Finite Group Databasebeta

← #27278953

Subgroups — #27278953

Order 13568.

578 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
1D4×#11719917C211yesmaximal, characteristic, complemented
2#20275635C211yesmaximal, characteristic, complemented
3C4×#13133138C211yesmaximal, characteristic, complemented
4#20273166C211yesmaximal, characteristic, complemented
5#20272617C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#3100C211yesmaximal, characteristic
8C22×#11719917—21no—
9#16810701—21no—
10#16810695—22no—
11D4×#88322—21no—
12#16810647C2211yescharacteristic
13C2×#13133144—21no—
14C2×#13133144—21no—
15C2×#13133138—21no—
16#16810587—21no—
17C4×#11719917C2211yescharacteristic
18C2×C4×#88322C2211yescharacteristic
19C4×C4⋊Dic53C2211yescharacteristic
20#16810385—21no—
21D4⋊#88322—22no—
22C53×#532C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#480—21no—
25C53×#474C411yescharacteristic, complemented
26C53×#474C411yescharacteristic, complemented
27C53×#459C411yescharacteristic, complemented
28C53×#459C411yescharacteristic, complemented
29C53×#447—21no—
30C4×D8×C53—21no—
31C4×D8×C53—21no—
32C53×#431C2211yescharacteristic
33C2×#11719917—22no—
34C2×#11719917—21no—
35C2×#11719917—42no—
36C2×#11719917—42no—
37C2×#11719917D411yescharacteristic
38C2×#11719917D411yescharacteristic
39C22×#88322—22no—
40C22×#88322—21no—
41C2×C4⋊Dic53—21no—
42C2×C4×Dic53—21no—
43C4⋊#88322—21no—
44C4⋊#88322D412yes—
45#13133144—42no—
46#13133144—42no—
47#13133138—24no—
48C22⋊#88322—22no—
49C22⋊#88322—22no—
50#13133128—21no—
51C4×#88322—21no—
52C4×#88322—21no—
53C4×#88322—21no—
54#13133124—21no—
55C2×#41×C53—21no—
56C22×D4×C53D411yescharacteristic
57C23×C4×C53—21no—
58C2×D8×C53—21no—
59C2×D8×C53—21no—
60C4⋊Q8×C53C2×C411yescharacteristic
61#127×C53C2×C411yescharacteristic
62#124×C53—21no—
63#121×C53—21no—
64#121×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53C2×C411yescharacteristic
70C4×D4×C53C2×C411yescharacteristic
71C4×D4×C53D411yescharacteristic
72C4×D4×C53D411yescharacteristic
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C22⋊C4×C53—21no—
75C2×C42×C53C2311yescharacteristic
76#107×C53—21no—
77#107×C53—21no—
78C4⋊C8×C53C2×C411yescharacteristic
79C4⋊C8×C53C2×C411yescharacteristic
80#102×C53—21no—
81#102×C53—21no—
82#102×C53—21no—
83#102×C53—21no—
84C22⋊C8×C53—21no—
85C4×C8×C53—21no—
86C24×C53—21no—
87#41×C53—41no—
88#41×C53—41no—
89#41×C53—41no—
90#41×C53—41no—
91C2×Q8×C53—21no—
92C2×D4×C53—22no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53C22⋊C411yescharacteristic
100C2×D4×C53C22⋊C411yescharacteristic
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—41no—
107C22×C4×C53—41no—
108C22×C4×C53—41no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×Dic53—41no—
115D8×C53—42no—
116D8×C53—42no—
117C2×C8×C53—21no—
118C2×C8×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53C22⋊C411yescharacteristic
123C4⋊C4×C53C22⋊C411yescharacteristic
124C22⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C22⋊C4×C53—21no—
127C22⋊C4×C53—21no—
128C22⋊C4×C53—41no—
129C22⋊C4×C53—41no—
130C42×C53C22×C411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53#4111yescharacteristic
134C4⋊Dic53—22no—
135C4⋊Dic53—41no—
136C4×Dic53—41no—
137C4×Dic53—41no—
138#11719917—21no—
139#11719917—21no—
140#11719917—21no—
141#11719917—44no—
142#11719917—41no—
143#11719917—41no—
144#11719917D812yescomplemented
145C2×#88322—21no—
146C2×#88322—22no—
147C2×#88322—21no—
148C2×#88322—22no—
149C2×#88322—44no—
150C2×#88322—41no—
151C2×#88322—41no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—42no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53—81no—
159C23×C53#12111yescharacteristic
160Q8×C53—41no—
161Q8×C53—41no—
162D4×C53—21no—
163D4×C53—21no—
164D4×C53—21no—
165D4×C53—42no—
166D4×C53—42no—
167D4×C53—41no—
168D4×C53—41no—
169D4×C53—41no—
170D4×C53—41no—
171D4×C53—81no—
172D4×C53—81no—
173D4×C53#10412yescomplemented
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—81no—
191C2×C4×C53—81no—
192C2×C4×C53C2×C22⋊C411yescharacteristic
193C2×C4×C53C4×D411yescharacteristic
194C2×C4×C53C4×D411yescharacteristic
195C2×C4×C53#12011yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12311yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53#13511yescharacteristic
200C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
202C2×Dic53—42no—
203C2×Dic53—81no—
204C2×Dic53—81no—
205C8×C53—41no—
206C8×C53—41no—
207#88322—22no—
208#88322—42no—
209#88322—42no—
210#88322—41no—
211#8991—531nomaximal, Sylow
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—22no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—82no—
221C22×C53—82no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53#38611yescharacteristic
226C22×C53#45911yescharacteristic
227C22×C53#46311yescharacteristic
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—81no—
234C4×C53C2×#10411yescharacteristic
235C4×C53#42111yescharacteristic
236C4×C53C4×D811yescharacteristic
237C4×C53#44211yescharacteristic
238C4×C53#44711yescharacteristic
239C4×C53#44911yescharacteristic
240Dic53—82no—
241#3100D5311yescharacteristic
242D4×M16—531no—
243#1980—531no—
244C4×#102—531no—
245#1492—531no—
246#1303—531no—
247#1278—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—42no—
251C2×C53—81no—
252C2×C53—161no—
253C2×C53#187111yescharacteristic
254C2×C53#187911yescharacteristic
255C2×C53#188111yescharacteristic
256C22×M16—1061no—
257#532C2×D5311yescharacteristic
258C2×C4×D4C2×D5311yescharacteristic
259#480—21no—
260#474Dic5311yescharacteristic, complemented
261#474Dic5311yescharacteristic, complemented
262#459Dic5311yescharacteristic, complemented
263#459Dic5311yescharacteristic, complemented
264#447—21no—
265C4×D8—21no—
266C4×D8—21no—
267#436—1061no—
268#435—1062no—
269C8×D4—1061no—
270#431C2×D5311yescharacteristic
271#423—531no—
272C2×#104—1061no—
273C2×#104—1061no—
274C2×#102—1061no—
275#407—1061no—
276C4×M16—531no—
277C2×C4×C8—531no—
278C4×C4⋊C4—531no—
279#340—1061no—
280D4⋊C8—1062no—
281C53#899111yescharacteristic, complemented, Sylow
282C2×#41—21no—
283C22×D4C22⋊D5311yescharacteristic
284C23×C4—21no—
285C2×D8—21no—
286C2×D8—21no—
287C2×M16—531no—
288C2×M16—531no—
289C2×M16—1062no—
290C2×M16—1061no—
291C2×M16—2122no—
292C2×M16—2122no—
293C22×C8—1062no—
294C22×C8—1061no—
295C4⋊Q8C2×Dic5311yescharacteristic
296#127C2×Dic5311yescharacteristic
297#124—21no—
298#121—21no—
299#121—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4C2×Dic5311yescharacteristic
305C4×D4C2×Dic5311yescharacteristic
306C4×D4C22⋊D5311yescharacteristic
307C4×D4C22⋊D5311yescharacteristic
308C2×C4⋊C4—1061no—
309C2×C4⋊C4C22⋊D5311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C42—1061no—
312C2×C42C22×D5311yescharacteristic
313#107—21no—
314#107—21no—
315C4⋊C8—532no—
316C4⋊C8—1061no—
317C4⋊C8C2×Dic5311yescharacteristic
318C4⋊C8C2×Dic5311yescharacteristic
319#104—2122no—
320#104—2122no—
321#102—21no—
322#102—21no—
323#102—21no—
324#102—21no—
325#102—1064no—
326C22⋊C8—21no—
327C22⋊C8—1062no—
328C22⋊C8—1062no—
329#97—1061no—
330C4×C8—21no—
331C4×C8—1061no—
332C4×C8—1061no—
333C4×C8—1061no—
334#95—1061no—
335C24—21no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340C2×Q8—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4C22⋊Dic5311yescharacteristic
349C2×D4C22⋊Dic5311yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—2121no—
361C22×C4D4×D5311yescharacteristic
362C22×C4#1171994711yescharacteristic
363C22×C4C2×C22⋊D5311yescharacteristic
364D8—42no—
365D8—42no—
366M16—532no—
367M16—1061no—
368M16—1061no—
369M16—1061no—
370M16—2124no—
371M16—2121no—
372M16—2121no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—1061no—
376C2×C8—1062no—
377C2×C8—1061no—
378C2×C8—1062no—
379C2×C8—2124no—
380C2×C8—2121no—
381C2×C8—2121no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—1062no—
386C4⋊C4—2121no—
387C4⋊C4C22⋊Dic5311yescharacteristic
388C4⋊C4C22⋊Dic5311yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—2121no—
396C42—2121no—
397C42D4×D5311yescharacteristic
398C42#1171994711yescharacteristic
399C42C22×Dic5311yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C23—21no—
402C23—21no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—81no—
408C23#1313318911yescharacteristic
409Q8—41no—
410Q8—41no—
411D4—21no—
412D4—21no—
413D4—21no—
414D4—42no—
415D4—42no—
416D4—41no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—81no—
421D4—81no—
422D4#1313314412yescomplemented
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—4241no—
442C2×C4—4241no—
443C2×C4—2122no—
444C2×C4C2×C22⋊Dic5311yescharacteristic
445C2×C4D4×Dic5311yescharacteristic
446C2×C4D4×Dic5311yescharacteristic
447C2×C4#1313318511yescharacteristic
448C2×C4#1313318911yescharacteristic
449C2×C4#1313319811yescharacteristic
450C2×C4D8×D5311yescharacteristic
451C2×C4#1313323611yescharacteristic
452C2×C4#1313324111yescharacteristic
453C2×C4#1313324111yescharacteristic
454C8—41no—
455C8—41no—
456C8—1062no—
457C8—2122no—
458C8—2122no—
459C8—2121no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—82no—
469C22—82no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22#1681050311yescharacteristic
474C22#1681081611yescharacteristic
475C22#1681083611yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—41no—
481C4—81no—
482C4—4242no—
483C4C2×#1313314411yescharacteristic
484C4#1681064011yescharacteristic
485C4D8×Dic5311yescharacteristic
486C4#1681073511yescharacteristic
487C4#1681075511yescharacteristic
488C4#1681076511yescharacteristic
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2—161no—
494C2#2027501811yescharacteristic
495C2#2027506511yescharacteristic
496C2#2027507911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3100C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3100D53
hypocenterC53#8991
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. D4×#11719917C2semidirect
2. #20275635C2semidirect
3. C4×#13133138C2semidirect
4. #20273166C2semidirect
5. #20272617C2semidirect
6. #20272551C2semidirect
25. C53×#474C4semidirect
26. C53×#474C4semidirect
27. C53×#459C4semidirect
28. C53×#459C4semidirect
144. #11719917D8semidirect
173. D4×C53#104semidirect
260. #474Dic53semidirect
261. #474Dic53semidirect
262. #459Dic53semidirect
263. #459Dic53semidirect
281. C53#8991semidirect
422. D4#13133144semidirect

Maximal subgroups

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

subgroupindex
D4×#117199172
#202756352
C4×#131331382
#202731662
#202726172
#202725512
C53×#31002
#899153