Finite Group Databasebeta

← #27472838

Subgroups — #27472838

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284508C211yesmaximal, characteristic, complemented
2#20283650C211yesmaximal, characteristic, complemented
3C2×C4×#11719922C211yesmaximal, characteristic
4#20276223C212yesmaximal, complemented
5#20275970C212yesmaximal
6#20275380C212yesmaximal, complemented
7#20275376C212yesmaximal, complemented
8#20274852C211yesmaximal, characteristic, complemented
9#20274707C211yesmaximal, characteristic, complemented
10#20274704C211yesmaximal, characteristic, complemented
11C53×#3077C211yesmaximal, characteristic, complemented
12C22×#11719922C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14#16810920—22no—
15#16810910—21no—
16#16810900—21no—
17#16810754—21no—
18#16810754—21no—
19Q8×#88322—21no—
20Q8×#88322—21no—
21C4×#11719922—21no—
22C4×#11719922—21no—
23C4×#11719922—21no—
24C4×#11719922—21no—
25#16810666C2212yes—
26C2×#13133150C2211yescharacteristic
27#16810650C2212yes—
28#16810647C2211yescharacteristic
29#16810647C2211yescharacteristic
30#16810627C2212yes—
31C2×#13133138C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133134C2212yes—
34C2×#13133133C2212yes—
35#16810579C2212yes—
36C2×C4×#88322C2211yescharacteristic
37#16810392C2212yes—
38#16810391C2212yes—
39#16810387C2212yes—
40#16810387C2212yes—
41C53×#533C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C53×#478—22no—
44C53×#476—21no—
45C53×#474—21no—
46C53×#428C2212yes—
47C53×#423C2211yescharacteristic
48C53×#418C2212yes—
49C22×D212C2311yescharacteristic
50C22×C4×D53—21no—
51C2×#11719922—21no—
52C2×#11719922—21no—
53C2×#11719922—22no—
54C2×#11719922—22no—
55C2×#11719917C2312yes—
56C2×#11719917C2312yes—
57C22×#88322C2311yescharacteristic
58C22×#88322C2311yescharacteristic
59C4×D212—21no—
60C4×D212—21no—
61C4×D212—21no—
62C4×D212—21no—
63C2×C4⋊Dic53C2311yescharacteristic
64C2×#11719921—21no—
65#13133150—22no—
66#13133150—21no—
67#13133150—21no—
68#13133148—22no—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322—21no—
73C4⋊#88322—22no—
74C4⋊#88322—22no—
75#13133138—21no—
76#13133138—21no—
77#13133137—22no—
78#13133137—22no—
79#13133137—21no—
80#13133137—21no—
81#13133134—24no—
82#13133133—24no—
83#13133128—22no—
84C4×#88322—22no—
85C4×#88322D411yescharacteristic
86C4×#88322D411yescharacteristic
87C4×#88322D411yescharacteristic
88C4×#88322D411yescharacteristic
89C22×D4×C53C2311yescharacteristic
90C23×C4×C53—21no—
91C2×M16×C53C2312yes—
92C4⋊Q8×C53D411yescharacteristic
93C4⋊Q8×C53D411yescharacteristic
94#125×C53D412yes—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53—21no—
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103#117×C53C2312yes—
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107#107×C53—22no—
108#106×C53—22no—
109C4⋊C8×C53—21no—
110C4⋊C8×C53—21no—
111#102×C53—22no—
112#102×C53—22no—
113C24×C53—21no—
114C23×D53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53#4112yes—
123C22×C4×C53—21no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53—82no—
127C22×C4×C53#4111yescharacteristic
128C22×C4×C53#4111yescharacteristic
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212#4112yes—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53—82no—
140M16×C53—22no—
141M16×C53—22no—
142C2×C8×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53C2×D412yes—
150C4⋊C4×C53C2×D412yes—
151C22⋊C4×C53—22no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C42×C53—22no—
155C42×C53C2×D411yescharacteristic
156C42×C53C2×D411yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159#11719922—24no—
160#11719922—24no—
161#11719921—41no—
162#11719921—41no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165#11719917—22no—
166#11719917—22no—
167#11719917—22no—
168#11719917—22no—
169C2×#88322—21no—
170C2×#88322—21no—
171C2×#88322—21no—
172C2×#88322—21no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322C2×D411yescharacteristic
176C2×#88322C2×D411yescharacteristic
177C2×#88322C2×D411yescharacteristic
178C2×#88322C2×D411yescharacteristic
179C23×C53—22no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—81no—
183C23×C53—81no—
184C23×C53C2×#4111yescharacteristic
185C22×D53—22no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—81no—
189C22×D53—81no—
190Q8×C53—42no—
191Q8×C53—42no—
192D4×C53—22no—
193D4×C53—22no—
194D4×C53—41no—
195D4×C53—41no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—82no—
202C2×C4×C53—82no—
203C2×C4×C53—81no—
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C532−1+411yescharacteristic
213C2×C4×C532−1+411yescharacteristic
214C2×Dic53—41no—
215C2×Dic53—81no—
216D212—22no—
217D212—22no—
218D212—41no—
219D212—41no—
220C4×D53—82no—
221C4×D53—82no—
222C8×C53—42no—
223#88322—21no—
224#88322—21no—
225#88322—21no—
226#88322—21no—
227#88322—42no—
228#88322—42no—
229#28217—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—42no—
233C22×C53—82no—
234C22×C53—82no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53C2×#12111yescharacteristic
238C22×C53#55211yescharacteristic
239C22×C53#55211yescharacteristic
240C2×D53—42no—
241C2×D53—82no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—82no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—42no—
249C4×C53—42no—
250C4×C53—81no—
251C4×C53#47212yes—
252C4×C53#52511yescharacteristic
253C4×C53#53611yescharacteristic
254C4×C53#54811yescharacteristic
255C4×C53#54811yescharacteristic
256Dic53—81no—
257#3077—531no—
258#3077D5311yescharacteristic, complemented
259#2960—531no—
260C2×C4×D8—531no—
261#2089—532no—
262#2045—532no—
263#1931—532no—
264#1931—532no—
265#1837—531no—
266#1800—531no—
267#1800—531no—
268C2×C53—21no—
269C2×C53—21no—
270C2×C53—82no—
271C2×C53#253611yescharacteristic
272C2×C53C2×#47211yescharacteristic
273C2×C53#306711yescharacteristic
274D53—82no—
275C22×D8—531no—
276#533C2×D5311yescharacteristic
277C2×C4×D4—531no—
278C2×C4×D4C2×D5311yescharacteristic
279#478—22no—
280#478—1062no—
281#476—21no—
282#476—1061no—
283#474—21no—
284#474—1061no—
285#446—1061no—
286#446—1061no—
287C8×Q8—1061no—
288C8×Q8—1061no—
289C4×D8—1061no—
290C4×D8—1061no—
291C4×D8—1061no—
292C4×D8—1061no—
293#428—532no—
294#428C2×D5312yes—
295C2×#107—531no—
296#424—532no—
297#423—531no—
298#423—531no—
299#423C2×D5311yescharacteristic
300#418—532no—
301#418C2×D5312yes—
302C2×#102—531no—
303C2×#102—531no—
304C2×#100—532no—
305C2×#100—532no—
306#405—532no—
307C2×C4×C8—531no—
308#343—532no—
309#343—532no—
310#341—532no—
311#341—532no—
312C53#2821711yescharacteristic, complemented, Sylow
313C22×D4—531no—
314C22×D4C22×D5311yescharacteristic
315C23×C4—21no—
316C23×C4—1061no—
317C2×D8—1061no—
318C2×D8—1061no—
319C2×D8—1062no—
320C2×D8—1062no—
321C2×M16—532no—
322C2×M16—532no—
323C2×M16C22×D5312yes—
324C22×C8—531no—
325C22×C8—531no—
326C4⋊Q8C22⋊D5311yescharacteristic
327C4⋊Q8C22⋊D5311yescharacteristic
328#125C22⋊D5312yes—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C4×Q8—21no—
332C4×Q8—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—1061no—
338C4×D4—1061no—
339C4×D4—1061no—
340C4×D4—1061no—
341#117C22×D5312yes—
342C2×C4⋊C4—531no—
343C2×C4⋊C4C22×D5311yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—1061no—
346C2×C42C22×D5311yescharacteristic
347#107—22no—
348#107—1061no—
349#107—1061no—
350#107—1062no—
351#106—22no—
352#106—1062no—
353C4⋊C8—21no—
354C4⋊C8—21no—
355C4⋊C8—1062no—
356C4⋊C8—1062no—
357C4⋊C8—1061no—
358C4⋊C8—1061no—
359C4⋊C8—1061no—
360C4⋊C8—1061no—
361#102—22no—
362#102—22no—
363#102—1061no—
364#102—1061no—
365#102—1061no—
366#102—1061no—
367#102—1062no—
368#102—1062no—
369#100—1064no—
370#100—1064no—
371#97—1062no—
372C4×C8—531no—
373C4×C8—531no—
374C4×C8—531no—
375C4×C8—531no—
376C4×C8—1062no—
377C24—21no—
378C24—1061no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—532no—
387C2×D4—1061no—
388C2×D4—1061no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—1061no—
392C2×D4#1171994912yes—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—82no—
397C22×C4—1061no—
398C22×C4—4242no—
399C22×C4—2121no—
400C22×C4—2121no—
401C22×C4#1171994711yescharacteristic
402C22×C4#1171994711yescharacteristic
403C22×C4C23×D5311yescharacteristic
404D8—1064no—
405D8—1064no—
406M16—22no—
407M16—22no—
408M16—1062no—
409M16—1062no—
410M16—1062no—
411M16—1062no—
412C2×C8—22no—
413C2×C8—531no—
414C2×C8—531no—
415C2×C8—531no—
416C2×C8—531no—
417C2×C8—1061no—
418C2×C8—1061no—
419C2×C8—1061no—
420C2×C8—1061no—
421C2×C8—1062no—
422C2×C8—1062no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4C2×C22⋊D5312yes—
432C4⋊C4C2×C22⋊D5312yes—
433C22⋊C4—22no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—2121no—
437C22⋊C4—2121no—
438C42—22no—
439C42D4×D5311yescharacteristic
440C42D4×D5311yescharacteristic
441C42C2×C22⋊D5311yescharacteristic
442C42C2×C22⋊D5311yescharacteristic
443C23—22no—
444C23—41no—
445C23—41no—
446C23—81no—
447C23—81no—
448C23—1062no—
449C23—4241no—
450C23—4241no—
451C23—2121no—
452C23—2121no—
453C23C2×#1171994711yescharacteristic
454Q8—42no—
455Q8—42no—
456D4—22no—
457D4—22no—
458D4—41no—
459D4—41no—
460D4—1062no—
461D4—1062no—
462D4—2121no—
463D4—2121no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—4242no—
473C2×C4—4242no—
474C2×C4—4241no—
475C2×C4—2121no—
476C2×C4#1313318911yescharacteristic
477C2×C4#1313318911yescharacteristic
478C2×C4#1313318911yescharacteristic
479C2×C4#1313318911yescharacteristic
480C2×C4C2×D4×D5311yescharacteristic
481C2×C4C22×C22⋊D5311yescharacteristic
482C2×C4#41×D5311yescharacteristic
483C2×C4C2×#1171994911yescharacteristic
484C2×C4#1313326411yescharacteristic
485C2×C4#1313326511yescharacteristic
486C8—42no—
487C8—1061no—
488C8—1061no—
489C8—1061no—
490C8—1061no—
491C8—2122no—
492C8—2122no—
493C22—21no—
494C22—21no—
495C22—42no—
496C22—82no—
497C22—82no—
498C22—81no—
499C22—81no—
500C22—4242no—
501C22—4242no—
502C22—4241no—
503C22—4241no—
504C22—2122no—
505C22C2×#1313318911yescharacteristic
506C22#1681130711yescharacteristic
507C22#1681131511yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—42no—
512C4—42no—
513C4—81no—
514C4—4241no—
515C4#1681089112yes—
516C4#1681113611yescharacteristic
517C4#1681119611yescharacteristic
518C4#1681128011yescharacteristic
519C4#1681128411yescharacteristic
520C2—21no—
521C2—21no—
522C2—82no—
523C2—4242no—
524C2#2027962211yescharacteristic
525C2C2×#1681089111yescharacteristic
526C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3077C2
socleC22×C53C2×#121
prime radicalC22×#11719922C22
hypercenter#3077D53
hypocenterC53#28217
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #20284508C2semidirect
2. #20283650C2semidirect
4. #20276223C2semidirect
6. #20275380C2semidirect
7. #20275376C2semidirect
8. #20274852C2semidirect
9. #20274707C2semidirect
10. #20274704C2semidirect
11. C53×#3077C2semidirect
258. #3077D53semidirect
312. C53#28217semidirect

Maximal subgroups

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

subgroupindex
#202845082
#202836502
C2×C4×#117199222
#202762232
#202759702
#202753802
#202753762
#202748522
#202747072
#202747042
C53×#30772
#2821753