Finite Group Databasebeta

← #27299067

Subgroups — #27299067

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810597C211yesmaximal, characteristic
2#20277307C211yesmaximal, characteristic
3#20272839C214yesmaximal, complemented
4C2×C53×#410C211yesmaximal, characteristic
5C2×#13133176—21no—
6C2×#13133176C411yescharacteristic, complemented
7C22×C4⋊Dic53C411yescharacteristic, complemented
8#16810597—22no—
9#16810597C414yescomplemented
10C2×#13133132—21no—
11C2×#13133132C2212yes—
12#16810530C412yescomplemented
13#16810526C2212yes—
14#16810503—21no—
15#16810503—21no—
16#16810483—21no—
17#16810483—21no—
18C22×#41×C53C2211yescharacteristic
19C2×#117×C53—21no—
20C53×#410—22no—
21C53×#410C414yescomplemented
22C2×#99×C53—21no—
23C2×#99×C53C2212yes—
24C23×Dic53—21no—
25#13133176—24no—
26#13133176—42no—
27#13133176—42no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—41no—
31C2×C4⋊Dic53—41no—
32C2×C4⋊Dic53—41no—
33C2×C4⋊Dic53C2×C412yes—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C4×Dic53—41no—
39C2×C4×Dic53C2×C412yes—
40#13133132—24no—
41#13133132—22no—
42#13133132C2×C414yescomplemented
43#13133124—22no—
44#13133124—22no—
45C2×#41×C53—41no—
46C2×#41×C53—41no—
47C2×#41×C53—42no—
48C2×#41×C53C2×C412yes—
49C2×#41×C53C2×C412yes—
50C2×#41×C53D412yes—
51C2×#41×C53Q812yes—
52C22×Q8×C53C2×C411yescharacteristic
53C22×D4×C53D411yescharacteristic
54C22×D4×C53D411yescharacteristic
55C22×D4×C53C2311yescharacteristic
56C23×C4×C53C2×C411yescharacteristic
57C23×C4×C53D411yescharacteristic
58C23×C4×C53D411yescharacteristic
59#117×C53—24no—
60#117×C53—42no—
61C2×C4⋊C4×C53—41no—
62C2×C22⋊C4×C53—22no—
63C2×C42×C53—41no—
64#99×C53—24no—
65#99×C53—22no—
66#99×C53C2×C414yescomplemented
67C24×C53C22⋊C411yescharacteristic
68C24×C53C2×D411yescharacteristic
69C24×C53C2×D411yescharacteristic
70#41×C53—42no—
71#41×C53—42no—
72#41×C53—42no—
73#41×C53—42no—
74#41×C53—42no—
75#41×C53—42no—
76#41×C53—42no—
77#41×C53—42no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53C4212yes—
81C2×Q8×C53C4⋊C412yes—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53C22⋊C412yes—
89C2×D4×C53C22⋊C412yes—
90C2×D4×C53C4⋊C412yes—
91C2×D4×C53C4⋊C412yes—
92C2×D4×C53C22×C411yescharacteristic
93C2×D4×C53C22×C411yescharacteristic
94C2×D4×C53C2×D411yescharacteristic
95C2×D4×C53C2×Q811yescharacteristic
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—42no—
106C22×C4×C53C4212yes—
107C22×C4×C53C22⋊C411yescharacteristic
108C22×C4×C53C22⋊C411yescharacteristic
109C22×C4×C53C22⋊C412yes—
110C22×C4×C53C22⋊C412yes—
111C22×C4×C53C22⋊C411yescharacteristic
112C22×C4×C53C4⋊C412yes—
113C22×C4×C53C4⋊C412yes—
114C22×C4×C53C4⋊C412yes—
115C22×C4×C53C22×C411yescharacteristic
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—42no—
119C22×Dic53—42no—
120C22×Dic53—82no—
121C4⋊C4×C53—44no—
122C22⋊C4×C53—24no—
123C22⋊C4×C53—42no—
124C42×C53—44no—
125C22⋊Dic53—24no—
126C22⋊Dic53—42no—
127C22⋊Dic53—42no—
128C22⋊Dic53—82no—
129C22⋊Dic53—82no—
130C4⋊Dic53—44no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133C4⋊Dic53—42no—
134C4⋊Dic53—42no—
135C4×Dic53—44no—
136C4×Dic53—42no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53#9512yes—
150C23×C53C2×C22⋊C411yescharacteristic
151C23×C53C2×C22⋊C411yescharacteristic
152C23×C53C2×C22⋊C411yescharacteristic
153C23×C53C2×C4⋊C411yescharacteristic
154C23×C53C2×C4⋊C411yescharacteristic
155Q8×C53—42no—
156Q8×C53—42no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—42no—
160D4×C53—42no—
161D4×C53—41no—
162D4×C53—41no—
163D4×C53—41no—
164D4×C53—41no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—22no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—22no—
182C2×C4×C53—22no—
183C2×C4×C53—44no—
184C2×C4×C53—82no—
185C2×C4×C53—82no—
186C2×C4×C53#9512yes—
187C2×C4×C53#9512yes—
188C2×C4×C53#9512yes—
189C2×C4×C53#9912yes—
190C2×C4×C53#9911yescharacteristic
191C2×C4×C53#9911yescharacteristic
192C2×C4×C53C2×C4211yescharacteristic
193C2×C4×C53C2×C4⋊C411yescharacteristic
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—44no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201C2×Dic53—82no—
202C2×Dic53—82no—
203C2×Dic53—84no—
204#12583—531nomaximal, Sylow
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—22no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—42no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—42no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53C2×#9511yescharacteristic
227C22×C53C2×#9911yescharacteristic
228C22×C53C2×#9911yescharacteristic
229C4×C53—22no—
230C4×C53—22no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—84no—
234C4×C53#34212yes—
235C4×C53#34212yes—
236Dic53—82no—
237Dic53—84no—
238Dic53—82no—
239C2×#410—531no—
240C2×#410D5311yescharacteristic
241#2311—531no—
242#1393—534no—
243C2×C53—21no—
244C2×C53—21no—
245C2×C53—41no—
246C2×C53—41no—
247C2×C53—42no—
248C2×C53—41no—
249C2×C53—41no—
250C2×C53C2×#34211yescharacteristic
251C2×C53#212811yescharacteristic
252C2×C53#212911yescharacteristic
253C22×#41C2×D5311yescharacteristic
254C2×#117—21no—
255C2×#117—531no—
256C2×#117—1061no—
257C22×C4⋊C4—531no—
258#410—22no—
259#410—534no—
260#410—1062no—
261#410Dic5314yescomplemented
262C2×#99—21no—
263C2×#99—532no—
264C2×#99—1061no—
265C2×#99C2×D5312yes—
266#391—532no—
267#390—532no—
268#386—1061no—
269#386—1061no—
270#380—1061no—
271#380—1061no—
272C53#1258311yescharacteristic, complemented, Sylow
273C2×#41—41no—
274C2×#41—41no—
275C2×#41—42no—
276C2×#41Dic10612yes—
277C2×#41C4×D5312yes—
278C2×#41C2×Dic5312yes—
279C2×#41C22⋊D5312yes—
280C22×Q8C4×D5311yescharacteristic
281C22×D4D21211yescharacteristic
282C22×D4C22⋊D5311yescharacteristic
283C22×D4C22×D5311yescharacteristic
284C23×C4—1061no—
285C23×C4C4×D5311yescharacteristic
286C23×C4D21211yescharacteristic
287C23×C4C22⋊D5311yescharacteristic
288#117—24no—
289#117—42no—
290#117—1064no—
291#117—2122no—
292#117—2122no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—532no—
295C2×C4⋊C4—1061no—
296C2×C4⋊C4—1061no—
297C2×C4⋊C4—2121no—
298C2×C4⋊C4—2121no—
299C2×C4⋊C4—2121no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—1062no—
302C2×C22⋊C4—1062no—
303C2×C22⋊C4—1062no—
304C2×C22⋊C4—1061no—
305C2×C42—41no—
306C2×C42—532no—
307C2×C42—2121no—
308#99—22no—
309#99—24no—
310#99—534no—
311#99—1062no—
312#99—1064no—
313#99C2×Dic5314yescomplemented
314#95—1062no—
315#95—1062no—
316C24#1171992111yescharacteristic
317C24C2×D21211yescharacteristic
318C24C2×C22⋊D5311yescharacteristic
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323#41—42no—
324#41—42no—
325#41—42no—
326#41—42no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8C4×Dic5312yes—
330C2×Q8#1171991912yes—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4#1171991912yes—
338C2×D4C4⋊Dic5312yes—
339C2×D4#1171992112yes—
340C2×D4C22⋊Dic5312yes—
341C2×D4C2×Dic10611yescharacteristic
342C2×D4C2×C4×D5311yescharacteristic
343C2×D4C22×Dic5311yescharacteristic
344C2×D4C2×C22⋊D5311yescharacteristic
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—1062no—
356C22×C4—1062no—
357C22×C4—4242no—
358C22×C4—2122no—
359C22×C4—2122no—
360C22×C4C4×Dic5312yes—
361C22×C4#1171991912yes—
362C22×C4#1171991912yes—
363C22×C4C4⋊Dic5312yes—
364C22×C4#1171992111yescharacteristic
365C22×C4#1171992111yescharacteristic
366C22×C4#1171992112yes—
367C22×C4#1171992111yescharacteristic
368C22×C4C22⋊Dic5312yes—
369C22×C4C2×C4×D5311yescharacteristic
370C4⋊C4—44no—
371C4⋊C4—2122no—
372C4⋊C4—2124no—
373C4⋊C4—2121no—
374C4⋊C4—2121no—
375C4⋊C4—2122no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—1064no—
379C22⋊C4—4242no—
380C22⋊C4—4242no—
381C22⋊C4—2122no—
382C22⋊C4—2122no—
383C42—44no—
384C42—2122no—
385C42—2124no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23#1313312412yes—
399C23C2×#1171992111yescharacteristic
400C23C2×#1171992111yescharacteristic
401C23C2×C22⋊Dic5311yescharacteristic
402C23C2×#1171991911yescharacteristic
403C23C2×C4⋊Dic5311yescharacteristic
404Q8—42no—
405Q8—42no—
406D4—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—42no—
411D4—42no—
412D4—42no—
413D4—42no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—44no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—4244no—
436C2×C4—4242no—
437C2×C4—4242no—
438C2×C4—2124no—
439C2×C4—2122no—
440C2×C4—2122no—
441C2×C4—2122no—
442C2×C4—2122no—
443C2×C4—2122no—
444C2×C4—2122no—
445C2×C4#1313312412yes—
446C2×C4#1313312412yes—
447C2×C4#1313312412yes—
448C2×C4#1313313112yes—
449C2×C4#1313313111yescharacteristic
450C2×C4#1313313111yescharacteristic
451C2×C4C2×C4×Dic5311yescharacteristic
452C2×C4C2×#1171991911yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22—81no—
474C22C2×#1313312411yescharacteristic
475C22C2×#1313313111yescharacteristic
476C22C2×#1313313111yescharacteristic
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—84no—
482C4—4242no—
483C4—4244no—
484C4—4242no—
485C4#1681038912yes—
486C4#1681038912yes—
487C2—21no—
488C2—21no—
489C2—41no—
490C2—41no—
491C2—42no—
492C2—41no—
493C2—41no—
494C2C2×#1681038911yescharacteristic
495C2#2027638611yescharacteristic
496C2#2027638911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC2×C53×#410C2
socleC22×C53C2×#99
prime radicalC22×#41×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#410D53
hypocenterC53#12583
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
3. #20272839C2semidirect
6. C2×#13133176C4semidirect
7. C22×C4⋊Dic53C4semidirect
9. #16810597C4semidirect
12. #16810530C4semidirect
21. C53×#410C4semidirect
42. #13133132C2×C4semidirect
66. #99×C53C2×C4semidirect
261. #410Dic53semidirect
272. C53#12583semidirect
313. #99C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105972
#202773072
#202728392
C2×C53×#4102
#1258353