Finite Group Databasebeta

← #27279430

Subgroups — #27279430

Order 13568.

610 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810674C211yesmaximal, characteristic
2C4×#13133193C211yesmaximal, characteristic
3#20274692C211yesmaximal, characteristic
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5#20272555C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#13133193C411yescharacteristic
9C2×C42×D53C2211yescharacteristic
10#16810674—22no—
11#16810674—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133146—21no—
14C2×#13133129—21no—
15C2×#13133129—21no—
16C2×C4×#88322C2211yescharacteristic
17#16810530C411yescharacteristic
18#16810519C411yescharacteristic
19#16810504—21no—
20#16810494—21no—
21#16810480C411yescharacteristic
22C4×#11719919—21no—
23C4×C4⋊Dic53C2211yescharacteristic
24C4×#11719921—21no—
25#16810377—21no—
26Q8⋊#88322—22no—
27C2×C4×Q8×C53C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C2×C4×C8×C53C2211yescharacteristic
30C53×Q8⋊C8—22no—
31C22×C4×D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34C22×#88322—21no—
35C22×#88322—21no—
36#13133193—22no—
37#13133193C814yescomplemented
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C42×D53—21no—
49C42×D53—21no—
50C42×D53—21no—
51C42×D53—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54#13133146—24no—
55#13133129—24no—
56#13133129—42no—
57C4×#88322—21no—
58C4×#88322D412yes—
59#13133124—21no—
60#13133124—21no—
61#13133124—21no—
62#13133124—21no—
63#13133124—21no—
64C22×Q8×C53C2×C411yescharacteristic
65C22×C8×C53—21no—
66C22×C8×C53—21no—
67C4×Q8×C53—21no—
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C4×Q8×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53C2×C411yescharacteristic
73C2×C42×C53—21no—
74C2×C42×C53C2311yescharacteristic
75C4⋊C8×C53—21no—
76C4⋊C8×C53—21no—
77C4×C8×C53—21no—
78C4×C8×C53D412yes—
79C23×D53C22⋊C411yescharacteristic
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53C2×C812yes—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53C22⋊C411yescharacteristic
94C22×Dic53C22⋊C411yescharacteristic
95C22×Dic53C22⋊C411yescharacteristic
96C2×C4×D53—21no—
97C2×C4×D53—21no—
98C2×C4×D53—21no—
99C2×C4×D53—21no—
100C2×C4×D53—21no—
101C2×C4×D53—21no—
102C2×C4×D53—21no—
103C2×C4×D53—21no—
104C2×C4×D53—21no—
105C2×C4×D53—21no—
106C2×C4×D53—21no—
107C2×C4×D53—21no—
108C2×C4×D53—41no—
109C2×C4×D53—41no—
110C2×C4×D53C2×C812yes—
111C2×C4×D53M1612yes—
112C2×C8×C53—22no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C2×C8×C53—41no—
117C2×C8×C53—41no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C42×C53#4111yescharacteristic
128#11719921—24no—
129#11719921—42no—
130C4⋊Dic53—21no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53C2×C812yes—
134#11719919—24no—
135#11719919—42no—
136C4×Dic53—21no—
137C4×Dic53—21no—
138C4×Dic53—21no—
139C4×Dic53—21no—
140C4×Dic53—21no—
141C4×Dic53—21no—
142C4×Dic53—41no—
143C4×Dic53—41no—
144C2×#88322—22no—
145C2×#88322—41no—
146C2×#88322—41no—
147C2×#88322—41no—
148C2×#88322—41no—
149C2×#88322—41no—
150C23×C53C2×C22⋊C411yescharacteristic
151C22×D53—21no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—21no—
157C22×D53C22⋊C812yes—
158Q8×C53—22no—
159Q8×C53—22no—
160Q8×C53—41no—
161Q8×C53—41no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—22no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53C4×D411yescharacteristic
171C2×C4×C53C4×D411yescharacteristic
172C2×C4×C53#12011yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12311yescharacteristic
176C2×C4×C53C22×C811yescharacteristic
177C2×C4×C53C2×M1611yescharacteristic
178C2×C4×C53#13111yescharacteristic
179C2×C4×C53#13111yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C53#13711yescharacteristic
182C2×C4×C53#13711yescharacteristic
183C2×C4×C53#13711yescharacteristic
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—22no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53—21no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53C22⋊C812yes—
199C4×D53—22no—
200C4×D53—22no—
201C4×D53—22no—
202C4×D53—22no—
203C4×D53—42no—
204C4×D53—42no—
205C4×D53—41no—
206C4×D53—41no—
207C4×D53—41no—
208C4×D53—41no—
209C8×C53—42no—
210C8×C53—81no—
211#88322—42no—
212#88322—81no—
213#9073—531nomaximal, Sylow
214C22×C53#38611yescharacteristic
215C22×C53C2×C22⋊C811yescharacteristic
216C22×C53#40811yescharacteristic
217C22×C53#41711yescharacteristic
218C22×C53#41911yescharacteristic
219C22×C53#42111yescharacteristic
220C22×C53#42111yescharacteristic
221C2×D53—22no—
222C2×D53—22no—
223C2×D53—22no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—42no—
231C4×C53—81no—
232C4×C53C8×D411yescharacteristic
233C4×C53C8×D411yescharacteristic
234C4×C53#43511yescharacteristic
235C4×C53#43511yescharacteristic
236C4×C53#46811yescharacteristic
237C4×C53#46811yescharacteristic
238C4×C53#47011yescharacteristic
239C4×C53#47011yescharacteristic
240Dic53—22no—
241Dic53—41no—
242Dic53—41no—
243Dic53—42no—
244Dic53—81no—
245C2×#431—531no—
246C4×C22⋊Q8—531no—
247#1797—531no—
248C2×Q8⋊C8—531no—
249C2×Q8⋊C8D5311yescharacteristic, complemented
250#1279—531no—
251#1279—531no—
252C2×C53#147711yescharacteristic
253C2×C53#147911yescharacteristic
254C2×C53#149511yescharacteristic
255C2×C53#149711yescharacteristic
256C2×C53#186911yescharacteristic
257C2×C53#189411yescharacteristic
258C2×C53#189711yescharacteristic
259D53—42no—
260C2×C22⋊Q8—531no—
261C2×C4×Q8C2×D5311yescharacteristic
262C22×C42—531no—
263#431—1062no—
264#431—1062no—
265C2×C4⋊C8—531no—
266C2×C4⋊C8—1061no—
267C2×C4⋊C8C2×D5311yescharacteristic
268C2×C22⋊C8—1061no—
269C2×C22⋊C8—1061no—
270C2×C4×C8—531no—
271C2×C4×C8C2×D5311yescharacteristic
272#391—531no—
273#389—531no—
274#387—1061no—
275#385—1061no—
276#380—531no—
277C4×C4⋊C4—531no—
278C4×C4⋊C4—1061no—
279C4×C22⋊C4—1061no—
280#336—1061no—
281Q8⋊C8—22no—
282Q8⋊C8—1062no—
283C53#907311yescharacteristic, complemented, Sylow
284C22×Q8C4×D5311yescharacteristic
285C23×C4—531no—
286C23×C4—531no—
287C23×C4—531no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8—1061no—
291C22×C8—1061no—
292C22⋊Q8—534no—
293C22⋊Q8—1062no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—1061no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4C4×D5311yescharacteristic
304C2×C22⋊C4—1061no—
305C2×C22⋊C4—1061no—
306C2×C42—21no—
307C2×C42—531no—
308C2×C42—531no—
309C2×C42—531no—
310C2×C42—1061no—
311C2×C42—1061no—
312C2×C42—1061no—
313C2×C42—1061no—
314C2×C42—1061no—
315C2×C42C22×D5311yescharacteristic
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—1064no—
319C4⋊C8—1061no—
320C4⋊C8—1061no—
321C22⋊C8—1064no—
322C22⋊C8—2122no—
323C4×C8—21no—
324C4×C8—532no—
325C4×C8—1061no—
326C4×C8C22⋊D5312yes—
327#95—1061no—
328#95—1061no—
329#95—1061no—
330#95—1061no—
331#95—1061no—
332C24—531no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8C8×D5312yes—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—531no—
342C22×C4—532no—
343C22×C4—532no—
344C22×C4—531no—
345C22×C4—531no—
346C22×C4—1061no—
347C22×C4—1061no—
348C22×C4—1061no—
349C22×C4—1061no—
350C22×C4—1061no—
351C22×C4—1061no—
352C22×C4—1061no—
353C22×C4—1061no—
354C22×C4—1061no—
355C22×C4—1061no—
356C22×C4—1061no—
357C22×C4—1061no—
358C22×C4—1061no—
359C22×C4—1061no—
360C22×C4—2121no—
361C22×C4—2121no—
362C22×C4C2×C4×D5311yescharacteristic
363C22×C4D4×D5311yescharacteristic
364C22×C4D4×D5311yescharacteristic
365C2×C8—22no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—1062no—
372C2×C8—2121no—
373C2×C8—2121no—
374C2×C8—2121no—
375C2×C8—2121no—
376C2×C8—2121no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—532no—
382C4⋊C4—1064no—
383C4⋊C4—1061no—
384C4⋊C4—1061no—
385C4⋊C4—1061no—
386C4⋊C4—2122no—
387C22⋊C4—1064no—
388C22⋊C4—2122no—
389C42—41no—
390C42—41no—
391C42—1061no—
392C42—1061no—
393C42—1061no—
394C42—1061no—
395C42—1061no—
396C42—1061no—
397C42—2121no—
398C42—2121no—
399C42#1171994511yescharacteristic
400C42D4×D5311yescharacteristic
401C42#1171994711yescharacteristic
402C42C2×C22⋊D5311yescharacteristic
403C23—532no—
404C23—1061no—
405C23—1061no—
406C23—1061no—
407C23—1061no—
408C23—1061no—
409C23—1061no—
410C23C22⋊C4×D5311yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—41no—
414Q8—41no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—532no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1062no—
427C2×C4—1062no—
428C2×C4—1062no—
429C2×C4—1062no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1062no—
433C2×C4—1061no—
434C2×C4—1061no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2121no—
448C2×C4#1313317711yescharacteristic
449C2×C4C4×C22⋊D5311yescharacteristic
450C2×C4#1313318511yescharacteristic
451C2×C4#1313318811yescharacteristic
452C2×C4#1313319011yescharacteristic
453C2×C4#1313319911yescharacteristic
454C2×C4C2×C8×D5311yescharacteristic
455C2×C4M16×D5311yescharacteristic
456C2×C4#1313322311yescharacteristic
457C2×C4#1313322511yescharacteristic
458C2×C4#1313324411yescharacteristic
459C2×C4#1313324611yescharacteristic
460C2×C4#1313324911yescharacteristic
461C2×C4#1313324911yescharacteristic
462C8—42no—
463C8—81no—
464C8—4241no—
465C8—2122no—
466C22—1062no—
467C22—1062no—
468C22—1062no—
469C22—2121no—
470C22—2121no—
471C22—2121no—
472C22—2121no—
473C22#1681049911yescharacteristic
474C22C22⋊C8×D5311yescharacteristic
475C22#1681058811yescharacteristic
476C22#1681062311yescharacteristic
477C22#1681063011yescharacteristic
478C22#1681063811yescharacteristic
479C22#1681063811yescharacteristic
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—1062no—
485C4—4241no—
486C4—2121no—
487C4—2122no—
488C4—2121no—
489C4#1681068611yescharacteristic
490C4C8×C22⋊D5311yescharacteristic
491C4#1681069211yescharacteristic
492C4#1681069711yescharacteristic
493C4#1681086611yescharacteristic
494C4#1681087211yescharacteristic
495C4#1681087811yescharacteristic
496C4#1681088011yescharacteristic
497C2—2122no—
498C2#2027310111yescharacteristic
499C2#2027310811yescharacteristic
500C2#2027317911yescharacteristic
501C2#2027318611yescharacteristic
502C2#2027500411yescharacteristic
503C2#2027514511yescharacteristic
504C2#2027516011yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×Q8⋊C8D53
hypocenterC53#9073
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊#88322C2semidirect
5. #20272555C2semidirect
6. #20272554C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
37. #13133193C8semidirect
249. C2×Q8⋊C8D53semidirect
283. C53#9073semidirect

Maximal subgroups

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

subgroupindex
C2×#168106742
C4×#131331932
#202746922
C2×Q8⋊#883222
#202725552
#202725542
C2×C53×Q8⋊C82
#907353