Finite Group Databasebeta

← #27271212

Subgroups — #27271212

Order 13568.

632 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
1C2×#16810347C211yesmaximal, characteristic
2#20272960C211yesmaximal, characteristic
3C2×C53×#342C211yesmaximal, characteristic
4C22×C4⋊Dic53C411yescharacteristic, complemented
5C22×C22⋊Dic53—21no—
6C2×C22⋊#88322—21no—
7C2×#13133124—21no—
8C2×#13133124C411yescharacteristic, complemented
9#16810347—21no—
10#16810347C412yescomplemented
11C22×C22⋊C4×C53—21no—
12C22×C22⋊C4×C53C2211yescharacteristic
13C53×#342—21no—
14C53×#342C412yescomplemented
15C23×Dic53—21no—
16C23×Dic53—21no—
17C22×#88322—41no—
18C2×C4⋊Dic53—42no—
19C2×C4⋊Dic53—41no—
20C2×C22⋊Dic53—42no—
21C2×C22⋊Dic53—42no—
22C2×C22⋊Dic53—41no—
23C2×C22⋊Dic53—41no—
24C22⋊#88322—22no—
25C22⋊#88322—41no—
26#13133124—42no—
27#13133124—41no—
28C25×C53D411yescharacteristic
29C23×C4×C53—41no—
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53D411yescharacteristic
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—41no—
36C2×C22⋊C4×C53—41no—
37C2×C22⋊C4×C53—41no—
38C2×C22⋊C4×C53—41no—
39C2×C22⋊C4×C53—41no—
40C2×C22⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53Q811yescharacteristic
44C24×C53—21no—
45C24×C53—21no—
46C24×C53—21no—
47C24×C53—21no—
48C24×C53—41no—
49C24×C53—41no—
50C24×C53—41no—
51C24×C53—41no—
52C24×C53C22⋊C411yescharacteristic
53C24×C53C22⋊C411yescharacteristic
54C24×C53C4⋊C411yescharacteristic
55C24×C53D811yescharacteristic
56C24×C53SD1611yescharacteristic
57C24×C53SD1611yescharacteristic
58C24×C53Q1611yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—81no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53—81no—
74C22×C4×C53C4211yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C4⋊C411yescharacteristic
77C22×C4×C53C4⋊C411yescharacteristic
78C22×Dic53—22no—
79C22×Dic53—22no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C22×Dic53—41no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22⋊C4×C53—24no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—42no—
100C22⋊Dic53—44no—
101C22⋊Dic53—44no—
102C4⋊Dic53—42no—
103C4⋊Dic53—42no—
104C2×#88322—42no—
105C2×#88322—81no—
106C2×#88322—81no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—81no—
140C23×C53—81no—
141C23×C53—81no—
142C23×C53—81no—
143C23×C53—81no—
144C23×C53—81no—
145C23×C53#9511yescharacteristic
146C23×C53#9911yescharacteristic
147C23×C53#9911yescharacteristic
148C23×C53#10211yescharacteristic
149C23×C53#10211yescharacteristic
150C23×C53#10311yescharacteristic
151C23×C53#10311yescharacteristic
152C23×C53#10411yescharacteristic
153C23×C53#10411yescharacteristic
154C23×C53#10611yescharacteristic
155C23×C53#10711yescharacteristic
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—44no—
170C2×C4×C53—81no—
171C2×C4×C53—82no—
172C2×C4×C53—82no—
173C2×C4×C53—82no—
174C2×C4×C53—82no—
175C2×C4×C53—82no—
176C2×C4×C53—82no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C2×Dic53—81no—
188C2×Dic53—81no—
189C2×Dic53—81no—
190C2×Dic53—81no—
191#88322—82no—
192#7250—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—81no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53—81no—
230C22×C53—81no—
231C22×C53#32711yescharacteristic
232C22×C53#32711yescharacteristic
233C22×C53#32811yescharacteristic
234C22×C53#32811yescharacteristic
235C22×C53#33911yescharacteristic
236C22×C53#34011yescharacteristic
237C22×C53#34211yescharacteristic
238C22×C53AΣL(1,𝔽16)11yescharacteristic
239C22×C53#35211yescharacteristic
240C22×C53#35311yescharacteristic
241C22×C53#35411yescharacteristic
242C4×C53—41no—
243C4×C53—41no—
244C4×C53—41no—
245C4×C53—41no—
246C4×C53—84no—
247Dic53—82no—
248Dic53—82no—
249C2×#342D5311yescharacteristic
250C2×#323—531no—
251#1439—531no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53—21no—
255C2×C53—21no—
256C2×C53—41no—
257C2×C53—41no—
258C2×C53—41no—
259C2×C53—41no—
260C2×C53#129411yescharacteristic
261C2×C53#134311yescharacteristic
262C2×C53#134411yescharacteristic
263C2×C53#135111yescharacteristic
264C2×C53#135211yescharacteristic
265C2×C53#139011yescharacteristic
266C2×C53#139311yescharacteristic
267C22×C4⋊C4—531no—
268C22×C22⋊C4—21no—
269C22×C22⋊C4—1061no—
270C22×C22⋊C4C2×D5311yescharacteristic
271C2×C22⋊C8—1061no—
272C2×#95—531no—
273C2×#95—1061no—
274#342—21no—
275#342Dic5312yescomplemented
276#323—532no—
277#323—1061no—
278C53#725011yescharacteristic, complemented, Sylow
279C25C22⋊D5311yescharacteristic
280C23×C4—41no—
281C23×C4—1061no—
282C23×C4—1061no—
283C23×C4C4×D5311yescharacteristic
284C23×C4D21211yescharacteristic
285C22×C8—2121no—
286C2×C4⋊C4—2121no—
287C2×C4⋊C4—2122no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—2122no—
297C2×C22⋊C4—2122no—
298C2×C22⋊C4—2121no—
299C2×C22⋊C4—2121no—
300C2×C22⋊C4Dic10611yescharacteristic
301C2×C22⋊C4C4×D5311yescharacteristic
302C2×C22⋊C4C2×Dic5311yescharacteristic
303C2×C22⋊C4C22⋊D5311yescharacteristic
304C22⋊C8—1062no—
305C22⋊C8—2121no—
306#95—2121no—
307#95—2122no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24#1171991911yescharacteristic
317C24#1171992111yescharacteristic
318C24#1171992211yescharacteristic
319C24#1171992311yescharacteristic
320C24#1171992411yescharacteristic
321C24#1171992511yescharacteristic
322C24C22⋊Dic5311yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—1062no—
339C22×C4—1062no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4—2121no—
345C22×C4—2121no—
346C22×C4—2121no—
347C22×C4—2121no—
348C22×C4—2121no—
349C22×C4—2121no—
350C22×C4—2121no—
351C22×C4—2121no—
352C22×C4C4×Dic5311yescharacteristic
353C22×C4#1171991911yescharacteristic
354C22×C4C4⋊Dic5311yescharacteristic
355C22×C4#1171992111yescharacteristic
356C2×C8—4241no—
357C2×C8—4241no—
358C2×C8—2122no—
359C4⋊C4—2122no—
360C4⋊C4—2122no—
361C22⋊C4—24no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—2124no—
370C22⋊C4—2124no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23#1313312411yescharacteristic
410C23#1313313111yescharacteristic
411C23#1313313111yescharacteristic
412C23#1313313711yescharacteristic
413C23#1313313811yescharacteristic
414C23#1313314011yescharacteristic
415C23#1313314111yescharacteristic
416C23#1313314511yescharacteristic
417C23#1313314511yescharacteristic
418C23#1313314811yescharacteristic
419C23#1313315011yescharacteristic
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—4241no—
442C2×C4—4241no—
443C2×C4—4241no—
444C2×C4—4241no—
445C2×C4—4241no—
446C2×C4—4241no—
447C2×C4—4241no—
448C2×C4—4241no—
449C2×C4—2122no—
450C2×C4—2122no—
451C2×C4—2122no—
452C2×C4—2122no—
453C2×C4—2122no—
454C2×C4—2122no—
455C8—4242no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22#1681035611yescharacteristic
495C22#1681035611yescharacteristic
496C22#1681035911yescharacteristic
497C22#1681035911yescharacteristic
498C22#1681038311yescharacteristic
499C22#1681038511yescharacteristic
500C22#1681038911yescharacteristic
501C22#1681040911yescharacteristic
502C22#1681041211yescharacteristic
503C22#1681041511yescharacteristic
504C22#1681041711yescharacteristic
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—84no—
510C4—4242no—
511C4—4242no—
512C2—21no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—41no—
517C2—41no—
518C2—41no—
519C2—41no—
520C2#2027259411yescharacteristic
521C2#2027271111yescharacteristic
522C2#2027271411yescharacteristic
523C2#2027273511yescharacteristic
524C2#2027273811yescharacteristic
525C2#2027283211yescharacteristic
526C2#2027283911yescharacteristic
subgroupG/H
centerC23#13133131
commutatorC22×C4×C53C42
FrattiniC22×C22⋊C4C2×D53
FittingC2×C53×#342C2
socleC23×C53#99
prime radicalC25×C53D4
cosocleC22×C22⋊C4×C53C22
hypercenterC2×#342D53
hypocenterC53#7250
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊Dic53C4semidirect
8. C2×#13133124C4semidirect
10. #16810347C4semidirect
14. C53×#342C4semidirect
275. #342Dic53semidirect
278. C53#7250semidirect

Maximal subgroups

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

subgroupindex
C2×#168103472
#202729602
C2×C53×#3422
#725053