Finite Group Databasebeta

← #27436218

Subgroups — #27436218

Order 13568.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284508C211yesmaximal, characteristic
2C2×#16810910C211yesmaximal, characteristic
3#20283664C211yesmaximal, characteristic, complemented
4#20276228C212yesmaximal, complemented
5#20276072C212yesmaximal
6#20275705C211yesmaximal, characteristic, complemented
7#20275492C212yesmaximal, complemented
8#20275490C211yesmaximal, characteristic, complemented
9#20275376C212yesmaximal
10C4×#13133133C211yesmaximal, characteristic
11C53×#3165C211yesmaximal, characteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810920—22no—
14#16810910—21no—
15#16810910—22no—
16#16810910—21no—
17#16810910—21no—
18#16810900—21no—
19#16810754—22no—
20#16810754—21no—
21Q8×#88322—21no—
22C2×#13133152C2212yes—
23#16810666C2212yes—
24C2×#13133148C2212yes—
25#16810650C2212yes—
26#16810647C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28#16810627C2212yes—
29C2×#13133137C2212yes—
30C2×#13133133C2211yescharacteristic
31C2×#13133133C2211yescharacteristic
32#16810579C2212yes—
33C4×#11719917C2211yescharacteristic
34#16810392C2212yes—
35#16810392C2212yes—
36#16810391C2211yescharacteristic
37#16810391C2211yescharacteristic
38#16810387C2212yes—
39C53×#533C2211yescharacteristic
40C2×C4⋊Q8×C53C2211yescharacteristic
41C53×#501—22no—
42C53×#500—21no—
43C53×#498—21no—
44C53×#428C2212yes—
45C53×#428C2212yes—
46C4×M16×C53C2211yescharacteristic
47C22×D212C2311yescharacteristic
48C22×C4×D53—21no—
49C2×#11719917C2311yescharacteristic
50C2×#11719917C2312yes—
51C2×#11719917C2311yescharacteristic
52C22×#88322C2312yes—
53C4×D212—21no—
54C4×D212—21no—
55C4×D212—21no—
56C4×D212—21no—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×#11719921—21no—
59#13133152—24no—
60#13133150—22no—
61#13133148—22no—
62#13133148—22no—
63#13133148—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322—22no—
67C4⋊#88322—22no—
68C4⋊#88322D412yes—
69C4⋊#88322D411yescharacteristic
70C4⋊#88322D411yescharacteristic
71#13133137—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75#13133133—22no—
76#13133133Q814yescomplemented
77#13133128—21no—
78#13133128—22no—
79C4×#88322—21no—
80C4×#88322—22no—
81C22×Q8×C53—21no—
82C2×M16×C53C2311yescharacteristic
83C2×M16×C53C2311yescharacteristic
84C4⋊Q8×C53—21no—
85C4⋊Q8×C53—21no—
86C4⋊Q8×C53—21no—
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53D411yescharacteristic
89C4⋊Q8×C53D411yescharacteristic
90#125×C53D412yes—
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95#117×C53C2312yes—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53C2312yes—
98C2×C42×C53C2311yescharacteristic
99#107×C53—22no—
100#107×C53—22no—
101#106×C53—22no—
102#106×C53—22no—
103#97×C53—21no—
104C4×C8×C53—21no—
105C23×D53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—21no—
108C2×Q8×C53—41no—
109C2×Q8×C53—41no—
110C2×Q8×C53—82no—
111C22×C4×C53—22no—
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53C2411yescharacteristic
115C22×Dic53—21no—
116C2×D212—21no—
117C2×D212—21no—
118C2×D212—21no—
119C2×D212—21no—
120C2×D212—21no—
121C2×D212C2×Q812yes—
122C2×C4×D53—41no—
123C2×C4×D53—41no—
124C2×C4×D53—82no—
125M16×C53—21no—
126M16×C53—21no—
127M16×C53C2×Q812yes—
128M16×C53#4112yes—
129C2×C8×C53—21no—
130C2×C8×C53—21no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53C2×D412yes—
140C4⋊C4×C53C2×D412yes—
141C22⋊C4×C53—22no—
142C42×C53—22no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147#11719921—41no—
148#11719921—41no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151#11719917—21no—
152#11719917—21no—
153#11719917—22no—
154#11719917—21no—
155#11719917—22no—
156#11719917C2×Q812yes—
157C2×#88322—21no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—21no—
161C2×#88322—22no—
162C2×#88322C2×D412yes—
163C2×#88322C2×D412yes—
164C23×C53C2×#4111yescharacteristic
165C22×D53—22no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—81no—
169C22×D53—81no—
170Q8×C53—42no—
171Q8×C53—81no—
172Q8×C53—82no—
173Q8×C53—82no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—22no—
177C2×C4×C53—41no—
178C2×C4×C53—42no—
179C2×C4×C53—82no—
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53C22×D411yescharacteristic
185C2×C4×C53C22×D411yescharacteristic
186C2×C4×C53C22×Q811yescharacteristic
187C2×C4×C53C2×#4111yescharacteristic
188C2×C4×C532−1+411yescharacteristic
189C2×C4×C532−1+411yescharacteristic
190C2×Dic53—41no—
191C2×Dic53—81no—
192D212—22no—
193D212—22no—
194D212—41no—
195D212—41no—
196C4×D53—82no—
197C4×D53—82no—
198C8×C53—22no—
199C8×C53—22no—
200#88322—22no—
201#88322—22no—
202#88322—22no—
203#88322—41no—
204#88322—42no—
205#25490—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53C2×#12111yescharacteristic
209C22×C53#53311yescharacteristic
210C22×C53#54111yescharacteristic
211C2×D53—42no—
212C2×D53—82no—
213C2×D53—81no—
214C2×D53—81no—
215C2×D53—82no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—42no—
220C4×C53—42no—
221C4×C53—82no—
222C4×C53#47112yes—
223C4×C53#54711yescharacteristic
224C4×C53#54811yescharacteristic
225C4×C53D4×Q811yescharacteristic
226C4×C53D4×Q811yescharacteristic
227Dic53—81no—
228#3165D5311yescharacteristic, complemented
229#3077—531no—
230C2×#476—531no—
231#2962—531no—
232#2090—532no—
233#2063—532no—
234#1997—531no—
235#1951—532no—
236#1950—531no—
237#1931—532no—
238C4×#100—531no—
239C2×C53—21no—
240C2×C53—21no—
241C2×C53#242911yescharacteristic
242C2×C53C2×#47111yescharacteristic
243C2×C53#306811yescharacteristic
244D53—82no—
245#533C2×D5311yescharacteristic
246C2×C4⋊Q8C2×D5311yescharacteristic
247C2×C4×D4—531no—
248#501—22no—
249#500—21no—
250#498—21no—
251#478—1062no—
252#476—1061no—
253#476—1062no—
254#476—1061no—
255#476—1061no—
256#474—1061no—
257#446—1062no—
258#446—1061no—
259C8×Q8—1061no—
260C2×#108—532no—
261#428—532no—
262#428C2×D5312yes—
263#428C2×D5312yes—
264C2×#106—532no—
265#424—532no—
266#423—531no—
267C2×C4⋊C8—531no—
268#418—532no—
269C2×#102—532no—
270C2×#100—531no—
271C2×#100—531no—
272#405—532no—
273C4×M16—531no—
274C4×M16C2×D5311yescharacteristic
275#343—531no—
276#343—531no—
277#343—532no—
278#343—532no—
279#341—532no—
280C53#2549011yescharacteristic, complemented, Sylow
281C22×Q8—21no—
282C22×D4—531no—
283C23×C4—1061no—
284C2×M16—531no—
285C2×M16—532no—
286C2×M16—531no—
287C2×M16C22×D5311yescharacteristic
288C2×M16C22×D5311yescharacteristic
289C22×C8—532no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C22⋊D5311yescharacteristic
295C4⋊Q8C22⋊D5311yescharacteristic
296#125C22⋊D5312yes—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×D4—1061no—
302C4×D4—1061no—
303C4×D4—1061no—
304C4×D4—1061no—
305#117C22×D5312yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—531no—
308C2×C4⋊C4C22×D5312yes—
309C2×C22⋊C4—1061no—
310C2×C42C22×D5311yescharacteristic
311#108—1064no—
312#107—22no—
313#107—22no—
314#107—1062no—
315#106—22no—
316#106—22no—
317#106—1062no—
318#106—1062no—
319#106—1062no—
320C4⋊C8—531no—
321C4⋊C8—531no—
322C4⋊C8—532no—
323C4⋊C8—1062no—
324C4⋊C8—1062no—
325C4⋊C8—1061no—
326C4⋊C8—1061no—
327#102—1062no—
328#102—1062no—
329#102—1062no—
330#102—1062no—
331#100—534no—
332#100—1062no—
333#97—21no—
334#97—1062no—
335#97—1061no—
336C4×C8—21no—
337C4×C8—1062no—
338C4×C8—1061no—
339C24—1061no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×D4—532no—
346C2×D4—1061no—
347C2×D4—1061no—
348C2×D4—1061no—
349C2×D4—1061no—
350C2×D4—1061no—
351C22×C4—22no—
352C22×C4—1061no—
353C22×C4—4242no—
354C22×C4—2121no—
355C22×C4—2121no—
356C22×C4#1171994711yescharacteristic
357C22×C4#1171994711yescharacteristic
358C22×C4C23×D5311yescharacteristic
359M16—21no—
360M16—21no—
361M16—532no—
362M16—1061no—
363M16—1061no—
364M16—1062no—
365M16—1061no—
366M16—1062no—
367M16Q8×D5312yes—
368M16#1171994912yes—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—532no—
372C2×C8—532no—
373C2×C8—1061no—
374C2×C8—1062no—
375C2×C8—1062no—
376C2×C8—1061no—
377C2×C8—1062no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—1061no—
387C4⋊C4—1061no—
388C4⋊C4C2×C22⋊D5312yes—
389C4⋊C4C2×C22⋊D5312yes—
390C22⋊C4—22no—
391C22⋊C4—2121no—
392C22⋊C4—2121no—
393C42—22no—
394C42D4×D5311yescharacteristic
395C42D4×D5311yescharacteristic
396C42C2×C22⋊D5311yescharacteristic
397C42C2×C22⋊D5311yescharacteristic
398C23—1062no—
399C23—4241no—
400C23—4241no—
401C23—2121no—
402C23—2121no—
403C23C2×#1171994711yescharacteristic
404Q8—42no—
405Q8—81no—
406Q8—82no—
407Q8—82no—
408D4—1062no—
409D4—1062no—
410D4—2121no—
411D4—2121no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—4242no—
419C2×C4—4242no—
420C2×C4—4241no—
421C2×C4—2121no—
422C2×C4#1313318911yescharacteristic
423C2×C4#1313318911yescharacteristic
424C2×C4#1313318911yescharacteristic
425C2×C4#1313318911yescharacteristic
426C2×C4C2×D4×D5311yescharacteristic
427C2×C4C22×C22⋊D5311yescharacteristic
428C2×C4C2×Q8×D5311yescharacteristic
429C2×C4C2×#1171994911yescharacteristic
430C2×C4#1313326511yescharacteristic
431C2×C4#1313326511yescharacteristic
432C8—22no—
433C8—22no—
434C8—1062no—
435C8—1062no—
436C8—1062no—
437C8—2121no—
438C8—2122no—
439C22—21no—
440C22—21no—
441C22—4242no—
442C22—4242no—
443C22—4241no—
444C22—4241no—
445C22—2122no—
446C22C2×#1313318911yescharacteristic
447C22#1681117811yescharacteristic
448C22#1681123411yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—4241no—
456C4#1681088812yes—
457C4#1681127611yescharacteristic
458C4#1681128411yescharacteristic
459C4Q8×C22⋊D5311yescharacteristic
460C4#1681128811yescharacteristic
461C2—21no—
462C2—21no—
463C2—4242no—
464C2#2027831711yescharacteristic
465C2C2×#1681088811yescharacteristic
466C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3165C2
socleC22×C53C2×#121
prime radicalC22×D212C23
hypercenter#3165D53
hypocenterC53#25490
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. #20283664C2semidirect
4. #20276228C2semidirect
6. #20275705C2semidirect
7. #20275492C2semidirect
8. #20275490C2semidirect
11. C53×#3165C2semidirect
76. #13133133Q8semidirect
228. #3165D53semidirect
280. C53#25490semidirect

Maximal subgroups

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

subgroupindex
#202845082
C2×#168109102
#202836642
#202762282
#202760722
#202757052
#202754922
#202754902
#202753762
C4×#131331332
C53×#31652
#2549053