Finite Group Databasebeta

← #27278939

Subgroups — #27278939

Order 13568.

540 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#11719917C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic, complemented
3C4×#13133138C211yesmaximal, characteristic
4#20273170C211yesmaximal, characteristic, complemented
5#20272617C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#3101C211yesmaximal, characteristic
8C22×#11719917—21no—
9#16810701—21no—
10#16810695—22no—
11D4×#88322—21no—
12#16810647C2211yescharacteristic
13C2×#13133138—21no—
14#16810587—21no—
15C4×#11719917C2211yescharacteristic
16C2×C4×#88322C2211yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18#16810385—21no—
19#16810382—21no—
20#16810382—21no—
21Q8⋊#88322—22no—
22C53×#533C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#483—21no—
25C53×#476C411yescharacteristic, complemented
26C53×#476C411yescharacteristic, complemented
27C53×#461C411yescharacteristic, complemented
28C53×#461C411yescharacteristic, complemented
29C53×#450—21no—
30C4×SD16×C53—21no—
31C4×SD16×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#13133138—24no—
46C22⋊#88322—22no—
47C22⋊#88322—22no—
48#13133128—21no—
49C4×#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52#13133124—21no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C2×SD16×C53—21no—
56C2×SD16×C53—21no—
57C4⋊Q8×C53C2×C411yescharacteristic
58C4⋊Q8×C53C2×C411yescharacteristic
59#125×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53C2×C411yescharacteristic
67C4×D4×C53C2×C411yescharacteristic
68C4×D4×C53D411yescharacteristic
69C4×D4×C53D411yescharacteristic
70#117×C53—21no—
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53—21no—
73C2×C42×C53C2311yescharacteristic
74#106×C53—21no—
75#106×C53—21no—
76C4⋊C8×C53C2×C411yescharacteristic
77C4⋊C8×C53C2×C411yescharacteristic
78#103×C53—21no—
79#103×C53—21no—
80#102×C53—21no—
81#102×C53—21no—
82C22⋊C8×C53—21no—
83C4×C8×C53—21no—
84C24×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×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
104C22×Dic53—41no—
105SD16×C53—42no—
106SD16×C53—42no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53C22⋊C411yescharacteristic
117C4⋊C4×C53C22⋊C411yescharacteristic
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C22⋊C4×C53—21no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—41no—
123C42×C53—41no—
124C42×C53C22×C411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C4⋊Dic53—22no—
129C4⋊Dic53—41no—
130C4×Dic53—41no—
131C4×Dic53—41no—
132#11719917—21no—
133#11719917—21no—
134#11719917—21no—
135#11719917—44no—
136#11719917—41no—
137#11719917—41no—
138#11719917SD1612yescomplemented
139C2×#88322—21no—
140C2×#88322—22no—
141C2×#88322—21no—
142C2×#88322—22no—
143C2×#88322—44no—
144C2×#88322—41no—
145C2×#88322—41no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—42no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—81no—
152C23×C53#12111yescharacteristic
153Q8×C53—42no—
154Q8×C53—41no—
155Q8×C53—41no—
156D4×C53—21no—
157D4×C53—21no—
158D4×C53—21no—
159D4×C53—42no—
160D4×C53#10412yes—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
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×C53C2×C22⋊C411yescharacteristic
179C2×C4×C53C4×D411yescharacteristic
180C2×C4×C53C4×D411yescharacteristic
181C2×C4×C53#12011yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12311yescharacteristic
184C2×C4×C53C2×SD1611yescharacteristic
185C2×C4×C53#13511yescharacteristic
186C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C53#13711yescharacteristic
188C2×Dic53—42no—
189C2×Dic53—81no—
190C2×Dic53—81no—
191C8×C53—41no—
192C8×C53—41no—
193#88322—22no—
194#88322—42no—
195#88322—42no—
196#88322—41no—
197#8989—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—22no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—82no—
207C22×C53—82no—
208C22×C53#38611yescharacteristic
209C22×C53#46111yescharacteristic
210C22×C53#46311yescharacteristic
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—41no—
215C4×C53—41no—
216C4×C53—81no—
217C4×C53—81no—
218C4×C53C2×#10411yescharacteristic
219C4×C53#42111yescharacteristic
220C4×C53C4×SD1611yescharacteristic
221C4×C53#44011yescharacteristic
222C4×C53#45011yescharacteristic
223C4×C53#45211yescharacteristic
224Dic53—82no—
225#3101D5311yescharacteristic
226D4×M16—531no—
227#1986—531no—
228C4×#102—531no—
229#1493—531no—
230#1303—531no—
231#1278—531no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—42no—
235C2×C53—81no—
236C2×C53#187111yescharacteristic
237C2×C53#187711yescharacteristic
238C2×C53#188111yescharacteristic
239C22×M16—1061no—
240#533C2×D5311yescharacteristic
241C2×C4×D4C2×D5311yescharacteristic
242#483—21no—
243#476Dic5311yescharacteristic, complemented
244#476Dic5311yescharacteristic, complemented
245#461Dic5311yescharacteristic, complemented
246#461Dic5311yescharacteristic, complemented
247#450—21no—
248C4×SD16—21no—
249C4×SD16—21no—
250#436—1061no—
251#435—1062no—
252C8×D4—1061no—
253#431C2×D5311yescharacteristic
254#423—531no—
255C2×#102—1061no—
256#407—1061no—
257C4×M16—531no—
258C2×C4×C8—531no—
259C4×C4⋊C4—531no—
260#340—1061no—
261#339—1061no—
262#339—1061no—
263Q8⋊C8—1062no—
264C53#898911yescharacteristic, complemented, Sylow
265C22×D4C22⋊D5311yescharacteristic
266C23×C4—21no—
267C2×SD16—21no—
268C2×SD16—21no—
269C2×M16—531no—
270C2×M16—531no—
271C2×M16—1062no—
272C2×M16—1061no—
273C2×M16—2122no—
274C2×M16—2122no—
275C22×C8—1062no—
276C22×C8—1061no—
277C4⋊Q8C2×Dic5311yescharacteristic
278C4⋊Q8C2×Dic5311yescharacteristic
279#125—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C2×Dic5311yescharacteristic
287C4×D4C2×Dic5311yescharacteristic
288C4×D4C22⋊D5311yescharacteristic
289C4×D4C22⋊D5311yescharacteristic
290#117—21no—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4C22⋊D5311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C42—1061no—
295C2×C42C22×D5311yescharacteristic
296#106—21no—
297#106—21no—
298C4⋊C8—532no—
299C4⋊C8—1061no—
300C4⋊C8C2×Dic5311yescharacteristic
301C4⋊C8C2×Dic5311yescharacteristic
302#103—21no—
303#103—21no—
304#102—21no—
305#102—21no—
306#102—1064no—
307C22⋊C8—21no—
308C22⋊C8—1062no—
309C22⋊C8—1062no—
310#97—1061no—
311C4×C8—21no—
312C4×C8—1061no—
313C4×C8—1061no—
314C4×C8—1061no—
315#95—1061no—
316C24—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4C22⋊Dic5311yescharacteristic
323C2×D4C22⋊Dic5311yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—2121no—
334C22×C4D4×D5311yescharacteristic
335C22×C4#1171994711yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337SD16—42no—
338SD16—42no—
339M16—532no—
340M16—1061no—
341M16—1061no—
342M16—1061no—
343M16—2124no—
344M16—2121no—
345M16—2121no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—1061no—
349C2×C8—1062no—
350C2×C8—1061no—
351C2×C8—1062no—
352C2×C8—2124no—
353C2×C8—2121no—
354C2×C8—2121no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—1062no—
363C4⋊C4—2121no—
364C4⋊C4C22⋊Dic5311yescharacteristic
365C4⋊C4C22⋊Dic5311yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—41no—
371C42—41no—
372C42—2121no—
373C42—2121no—
374C42D4×D5311yescharacteristic
375C42#1171994711yescharacteristic
376C42C22×Dic5311yescharacteristic
377C42C2×C22⋊D5311yescharacteristic
378C23—21no—
379C23—21no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23#1313318911yescharacteristic
385Q8—42no—
386Q8—41no—
387Q8—41no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—42no—
392D4#1313314412yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—2122no—
413C2×C4C2×C22⋊Dic5311yescharacteristic
414C2×C4D4×Dic5311yescharacteristic
415C2×C4D4×Dic5311yescharacteristic
416C2×C4#1313318511yescharacteristic
417C2×C4#1313318911yescharacteristic
418C2×C4#1313319811yescharacteristic
419C2×C4SD16×D5311yescharacteristic
420C2×C4#1313323811yescharacteristic
421C2×C4#1313324411yescharacteristic
422C2×C4#1313324611yescharacteristic
423C8—41no—
424C8—41no—
425C8—1062no—
426C8—2122no—
427C8—2122no—
428C8—2121no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—82no—
438C22—82no—
439C22#1681050311yescharacteristic
440C22#1681082611yescharacteristic
441C22#1681084011yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—81no—
448C4—81no—
449C4—4242no—
450C4C2×#1313314411yescharacteristic
451C4#1681064011yescharacteristic
452C4SD16×Dic5311yescharacteristic
453C4#1681072411yescharacteristic
454C4#1681077211yescharacteristic
455C4#1681078211yescharacteristic
456C2—21no—
457C2—21no—
458C2—42no—
459C2—81no—
460C2#2027501811yescharacteristic
461C2#2027505111yescharacteristic
462C2#2027507911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3101C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#3101D53
hypocenterC53#8989
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
2. #20275660C2semidirect
4. #20273170C2semidirect
5. #20272617C2semidirect
6. #20272551C2semidirect
25. C53×#476C4semidirect
26. C53×#476C4semidirect
27. C53×#461C4semidirect
28. C53×#461C4semidirect
138. #11719917SD16semidirect
243. #476Dic53semidirect
244. #476Dic53semidirect
245. #461Dic53semidirect
246. #461Dic53semidirect
264. C53#8989semidirect

Maximal subgroups

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

subgroupindex
D4×#117199172
#202756602
C4×#131331382
#202731702
#202726172
#202725512
C53×#31012
#898953