Finite Group Databasebeta

← C53×#61319 #27268831

Subgroups — C53×#61319 #27268831

Order 13568.

1836 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3480C211yesmaximal, characteristic
2C2×Q82×C53C211yesmaximal, characteristic, complemented
3C53×#3375C216yesmaximal, complemented
4C53×#3369C216yesmaximal, complemented
5C53×#3363C212yesmaximal, complemented
6C53×#3362C212yesmaximal, complemented
7C53×#3141C213yesmaximal, complemented
8C53×#3079C213yesmaximal, complemented
9C53×#3037C213yesmaximal, complemented
10Q8×M16×C53C211yesmaximal, characteristic, complemented
11C4×#137×C53C213yesmaximal, complemented
12C2×#137×C53C2213yes—
13C53×#559C2213yes—
14Q82×C53—21no—
15Q82×C53—23no—
16Q82×C53C2212yes—
17Q82×C53C2216yes—
18C53×#557C2211yescharacteristic
19C53×#556C2213yes—
20C53×#553—23no—
21C53×#550C2211yescharacteristic
22C53×#548—21no—
23C53×#532C2213yes—
24C2×C4⋊Q8×C53—23no—
25C2×C4⋊Q8×C53C2213yes—
26C53×#529—23no—
27C4×#41×C53—21no—
28C4×#41×C53C2213yes—
29C2×C4×Q8×C53—21no—
30C2×C4×Q8×C53C2213yes—
31C2×C4×Q8×C53C2211yescharacteristic
32C53×#489C22112yes—
33C53×#487C2216yes—
34C53×#486C2216yes—
35C53×#477C2216yes—
36C53×#476C2216yes—
37C53×#475C2216yes—
38C53×#474C2216yes—
39C53×#464C22112yes—
40C53×#462C2216yes—
41C53×#460C2216yes—
42C53×#446C2216yes—
43C8×Q8×C53C2212yes—
44C53×#441C2216yes—
45C53×#440C2216yes—
46C4×Q16×C53C2216yes—
47C4×SD16×C53C2216yes—
48C53×#428C2213yes—
49C53×#423C2213yes—
50C53×#419C2211yescharacteristic
51C53×#417C2213yes—
52C53×#417C2216yes—
53C4×M16×C53C2213yes—
54C2×#41×C53—21no—
55C2×#41×C53C2313yes—
56C22×Q8×C53—23no—
57C22×Q8×C53D411yescharacteristic
58C22×Q8×C53C2311yescharacteristic
59#137×C53—212no—
60C2×Q16×C53C2316yes—
61C2×SD16×C53C2316yes—
62C2×M16×C53C2313yes—
63C2×M16×C53C2311yescharacteristic
64C4⋊Q8×C53—26no—
65C4⋊Q8×C53—26no—
66C4⋊Q8×C53—212no—
67C4⋊Q8×C53—23no—
68C4⋊Q8×C53—23no—
69C4⋊Q8×C53C2316yes—
70C4⋊Q8×C53C2316yes—
71C4⋊Q8×C53C2313yes—
72#127×C53C2313yes—
73#126×C53—26no—
74#125×C53—23no—
75#124×C53—23no—
76#124×C53C2316yes—
77#123×C53—26no—
78C22⋊Q8×C53—23no—
79#121×C53—23no—
80#121×C53—23no—
81#121×C53—23no—
82C4×Q8×C53—23no—
83C4×Q8×C53—23no—
84C4×Q8×C53—26no—
85C4×Q8×C53—22no—
86C4×Q8×C53—21no—
87C4×Q8×C53D411yescharacteristic
88C4×Q8×C53D413yes—
89C4×Q8×C53C2311yescharacteristic
90C4×Q8×C53C2313yes—
91C4×Q8×C53C2316yes—
92C4×Q8×C53C2316yes—
93C4×Q8×C53C2313yes—
94C4×Q8×C53C2313yes—
95C4×D4×C53—23no—
96C4×D4×C53—23no—
97C4×D4×C53—23no—
98C4×D4×C53—23no—
99C4×D4×C53—23no—
100C4×D4×C53C2313yes—
101C4×D4×C53C2313yes—
102#117×C53—23no—
103#117×C53—23no—
104#117×C53C2311yescharacteristic
105C2×C4⋊C4×C53—23no—
106C2×C4⋊C4×C53—23no—
107C2×C4⋊C4×C53D413yes—
108C2×C4⋊C4×C53C2313yes—
109C2×C42×C53—23no—
110C2×C42×C53—23no—
111C2×C42×C53C2313yes—
112#107×C53C2316yes—
113#106×C53C2316yes—
114C4⋊C8×C53C2316yes—
115C4⋊C8×C53C2316yes—
116#103×C53C2316yes—
117#103×C53C23112yes—
118#103×C53C2312yes—
119#103×C53C2312yes—
120#102×C53C2316yes—
121#102×C53C23112yes—
122#97×C53C2316yes—
123C4×C8×C53C2316yes—
124#41×C53—26no—
125#41×C53—26no—
126#41×C53—44no—
127C2×Q8×C53—24no—
128C2×Q8×C53—21no—
129C2×Q8×C53—26no—
130C2×Q8×C53—26no—
131C2×Q8×C53—22no—
132C2×Q8×C53—86no—
133C2×Q8×C53C2×D413yes—
134C2×Q8×C53C2×D411yescharacteristic
135C2×Q8×C53#4114yescomplemented
136C2×Q8×C53C2413yes—
137C2×Q8×C53C2412yes—
138C2×D4×C53—23no—
139C2×D4×C53—23no—
140C2×D4×C53—23no—
141C2×D4×C53C2413yes—
142C22×C4×C53—21no—
143C22×C4×C53—23no—
144C22×C4×C53—23no—
145C22×C4×C53—23no—
146C22×C4×C53—23no—
147C22×C4×C53C2×D413yes—
148C22×C4×C53C2×D413yes—
149C22×C4×C53C2411yescharacteristic
150Q16×C53—212no—
151SD16×C53—212no—
152M16×C53—26no—
153M16×C53—22no—
154C2×C8×C53C2412yes—
155C2×C8×C53C2416yes—
156C4⋊C4×C53—26no—
157C4⋊C4×C53—26no—
158C4⋊C4×C53—26no—
159C4⋊C4×C53—26no—
160C4⋊C4×C53—23no—
161C4⋊C4×C53—23no—
162C4⋊C4×C53—23no—
163C4⋊C4×C53—23no—
164C4⋊C4×C53—23no—
165C4⋊C4×C53C2×D413yes—
166C4⋊C4×C53C2×D413yes—
167C4⋊C4×C53C2×D416yes—
168C4⋊C4×C53C2416yes—
169C4⋊C4×C53C2411yescharacteristic
170C4⋊C4×C53C2411yescharacteristic
171C22⋊C4×C53—21no—
172C22⋊C4×C53—26no—
173C22⋊C4×C53—23no—
174C22⋊C4×C53—23no—
175C22⋊C4×C53—26no—
176C42×C53—21no—
177C42×C53—23no—
178C42×C53—23no—
179C42×C53—26no—
180C42×C53—26no—
181C42×C53C2×D413yes—
182C42×C53C2×D413yes—
183C42×C53C2413yes—
184C42×C53C2413yes—
185C23×C53—23no—
186C23×C53C22×D411yescharacteristic
187Q8×C53—24no—
188Q8×C53—26no—
189Q8×C53—22no—
190Q8×C53—23no—
191Q8×C53—23no—
192Q8×C53—86no—
193Q8×C53—86no—
194Q8×C53#13714yescomplemented
195Q8×C53C2×#4114yes—
196D4×C53—23no—
197D4×C53—23no—
198D4×C53—46no—
199D4×C53—46no—
200D4×C53—83no—
201C2×C4×C53—24no—
202C2×C4×C53—22no—
203C2×C4×C53—26no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—23no—
207C2×C4×C53—26no—
208C2×C4×C53—43no—
209C2×C4×C53—46no—
210C2×C4×C53—46no—
211C2×C4×C53—46no—
212C2×C4×C53—83no—
213C2×C4×C53C22×D413yes—
214C2×C4×C53C22×D413yes—
215C2×C4×C53C22×D413yes—
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C22×D413yes—
218C2×C4×C53C2×#4112yes—
219C2×C4×C532−1+412yes—
220C2×C4×C53C2511yescharacteristic
221C8×C53—26no—
222C8×C53—22no—
223#61319C5311yesmaximal, characteristic, direct product factor, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—43no—
227C22×C53—43no—
228C22×C53—43no—
229C22×C53#54812yes—
230C22×C53C23×D411yescharacteristic
231C4×C53—23no—
232C4×C53—23no—
233C4×C53—24no—
234C4×C53—23no—
235C4×C53—46no—
236C4×C53—41no—
237C4×C53—43no—
238C4×C53—41no—
239C4×C53#54812yes—
240C4×C53C2×#13716yes—
241C4×C53C22×#4111yescharacteristic
242C4×C53C2×2−1+411yescharacteristic
243#3480C2×C5311yescharacteristic
244C2×Q82C2×C5311yescharacteristic, complemented
245#3375C2×C5316yescomplemented
246#3369C2×C5316yescomplemented
247#3363C2×C5312yescomplemented
248#3362C2×C5312yescomplemented
249#3141C2×C5313yescomplemented
250#3079C2×C5313yescomplemented
251#3037C2×C5313yescomplemented
252Q8×M16C2×C5311yescharacteristic, complemented
253C4×#137C2×C5313yescomplemented
254C2×C53—22no—
255C2×C53—83no—
256C2×C53C2×#54811yescharacteristic
257C2×C53C22×#13711yescharacteristic
258C2×C53#358511yescharacteristic
259C2×#137C22×C5313yes—
260#559C22×C5313yes—
261Q82—21no—
262Q82—23no—
263Q82C22×C5312yes—
264Q82C22×C5316yes—
265#557C22×C5311yescharacteristic
266#556C22×C5313yes—
267#553—23no—
268#550C22×C5311yescharacteristic
269#548—21no—
270#532C22×C5313yes—
271C2×C4⋊Q8—23no—
272C2×C4⋊Q8C22×C5313yes—
273#529—23no—
274C4×#41—21no—
275C4×#41C22×C5313yes—
276C2×C4×Q8—21no—
277C2×C4×Q8C22×C5313yes—
278C2×C4×Q8C22×C5311yescharacteristic
279#489C22×C53112yes—
280#487C22×C5316yes—
281#486C22×C5316yes—
282#477C22×C5316yes—
283#476C22×C5316yes—
284#475C22×C5316yes—
285#474C22×C5316yes—
286#464C22×C53112yes—
287#462C22×C5316yes—
288#460C22×C5316yes—
289#446C22×C5316yes—
290C8×Q8C22×C5312yes—
291#441C22×C5316yes—
292#440C22×C5316yes—
293C4×Q16C22×C5316yes—
294C4×SD16C22×C5316yes—
295#428C22×C5313yes—
296#423C22×C5313yes—
297#419C22×C5311yescharacteristic
298#417C22×C5316yes—
299#417C22×C5313yes—
300C4×M16C22×C5313yes—
301C53#6131911yescharacteristic, direct product factor, Sylow
302C2×#41—21no—
303C2×#41C23×C5313yes—
304C22×Q8—23no—
305C22×Q8D4×C5311yescharacteristic
306C22×Q8C23×C5311yescharacteristic
307#137—212no—
308C2×Q16C23×C5316yes—
309C2×SD16C23×C5316yes—
310C2×M16C23×C5313yes—
311C2×M16C23×C5311yescharacteristic
312C4⋊Q8—26no—
313C4⋊Q8—26no—
314C4⋊Q8—212no—
315C4⋊Q8—23no—
316C4⋊Q8—23no—
317C4⋊Q8C23×C5313yes—
318C4⋊Q8C23×C5316yes—
319C4⋊Q8C23×C5316yes—
320#127C23×C5313yes—
321#126—26no—
322#125—23no—
323#124—23no—
324#124C23×C5316yes—
325#123—26no—
326C22⋊Q8—23no—
327#121—23no—
328#121—23no—
329#121—23no—
330C4×Q8—26no—
331C4×Q8—22no—
332C4×Q8—21no—
333C4×Q8—23no—
334C4×Q8—23no—
335C4×Q8D4×C5311yescharacteristic
336C4×Q8D4×C5313yes—
337C4×Q8C23×C5316yes—
338C4×Q8C23×C5316yes—
339C4×Q8C23×C5313yes—
340C4×Q8C23×C5313yes—
341C4×Q8C23×C5311yescharacteristic
342C4×Q8C23×C5313yes—
343C4×D4—23no—
344C4×D4—23no—
345C4×D4—23no—
346C4×D4—23no—
347C4×D4—23no—
348C4×D4C23×C5313yes—
349C4×D4C23×C5313yes—
350#117—23no—
351#117—23no—
352#117C23×C5311yescharacteristic
353C2×C4⋊C4—23no—
354C2×C4⋊C4—23no—
355C2×C4⋊C4D4×C5313yes—
356C2×C4⋊C4C23×C5313yes—
357C2×C42—23no—
358C2×C42—23no—
359C2×C42C23×C5313yes—
360#107C23×C5316yes—
361#106C23×C5316yes—
362C4⋊C8C23×C5316yes—
363C4⋊C8C23×C5316yes—
364#103C23×C5316yes—
365#103C23×C53112yes—
366#103C23×C5312yes—
367#103C23×C5312yes—
368#102C23×C5316yes—
369#102C23×C53112yes—
370#97C23×C5316yes—
371C4×C8C23×C5316yes—
372#41—26no—
373#41—26no—
374#41—44no—
375C2×Q8—24no—
376C2×Q8—21no—
377C2×Q8—26no—
378C2×Q8—26no—
379C2×Q8—22no—
380C2×Q8—86no—
381C2×Q8C2×D4×C5313yes—
382C2×Q8C2×D4×C5311yescharacteristic
383C2×Q8#41×C5314yescomplemented
384C2×Q8C24×C5313yes—
385C2×Q8C24×C5312yes—
386C2×D4—23no—
387C2×D4—23no—
388C2×D4—23no—
389C2×D4C24×C5313yes—
390C22×C4—21no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4C2×D4×C5313yes—
396C22×C4C2×D4×C5313yes—
397C22×C4C24×C5311yescharacteristic
398Q16—212no—
399SD16—212no—
400M16—26no—
401M16—22no—
402C2×C8C24×C5312yes—
403C2×C8C24×C5316yes—
404C4⋊C4—26no—
405C4⋊C4—26no—
406C4⋊C4—26no—
407C4⋊C4—26no—
408C4⋊C4—23no—
409C4⋊C4—23no—
410C4⋊C4—23no—
411C4⋊C4—23no—
412C4⋊C4—23no—
413C4⋊C4C2×D4×C5313yes—
414C4⋊C4C2×D4×C5313yes—
415C4⋊C4C2×D4×C5316yes—
416C4⋊C4C24×C5316yes—
417C4⋊C4C24×C5311yescharacteristic
418C4⋊C4C24×C5311yescharacteristic
419C22⋊C4—21no—
420C22⋊C4—26no—
421C22⋊C4—23no—
422C22⋊C4—23no—
423C22⋊C4—26no—
424C42—21no—
425C42—23no—
426C42—23no—
427C42—26no—
428C42—26no—
429C42C2×D4×C5313yes—
430C42C2×D4×C5313yes—
431C42C24×C5313yes—
432C42C24×C5313yes—
433C23—23no—
434C23C22×D4×C5311yescharacteristic
435Q8—24no—
436Q8—26no—
437Q8—22no—
438Q8—23no—
439Q8—23no—
440Q8—86no—
441Q8—86no—
442Q8#137×C5314yescomplemented
443Q8C2×#41×C5314yes—
444D4—23no—
445D4—23no—
446D4—46no—
447D4—46no—
448D4—83no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—26no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—23no—
455C2×C4—26no—
456C2×C4—43no—
457C2×C4—46no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—83no—
461C2×C4C22×D4×C5313yes—
462C2×C4C22×D4×C5313yes—
463C2×C4C22×D4×C5313yes—
464C2×C4C22×D4×C5311yescharacteristic
465C2×C4C22×D4×C5313yes—
466C2×C4C2×#41×C5312yes—
467C2×C42−1+4×C5312yes—
468C2×C4C25×C5311yescharacteristic
469C8—26no—
470C8—22no—
471C22—21no—
472C22—21no—
473C22—43no—
474C22—43no—
475C22—43no—
476C22C53×#54812yes—
477C22C23×D4×C5311yescharacteristic
478C4—24no—
479C4—23no—
480C4—23no—
481C4—23no—
482C4—43no—
483C4—41no—
484C4—46no—
485C4—41no—
486C4C53×#54812yes—
487C4C2×#137×C5316yes—
488C4C22×#41×C5311yescharacteristic
489C4C2×2−1+4×C5311yescharacteristic
490C2—22no—
491C2—83no—
492C2C2×C53×#54811yescharacteristic
493C2C22×#137×C5311yescharacteristic
494C2C53×#358511yescharacteristic
subgroupG/H
center, socleC22×C53C23×D4
commutator, Frattini, cosocleC2×C4C25×C53
prime radicalC53×#3480C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C4×Q8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×Q82×C53C2semidirect
3. C53×#3375C2semidirect
4. C53×#3369C2semidirect
5. C53×#3363C2semidirect
6. C53×#3362C2semidirect
7. C53×#3141C2semidirect
8. C53×#3079C2semidirect
9. C53×#3037C2semidirect
10. Q8×M16×C53C2semidirect
11. C4×#137×C53C2semidirect
135. C2×Q8×C53#41semidirect
194. Q8×C53#137semidirect
223. #61319C53direct (N × H)
244. C2×Q82C2×C53semidirect
245. #3375C2×C53semidirect
246. #3369C2×C53semidirect
247. #3363C2×C53semidirect
248. #3362C2×C53semidirect
249. #3141C2×C53semidirect
250. #3079C2×C53semidirect
251. #3037C2×C53semidirect
252. Q8×M16C2×C53semidirect
253. C4×#137C2×C53semidirect
301. C53#61319direct (N × H)
383. C2×Q8#41×C53semidirect
442. Q8#137×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#34802
C2×Q82×C532
C53×#33752
C53×#33692
C53×#33632
C53×#33622
C53×#31412
C53×#30792
C53×#30372
Q8×M16×C532
C4×#137×C532
#6131953