Finite Group Databasebeta

← #27472865

Subgroups — #27472865

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284508C211yesmaximal, characteristic, complemented
2#20283656C211yesmaximal, characteristic, complemented
3C2×#16810733C211yesmaximal, characteristic
4#20276228C212yesmaximal, complemented
5#20275970C212yesmaximal
6#20275380C212yesmaximal, complemented
7#20275376C212yesmaximal, complemented
8#20274855C211yesmaximal, characteristic, complemented
9#20274707C211yesmaximal, characteristic, complemented
10#20274704C211yesmaximal, characteristic, complemented
11C53×#3077C211yesmaximal, characteristic, complemented
12C22×#11719922C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14#16810920—22no—
15#16810910—21no—
16#16810900—21no—
17#16810754—22no—
18#16810754—22no—
19#16810733—22no—
20#16810733—22no—
21#16810666C2212yes—
22C2×#13133148C2211yescharacteristic
23#16810650C2212yes—
24#16810647C2211yescharacteristic
25#16810647C2211yescharacteristic
26#16810627C2212yes—
27C2×#13133138C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133134C2212yes—
30C2×#13133133C2212yes—
31#16810579C2212yes—
32C2×#13133128C2211yescharacteristic
33#16810392C2212yes—
34#16810391C2212yes—
35#16810387C2212yes—
36#16810387C2212yes—
37C53×#533C2211yescharacteristic
38C2×C4×D4×C53C2211yescharacteristic
39C53×#478—22no—
40C53×#476—21no—
41C53×#474—21no—
42C53×#428C2212yes—
43C53×#423C2211yescharacteristic
44C53×#418C2212yes—
45C22×D212C2311yescharacteristic
46C22×C4×D53—21no—
47C2×#11719922—22no—
48C2×#11719922—22no—
49C2×#11719922—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#13133148—22no—
62#13133148—22no—
63C4⋊#88322—21no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322—21no—
67C4⋊#88322—24no—
68#13133138—22no—
69#13133137—22no—
70#13133137—22no—
71#13133137—22no—
72#13133134—24no—
73#13133133—24no—
74#13133128—22no—
75#13133128D412yes—
76#13133128D412yes—
77C4×#88322—22no—
78C22×D4×C53C2311yescharacteristic
79C23×C4×C53—21no—
80C2×M16×C53C2312yes—
81C4⋊Q8×C53D411yescharacteristic
82C4⋊Q8×C53D411yescharacteristic
83#125×C53D412yes—
84C22⋊Q8×C53—21no—
85C22⋊Q8×C53—21no—
86C4×Q8×C53—21no—
87C4×Q8×C53—21no—
88C4×D4×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92#117×C53C2312yes—
93C2×C4⋊C4×C53C2311yescharacteristic
94C2×C22⋊C4×C53—21no—
95C2×C42×C53C2311yescharacteristic
96#107×C53—22no—
97#106×C53—22no—
98C4⋊C8×C53—21no—
99C4⋊C8×C53—21no—
100#102×C53—22no—
101#102×C53—22no—
102C24×C53—21no—
103C23×D53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53#4112yes—
112C22×C4×C53—21no—
113C22×C4×C53—41no—
114C22×C4×C53—41no—
115C22×C4×C53—82no—
116C22×C4×C53#4111yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C22×C4×C53C2411yescharacteristic
119C22×Dic53—21no—
120C2×D212—21no—
121C2×D212—21no—
122C2×D212—21no—
123C2×D212—21no—
124C2×D212—21no—
125C2×D212#4112yes—
126C2×C4×D53—41no—
127C2×C4×D53—41no—
128C2×C4×D53—82no—
129M16×C53—22no—
130M16×C53—22no—
131C2×C8×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53C2×D412yes—
139C4⋊C4×C53C2×D412yes—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—41no—
142C22⋊C4×C53—41no—
143C42×C53—22no—
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148#11719922—28no—
149#11719921—41no—
150#11719921—41no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153#11719917—22no—
154#11719917—22no—
155#11719917—22no—
156#11719917—22no—
157C2×#88322—21no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—21no—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322C2×D412yes—
164C2×#88322C2×D412yes—
165C23×C53—22no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53—81no—
169C23×C53—81no—
170C23×C53C2×#4111yescharacteristic
171C22×D53—22no—
172C22×D53—41no—
173C22×D53—41no—
174C22×D53—81no—
175C22×D53—81no—
176Q8×C53—81no—
177Q8×C53—81no—
178D4×C53—22no—
179D4×C53—22no—
180D4×C53—41no—
181D4×C53—41no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—82no—
188C2×C4×C53—82no—
189C2×C4×C53—81no—
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53C22×D411yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C2×C4×C53C2×#4111yescharacteristic
197C2×C4×C53C2×#4111yescharacteristic
198C2×C4×C532−1+411yescharacteristic
199C2×C4×C532−1+411yescharacteristic
200C2×Dic53—41no—
201C2×Dic53—81no—
202D212—22no—
203D212—22no—
204D212—41no—
205D212—41no—
206C4×D53—82no—
207C4×D53—82no—
208C8×C53—42no—
209#88322—22no—
210#88322—22no—
211#88322—42no—
212#88322—42no—
213#28220—531nomaximal, Sylow
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—42no—
217C22×C53—82no—
218C22×C53—82no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53C2×#12111yescharacteristic
222C22×C53#55211yescharacteristic
223C22×C53#55211yescharacteristic
224C2×D53—42no—
225C2×D53—82no—
226C2×D53—81no—
227C2×D53—81no—
228C2×D53—82no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—42no—
234C4×C53—42no—
235C4×C53—81no—
236C4×C53#52511yescharacteristic
237C4×C53#53611yescharacteristic
238C4×C53#54811yescharacteristic
239C4×C53#54811yescharacteristic
240Dic53—81no—
241#3077—531no—
242#3077D5311yescharacteristic, complemented
243#2961—531no—
244C2×#442—531no—
245#2090—532no—
246#2045—532no—
247#1931—532no—
248#1931—532no—
249#1838—531no—
250#1800—531no—
251#1800—531no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53—82no—
255C2×C53#253611yescharacteristic
256C2×C53#306811yescharacteristic
257C2×C53#306811yescharacteristic
258D53—82no—
259C22×D8—531no—
260#533C2×D5311yescharacteristic
261C2×C4×D4—531no—
262C2×C4×D4C2×D5311yescharacteristic
263#478—22no—
264#478—1062no—
265#476—21no—
266#476—1061no—
267#474—21no—
268#474—1061no—
269#446—1062no—
270#446—1062no—
271#442—1062no—
272#442—1062no—
273#428—532no—
274#428C2×D5312yes—
275C2×#106—531no—
276#424—532no—
277#423—531no—
278#423—531no—
279#423C2×D5311yescharacteristic
280#418—532no—
281#418C2×D5312yes—
282C2×#102—531no—
283C2×#102—531no—
284C2×#100—532no—
285C2×#100—532no—
286#405—532no—
287C2×#97—531no—
288#343—532no—
289#343—532no—
290#341—532no—
291#341—532no—
292C53#2822011yescharacteristic, complemented, Sylow
293C22×D4—531no—
294C22×D4C22×D5311yescharacteristic
295C23×C4—21no—
296C23×C4—1061no—
297C2×D8—1062no—
298C2×D8—1062no—
299C2×D8—1062no—
300C2×M16—532no—
301C2×M16—532no—
302C2×M16C22×D5312yes—
303C22×C8—531no—
304C22×C8—531no—
305C4⋊Q8C22⋊D5311yescharacteristic
306C4⋊Q8C22⋊D5311yescharacteristic
307#125C22⋊D5312yes—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—1061no—
317C4×D4—1061no—
318C4×D4—1061no—
319C4×D4—1061no—
320#117C22×D5312yes—
321C2×C4⋊C4—531no—
322C2×C4⋊C4C22×D5311yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—1061no—
325C2×C42C22×D5311yescharacteristic
326#107—22no—
327#107—1062no—
328#106—22no—
329#106—1062no—
330#106—1062no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—1064no—
334C4⋊C8—1061no—
335C4⋊C8—1061no—
336C4⋊C8—1061no—
337C4⋊C8—1061no—
338#102—22no—
339#102—22no—
340#102—1062no—
341#102—1062no—
342#102—1062no—
343#102—1062no—
344#100—1064no—
345#100—1064no—
346#97—532no—
347#97—532no—
348#97—1062no—
349C4×C8—1062no—
350C24—21no—
351C24—1061no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—532no—
360C2×D4—1061no—
361C2×D4—1061no—
362C2×D4—1061no—
363C2×D4—1061no—
364C2×D4—1061no—
365C2×D4#1171994912yes—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—82no—
370C22×C4—1061no—
371C22×C4—4242no—
372C22×C4—2121no—
373C22×C4—2121no—
374C22×C4#1171994711yescharacteristic
375C22×C4#1171994711yescharacteristic
376C22×C4C23×D5311yescharacteristic
377D8—1068no—
378M16—22no—
379M16—22no—
380M16—1062no—
381M16—1062no—
382M16—1062no—
383M16—1062no—
384C2×C8—22no—
385C2×C8—532no—
386C2×C8—532no—
387C2×C8—1061no—
388C2×C8—1061no—
389C2×C8—1061no—
390C2×C8—1061no—
391C2×C8—1062no—
392C2×C8—1062no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—1061no—
400C4⋊C4—1061no—
401C4⋊C4C2×C22⋊D5312yes—
402C4⋊C4C2×C22⋊D5312yes—
403C22⋊C4—22no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—2121no—
407C22⋊C4—2121no—
408C42—22no—
409C42D4×D5311yescharacteristic
410C42D4×D5311yescharacteristic
411C42C2×C22⋊D5311yescharacteristic
412C42C2×C22⋊D5311yescharacteristic
413C23—22no—
414C23—41no—
415C23—41no—
416C23—81no—
417C23—81no—
418C23—1062no—
419C23—4241no—
420C23—4241no—
421C23—2121no—
422C23—2121no—
423C23C2×#1171994711yescharacteristic
424Q8—81no—
425Q8—81no—
426D4—22no—
427D4—22no—
428D4—41no—
429D4—41no—
430D4—1062no—
431D4—1062no—
432D4—2121no—
433D4—2121no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—4242no—
443C2×C4—4242no—
444C2×C4—4241no—
445C2×C4—2121no—
446C2×C4#1313318911yescharacteristic
447C2×C4#1313318911yescharacteristic
448C2×C4#1313318911yescharacteristic
449C2×C4#1313318911yescharacteristic
450C2×C4C2×D4×D5311yescharacteristic
451C2×C4C22×C22⋊D5311yescharacteristic
452C2×C4#41×D5311yescharacteristic
453C2×C4C2×#1171994911yescharacteristic
454C2×C4#1313326411yescharacteristic
455C2×C4#1313326511yescharacteristic
456C8—42no—
457C8—1062no—
458C8—1062no—
459C8—2122no—
460C8—2122no—
461C22—21no—
462C22—21no—
463C22—42no—
464C22—82no—
465C22—82no—
466C22—81no—
467C22—81no—
468C22—4242no—
469C22—4242no—
470C22—4241no—
471C22—4241no—
472C22—2122no—
473C22C2×#1313318911yescharacteristic
474C22#1681130711yescharacteristic
475C22#1681131511yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—42no—
481C4—42no—
482C4—81no—
483C4—4241no—
484C4#1681113611yescharacteristic
485C4#1681119611yescharacteristic
486C4#1681128011yescharacteristic
487C4#1681128411yescharacteristic
488C2—21no—
489C2—21no—
490C2—82no—
491C2—4242no—
492C2#2027962211yescharacteristic
493C2#2028444011yescharacteristic
494C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3077C2
socleC22×C53C2×#121
prime radicalC22×#11719922C22
hypercenter#3077D53
hypocenterC53#28220
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #20284508C2semidirect
2. #20283656C2semidirect
4. #20276228C2semidirect
6. #20275380C2semidirect
7. #20275376C2semidirect
8. #20274855C2semidirect
9. #20274707C2semidirect
10. #20274704C2semidirect
11. C53×#3077C2semidirect
242. #3077D53semidirect
292. C53#28220semidirect

Maximal subgroups

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

subgroupindex
#202845082
#202836562
C2×#168107332
#202762282
#202759702
#202753802
#202753762
#202748552
#202747072
#202747042
C53×#30772
#2822053