Finite Group Databasebeta

← #27430263

Subgroups — #27430263

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280073C211yesmaximal, characteristic
2Q8×C4⋊Dic53C211yesmaximal, characteristic, complemented
3#20276247C211yesmaximal, characteristic
4#20276242C211yesmaximal, characteristic
5#20276193C211yesmaximal, characteristic, complemented
6#20276188C211yesmaximal, characteristic, complemented
7#20275485C211yesmaximal, characteristic, complemented
8#20275343C211yesmaximal, characteristic
9#20275336C211yesmaximal, characteristic
10#20275308C211yesmaximal, characteristic, complemented
11#20275301C211yesmaximal, characteristic, complemented
12#20274883C211yesmaximal, characteristic, complemented
13#20274716C211yesmaximal, characteristic
14#20274541C211yesmaximal, characteristic
15C53×#1985C211yesmaximal, characteristic, complemented
16C2×Q8×Dic53—21no—
17C2×C4×D212C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28#16810558—21no—
29#16810534—21no—
30#16810525C2211yescharacteristic
31#16810525C2211yescharacteristic
32#16810522—21no—
33#16810522—21no—
34#16810522—21no—
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810494—21no—
38#16810494—21no—
39#16810493C2211yescharacteristic
40#16810493C2211yescharacteristic
41#16810493C2211yescharacteristic
42#16810489C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4×Q8×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×#97×C53C2211yescharacteristic
55C53×#336C2211yescharacteristic
56C53×#336C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×C4×D53—21no—
61Q8×Dic53—44no—
62C4×D212—22no—
63C4×D212—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921—21no—
83C2×#11719921—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87#13133151—22no—
88#13133151—41no—
89#13133151—41no—
90#13133149—22no—
91#13133149—41no—
92#13133149—41no—
93#13133139—22no—
94#13133139—41no—
95#13133139—41no—
96#13133139Q814yescomplemented
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×Q8×C53D411yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C4×Q8×C53—22no—
110C4×Q8×C53—22no—
111C2×C4⋊C4×C53D411yescharacteristic
112C2×C4⋊C4×C53D411yescharacteristic
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C4⋊C8×C53—22no—
118#97×C53—22no—
119C23×D53—21no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53—22no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2411yescharacteristic
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212C2×Q812yes—
143C2×C4×D53—42no—
144C2×C4×D53—81no—
145C2×C4×D53—81no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53C2×Q812yes—
157C2×C8×C53#4112yes—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53#4112yes—
166C42×C53#4112yes—
167#11719921—42no—
168#11719921—42no—
169#11719921—42no—
170#11719921—81no—
171#11719921—81no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53—81no—
189C4⋊Dic53—81no—
190C4⋊Dic53—81no—
191C4⋊Dic53C2×Q812yes—
192#11719919—81no—
193#11719919—81no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C23×C53C22×D411yescharacteristic
198C22×D53—22no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205Q8×C53—28no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53#12312yes—
221C2×C4×C53#12312yes—
222C2×C4×C53AGL(1,ℤ/8)12yes—
223C2×C4×C53#13712yes—
224C2×C4×C53C22×Q811yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53H(5,𝔽2)11yescharacteristic
229C2×C4×C532−1+411yescharacteristic
230C2×Dic53—22no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251D212—22no—
252D212—22no—
253D212—41no—
254D212—41no—
255C4×D53—84no—
256C8×C53—22no—
257C8×C53—22no—
258C8×C53—41no—
259C8×C53—41no—
260#25068—531nomaximal, Sylow
261C22×C53C2×#12311yescharacteristic
262C22×C53D4×Q811yescharacteristic
263C22×C53#55011yescharacteristic
264C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C53C2×#13711yescharacteristic
266C22×C53#57611yescharacteristic
267C22×C53#57711yescharacteristic
268C2×D53—42no—
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—81no—
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—22no—
279C4×C53—41no—
280C4×C53—41no—
281C4×C53#53311yescharacteristic
282C4×C53#54211yescharacteristic
283C4×C53#55211yescharacteristic
284C4×C53#55411yescharacteristic
285Dic53—42no—
286Dic53—81no—
287Dic53—81no—
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2580—531no—
293Q8×C4⋊C4—531no—
294#2094—531no—
295#2093—531no—
296#2084—531no—
297#2083—531no—
298#1985D5311yescharacteristic, complemented
299#1949—531no—
300#1926—531no—
301#1925—531no—
302#1921—531no—
303#1920—531no—
304#1847—531no—
305#1802—531no—
306#1762—531no—
307C2×C53#242311yescharacteristic
308C2×C53#309211yescharacteristic
309C2×C53#309611yescharacteristic
310C2×C53#338411yescharacteristic
311C2×C53#338511yescharacteristic
312C2×C53#339911yescharacteristic
313C2×C53#340011yescharacteristic
314D53—82no—
315C2×C4×Q8—1061no—
316C2×C4×Q8C2×D5311yescharacteristic
317C2×C4×D4—531no—
318C2×#107—531no—
319C2×#107—531no—
320C2×#107—531no—
321C2×#106—531no—
322C2×#106—531no—
323C2×#106—531no—
324C2×C4⋊C8C2×D5311yescharacteristic
325C2×#102—531no—
326C2×#102—531no—
327C2×#102—531no—
328C2×#102—531no—
329C2×#97C2×D5311yescharacteristic
330#399—1061no—
331#393—1061no—
332#390—531no—
333#390—531no—
334#389—531no—
335#389—531no—
336#389—1061no—
337#389—1061no—
338#389—1061no—
339#385—1061no—
340#385—1061no—
341#384—531no—
342#384—531no—
343#384—531no—
344#382—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#336C2×D5311yescharacteristic
355#336C2×D5311yescharacteristic
356#336C2×D5311yescharacteristic
357#336C2×D5311yescharacteristic
358C53#2506811yescharacteristic, complemented, Sylow
359C22×Q8D21211yescharacteristic
360C22×D4—531no—
361C23×C4—1061no—
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C22×C8C22×D5311yescharacteristic
365C22×C8C22×D5311yescharacteristic
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—2124no—
369C4×D4—1062no—
370C4×D4—1062no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4D21211yescharacteristic
387C2×C4⋊C4D21211yescharacteristic
388C2×C4⋊C4D21211yescharacteristic
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42C22×D5311yescharacteristic
398C2×C42C22×D5311yescharacteristic
399C2×C42C22×D5311yescharacteristic
400#107—1062no—
401#107—2121no—
402#107—2121no—
403#106—1062no—
404#106—2121no—
405#106—2121no—
406C4⋊C8—22no—
407#102—534no—
408#102—1062no—
409#102—2121no—
410#102—2121no—
411#97—22no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419C24—1061no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×D4—532no—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—1061no—
427C2×D4—1061no—
428C2×D4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—4241no—
437C22×C4—4241no—
438C22×C4—2122no—
439C22×C4C2×D21211yescharacteristic
440C22×C4C2×D21211yescharacteristic
441C22×C4C2×D21211yescharacteristic
442C22×C4C2×D21211yescharacteristic
443C22×C4C2×D21211yescharacteristic
444C22×C4C2×D21211yescharacteristic
445C22×C4C23×D5311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8Q8×D5312yes—
457C2×C8#1171994912yes—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—532no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—4241no—
476C4⋊C4—4241no—
477C4⋊C4—4241no—
478C4⋊C4—4241no—
479C4⋊C4—4241no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C22⋊C4—4241no—
484C22⋊C4—4241no—
485C22⋊C4—2122no—
486C22⋊C4—2122no—
487C22⋊C4—2122no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—2122no—
493C42—2122no—
494C42—2122no—
495C42#1171994712yes—
496C42#1171994712yes—
497C23—1062no—
498C23—2121no—
499C23—2121no—
500C23—2121no—
501C23—2121no—
502C23—2121no—
503C23—2121no—
504C23C22×D21211yescharacteristic
505Q8—28no—
506D4—1062no—
507D4—1062no—
508D4—2121no—
509D4—2121no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—1062no—
525C2×C4—4244no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4#1313320112yes—
547C2×C4#1313320112yes—
548C2×C4#1313324512yes—
549C2×C4#1313325012yes—
550C2×C4C2×Q8×D5311yescharacteristic
551C2×C4C2×#1171994711yescharacteristic
552C2×C4C2×#1171994711yescharacteristic
553C2×C4C2×#1171994911yescharacteristic
554C2×C4#1313326311yescharacteristic
555C2×C4#1313326411yescharacteristic
556C8—22no—
557C8—22no—
558C8—41no—
559C8—41no—
560C22—4241no—
561C22—4241no—
562C22—4241no—
563C22—4241no—
564C22—2122no—
565C22—2122no—
566C22—2122no—
567C22C2×#1313320111yescharacteristic
568C22Q8×D21211yescharacteristic
569C22#1681129511yescharacteristic
570C22C2×#1313324511yescharacteristic
571C22C2×#1313325011yescharacteristic
572C22#1681142811yescharacteristic
573C22#1681143511yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—4241no—
581C4—4241no—
582C4—4241no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—2122no—
587C4#1681117811yescharacteristic
588C4#1681124211yescharacteristic
589C4#1681131511yescharacteristic
590C4#1681133111yescharacteristic
591C2—4242no—
592C2#2027823811yescharacteristic
593C2#2028464011yescharacteristic
594C2#2028467411yescharacteristic
595C2#2028761611yescharacteristic
596C2#2028762711yescharacteristic
597C2#2028775911yescharacteristic
598C2#2028777011yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1985C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1985D53
hypocenterC53#25068
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic53C2semidirect
5. #20276193C2semidirect
6. #20276188C2semidirect
7. #20275485C2semidirect
10. #20275308C2semidirect
11. #20275301C2semidirect
12. #20274883C2semidirect
15. C53×#1985C2semidirect
96. #13133139Q8semidirect
298. #1985D53semidirect
358. C53#25068semidirect

Maximal subgroups

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

subgroupindex
#202800732
Q8×C4⋊Dic532
#202762472
#202762422
#202761932
#202761882
#202754852
#202753432
#202753362
#202753082
#202753012
#202748832
#202747162
#202745412
C53×#19852
#2506853