Finite Group Databasebeta

← #27280272

Subgroups — #27280272

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810819C211yesmaximal, characteristic
2C2×C8×D212C211yesmaximal, characteristic, complemented
3#20275521C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1277C211yesmaximal, characteristic, complemented
8C22×D424—21no—
9C22×C8×D53—21no—
10C2×#13133214C2211yescharacteristic
11C2×C4×D212C2211yescharacteristic
12#16810819—22no—
13#16810819C414yescomplemented
14C8×D212—22no—
15C8×D212—22no—
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133139—21no—
18C2×#13133139—21no—
19C2×#13133139—21no—
20C2×#13133139—21no—
21C2×#13133129—21no—
22#16810493C2211yescharacteristic
23#16810386—21no—
24#16810366—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212D411yescharacteristic
32C22×C4×D53—21no—
33C2×D424—24no—
34C2×D424—42no—
35C2×D424—42no—
36C22×#88322—21no—
37C2×C8×D53—24no—
38C2×C8×D53—42no—
39C2×C8×D53—42no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214C2×C412yes—
44C4×D212—21no—
45C4×D212—21no—
46C4×D212C2×C412yes—
47C4×D212D412yes—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53D411yescharacteristic
52C2×#11719921—21no—
53C4⋊#88322—21no—
54C4⋊#88322D412yes—
55#13133139—24no—
56#13133139—42no—
57#13133139—42no—
58#13133139—42no—
59#13133129—24no—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C22×C8×C53D411yescharacteristic
63C22×C8×C53D411yescharacteristic
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4⋊C8×C53C2×C412yes—
69C4×C8×C53—21no—
70C4×C8×C53—21no—
71C23×D53—21no—
72C23×D53—41no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4111yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—41no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—22no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—81no—
96C2×D212—81no—
97C2×D212C22⋊C412yes—
98C2×D212D812yes—
99C2×D212SD1612yes—
100C2×C4×D53—22no—
101C2×C4×D53—22no—
102C2×C4×D53—41no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C4×D53—41no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—22no—
109C2×C8×C53—21no—
110C2×C8×C53—21no—
111C2×C8×C53—21no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C2×C8×C53—41no—
117C2×C8×C53D812yescomplemented
118C42×C53C22×C411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122#11719921—22no—
123#11719921—22no—
124C4⋊Dic53—21no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C4⋊Dic53—81no—
130C4⋊Dic53C22⋊C412yes—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—41no—
134D424—48no—
135C8×D53—48no—
136C23×C53#12111yescharacteristic
137C22×D53—22no—
138C22×D53—22no—
139C22×D53—22no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—42no—
145C22×D53—81no—
146C22×D53—81no—
147C22×D53—81no—
148C22×D53—81no—
149C22×D53—81no—
150C22×D53—81no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C2×C4×C53C2×D811yescharacteristic
160C2×C4×C53C2×D811yescharacteristic
161C2×C4×C53C2×SD1611yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53#13511yescharacteristic
164C2×C4×C53#13511yescharacteristic
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
167C2×Dic53—22no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—81no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173D212—22no—
174D212—22no—
175D212—22no—
176D212—42no—
177D212—42no—
178D212—42no—
179D212—42no—
180D212—82no—
181D212—82no—
182D212—81no—
183D212—81no—
184D212—81no—
185D212—81no—
186D212#10214yescomplemented
187C4×D53—42no—
188C4×D53—42no—
189C4×D53—42no—
190C4×D53—42no—
191C8×C53—22no—
192C8×C53—42no—
193C8×C53—41no—
194C8×C53—81no—
195#88322—42no—
196#9221—531nomaximal, Sylow
197C22×C53#38611yescharacteristic
198C22×C53#45911yescharacteristic
199C22×C53#46311yescharacteristic
200C22×C53#46511yescharacteristic
201C22×C53#46611yescharacteristic
202C22×C53#47111yescharacteristic
203C22×C53#47211yescharacteristic
204C2×D53—24no—
205C2×D53—42no—
206C2×D53—42no—
207C2×D53—42no—
208C2×D53—42no—
209C2×D53—42no—
210C2×D53—42no—
211C2×D53—82no—
212C2×D53—82no—
213C2×D53—82no—
214C2×D53—161no—
215C2×D53—161no—
216C2×D53—161no—
217C2×D53—161no—
218C4×C53—21no—
219C4×C53—21no—
220C4×C53C2×#10211yescharacteristic
221C4×C53#41611yescharacteristic
222C4×C53C4×D811yescharacteristic
223C4×C53#44211yescharacteristic
224C4×C53#44711yescharacteristic
225C4×C53#44911yescharacteristic
226C4×C53#46511yescharacteristic
227C4×C53#46611yescharacteristic
228Dic53—42no—
229Dic53—161no—
230C2×#459—531no—
231C2×C8×D4—531no—
232#1957—531no—
233#1802—531no—
234C2×#331—531no—
235#1296—531no—
236#1277D5311yescharacteristic, complemented
237C2×C53#166511yescharacteristic
238C2×C53#166711yescharacteristic
239C2×C53#167711yescharacteristic
240C2×C53#168111yescharacteristic
241C2×C53#187911yescharacteristic
242C2×C53#189311yescharacteristic
243C2×C53#190711yescharacteristic
244D53—44no—
245D53—162no—
246C22×D8—1061no—
247C23×C8—1061no—
248C2×#127—531no—
249C2×C4×D4—531no—
250#459—534no—
251#459—1062no—
252C8×D4—1062no—
253C8×D4—1062no—
254C2×C4⋊C8—531no—
255C2×C4⋊C8C2×D5311yescharacteristic
256C2×C4⋊C8C2×D5311yescharacteristic
257C2×#102—1061no—
258C2×#102—1061no—
259C2×#102—1061no—
260C2×#102—1061no—
261C2×C22⋊C8—1061no—
262C2×C4×C8C2×D5311yescharacteristic
263#384—531no—
264#340—1061no—
265#331—1062no—
266C53#922111yescharacteristic, complemented, Sylow
267C22×D4—531no—
268C22×D4—1061no—
269C22×D4—1061no—
270C22×D4—1061no—
271C23×C4—1061no—
272C2×D8—1064no—
273C2×D8—2122no—
274C2×D8—2122no—
275C22×C8—21no—
276C22×C8—21no—
277C22×C8—1064no—
278C22×C8—1061no—
279C22×C8—2122no—
280C22×C8—2122no—
281C22×C8D21211yescharacteristic
282C22×C8D21211yescharacteristic
283#127—532no—
284#127—1061no—
285#127—1061no—
286#127—1061no—
287C4×D4—532no—
288C4×D4—532no—
289C4×D4—1061no—
290C4×D4—1061no—
291C2×C4⋊C4—531no—
292C2×C4⋊C4—1061no—
293C2×C4⋊C4—1061no—
294C2×C4⋊C4—1061no—
295C2×C22⋊C4—1061no—
296C2×C42C22×D5311yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—532no—
301C4⋊C8—1061no—
302C4⋊C8C4×D5312yes—
303#102—1064no—
304#102—2122no—
305#102—2122no—
306#102—2122no—
307C22⋊C8—1064no—
308C4×C8—21no—
309C4×C8—21no—
310C24—1061no—
311C24—2121no—
312C2×D4—532no—
313C2×D4—532no—
314C2×D4—532no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—1062no—
318C2×D4—1061no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—4241no—
322C2×D4—4241no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4—2121no—
328C2×D4—2121no—
329C2×D4—2121no—
330C2×D4—2121no—
331C2×D4—2121no—
332C2×D4—2121no—
333C2×D4—2121no—
334C22×C4—1061no—
335C22×C4—1062no—
336C22×C4—1062no—
337C22×C4—2121no—
338C22×C4—2121no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4C2×D21211yescharacteristic
343C22×C4D4×D5311yescharacteristic
344C22×C4#1171994911yescharacteristic
345D8—2128no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—1062no—
358C2×C8—2128no—
359C2×C8—2121no—
360C2×C8—2121no—
361C2×C8D42412yescomplemented
362C4⋊C4—532no—
363C4⋊C4—1061no—
364C4⋊C4—4241no—
365C4⋊C4—2121no—
366C4⋊C4—2121no—
367C4⋊C4—2121no—
368C4⋊C4—2121no—
369C22⋊C4—1062no—
370C22⋊C4—1062no—
371C42C2×C4×D5311yescharacteristic
372C42#1171994511yescharacteristic
373C42D4×D5311yescharacteristic
374C42D4×D5311yescharacteristic
375C23—1062no—
376C23—1062no—
377C23—1062no—
378C23—4241no—
379C23—4241no—
380C23—4241no—
381C23—4241no—
382C23—4241no—
383C23—4241no—
384C23—2122no—
385C23—2121no—
386C23—2121no—
387C23—2121no—
388C23—2121no—
389C23#1313319211yescharacteristic
390D4—534no—
391D4—1062no—
392D4—1062no—
393D4—1062no—
394D4—4242no—
395D4—4242no—
396D4—4241no—
397D4—4241no—
398D4—4241no—
399D4—4241no—
400D4—2122no—
401D4—2122no—
402D4—2122no—
403D4—2122no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—1062no—
407C2×C4—4241no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2122no—
413C2×C4—2122no—
414C2×C4—2122no—
415C2×C4—2122no—
416C2×C4C22⋊C4×D5311yescharacteristic
417C2×C4#1313317911yescharacteristic
418C2×C4C4×D21211yescharacteristic
419C2×C4#1313318711yescharacteristic
420C2×C4#1313319011yescharacteristic
421C2×C4#1313320211yescharacteristic
422C2×C4D8×D5311yescharacteristic
423C2×C4C2×D42411yescharacteristic
424C2×C4SD16×D5311yescharacteristic
425C2×C4#1313323811yescharacteristic
426C2×C4#1313323911yescharacteristic
427C2×C4#1313324011yescharacteristic
428C2×C4#1313324511yescharacteristic
429C2×C4#1313324511yescharacteristic
430C8—22no—
431C8—41no—
432C8—42no—
433C8—81no—
434C8—2122no—
435C22—8481no—
436C22—8481no—
437C22—8481no—
438C22—8481no—
439C22—1064no—
440C22—4242no—
441C22—4242no—
442C22—4242no—
443C22—2122no—
444C22—2122no—
445C22—2122no—
446C22—2122no—
447C22—2122no—
448C22—2122no—
449C22#1681050111yescharacteristic
450C22#1681081911yescharacteristic
451C22#1681083911yescharacteristic
452C22#1681085311yescharacteristic
453C22#1681086011yescharacteristic
454C22#1681088711yescharacteristic
455C22#1681089411yescharacteristic
456C4—21no—
457C4—21no—
458C4—8481no—
459C4—2122no—
460C4#102×D5311yescharacteristic
461C4#1681061811yescharacteristic
462C4C4×D42411yescharacteristic
463C4#1681073611yescharacteristic
464C4#1681075911yescharacteristic
465C4#1681077011yescharacteristic
466C4#1681085511yescharacteristic
467C4#1681085811yescharacteristic
468C2—8482no—
469C2—2124no—
470C2#2027405711yescharacteristic
471C2#2027407111yescharacteristic
472C2#2027412511yescharacteristic
473C2#2027415311yescharacteristic
474C2#2027506211yescharacteristic
475C2#2027514211yescharacteristic
476C2#2027522211yescharacteristic
subgroupG/H
centerC23#13133192
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1277C2
socleC23×C53#121
prime radicalC2×#16810819C2
cosocleC2×C42×C53C23
hypercenter#1277D53
hypocenterC53#9221
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. C2×C8×D212C2semidirect
3. #20275521C2semidirect
4. #20274716C2semidirect
5. C2×#16810366C2semidirect
6. #20272599C2semidirect
7. C53×#1277C2semidirect
13. #16810819C4semidirect
117. C2×C8×C53D8semidirect
186. D212#102semidirect
236. #1277D53semidirect
266. C53#9221semidirect
361. C2×C8D424semidirect

Maximal subgroups

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

subgroupindex
C2×#168108192
C2×C8×D2122
#202755212
#202747162
C2×#168103662
#202725992
C53×#12772
#922153