Finite Group Databasebeta

← C2×#20285236 #28253705

Subgroups — C2×#20285236 #28253705

Order 13568.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811179C211yesmaximal, characteristic
2C2×#16811178C211yesmaximal, characteristic
3#20285236C2116yesmaximal, direct product factor
4C2×#16811033C211yesmaximal, characteristic, complemented
5C2×#16811032C211yesmaximal, characteristic, complemented
6C2×#16811031C212yesmaximal, complemented
7C2×#16811030C212yesmaximal, complemented
8C2×#16811029C211yesmaximal, characteristic, complemented
9C2×#16810660C212yesmaximal
10C2×#16810658C212yesmaximal
11C2×#97×D53C211yesmaximal, characteristic
12C2×C53×#501C211yesmaximal, characteristic, complemented
13C22×C8×D53C2212yes—
14#16811179—24no—
15#16811179C2218yes—
16#16811178—24no—
17#16811178C2218yes—
18C2×#13133217C2211yescharacteristic
19C2×#13133216C2211yescharacteristic
20C2×#13133216C2211yescharacteristic
21C2×#13133209C2211yescharacteristic
22C2×#13133208C2212yes—
23C2×#13133197—21no—
24C2×#13133195—21no—
25C2×#13133195—21no—
26C2×#13133193—21no—
27C2×Q8×Dic53—21no—
28C2×#13133182—21no—
29C2×#13133182—21no—
30C2×#13133182—21no—
31C2×#13133175C2212yes—
32C2×#13133175C2211yescharacteristic
33C2×#13133175C2211yescharacteristic
34C2×C42×D53C2211yescharacteristic
35#16811033C2218yescomplemented
36#16811032C2218yescomplemented
37#16811031C22116yescomplemented
38#16811030C22116yescomplemented
39#16811029C2218yescomplemented
40#16810660C22116yes—
41#16810658C22116yes—
42C2×#13133151C2212yes—
43C2×#13133150C2212yes—
44C2×#13133150C2212yes—
45C2×#13133149C2212yes—
46C2×#13133148C2212yes—
47C2×#13133148C2212yes—
48C2×#13133128C2211yescharacteristic
49C2×#13133127C2212yes—
50#97×D53C2218yes—
51C2×C4⋊Q8×C53C2211yescharacteristic
52C2×#125×C53C2211yescharacteristic
53C53×#501C2218yescomplemented
54C2×#107×C53C2212yes—
55C2×#106×C53C2212yes—
56C2×#97×C53C2211yescharacteristic
57C22×Dic106—21no—
58C22×Dic106—21no—
59C22×Dic106—21no—
60C22×C4×D53D411yescharacteristic
61C22×C4×D53D411yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2312yes—
64C2×C8×D53Q8116yescomplemented
65C2×C8×D53C2318yes—
66#13133217—22no—
67#13133217C2314yes—
68#13133216—22no—
69#13133216—22no—
70#13133216C2314yes—
71#13133216C2314yes—
72#13133209—22no—
73#13133209C2314yes—
74#13133208—24no—
75#13133208C2318yes—
76#13133197—28no—
77#13133195—28no—
78#13133195—28no—
79#13133193—28no—
80Q8×Dic53—28no—
81#13133182—28no—
82#13133182—28no—
83#13133182—28no—
84#13133175—24no—
85#13133175—22no—
86#13133175—22no—
87#13133175C2314yes—
88#13133175C2318yes—
89#13133175C2314yes—
90C2×C4⋊Dic53—21no—
91C2×C4⋊Dic53—22no—
92C2×C4⋊Dic53C2312yes—
93C2×C4⋊Dic53C2311yescharacteristic
94C2×C4⋊Dic53C2311yescharacteristic
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—22no—
98C2×#11719919—21no—
99C2×#11719919—21no—
100C2×#11719921—22no—
101C2×#11719921—21no—
102C2×#11719921—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53—22no—
106C2×C4×Dic53D411yescharacteristic
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109C42×D53D414yes—
110C42×D53C2314yes—
111#13133151C2318yes—
112#13133150C2318yes—
113#13133150C2318yes—
114#13133149C2318yes—
115#13133148C2318yes—
116#13133148C2318yes—
117#13133128C2314yes—
118#13133127C2318yes—
119C22×Q8×C53—21no—
120C22×C8×C53C2312yes—
121C4⋊Q8×C53—22no—
122C4⋊Q8×C53C2314yes—
123#125×C53—22no—
124#125×C53C2314yes—
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2312yes—
131C2×C42×C53C2311yescharacteristic
132#107×C53C2318yes—
133#106×C53C2318yes—
134#97×C53C2314yes—
135C23×D53C2×D411yescharacteristic
136C2×Q8×C53—24no—
137C2×Q8×C53—44no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—22no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53C2×D414yes—
153C2×C4×D53C2×D414yes—
154C2×C4×D53C2×D412yes—
155C2×C4×D53C2×Q818yes—
156C2×C4×D53C2412yes—
157C2×Dic106—24no—
158C2×Dic106—24no—
159C2×Dic106—24no—
160C2×Dic106—44no—
161C2×Dic106—44no—
162C2×Dic106—44no—
163C2×C8×C53C2×Q818yes—
164C2×C8×C53C2414yes—
165C4⋊C4×C53—24no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—24no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53C2412yes—
172C4⋊C4×C53C2414yes—
173C4⋊C4×C53C2412yes—
174C42×C53C2×D412yes—
175C42×C53C2412yes—
176#11719921—28no—
177#11719921—24no—
178#11719921—24no—
179C4⋊Dic53—24no—
180C4⋊Dic53—22no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—28no—
184C4⋊Dic53C2412yes—
185C4⋊Dic53C2414yes—
186C4⋊Dic53C2412yes—
187#11719919—24no—
188#11719919—24no—
189#11719919—28no—
190#11719919—24no—
191#11719919—24no—
192C4×Dic53—24no—
193C4×Dic53—24no—
194C4×Dic53—28no—
195C4×Dic53C2×D412yes—
196C4×Dic53C2×D414yes—
197C4×Dic53C2×D414yes—
198C4×Dic53C2412yes—
199C2×#88322C2×Q818yes—
200C2×#88322C2414yes—
201C8×D53C2×Q8132yescomplemented
202C23×C53C22×D411yescharacteristic
203C22×D53—22no—
204C22×D53—22no—
205C22×D53C4⋊Q814yes—
206C22×D53C22×D412yes—
207Q8×C53—48no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—24no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—44no—
216C2×C4×C53C22×D412yes—
217C2×C4×C53C22×D412yes—
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C22×Q814yes—
220C2×C4×C53C2511yescharacteristic
221C2×Dic53—22no—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—24no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—44no—
229C2×Dic53C4⋊Q814yes—
230C2×Dic53C22×D412yes—
231C2×Dic53C22×D412yes—
232C2×Dic53C22×D412yes—
233C4×D53—44no—
234C4×D53—44no—
235C4×D53C4⋊Q818yes—
236C4×D53C22×Q818yes—
237Dic106—48no—
238Dic106—48no—
239Dic106—48no—
240C8×C53C22×Q818yes—
241#88322C22×Q818yes—
242C2×#3162—531nomaximal, Sylow
243C22×C53C2×C4⋊Q812yes—
244C22×C53#57512yes—
245C22×C53#57512yes—
246C22×C53C23×D411yescharacteristic
247C2×D53—28no—
248C2×D53—22no—
249C2×D53—22no—
250C2×D53C2×C4⋊Q814yes—
251C4×C53—22no—
252C4×C53—22no—
253C4×C53—42no—
254C4×C53—42no—
255C4×C53—44no—
256C4×C53C2×C4⋊Q812yes—
257C4×C53C23×Q812yes—
258Dic53—22no—
259Dic53—22no—
260Dic53—42no—
261Dic53—42no—
262Dic53—44no—
263Dic53C2×C4⋊Q814yes—
264C2×#533—531no—
265C2×#533—531no—
266#3162—5316no—
267C2×#501—532no—
268C2×#501—532no—
269C2×#501—531no—
270C2×#501—531no—
271C2×#501—531no—
272C2×#501D5311yescharacteristic, complemented
273C2×#427—532no—
274C2×#427—532no—
275C22×#97—531no—
276C2×C53#316214yescomplemented
277C2×C53C22×C4⋊Q811yescharacteristic
278C2×C53C2×#57511yescharacteristic
279C2×C53C2×#57511yescharacteristic
280D53—24no—
281C23×C8—532no—
282#533—538no—
283#533—538no—
284#533—1064no—
285#533—1064no—
286C2×C4⋊Q8—531no—
287C2×C4⋊Q8—531no—
288C2×C4⋊Q8—531no—
289C2×C4⋊Q8C2×D5311yescharacteristic
290C2×#125—532no—
291C2×#125—531no—
292C2×#125C2×D5311yescharacteristic
293C2×C22⋊Q8—1061no—
294C2×C22⋊Q8—1061no—
295C2×C22⋊Q8—1061no—
296C2×C22⋊Q8—1061no—
297C2×C4×Q8—1061no—
298C2×C4×Q8—1061no—
299C2×C4×Q8—1061no—
300C2×C4×Q8—1061no—
301C2×#117—532no—
302C2×#117—531no—
303C2×#117—531no—
304C22×C42—531no—
305#501—538no—
306#501—538no—
307#501—5316no—
308#501—5316no—
309#501—538no—
310#501C2×D5318yescomplemented
311#427—5316no—
312#427—5316no—
313C2×#107—532no—
314C2×#107—532no—
315C2×#107—532no—
316C2×#107C2×D5312yes—
317C2×#106—532no—
318C2×#106—532no—
319C2×#106—532no—
320C2×#106C2×D5312yes—
321C2×#97—532no—
322C2×#97—531no—
323C2×#97—538no—
324C2×#97C2×D5311yescharacteristic
325C53C2×#316211yescharacteristic, complemented, Sylow
326C22×Q8—21no—
327C22×Q8—1061no—
328C22×Q8—1061no—
329C22×Q8—1061no—
330C23×C4—531no—
331C23×C4—531no—
332C23×C4—531no—
333C22×C8—532no—
334C22×C8—5316no—
335C22×C8—538no—
336C22×C8C22×D5312yes—
337C4⋊Q8—22no—
338C4⋊Q8—534no—
339C4⋊Q8—534no—
340C4⋊Q8—534no—
341C4⋊Q8—1062no—
342C4⋊Q8—1062no—
343C4⋊Q8—1062no—
344C4⋊Q8C22×D5314yes—
345#125—22no—
346#125—538no—
347#125—534no—
348#125—1064no—
349#125—1062no—
350#125C22×D5314yes—
351C22⋊Q8—1068no—
352C22⋊Q8—1068no—
353C22⋊Q8—1068no—
354C22⋊Q8—1068no—
355C4×Q8—1068no—
356C4×Q8—1068no—
357C4×Q8—1068no—
358C4×Q8—1068no—
359#117—534no—
360#117—538no—
361#117—534no—
362#117—1062no—
363#117—1062no—
364#117—1064no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—532no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4C22×D5312yes—
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1062no—
384C2×C42—531no—
385C2×C42—531no—
386C2×C42—531no—
387C2×C42—534no—
388C2×C42—534no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42—1062no—
392C2×C42C22×D5311yescharacteristic
393#107—538no—
394#107—538no—
395#107—538no—
396#107C22×D5318yes—
397#106—538no—
398#106—538no—
399#106—538no—
400#106C22×D5318yes—
401#97—538no—
402#97—534no—
403#97C22×D5314yes—
404C24—531no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—1064no—
408C2×Q8—1064no—
409C2×Q8—1064no—
410C2×Q8—2124no—
411C2×Q8—2124no—
412C2×Q8—2124no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—531no—
417C22×C4—532no—
418C22×C4—538no—
419C22×C4—532no—
420C22×C4—531no—
421C22×C4—531no—
422C22×C4—534no—
423C22×C4—534no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1062no—
427C22×C4—2122no—
428C22×C4—2122no—
429C22×C4D4×D5311yescharacteristic
430C22×C4D4×D5311yescharacteristic
431C22×C4C23×D5311yescharacteristic
432C2×C8—538no—
433C2×C8—534no—
434C2×C8—5332no—
435C2×C8Q8×D5318yes—
436C2×C8C23×D5314yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—532no—
444C4⋊C4—534no—
445C4⋊C4—532no—
446C4⋊C4—1068no—
447C4⋊C4—1068no—
448C4⋊C4—1064no—
449C4⋊C4—1064no—
450C4⋊C4—1062no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—1064no—
454C4⋊C4—1064no—
455C4⋊C4—1064no—
456C4⋊C4C23×D5312yes—
457C4⋊C4C23×D5314yes—
458C4⋊C4C23×D5312yes—
459C22⋊C4—1064no—
460C22⋊C4—1064no—
461C22⋊C4—1068no—
462C42—534no—
463C42—534no—
464C42—532no—
465C42—532no—
466C42—1064no—
467C42—1064no—
468C42—1068no—
469C42D4×D5312yes—
470C42C23×D5312yes—
471C23—534no—
472C23—532no—
473C23—1062no—
474C23—1062no—
475C23C2×D4×D5311yescharacteristic
476Q8—48no—
477Q8—2128no—
478Q8—2128no—
479Q8—2128no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—534no—
489C2×C4—532no—
490C2×C4—538no—
491C2×C4—538no—
492C2×C4—532no—
493C2×C4—532no—
494C2×C4—1062no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—1064no—
499C2×C4—2124no—
500C2×C4—2124no—
501C2×C4—2122no—
502C2×C4—2124no—
503C2×C4—2122no—
504C2×C4C2×D4×D5312yes—
505C2×C4C2×D4×D5312yes—
506C2×C4C2×D4×D5311yescharacteristic
507C2×C4C2×Q8×D5314yes—
508C2×C4C24×D5311yescharacteristic
509C8—538no—
510C8C2×Q8×D5318yes—
511C22—534no—
512C22—1068no—
513C22—1062no—
514C22—1062no—
515C22C4⋊Q8×D5312yes—
516C22#1681141912yes—
517C22#1681142212yes—
518C22C22×D4×D5311yescharacteristic
519C4—22no—
520C4—22no—
521C4—44no—
522C4—42no—
523C4—42no—
524C4—534no—
525C4—1062no—
526C4—1062no—
527C4—2122no—
528C4—2124no—
529C4—2122no—
530C4C4⋊Q8×D5312yes—
531C4C22×Q8×D5312yes—
532C2—1064no—
533C2#2028523614yesdirect product factor
534C2C2×C4⋊Q8×D5311yescharacteristic
535C2C2×#1681141911yescharacteristic
536C2C2×#1681142211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#501C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#501D53
hypocenterC53C2×#3162
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #20285236C2direct (N × H)
4. C2×#16811033C2semidirect
5. C2×#16811032C2semidirect
6. C2×#16811031C2semidirect
7. C2×#16811030C2semidirect
8. C2×#16811029C2semidirect
12. C2×C53×#501C2semidirect
35. #16811033C22semidirect
36. #16811032C22semidirect
37. #16811031C22semidirect
38. #16811030C22semidirect
39. #16811029C22semidirect
53. C53×#501C22semidirect
64. C2×C8×D53Q8semidirect
201. C8×D53C2×Q8semidirect
272. C2×#501D53semidirect
276. C2×C53#3162semidirect
310. #501C2×D53semidirect
325. C53C2×#3162semidirect
533. C2#20285236direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111792
C2×#168111782
#202852362
C2×#168110332
C2×#168110322
C2×#168110312
C2×#168110302
C2×#168110292
C2×#168106602
C2×#168106582
C2×#97×D532
C2×C53×#5012
C2×#316253