristic
49C4×Q8×D53C2212yes—
50C4×Q8×D53C2213yes—
51C53×#557C411yescharacteristic, complemented
52Q8×#6157—28no—
53Q8×#6157—24no—
54Q8×#6157—212no—
55Q8×#6157—212no—
56Q8×#6157Q818yescomplemented
57C2×C4⋊#6157C2313yes—
58C2×C4⋊#6157C2316yes—
59C2×C4⋊#6157C2313yes—
60C2×C4⋊#6157C2316yes—
61C2×C4⋊#6157C2316yes—
62C2×C4⋊#6157C2316yes—
63C2×C4⋊#6157C2313yes—
64C2×C4⋊#6157C2311yescharacteristic
65C2×C4×#6157C2312yes—
66C2×C4×#6157C2313yes—
67C2×C4×#6157C2316yes—
68C2×C4×#6157C2313yes—
69C2×C4×#6157C2313yes—
70C2×C4×#6157C2316yes—
71C2×C4×#6157C2316yes—
72C2×C4×#6157C2316yes—
73C2×C4×#6157C2313yes—
74C2×C4×#6157C2313yes—
75C2×C4×#6157C2311yescharacteristic
76(C2×#6157)⋊C4C2312yes—
77(C2×#6157)⋊C4C2316yes—
78(C2×#6157)⋊C4C2316yes—
79(C2×#6157)⋊C4C2316yes—
80(C2×#6157)⋊C4C23112yes—
81(C2×#6157)⋊C4C23112yes—
82C2×Q8×D53C2313yes—
83C2×Q8×D53C2311yescharacteristic
84#13133217C2×C413yes—
85#13133216C2×C416yes—
86#13133216C2×C416yes—
87#13133215C2×C416yes—
88#13133209C2×C416yes—
89#13133208C2×C416yes—
90#13133208C2×C416yes—
91#13133207C2×C4112yes—
92#13133207C2×C4112yes—
93Q8×Dic53C2×C413yes—
94Q8×Dic53C2×C411yescharacteristic
95Q8×Dic53C2×C412yes—
96C4×Dic106C2×C413yes—
97C4×Dic106C2×C413yes—
98C4×Dic106C2×C416yes—
99C4×Dic106C2×C416yes—
100#13133182C2×C416yes—
101#13133182C2×C416yes—
102#13133182C2×C413yes—
103#13133182C2×C413yes—
104C4⋊C4×D53C2311yescharacteristic
105C4⋊C4×D53C2316yes—
106C4⋊C4×D53C2313yes—
107C4⋊C4×D53C2316yes—
108C4⋊C4×D53C2316yes—
109C42×D53C2316yes—
110C42×D53C2313yes—
111C4⋊Q8×C53C2×C413yes—
112#125×C53C2×C416yes—
113C4×Q8×C53C2×C412yes—
114C4×Q8×C53C2×C413yes—
115C4×Q8×C53C2×C411yescharacteristic
116C22×#6157C2413yes—
117C22×#6157C2416yes—
118C22×#6157C2413yes—
119C22×#6157C2412yes—
120C22×#6157C2411yescharacteristic
121C22×#6157C2411yescharacteristic
122C2×Q8×C53C22×C413yes—
123C2×Q8×C53C22×C411yescharacteristic
124Q8×D53—26no—
125Q8×D53—26no—
126Q8×D53C2×Q814yes—
127Q8×D53#4114yes—
128C2×C4×D53C2412yes—
129C2×C4×D53C2413yes—
130C2×C4×D53C2416yes—
131C2×C4×D53C2413yes—
132C2×C4×D53C2411yescharacteristic
133C2×Dic106C22×C413yes—
134C2×Dic106C22×C416yes—
135C2×Dic106C22×C413yes—
136C4⋊#6157—212no—
137C4⋊#6157—212no—
138C4⋊#6157—26no—
139C4⋊#6157—212no—
140C4⋊#6157—26no—
141C4⋊#6157—46no—
142C4⋊#6157C2×Q8112yes—
143C4⋊#6157#4114yes—
144C4×#6157—212no—
145C4×#6157—212no—
146C4×#6157—24no—
147C4×#6157—26no—
148C4×#6157—26no—
149C4×#6157—212no—
150C4×#6157—26no—
151C4×#6157—46no—
152C4×#6157—43no—
153C4×#6157C2×Q8112yes—
154C4×#6157#4114yes—
155C4⋊C4×C53C22×C411yescharacteristic
156C4⋊C4×C53C22×C416yes—
157C4⋊C4×C53C22×C413yes—
158C4⋊C4×C53C22×C416yes—
159C4⋊C4×C53C22×C416yes—
160C42×C53C22×C416yes—
161C42×C53C22×C413yes—
162C4⋊Dic53C22×C411yescharacteristic
163C4⋊Dic53C22×C416yes—
164C4⋊Dic53C22×C413yes—
165C4⋊Dic53C22×C416yes—
166C4⋊Dic53C22×C416yes—
167#11719919C22×C412yes—
168#11719919C22×C416yes—
169#11719919C22×C416yes—
170#11719919C22×C416yes—
171#11719919C22×C4112yes—
172#11719919C22×C4112yes—
173C4×Dic53C22×C413yes—
174C4×Dic53C22×C416yes—
175C4×Dic53C22×C416yes—
176C4×Dic53C22×C416yes—
177C4×Dic53C22×C413yes—
178C4×Dic53C22×C413yes—
179C22×D53C2511yescharacteristic
180C2×#6157—23no—
181C2×#6157—21no—
182C2×#6157—22no—
183C2×#6157—22no—
184C2×#6157—23no—
185C2×#6157—21no—
186C2×#6157—26no—
187C2×#6157—26no—
188C2×#6157—23no—
189C2×#6157—23no—
190C2×#6157—23no—
191C2×#6157—26no—
192C2×#6157C22×Q816yes—
193C2×#6157C22×Q812yes—
194C2×#6157C2×#4114yes—
195C2×#6157C2×#4112yes—
196C2×#6157C2×#4112yes—
197C2×#61572−1+412yes—
198Q8×C53—26no—
199Q8×C53C4×Q814yescomplemented
200C2×C4×C53C23×C416yes—
201C2×C4×C53C23×C412yes—
202C2×C4×C53C23×C413yes—
203C2×C4×C53C23×C413yes—
204C2×C4×C53C23×C411yescharacteristic
205C2×Dic53C23×C416yes—
206C2×Dic53C23×C412yes—
207C2×Dic53C23×C413yes—
208C2×Dic53C23×C413yes—
209C2×Dic53C23×C411yescharacteristic
210C4×D53—26no—
211C4×D53—26no—
212C4×D53—22no—
213C4×D53—23no—
214C4×D53—22no—
215C4×D53—23no—
216C4×D53C22×Q816yes—
217C4×D53C2×#4116yes—
218C4×D53C2×#4112yes—
219C4×D53C2×#4112yes—
220Dic106—26no—
221Dic106—212no—
222Dic106C4×Q8112yescomplemented
223C4×#557—531nomaximal, Sylow
224C22×C53C24×C411yescharacteristic
225C2×D53C23×Q811yescharacteristic
226C2×D53C22×#4111yescharacteristic
227C2×D53C22×#4111yescharacteristic
228C2×D53C22×#4111yescharacteristic
229C2×D53C2×2−1+411yescharacteristic
230C2×D53#58511yescharacteristic
231#6157—26no—
232#6157—22no—
233#6157—24no—
234#6157—46no—
235#6157—43no—
236#6157#55714yesdirect product factor
237C4×C53—26no—
238C4×C53—22no—
239C4×C53—23no—
240C4×C53C2×C4×Q816yes—
241C4×C53C4×#4112yes—
242Dic53—26no—
243Dic53—22no—
244Dic53—23no—
245Dic53C2×C4×Q816yes—
246Dic53C4×#4112yes—
247C2×#557—531no—
248#2370—536no—
249#2365—533no—
250#2355—533no—
251#2351—532no—
252Q8×C4⋊C4—531no—
253C4×C4⋊Q8—533no—
254C4×#125—536no—
255C42×Q8—531no—
256C42×Q8—532no—
257C42×Q8—533no—
258C2×C53C22×C4×Q811yescharacteristic
259C2×C53C2×C4×#4111yescharacteristic
260C2×C53C4×2−1+411yescharacteristic
261D53C2×#55711yescharacteristic
262D53Q8×#4111yescharacteristic
263D53#348011yescharacteristic
264D53#348211yescharacteristic
265#557—533no—
266#557—536no—
267#557—532no—
268#557—533no—
269#557—531no—
270#557#615711yescharacteristic, direct product factor
271C2×C4⋊Q8—533no—
272C2×#125—536no—
273C2×C4×Q8—531no—
274C2×C4×Q8—531no—
275C2×C4×Q8—532no—
276C2×C4×Q8—533no—
277C2×C4×Q8—533no—
278C2×C4×Q8—532no—
279C2×C4×Q8—531no—
280C2×C4×Q8—533no—
281#391—536no—
282#389—536no—
283#389—536no—
284#389—536no—
285#389—536no—
286#387—5312no—
287#387—5312no—
288#384—533no—
289#382—536no—
290C4×C4⋊C4—536no—
291C4×C4⋊C4—536no—
292C4×C4⋊C4—533no—
293C4×C4⋊C4—536no—
294C4×C4⋊C4—533no—
295C4×C4⋊C4—536no—
296C4×C4⋊C4—533no—
297C4×C4⋊C4—536no—
298C4×C4⋊C4—533no—
299C4×C4⋊C4—536no—
300C4×C4⋊C4—536no—
301C4×C4⋊C4—533no—
302C4×C4⋊C4—531no—
303C43—536no—
304C43—533no—
305C53C4×#55711yescharacteristic, complemented, Sylow
306C22×Q8—531no—
307C22×Q8—533no—
308C4⋊Q8—536no—
309C4⋊Q8—536no—
310C4⋊Q8—536no—
311C4⋊Q8—533no—
312C4⋊Q8C2×#615713yes—
313#125—5312no—
314#125—5312no—
315#125—536no—
316#125—536no—
317#125—536no—
318#125C2×#615716yes—
319C4×Q8—538no—
320C4×Q8—532no—
321C4×Q8—533no—
322C4×Q8—531no—
323C4×Q8—536no—
324C4×Q8—536no—
325C4×Q8—533no—
326C4×Q8—533no—
327C4×Q8—536no—
328C4×Q8—533no—
329C4×Q8—536no—
330C4×Q8—533no—
331C4×Q8—10612no—
332C4×Q8—10612no—
333C4×Q8—1068no—
334C4×Q8—1064no—
335C4×Q8C2×#615712yes—
336C4×Q8C2×#615713yes—
337C4×Q8C2×#615711yescharacteristic
338C2×C4⋊C4—531no—
339C2×C4⋊C4—533no—
340C2×C4⋊C4—531no—
341C2×C4⋊C4—536no—
342C2×C4⋊C4—533no—
343C2×C4⋊C4—536no—
344C2×C4⋊C4—536no—
345C2×C4⋊C4—533no—
346C2×C4⋊C4—536no—
347C2×C4⋊C4—533no—
348C2×C4⋊C4—536no—
349C2×C4⋊C4—536no—
350C2×C4⋊C4—536no—
351C2×C42—531no—
352C2×C42—533no—
353C2×C42—536no—
354C2×C42—533no—
355C2×C42—536no—
356C2×C42—536no—
357C2×C42—536no—
358C2×C42—533no—
359C2×C42—533no—
360C2×C42—532no—
361C2×C42—533no—
362C2×C42—536no—
363C2×C42—533no—
364#95—532no—
365#95—536no—
366#95—536no—
367#95—536no—
368#95—5312no—
369#95—5312no—
370C2×Q8—534no—
371C2×Q8—534no—
372C2×Q8—533no—
373C2×Q8—536no—
374C2×Q8—533no—
375C2×Q8—1066no—
376C2×Q8—1066no—
377C2×Q8C22×#615713yes—
378C2×Q8C22×#615711yescharacteristic
379C22×C4—531no—
380C22×C4—533no—
381C22×C4—531no—
382C22×C4—533no—
383C22×C4—532no—
384C22×C4—531no—
385C22×C4—532no—
386C22×C4—533no—
387C22×C4—536no—
388C22×C4—533no—
389C22×C4—536no—
390C4⋊C4—5312no—
391C4⋊C4—534no—
392C4⋊C4—532no—
393C4⋊C4—531no—
394C4⋊C4—536no—
395C4⋊C4—536no—
396C4⋊C4—536no—
397C4⋊C4—536no—
398C4⋊C4—533no—
399C4⋊C4—5312no—
400C4⋊C4—536no—
401C4⋊C4—5312no—
402C4⋊C4—536no—
403C4⋊C4—1066no—
404C4⋊C4—10612no—
405C4⋊C4—1066no—
406C4⋊C4—10612no—
407C4⋊C4—10612no—
408C4⋊C4—2126no—
409C4⋊C4C22×#615711yescharacteristic
410C4⋊C4C22×#615716yes—
411C4⋊C4C22×#615713yes—
412C4⋊C4C22×#615716yes—
413C4⋊C4C22×#615716yes—
414C42—5312no—
415C42—534no—
416C42—536no—
417C42—536no—
418C42—536no—
419C42—533no—
420C42—533no—
421C42—533no—
422C42—1064no—
423C42—1066no—
424C42—10612no—
425C42—10612no—
426C42—1066no—
427C42—1066no—
428C42—10612no—
429C42—2126no—
430C42—2123no—
431C42C22×#615716yes—
432C42C22×#615713yes—
433C23—531no—
434Q8—26no—
435Q8—5312no—
436Q8—10612no—
437Q8—1066no—
438Q8Q8×#615714yescomplemented
439C2×C4—532no—
440C2×C4—532no—
441C2×C4—532no—
442C2×C4—533no—
443C2×C4—531no—
444C2×C4—536no—
445C2×C4—536no—
446C2×C4—532no—
447C2×C4—532no—
448C2×C4—536no—
449C2×C4—532no—
450C2×C4—534no—
451C2×C4—532no—
452C2×C4—533no—
453C2×C4—536no—
454C2×C4—1066no—
455C2×C4—1066no—
456C2×C4—1062no—
457C2×C4—1063no—
458C2×C4—1062no—
459C2×C4—1063no—
460C2×C4—1062no—
461C2×C4—1061no—
462C2×C4—1063no—
463C2×C4—1062no—
464C2×C4—1063no—
465C2×C4—1061no—
466C2×C4—1063no—
467C2×C4—1066no—
468C2×C4—1063no—
469C2×C4—1066no—
470C2×C4—1063no—
471C2×C4—1066no—
472C2×C4C23×#615712yes—
473C2×C4C23×#615713yes—
474C2×C4C23×#615716yes—
475C2×C4C23×#615713yes—
476C2×C4C23×#615711yescharacteristic
477C22—531no—
478C22—531no—
479C22—531no—
480C22—531no—
481C22—531no—
482C22—531no—
483C22C24×#615711yescharacteristic
484C4—26no—
485C4—22no—
486C4—23no—
487C4—534no—
488C4—536no—
489C4—532no—
490C4—1066no—
491C4—1064no—
492C4—1062no—
493C4—1066no—
494C4—1062no—
495C4—1063no—
496C4—2123no—
497C4—2126no—
498C4C2×Q8×#615716yes—
499C4#41×#615712yes—
500C2—531no—
501C2—531no—
502C2—531no—
503C2—531no—
504C2C22×Q8×#615711yescharacteristic
505C2C2×#41×#615711yescharacteristic
506C22−1+4×#615711yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6157
commutator, socleC22×C53C24×C4
FittingC53×#557C4
prime radical, cosocleC22×D53C25
hypercenter#557#6157
hypocenterC53C4×#557
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
41. #16811352C4semidirect
42. #16811351C4semidirect
43. #16811350C4semidirect
44. #16811349C4semidirect
45. #16811348C4semidirect
51. C53×#557C4semidirect
56. Q8×#6157Q8semidirect
199. Q8×C53C4×Q8semidirect
222. Dic106C4×Q8semidirect
236. #6157#557direct (N × H)
270. #557#6157direct (N × H)
305. C53C4×#557semidirect
438. Q8Q8×#6157semidirect

Maximal subgroups

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

subgroupindex
#202912492
#202912442
#202912342
#202912302
Q8×C4⋊#61572
C4⋊Q8×#61572
#125×#61572
C4×Q8×#61572
#557×D532
C4×#55753