Finite Group Databasebeta

← #27472875

Subgroups — #27472875

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284508C211yesmaximal, characteristic
2#20283656C211yesmaximal, characteristic, complemented
3C2×#16810721C211yesmaximal, characteristic
4#20276223C212yesmaximal, complemented
5#20275975C212yesmaximal
6#20275376C212yesmaximal
7#20275373C212yesmaximal, complemented
8#20274855C211yesmaximal, characteristic, complemented
9#20274711C211yesmaximal, characteristic, complemented
10#20274704C211yesmaximal, characteristic
11C53×#3078C211yesmaximal, characteristic, complemented
12C22×#11719924C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14#16810920—22no—
15#16810910—21no—
16#16810900—21no—
17#16810754—22no—
18#16810754—22no—
19#16810721—22no—
20#16810721—22no—
21#16810666C2212yes—
22C2×#13133150C2211yescharacteristic
23#16810650C2212yes—
24#16810647C2211yescharacteristic
25#16810647C2211yescharacteristic
26#16810627C2212yes—
27C2×#13133141C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133136C2212yes—
30C2×#13133133C2212yes—
31#16810579C2212yes—
32C2×#13133128C2211yescharacteristic
33#16810392C2212yes—
34#16810391C2212yes—
35#16810387C2212yes—
36#16810387C2212yes—
37C53×#533C2211yescharacteristic
38C2×C4×Q8×C53C2211yescharacteristic
39C53×#479—22no—
40C53×#477—21no—
41C53×#475—21no—
42C53×#428C2212yes—
43C53×#423C2211yescharacteristic
44C53×#419C2212yes—
45C22×D212C2311yescharacteristic
46C22×C4×D53—21no—
47C2×#11719924—22no—
48C2×#11719924—22no—
49C2×#11719924—22no—
50C2×#11719917C2312yes—
51C2×#11719917C2312yes—
52C22×#88322C2311yescharacteristic
53C22×#88322C2311yescharacteristic
54C4×D212—21no—
55C4×D212—21no—
56C4×D212—21no—
57C4×D212—21no—
58C2×C4⋊Dic53C2311yescharacteristic
59C2×#11719921—21no—
60#13133150—22no—
61#13133150—22no—
62#13133148—22no—
63C4⋊#88322—21no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322—21no—
67C4⋊#88322—24no—
68#13133141—22no—
69#13133137—22no—
70#13133137—22no—
71#13133137—22no—
72#13133136—24no—
73#13133133—24no—
74#13133128—22no—
75#13133128D412yes—
76#13133128D412yes—
77C4×#88322—22no—
78C22×Q8×C53C2311yescharacteristic
79C2×M16×C53C2312yes—
80C4⋊Q8×C53D411yescharacteristic
81C4⋊Q8×C53D411yescharacteristic
82#125×C53D412yes—
83C22⋊Q8×C53—21no—
84C22⋊Q8×C53—21no—
85C4×Q8×C53—21no—
86C4×Q8×C53—21no—
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×Q8×C53—21no—
90C4×Q8×C53—21no—
91#117×C53C2312yes—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53C2311yescharacteristic
94C2×C42×C53—21no—
95C2×C42×C53C2311yescharacteristic
96#107×C53—22no—
97#106×C53—22no—
98C4⋊C8×C53—21no—
99C4⋊C8×C53—21no—
100#103×C53—22no—
101#103×C53—22no—
102C23×D53—21no—
103C2×Q8×C53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—21no—
108C2×Q8×C53—21no—
109C2×Q8×C53—21no—
110C2×Q8×C53#4112yes—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53C2411yescharacteristic
116C22×Dic53—21no—
117C2×D212—21no—
118C2×D212—21no—
119C2×D212—21no—
120C2×D212—21no—
121C2×D212—21no—
122C2×D212#4112yes—
123C2×C4×D53—41no—
124C2×C4×D53—41no—
125C2×C4×D53—82no—
126M16×C53—22no—
127M16×C53—22no—
128C2×C8×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53C2×D412yes—
138C4⋊C4×C53C2×D412yes—
139C22⋊C4×C53—22no—
140C42×C53—22no—
141C42×C53—41no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147#11719924—28no—
148#11719921—41no—
149#11719921—41no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152#11719917—22no—
153#11719917—22no—
154#11719917—22no—
155#11719917—22no—
156C2×#88322—21no—
157C2×#88322—21no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—22no—
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—22no—
171Q8×C53—22no—
172Q8×C53—41no—
173Q8×C53—41no—
174Q8×C53—81no—
175Q8×C53—81no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—81no—
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C2×#4111yescharacteristic
192C2×C4×C53C2×#4111yescharacteristic
193C2×C4×C532−1+411yescharacteristic
194C2×C4×C532−1+411yescharacteristic
195C2×Dic53—41no—
196C2×Dic53—81no—
197D212—22no—
198D212—22no—
199D212—41no—
200D212—41no—
201C4×D53—82no—
202C4×D53—82no—
203C8×C53—42no—
204#88322—22no—
205#88322—22no—
206#88322—42no—
207#88322—42no—
208#28221—531nomaximal, Sylow
209C22×C53—21no—
210C22×C53—21no—
211C22×C53C2×#12111yescharacteristic
212C22×C53#55211yescharacteristic
213C22×C53#55211yescharacteristic
214C2×D53—42no—
215C2×D53—82no—
216C2×D53—81no—
217C2×D53—81no—
218C2×D53—82no—
219C4×C53—21no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—42no—
224C4×C53—42no—
225C4×C53—42no—
226C4×C53—81no—
227C4×C53#52511yescharacteristic
228C4×C53#53611yescharacteristic
229C4×C53#54811yescharacteristic
230C4×C53#54811yescharacteristic
231Dic53—81no—
232#3078D5311yescharacteristic, complemented
233#3077—531no—
234#2961—531no—
235C2×#440—531no—
236#2089—532no—
237#2046—532no—
238#1931—532no—
239#1930—532no—
240#1838—531no—
241#1801—531no—
242#1800—531no—
243C2×C53—21no—
244C2×C53—21no—
245C2×C53#253611yescharacteristic
246C2×C53#306811yescharacteristic
247C2×C53#306911yescharacteristic
248D53—82no—
249C22×SD16—531no—
250#533C2×D5311yescharacteristic
251C2×C4×Q8C2×D5311yescharacteristic
252C2×C4×D4—531no—
253#479—22no—
254#478—1062no—
255#477—21no—
256#476—1061no—
257#475—21no—
258#474—1061no—
259#446—1062no—
260#446—1062no—
261#440—1062no—
262#440—1062no—
263#428—532no—
264#428C2×D5312yes—
265C2×#107—531no—
266#424—532no—
267#423—531no—
268#423—531no—
269#423C2×D5311yescharacteristic
270#419C2×D5312yes—
271#418—532no—
272C2×#103—531no—
273C2×#102—531no—
274C2×#101—532no—
275C2×#100—532no—
276#405—532no—
277C2×#97—531no—
278#343—532no—
279#343—532no—
280#341—532no—
281#341—532no—
282C53#2822111yescharacteristic, complemented, Sylow
283C22×Q8C22×D5311yescharacteristic
284C22×D4—531no—
285C23×C4—1061no—
286C2×SD16—1062no—
287C2×SD16—1062no—
288C2×SD16—1062no—
289C2×M16—532no—
290C2×M16—532no—
291C2×M16C22×D5312yes—
292C22×C8—531no—
293C22×C8—531no—
294C4⋊Q8C22⋊D5311yescharacteristic
295C4⋊Q8C22⋊D5311yescharacteristic
296#125C22⋊D5312yes—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×D4—1061no—
306C4×D4—1061no—
307C4×D4—1061no—
308C4×D4—1061no—
309#117C22×D5312yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—531no—
312C2×C4⋊C4C22×D5311yescharacteristic
313C2×C22⋊C4—1061no—
314C2×C42—21no—
315C2×C42C22×D5311yescharacteristic
316#107—22no—
317#107—1062no—
318#107—1062no—
319#106—22no—
320#106—1062no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—1064no—
324C4⋊C8—1061no—
325C4⋊C8—1061no—
326C4⋊C8—1061no—
327C4⋊C8—1061no—
328#103—22no—
329#103—22no—
330#103—1062no—
331#102—1062no—
332#102—1062no—
333#102—1062no—
334#101—1064no—
335#100—1064no—
336#97—532no—
337#97—532no—
338#97—1062no—
339C4×C8—1062no—
340C24—1061no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8#1171994912yes—
349C2×D4—532no—
350C2×D4—1061no—
351C2×D4—1061no—
352C2×D4—1061no—
353C2×D4—1061no—
354C2×D4—1061no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—1061no—
358C22×C4—4242no—
359C22×C4—2121no—
360C22×C4—2121no—
361C22×C4#1171994711yescharacteristic
362C22×C4#1171994711yescharacteristic
363C22×C4C23×D5311yescharacteristic
364SD16—1068no—
365M16—22no—
366M16—22no—
367M16—1062no—
368M16—1062no—
369M16—1062no—
370M16—1062no—
371C2×C8—22no—
372C2×C8—532no—
373C2×C8—532no—
374C2×C8—1061no—
375C2×C8—1061no—
376C2×C8—1061no—
377C2×C8—1061no—
378C2×C8—1062no—
379C2×C8—1062no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—1061no—
389C4⋊C4—1061no—
390C4⋊C4C2×C22⋊D5312yes—
391C4⋊C4C2×C22⋊D5312yes—
392C22⋊C4—22no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C42—22no—
396C42—41no—
397C42—41no—
398C42D4×D5311yescharacteristic
399C42D4×D5311yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C42C2×C22⋊D5311yescharacteristic
402C23—1062no—
403C23—4241no—
404C23—4241no—
405C23—2121no—
406C23—2121no—
407C23C2×#1171994711yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—41no—
411Q8—41no—
412Q8—81no—
413Q8—81no—
414D4—1062no—
415D4—1062no—
416D4—2121no—
417D4—2121no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—4242no—
428C2×C4—4242no—
429C2×C4—4241no—
430C2×C4—2121no—
431C2×C4#1313318911yescharacteristic
432C2×C4#1313318911yescharacteristic
433C2×C4#1313318911yescharacteristic
434C2×C4#1313318911yescharacteristic
435C2×C4C2×D4×D5311yescharacteristic
436C2×C4C22×C22⋊D5311yescharacteristic
437C2×C4#41×D5311yescharacteristic
438C2×C4C2×#1171994911yescharacteristic
439C2×C4#1313326411yescharacteristic
440C2×C4#1313326511yescharacteristic
441C8—42no—
442C8—1062no—
443C8—1062no—
444C8—2122no—
445C8—2122no—
446C22—21no—
447C22—21no—
448C22—4242no—
449C22—4242no—
450C22—4241no—
451C22—4241no—
452C22—2122no—
453C22C2×#1313318911yescharacteristic
454C22#1681130711yescharacteristic
455C22#1681131511yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—81no—
464C4—4241no—
465C4#1681113611yescharacteristic
466C4#1681119611yescharacteristic
467C4#1681128011yescharacteristic
468C4#1681128411yescharacteristic
469C2—21no—
470C2—21no—
471C2—4242no—
472C2#2027962211yescharacteristic
473C2#2028444511yescharacteristic
474C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3078C2
socleC22×C53C2×#121
prime radicalC22×D212C23
hypercenter#3078D53
hypocenterC53#28221
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
2. #20283656C2semidirect
4. #20276223C2semidirect
7. #20275373C2semidirect
8. #20274855C2semidirect
9. #20274711C2semidirect
11. C53×#3078C2semidirect
232. #3078D53semidirect
282. C53#28221semidirect

Maximal subgroups

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

subgroupindex
#202845082
#202836562
C2×#168107212
#202762232
#202759752
#202753762
#202753732
#202748552
#202747112
#202747042
C53×#30782
#2822153