Finite Group Databasebeta

← #27436209

Subgroups — #27436209

Order 13568.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285262C211yesmaximal, characteristic, complemented
2#20284508C211yesmaximal, characteristic
3C2×#16810914C211yesmaximal, characteristic
4#20276225C212yesmaximal, complemented
5#20276074C212yesmaximal
6#20275704C211yesmaximal, characteristic, complemented
7#20275495C212yesmaximal, complemented
8#20275491C211yesmaximal, characteristic, complemented
9#20275377C212yesmaximal
10C4×#13133133C211yesmaximal, characteristic
11C53×#2962C211yesmaximal, characteristic, complemented
12C2×#13133217C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14#16811033—22no—
15#16811026—21no—
16#16811019—21no—
17#16810920—22no—
18#16810914—21no—
19#16810914—21no—
20#16810914—22no—
21#16810910—21no—
22#16810900—21no—
23C2×#13133153C2212yes—
24#16810666C2212yes—
25#16810666C2212yes—
26#16810666C2212yes—
27C2×#13133149C2212yes—
28#16810647C2211yescharacteristic
29#16810627C2212yes—
30C2×#13133139C2212yes—
31C2×#13133133C2211yescharacteristic
32C2×#13133133C2211yescharacteristic
33C4×#11719917C2211yescharacteristic
34#16810391C2211yescharacteristic
35#16810391C2212yes—
36#16810391C2211yescharacteristic
37#16810391C2212yes—
38#16810388C2212yes—
39C53×#533C2211yescharacteristic
40C53×#446—22no—
41C53×#446—21no—
42C8×Q8×C53—21no—
43C53×#424C2212yes—
44C2×C4⋊C8×C53C2211yescharacteristic
45C53×#405C2212yes—
46C4×M16×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×D212C2311yescharacteristic
49C22×C4×D53—21no—
50C2×#11719917C2312yes—
51C2×#11719917C2311yescharacteristic
52C2×#11719917C2311yescharacteristic
53#13133217—21no—
54#13133217—21no—
55#13133217—21no—
56#13133217—21no—
57C4×D212—21no—
58C4×D212—21no—
59C4×D212—21no—
60C4×D212—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53C2311yescharacteristic
63C2×C4⋊Dic53C2312yes—
64C2×#11719921—21no—
65#13133153—24no—
66#13133150—22no—
67#13133150—22no—
68#13133150—22no—
69#13133149—22no—
70#13133149—22no—
71#13133148—22no—
72#13133148—22no—
73#13133148—22no—
74C4⋊#88322—21no—
75C4⋊#88322—21no—
76#13133139—22no—
77#13133139—22no—
78#13133137—22no—
79#13133137—22no—
80#13133133—22no—
81#13133133Q814yescomplemented
82#13133128—21no—
83C4×#88322—21no—
84C2×M16×C53C2311yescharacteristic
85C2×M16×C53C2311yescharacteristic
86C22×C8×C53C2312yes—
87C4⋊Q8×C53D411yescharacteristic
88C4⋊Q8×C53D411yescharacteristic
89#125×C53D412yes—
90C22⋊Q8×C53—21no—
91C22⋊Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94#117×C53C2312yes—
95C2×C42×C53C2311yescharacteristic
96C4⋊C8×C53—22no—
97C4⋊C8×C53—22no—
98C4⋊C8×C53D411yescharacteristic
99C4⋊C8×C53D411yescharacteristic
100C4⋊C8×C53D412yes—
101#97×C53—22no—
102#97×C53—21no—
103C4×C8×C53—22no—
104C4×C8×C53—21no—
105C23×D53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—21no—
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53C2411yescharacteristic
111C22×Dic53—21no—
112C22×Dic53—22no—
113C2×D212—21no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212—21no—
117C2×D212—21no—
118C2×D212C2×Q812yes—
119C2×C4×D53—41no—
120C2×C4×D53—41no—
121C2×C4×D53—82no—
122C2×Dic106—41no—
123C2×Dic106—41no—
124C2×Dic106—82no—
125M16×C53—21no—
126M16×C53—21no—
127M16×C53—21no—
128M16×C53C2×Q812yes—
129C2×C8×C53—21no—
130C2×C8×C53—22no—
131C2×C8×C53—22no—
132C2×C8×C53—21no—
133C2×C8×C53C2×D412yes—
134C2×C8×C53C2×D412yes—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53C2×D412yes—
140C4⋊C4×C53C2×D412yes—
141C22⋊C4×C53—22no—
142C42×C53—22no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147#11719921—41no—
148#11719921—41no—
149C4⋊Dic53—22no—
150C4⋊Dic53—21no—
151C4⋊Dic53—22no—
152C4⋊Dic53—21no—
153C4⋊Dic53—41no—
154C4⋊Dic53—41no—
155#11719917—21no—
156#11719917—21no—
157#11719917—22no—
158#11719917—22no—
159#11719917C2×Q812yes—
160#11719917#4112yes—
161C2×#88322—21no—
162C2×#88322—21no—
163C2×#88322—22no—
164C23×C53C2×#4111yescharacteristic
165C22×D53—22no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—81no—
169C22×D53—81no—
170Q8×C53—42no—
171Q8×C53—81no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—41no—
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C22×Q811yescharacteristic
183C2×C4×C53C2×#4111yescharacteristic
184C2×C4×C532−1+411yescharacteristic
185C2×C4×C532−1+411yescharacteristic
186C2×Dic53—42no—
187C2×Dic53—41no—
188C2×Dic53—82no—
189C2×Dic53—81no—
190D212—22no—
191D212—22no—
192D212—41no—
193D212—41no—
194C4×D53—82no—
195C4×D53—82no—
196Dic106—82no—
197Dic106—82no—
198C8×C53—22no—
199C8×C53—22no—
200C8×C53—22no—
201C8×C53—41no—
202#88322—22no—
203#88322—22no—
204#88322—42no—
205#25490—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53C2×#12111yescharacteristic
209C22×C53#53311yescharacteristic
210C22×C53#54111yescharacteristic
211C2×D53—42no—
212C2×D53—82no—
213C2×D53—81no—
214C2×D53—81no—
215C2×D53—82no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—42no—
220C4×C53—42no—
221C4×C53#47112yes—
222C4×C53#54711yescharacteristic
223C4×C53#54811yescharacteristic
224C4×C53D4×Q811yescharacteristic
225C4×C53D4×Q811yescharacteristic
226Dic53—81no—
227Dic53—82no—
228#3165—531no—
229#3077—531no—
230C2×#476—531no—
231#2962D5311yescharacteristic, complemented
232#2090—532no—
233#2063—532no—
234#1997—531no—
235#1951—532no—
236#1950—531no—
237#1931—532no—
238C4×#100—531no—
239C2×C53—21no—
240C2×C53—21no—
241C2×C53#242911yescharacteristic
242C2×C53C2×#47111yescharacteristic
243C2×C53#306811yescharacteristic
244D53—82no—
245#533C2×D5311yescharacteristic
246C2×C4⋊Q8—531no—
247C2×C4×D4—531no—
248#501—1062no—
249#500—1061no—
250#498—1061no—
251#478—1062no—
252#476—1061no—
253#476—1061no—
254#476—1062no—
255#476—1061no—
256#474—1061no—
257#446—22no—
258#446—21no—
259C8×Q8—21no—
260C2×#108—532no—
261#428—532no—
262#428—532no—
263#428—532no—
264C2×#106—532no—
265#424C2×D5312yes—
266#423—531no—
267C2×C4⋊C8C2×D5311yescharacteristic
268#418—532no—
269C2×#102—532no—
270C2×#100—531no—
271C2×#100—531no—
272#405C2×D5312yes—
273C4×M16—531no—
274C4×M16C2×D5311yescharacteristic
275#343—532no—
276#343—531no—
277#343—532no—
278#343—531no—
279#341—532no—
280C53#2549011yescharacteristic, complemented, Sylow
281C22×Q8—1061no—
282C22×D4—531no—
283C23×C4—1061no—
284C2×M16—531no—
285C2×M16—531no—
286C2×M16—532no—
287C2×M16C22×D5311yescharacteristic
288C2×M16C22×D5311yescharacteristic
289C22×C8C22×D5312yes—
290C4⋊Q8—1061no—
291C4⋊Q8—1061no—
292C4⋊Q8—1061no—
293C4⋊Q8—1061no—
294C4⋊Q8D21211yescharacteristic
295C4⋊Q8D21211yescharacteristic
296#125D21212yes—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×D4—1061no—
302C4×D4—1061no—
303C4×D4—1061no—
304C4×D4—1061no—
305#117C22×D5312yes—
306C2×C4⋊C4—531no—
307C2×C4⋊C4—532no—
308C2×C4⋊C4—1061no—
309C2×C22⋊C4—1061no—
310C2×C42C22×D5311yescharacteristic
311#108—1064no—
312#107—1062no—
313#107—1062no—
314#107—1062no—
315#106—1062no—
316#106—1062no—
317#106—1062no—
318#106—1062no—
319#106—1062no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—1061no—
323C4⋊C8—1061no—
324C4⋊C8C22⋊D5311yescharacteristic
325C4⋊C8C22⋊D5311yescharacteristic
326C4⋊C8C22⋊D5312yes—
327#102—1062no—
328#102—1062no—
329#102—1062no—
330#102—1062no—
331#100—534no—
332#100—1062no—
333#97—22no—
334#97—21no—
335#97—1061no—
336C4×C8—22no—
337C4×C8—21no—
338C4×C8—1061no—
339C24—1061no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—4242no—
343C2×Q8—2121no—
344C2×Q8—2121no—
345C2×D4—532no—
346C2×D4—1061no—
347C2×D4—1061no—
348C2×D4—1061no—
349C2×D4—1061no—
350C2×D4—1061no—
351C22×C4—1061no—
352C22×C4—1062no—
353C22×C4—4242no—
354C22×C4—2121no—
355C22×C4—2121no—
356C22×C4#1171994511yescharacteristic
357C22×C4#1171994511yescharacteristic
358C22×C4C23×D5311yescharacteristic
359M16—21no—
360M16—21no—
361M16—21no—
362M16—532no—
363M16—532no—
364M16—1061no—
365M16—1061no—
366M16—1062no—
367M16—1062no—
368M16Q8×D5312yes—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—1061no—
374C2×C8—1061no—
375C2×C8—1062no—
376C2×C8C2×C22⋊D5312yes—
377C2×C8C2×C22⋊D5312yes—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—1062no—
383C4⋊C4—1061no—
384C4⋊C4—1061no—
385C4⋊C4—1062no—
386C4⋊C4—2121no—
387C4⋊C4—2121no—
388C4⋊C4C2×D21212yes—
389C4⋊C4C2×D21212yes—
390C22⋊C4—22no—
391C22⋊C4—2121no—
392C22⋊C4—2121no—
393C42—22no—
394C42C2×D21211yescharacteristic
395C42C2×D21211yescharacteristic
396C42C2×C22⋊D5311yescharacteristic
397C42C2×C22⋊D5311yescharacteristic
398C23—1062no—
399C23—4241no—
400C23—4241no—
401C23—2121no—
402C23—2121no—
403C23C2×#1171994511yescharacteristic
404Q8—42no—
405Q8—81no—
406Q8—4242no—
407Q8—4242no—
408D4—1062no—
409D4—1062no—
410D4—2121no—
411D4—2121no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—4242no—
417C2×C4—4242no—
418C2×C4—4241no—
419C2×C4—4242no—
420C2×C4—2121no—
421C2×C4—2122no—
422C2×C4#1313319111yescharacteristic
423C2×C4#1313319111yescharacteristic
424C2×C4#1313319111yescharacteristic
425C2×C4#1313319111yescharacteristic
426C2×C4C22×C22⋊D5311yescharacteristic
427C2×C4C22×D21211yescharacteristic
428C2×C4C2×Q8×D5311yescharacteristic
429C2×C4#41×D5311yescharacteristic
430C2×C4#1313326411yescharacteristic
431C2×C4#1313326511yescharacteristic
432C8—22no—
433C8—22no—
434C8—22no—
435C8—41no—
436C8—1062no—
437C8—1062no—
438C8—2122no—
439C22—21no—
440C22—21no—
441C22—4242no—
442C22—4242no—
443C22—4241no—
444C22—4241no—
445C22—2122no—
446C22C2×#1313319111yescharacteristic
447C22#1681117711yescharacteristic
448C22#1681123111yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—4242no—
455C4—4241no—
456C4#1681088612yes—
457C4#1681127211yescharacteristic
458C4#1681128511yescharacteristic
459C4Q8×C22⋊D5311yescharacteristic
460C4Q8×D21211yescharacteristic
461C2—21no—
462C2—21no—
463C2—4242no—
464C2#2027830811yescharacteristic
465C2C2×#1681088611yescharacteristic
466C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2962C2
socleC22×C53C2×#121
prime radicalC22×D212C23
hypercenter#2962D53
hypocenterC53#25490
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #20285262C2semidirect
4. #20276225C2semidirect
6. #20275704C2semidirect
7. #20275495C2semidirect
8. #20275491C2semidirect
11. C53×#2962C2semidirect
81. #13133133Q8semidirect
231. #2962D53semidirect
280. C53#25490semidirect

Maximal subgroups

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

subgroupindex
#202852622
#202845082
C2×#168109142
#202762252
#202760742
#202757042
#202754952
#202754912
#202753772
C4×#131331332
C53×#29622
#2549053