Finite Group Databasebeta

← #27278865

Subgroups — #27278865

Order 13568.

660 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
1C2×#16810823C211yesmaximal, characteristic
2Q8×C4⋊Dic53C211yesmaximal, characteristic, complemented
3#20275641C211yesmaximal, characteristic
4#20275521C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133214C2211yescharacteristic
10C2×Q8×Dic53—21no—
11#16810823—22no—
12#16810823C414yescomplemented
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133139—22no—
15C2×#13133139—21no—
16C2×#13133139—21no—
17C2×#13133137—21no—
18#16810522—22no—
19#16810522—21no—
20#16810493—21no—
21#16810493C2211yescharacteristic
22C4×C4⋊Dic53—21no—
23C4×C4⋊Dic53C2211yescharacteristic
24C2×C4×Q8×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4×C8×C53C2211yescharacteristic
27C53×Q8⋊C8—22no—
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719924—24no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C22×#88322—21no—
35#13133214—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214C2×C412yes—
39Q8×Dic53—42no—
40Q8×Dic53—42no—
41C2×C4⋊Dic53—22no—
42C2×C4⋊Dic53—22no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53D411yescharacteristic
46C2×C4⋊Dic53D411yescharacteristic
47C2×C4⋊Dic53D412yes—
48C2×C4×Dic53—22no—
49C2×C4×Dic53—21no—
50C4⋊#88322—21no—
51C4⋊#88322C2×C412yes—
52#13133139—42no—
53#13133139—44no—
54#13133139—42no—
55#13133137—24no—
56#13133124—22no—
57#13133124—21no—
58C22×Q8×C53D411yescharacteristic
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53C2×C412yes—
64C4×Q8×C53D412yes—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53—21no—
73C23×D53—41no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53C22⋊C412yes—
78C2×Q8×C53D812yes—
79C2×Q8×C53SD1612yes—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—42no—
92C2×D212—42no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×C8×C53—22no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53C22⋊C412yes—
114C42×C53—22no—
115C42×C53—22no—
116C42×C53C22×C411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119C42×C53#4111yescharacteristic
120#11719924—48no—
121C4⋊Dic53—21no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—44no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C4⋊Dic53—42no—
130C4⋊Dic53—82no—
131C4⋊Dic53SD1612yescomplemented
132C4×Dic53—44no—
133C4×Dic53—41no—
134C4×Dic53—41no—
135C2×#88322—22no—
136C2×#88322—41no—
137C2×#88322—41no—
138C23×C53#12011yescharacteristic
139C22×D53—42no—
140C22×D53—81no—
141C22×D53—81no—
142C22×D53—81no—
143C22×D53—81no—
144C22×D53—81no—
145C22×D53—81no—
146Q8×C53—22no—
147Q8×C53—22no—
148Q8×C53—22no—
149Q8×C53#10214yescomplemented
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—22no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53C2×C22⋊C411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12311yescharacteristic
164C2×C4×C53C2×D811yescharacteristic
165C2×C4×C53C2×SD1611yescharacteristic
166C2×C4×C53C2×SD1611yescharacteristic
167C2×C4×C53#13511yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C53#13711yescharacteristic
171C2×C4×C53#13711yescharacteristic
172C2×Dic53—22no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—41no—
181D212—44no—
182D212—42no—
183D212—42no—
184D212—42no—
185D212—42no—
186D212—82no—
187D212—81no—
188D212—81no—
189D212—81no—
190D212—81no—
191C8×C53—42no—
192C8×C53—81no—
193#88322—42no—
194#8979—531nomaximal, Sylow
195C22×C53#38611yescharacteristic
196C22×C53#44711yescharacteristic
197C22×C53#44811yescharacteristic
198C22×C53#44811yescharacteristic
199C22×C53#44911yescharacteristic
200C22×C53#45311yescharacteristic
201C22×C53#45411yescharacteristic
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—82no—
205C2×D53—161no—
206C2×D53—161no—
207C2×D53—161no—
208C2×D53—161no—
209C4×C53—24no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—42no—
213C4×C53C2×#10211yescharacteristic
214C4×C53#41711yescharacteristic
215C4×C53C4×SD1611yescharacteristic
216C4×C53#44011yescharacteristic
217C4×C53#46011yescharacteristic
218C4×C53#46411yescharacteristic
219C4×C53#48011yescharacteristic
220C4×C53#48111yescharacteristic
221Dic53—42no—
222Dic53—81no—
223Dic53—82no—
224C2×#460—531no—
225Q8×C4⋊C4—531no—
226#1982—531no—
227#1957—531no—
228C2×Q8⋊C8D5311yescharacteristic, complemented
229#1296—531no—
230#1279—531no—
231C2×C53#162111yescharacteristic
232C2×C53#162311yescharacteristic
233C2×C53#162611yescharacteristic
234C2×C53#162711yescharacteristic
235C2×C53#187411yescharacteristic
236C2×C53#188011yescharacteristic
237C2×C53#188311yescharacteristic
238D53—162no—
239C22×SD16—1061no—
240C2×#127—531no—
241C2×C4×Q8—1061no—
242C2×C4×Q8C2×D5311yescharacteristic
243#460—534no—
244#460—1062no—
245C2×C4⋊C8—531no—
246C2×C4⋊C8C2×D5311yescharacteristic
247C2×#102—1061no—
248C2×#102—1062no—
249C2×#102—1061no—
250C2×#102—1061no—
251C2×C4×C8C2×D5311yescharacteristic
252#389—1062no—
253#389—1061no—
254#384—531no—
255#384—1061no—
256C4×C4⋊C4—531no—
257C4×C4⋊C4—1061no—
258Q8⋊C8—22no—
259C53#897911yescharacteristic, complemented, Sylow
260C22×Q8D21211yescharacteristic
261C22×D4—1061no—
262C22×D4—1061no—
263C22×D4—1061no—
264C2×SD16—1064no—
265C2×SD16—2122no—
266C2×SD16—2122no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—1061no—
270#127—532no—
271#127—1061no—
272#127—1061no—
273#127—1061no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—2122no—
277C4×Q8—2122no—
278C4×Q8C4×D5312yes—
279C4×Q8C22⋊D5312yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—531no—
282C2×C4⋊C4—531no—
283C2×C4⋊C4—532no—
284C2×C4⋊C4—1062no—
285C2×C4⋊C4—1062no—
286C2×C4⋊C4—1061no—
287C2×C4⋊C4—1061no—
288C2×C4⋊C4D21211yescharacteristic
289C2×C42—21no—
290C2×C42—1062no—
291C2×C42—1061no—
292C2×C42C22×D5311yescharacteristic
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—532no—
296C4⋊C8—1061no—
297#102—1064no—
298#102—2122no—
299#102—2122no—
300#102—2124no—
301C4×C8—21no—
302C4×C8—21no—
303#95—1062no—
304#95—1061no—
305C24—2121no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8#1171991312yes—
310C2×Q8D42412yes—
311C2×Q8#1171992112yes—
312C2×D4—1062no—
313C2×D4—1062no—
314C2×D4—1062no—
315C2×D4—2122no—
316C2×D4—2122no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4—2121no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—1061no—
331C22×C4—1061no—
332C22×C4—1062no—
333C22×C4C2×D21211yescharacteristic
334C22×C4D4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336SD16—2128no—
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—1062no—
344C2×C8—2121no—
345C2×C8—2121no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—532no—
350C4⋊C4—1061no—
351C4⋊C4—1061no—
352C4⋊C4—1061no—
353C4⋊C4—4242no—
354C4⋊C4—2122no—
355C4⋊C4—2124no—
356C4⋊C4—2121no—
357C4⋊C4—2121no—
358C4⋊C4—2121no—
359C4⋊C4—2121no—
360C4⋊C4#1171992112yes—
361C42—22no—
362C42—22no—
363C42—2124no—
364C42—2121no—
365C42—2121no—
366C42C2×C4×D5311yescharacteristic
367C42#1171994711yescharacteristic
368C42#1171994711yescharacteristic
369C42C2×C22⋊D5311yescharacteristic
370C23—4241no—
371C23—4241no—
372C23—4241no—
373C23—4241no—
374C23—4241no—
375C23—4241no—
376C23—2122no—
377C23#1313318711yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—22no—
381Q8#1313313914yescomplemented
382D4—4242no—
383D4—4241no—
384D4—4241no—
385D4—4241no—
386D4—4241no—
387D4—2124no—
388D4—2122no—
389D4—2122no—
390D4—2122no—
391D4—2122no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—1062no—
401C2×C4—2122no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4—2122no—
405C2×C4—2121no—
406C2×C4—2122no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4C2×#1171992111yescharacteristic
410C2×C4#1313317711yescharacteristic
411C2×C4#1313317711yescharacteristic
412C2×C4#1313318911yescharacteristic
413C2×C4#1313318911yescharacteristic
414C2×C4#1313320111yescharacteristic
415C2×C4C2×D42411yescharacteristic
416C2×C4SD16×D5311yescharacteristic
417C2×C4C2×#1171991311yescharacteristic
418C2×C4#1313324011yescharacteristic
419C2×C4#1313324411yescharacteristic
420C2×C4#1313324511yescharacteristic
421C2×C4#1313324911yescharacteristic
422C2×C4#1313325011yescharacteristic
423C8—42no—
424C8—81no—
425C8—2122no—
426C22—8481no—
427C22—8481no—
428C22—8481no—
429C22—8481no—
430C22—4242no—
431C22—4242no—
432C22—4242no—
433C22#1681050011yescharacteristic
434C22#1681075911yescharacteristic
435C22#1681076311yescharacteristic
436C22#1681076411yescharacteristic
437C22#1681077111yescharacteristic
438C22#1681079211yescharacteristic
439C22#1681079711yescharacteristic
440C4—24no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—4241no—
445C4—4242no—
446C4—2122no—
447C4C2×#1313313911yescharacteristic
448C4#1681062511yescharacteristic
449C4#1681070811yescharacteristic
450C4#1681072611yescharacteristic
451C4#1681082111yescharacteristic
452C4#1681084711yescharacteristic
453C4#1681093711yescharacteristic
454C4#1681094211yescharacteristic
455C2—8482no—
456C2#2027378211yescharacteristic
457C2#2027379511yescharacteristic
458C2#2027381411yescharacteristic
459C2#2027382011yescharacteristic
460C2#2027503011yescharacteristic
461C2#2027506811yescharacteristic
462C2#2027508911yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53#120
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterC2×Q8⋊C8D53
hypocenterC53#8979
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic53C2semidirect
5. #20272599C2semidirect
6. #20272555C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
12. #16810823C4semidirect
131. C4⋊Dic53SD16semidirect
149. Q8×C53#102semidirect
228. C2×Q8⋊C8D53semidirect
259. C53#8979semidirect
381. Q8#13133139semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
Q8×C4⋊Dic532
#202756412
#202755212
#202725992
#202725552
C2×C53×Q8⋊C82
#897953