Finite Group Databasebeta

← #27278951

Subgroups — #27278951

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
1#20284715C211yesmaximal, characteristic
2#20275634C211yesmaximal, characteristic, complemented
3C4×#13133138C211yesmaximal, characteristic, complemented
4#20273165C211yesmaximal, characteristic, complemented
5#20272616C211yesmaximal, characteristic, complemented
6#20272553C211yesmaximal, characteristic, complemented
7D4×M16×C53C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810936—21no—
10#16810903C411yescharacteristic, complemented
11#16810903C411yescharacteristic, complemented
12#16810818C411yescharacteristic, complemented
13#16810818C411yescharacteristic, complemented
14#16810757—21no—
15C4×#11719922—21no—
16C4×#11719922—21no—
17#16810676C2211yescharacteristic
18C2×#13133143—21no—
19C2×#13133143—21no—
20C2×#13133138—21no—
21C2×C4×#88322C2211yescharacteristic
22C4×C4⋊Dic53C2211yescharacteristic
23#16810385—21no—
24#16810350—22no—
25C22×M16×C53—21no—
26C2×C4×D4×C53C2211yescharacteristic
27C53×#436—21no—
28C53×#435—22no—
29C8×D4×C53—21no—
30C53×#423C2211yescharacteristic
31C53×#407—21no—
32C4×M16×C53C2211yescharacteristic
33C2×#11719945—21no—
34C2×#11719922—21no—
35C2×#11719922—21no—
36C22×#88322—21no—
37#13133217C2×C411yescharacteristic
38#13133214C2×C411yescharacteristic
39#13133205—21no—
40#13133191—21no—
41#13133191—21no—
42C4×D212—21no—
43C4×D212—21no—
44C2×C4⋊Dic53—21no—
45C2×C4×Dic53—21no—
46#13133150—21no—
47#13133150—21no—
48C4⋊#88322C2×C411yescharacteristic
49C4⋊#88322C2×C411yescharacteristic
50#13133143—42no—
51#13133143—42no—
52#13133138—24no—
53#13133138—21no—
54#13133138—21no—
55#13133137—21no—
56#13133137—21no—
57C22⋊#88322—21no—
58C4×#88322—21no—
59C4×#88322—21no—
60C4×#88322—21no—
61#13133124—21no—
62C22×D4×C53D411yescharacteristic
63C23×C4×C53—21no—
64C2×M16×C53—22no—
65C2×M16×C53—21no—
66C2×M16×C53—42no—
67C2×M16×C53—42no—
68C2×M16×C53D411yescharacteristic
69C2×M16×C53D411yescharacteristic
70C22×C8×C53—22no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53C2×C411yescharacteristic
74C4×D4×C53C2×C411yescharacteristic
75C4×D4×C53D411yescharacteristic
76C4×D4×C53D411yescharacteristic
77C2×C4⋊C4×C53D411yescharacteristic
78C2×C22⋊C4×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81C4⋊C8×C53D412yes—
82C22⋊C8×C53—22no—
83C22⋊C8×C53—22no—
84#97×C53—21no—
85C4×C8×C53—21no—
86C24×C53—21no—
87C2×D4×C53—22no—
88C2×D4×C53—21no—
89C2×D4×C53—41no—
90C2×D4×C53C22⋊C411yescharacteristic
91C2×D4×C53C22⋊C411yescharacteristic
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C2×C22⋊D53—41no—
105C22×Dic53—41no—
106#11719945—41no—
107#11719945—41no—
108#11719945—41no—
109#11719945—41no—
110C2×D212—21no—
111C2×D212—21no—
112C2×D212—21no—
113C2×C4×D53—41no—
114C2×Dic106—21no—
115M16×C53—21no—
116M16×C53—21no—
117M16×C53—21no—
118M16×C53—44no—
119M16×C53—41no—
120M16×C53—41no—
121M16×C53D812yescomplemented
122C2×C8×C53—21no—
123C2×C8×C53—21no—
124C2×C8×C53—22no—
125C2×C8×C53—44no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53C22⋊C411yescharacteristic
128C4⋊C4×C53C22⋊C411yescharacteristic
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C42×C53C22×C411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53C2×D411yescharacteristic
136C42×C53#4111yescharacteristic
137#11719922—42no—
138#11719922—42no—
139#11719921—41no—
140#11719921—41no—
141C4⋊Dic53—22no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144C4⋊Dic53—41no—
145C4×Dic53—41no—
146C4×Dic53—41no—
147C2×#88322—22no—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—41no—
151C2×#88322—41no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—42no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—81no—
158C23×C53#12111yescharacteristic
159C22×D53—41no—
160D4×C53—21no—
161D4×C53—21no—
162D4×C53—21no—
163D4×C53—42no—
164D4×C53#10412yescomplemented
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53C2×C22⋊C411yescharacteristic
181C2×C4×C53C4×D411yescharacteristic
182C2×C4×C53C4×D411yescharacteristic
183C2×C4×C53#12011yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12311yescharacteristic
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53#13511yescharacteristic
188C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
190C22⋊D53—81no—
191C22⋊D53—81no—
192C2×Dic53—42no—
193C2×Dic53—41no—
194C2×Dic53—81no—
195C2×Dic53—81no—
196D212—42no—
197D212—41no—
198D212—41no—
199D212—41no—
200D212—41no—
201C4×D53—81no—
202C4×D53—81no—
203Dic106—41no—
204Dic106—41no—
205C8×C53—22no—
206C8×C53—42no—
207C8×C53—41no—
208#88322—42no—
209#88322—41no—
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#38611yescharacteristic
223C22×C53#45911yescharacteristic
224C22×C53#46311yescharacteristic
225C2×D53—81no—
226C2×D53—81no—
227C2×D53—81no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53C2×#10411yescharacteristic
234C4×C53#42111yescharacteristic
235C4×C53C4×D811yescharacteristic
236C4×C53#44211yescharacteristic
237C4×C53#44711yescharacteristic
238C4×C53#44911yescharacteristic
239Dic53—82no—
240Dic53—81no—
241#3100—531no—
242D4×M16D5311yescharacteristic, complemented
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#187111yescharacteristic
253C2×C53#187911yescharacteristic
254C2×C53#188111yescharacteristic
255D53—161no—
256C22×M16—21no—
257#532—531no—
258C2×C4×D4C2×D5311yescharacteristic
259#480—1061no—
260#474—531no—
261#474—531no—
262#459—531no—
263#459—531no—
264#447—1061no—
265C4×D8—1061no—
266C4×D8—1061no—
267#436—21no—
268#435—22no—
269C8×D4—21no—
270#431—531no—
271#423C2×D5311yescharacteristic
272C2×#104—1061no—
273C2×#104—1061no—
274C2×#102—1061no—
275#407—21no—
276C4×M16C2×D5311yescharacteristic
277C2×C4×C8—531no—
278C4×C4⋊C4—531no—
279#340—1061no—
280D4⋊C8—1062no—
281C53#899111yescharacteristic, complemented, Sylow
282C2×#41—1061no—
283C22×D4D21211yescharacteristic
284C23×C4—21no—
285C2×D8—1061no—
286C2×D8—1061no—
287C2×M16—22no—
288C2×M16—21no—
289C2×M16—42no—
290C2×M16—42no—
291C2×M16C22⋊D5311yescharacteristic
292C2×M16C22⋊D5311yescharacteristic
293C22×C8—22no—
294C22×C8—1061no—
295C4⋊Q8—531no—
296#127—531no—
297#124—1061no—
298#121—1061no—
299#121—1061no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—1061no—
303C4×D4—1061no—
304C4×D4C4×D5311yescharacteristic
305C4×D4C4×D5311yescharacteristic
306C4×D4C22⋊D5311yescharacteristic
307C4×D4C22⋊D5311yescharacteristic
308C2×C4⋊C4—1061no—
309C2×C4⋊C4D21211yescharacteristic
310C2×C22⋊C4—21no—
311C2×C42—1061no—
312C2×C42C22×D5311yescharacteristic
313#107—1061no—
314#107—1061no—
315C4⋊C8—21no—
316C4⋊C8—531no—
317C4⋊C8—531no—
318C4⋊C8C22⋊D5312yes—
319#104—2122no—
320#104—2122no—
321#102—1064no—
322#102—1061no—
323#102—1061no—
324#102—1061no—
325#102—1061no—
326C22⋊C8—22no—
327C22⋊C8—22no—
328C22⋊C8—1061no—
329#97—21no—
330C4×C8—21no—
331C4×C8—1061no—
332C4×C8—1061no—
333C4×C8—1061no—
334#95—1061no—
335C24—21no—
336#41—2121no—
337#41—2121no—
338#41—2121no—
339#41—2121no—
340C2×Q8—1061no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—41no—
344C2×D4—1061no—
345C2×D4—1061no—
346C2×D4—1061no—
347C2×D4—2121no—
348C2×D4#1171992111yescharacteristic
349C2×D4#1171992111yescharacteristic
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—2121no—
360C22×C4—2121no—
361C22×C4C2×D21211yescharacteristic
362C22×C4#1171994511yescharacteristic
363C22×C4C2×C22⋊D5311yescharacteristic
364D8—2122no—
365D8—2122no—
366M16—21no—
367M16—21no—
368M16—21no—
369M16—44no—
370M16—41no—
371M16—41no—
372M16#1171992212yescomplemented
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—44no—
377C2×C8—1062no—
378C2×C8—1061no—
379C2×C8—1061no—
380C2×C8—2121no—
381C2×C8—2121no—
382C4⋊C4—21no—
383C4⋊C4—1062no—
384C4⋊C4—1061no—
385C4⋊C4—1061no—
386C4⋊C4—2121no—
387C4⋊C4#1171992111yescharacteristic
388C4⋊C4#1171992111yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—21no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C42—2121no—
396C42—2121no—
397C42C2×C4×D5311yescharacteristic
398C42#1171994511yescharacteristic
399C42C2×C22⋊D5311yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C23—21no—
402C23—21no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—81no—
407C23—2121no—
408C23#1313319111yescharacteristic
409Q8—2121no—
410Q8—2121no—
411D4—21no—
412D4—21no—
413D4—21no—
414D4—42no—
415D4—4241no—
416D4—4241no—
417D4—2122no—
418D4—2121no—
419D4—2121no—
420D4—2121no—
421D4—2121no—
422D4#1313314312yescomplemented
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—4241no—
439C2×C4—4241no—
440C2×C4—4241no—
441C2×C4—4241no—
442C2×C4—2122no—
443C2×C4—2121no—
444C2×C4C2×#1171992111yescharacteristic
445C2×C4C4×C22⋊D5311yescharacteristic
446C2×C4C4×C22⋊D5311yescharacteristic
447C2×C4#1313318611yescharacteristic
448C2×C4#1313319111yescharacteristic
449C2×C4#1313320011yescharacteristic
450C2×C4C2×#1171992211yescharacteristic
451C2×C4#1313323711yescharacteristic
452C2×C4#1313324211yescharacteristic
453C2×C4#1313324311yescharacteristic
454C8—22no—
455C8—42no—
456C8—41no—
457C8—2122no—
458C8—2121no—
459C8—2121no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—82no—
469C22—82no—
470C22—4241no—
471C22—4241no—
472C22—4241no—
473C22#1681050211yescharacteristic
474C22#1681081811yescharacteristic
475C22#1681083811yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—41no—
481C4—4242no—
482C4—4241no—
483C4C2×#1313314311yescharacteristic
484C4#1681063911yescharacteristic
485C4C4×#1171992211yescharacteristic
486C4#1681073311yescharacteristic
487C4#1681075711yescharacteristic
488C4#1681076811yescharacteristic
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2—8481no—
494C2#2027501711yescharacteristic
495C2#2027506311yescharacteristic
496C2#2027507711yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingD4×M16×C53C2
socleC22×C53#386
prime radical#20284715C2
cosocleC2×C42×C53C23
hypercenterD4×M16D53
hypocenterC53#8991
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
2. #20275634C2semidirect
3. C4×#13133138C2semidirect
4. #20273165C2semidirect
5. #20272616C2semidirect
6. #20272553C2semidirect
7. D4×M16×C53C2semidirect
10. #16810903C4semidirect
11. #16810903C4semidirect
12. #16810818C4semidirect
13. #16810818C4semidirect
121. M16×C53D8semidirect
164. D4×C53#104semidirect
242. D4×M16D53semidirect
281. C53#8991semidirect
372. M16#11719922semidirect
422. D4#13133143semidirect

Maximal subgroups

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

subgroupindex
#202847152
#202756342
C4×#131331382
#202731652
#202726162
#202725532
D4×M16×C532
#899153