Finite Group Databasebeta

← #27279796

Subgroups — #27279796

Order 13568.

752 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810645C211yesmaximal, characteristic, complemented
2#20283061C211yesmaximal, characteristic
3#20274692C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C2×#13133214C2211yescharacteristic
10C2×#13133192—21no—
11C2×#13133192—21no—
12C2×#13133192C412yes—
13C2×C42×D53C2211yescharacteristic
14#16810645—22no—
15#16810645—22no—
16C2×C4⋊#88322C2211yescharacteristic
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133146—21no—
19C2×#13133129—21no—
20#16810524—21no—
21#16810524—21no—
22#16810524C412yes—
23#16810377—21no—
24#16810366—22no—
25#16810366—22no—
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#384C2211yescharacteristic
29C22×D212—22no—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212C2×C412yes—
33C22×C4×D53C2×C411yescharacteristic
34C22×C4×D53D411yescharacteristic
35C22×C4×D53D411yescharacteristic
36C2×#11719912—24no—
37C2×#11719912—42no—
38C2×#11719912—42no—
39C22×#88322—21no—
40C22×#88322—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214—21no—
44#13133214—21no—
45#13133192—42no—
46#13133192—42no—
47#13133192—44no—
48#13133192—42no—
49#13133192—42no—
50C2×#11719921—22no—
51C2×#11719921—22no—
52C2×#11719921—22no—
53C2×C4×Dic53C2×C411yescharacteristic
54C2×C4×Dic53D411yescharacteristic
55C2×C4×Dic53D411yescharacteristic
56C42×D53—21no—
57C42×D53—21no—
58C42×D53—21no—
59C42×D53—21no—
60C4⋊#88322—21no—
61C4⋊#88322—21no—
62C4⋊#88322—21no—
63C4⋊#88322D412yescomplemented
64#13133146—24no—
65#13133129—42no—
66C22×C8×C53—21no—
67C22×C8×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53C2×C412yes—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—21no—
73C4⋊C8×C53—21no—
74C4⋊C8×C53—21no—
75C4⋊C8×C53D412yescomplemented
76C23×D53—22no—
77C23×D53C22⋊C411yescharacteristic
78C22×C4×C53—22no—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C22⋊C411yescharacteristic
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—42no—
88C2×D212—42no—
89C2×D212—44no—
90C2×D212—44no—
91C2×D212—44no—
92C2×D212—41no—
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×C4×D53—21no—
105C2×C4×D53—21no—
106C2×C4×D53—21no—
107C2×C4×D53—21no—
108C2×C4×D53—21no—
109C2×C4×D53—21no—
110C2×C4×D53—21no—
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×C4×D53—21no—
114C2×C4×D53—21no—
115C2×C4×D53—21no—
116C2×C4×D53—22no—
117C2×C4×D53—41no—
118C2×C4×D53D812yes—
119C2×C4×D53SD1612yes—
120C2×C8×C53—22no—
121C2×C8×C53—41no—
122C2×C8×C53—41no—
123C2×C8×C53—41no—
124C2×C8×C53—41no—
125C2×C8×C53—41no—
126C4⋊C4×C53—42no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53—41no—
129C4⋊C4×C53—41no—
130C4⋊C4×C53—41no—
131C42×C53C2×D411yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53#4111yescharacteristic
134C42×C53#4111yescharacteristic
135#11719921—44no—
136#11719921—44no—
137#11719921—44no—
138C4×Dic53—21no—
139C4×Dic53—21no—
140C4×Dic53—21no—
141C4×Dic53—21no—
142C4×Dic53—21no—
143C4×Dic53—21no—
144C2×#88322—22no—
145C2×#88322—41no—
146C2×#88322—41no—
147C2×#88322—41no—
148C2×#88322—41no—
149C2×#88322—41no—
150#11719912—48no—
151C23×C53C2×C22⋊C411yescharacteristic
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—42no—
159C22×D53—42no—
160C22×D53—42no—
161C22×D53—42no—
162C22×D53—42no—
163C22×D53—42no—
164C22×D53—42no—
165C22×D53#10212yes—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53C4×D411yescharacteristic
172C2×C4×C53C4×D411yescharacteristic
173C2×C4×C53#12011yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12311yescharacteristic
177C2×C4×C53C2×D811yescharacteristic
178C2×C4×C53C2×SD1611yescharacteristic
179C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C53#13711yescharacteristic
184C2×C4×C53#13711yescharacteristic
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53#10012yes—
193C2×Dic53#10212yes—
194D212—82no—
195D212—82no—
196D212—88no—
197D212—81no—
198D212—81no—
199D212—81no—
200D212—81no—
201D212—81no—
202D212—81no—
203D212—81no—
204D212—81no—
205C4×D53—22no—
206C4×D53—22no—
207C4×D53—22no—
208C4×D53—22no—
209C4×D53—42no—
210C4×D53—42no—
211C4×D53—41no—
212C4×D53—41no—
213C4×D53—41no—
214C4×D53—41no—
215C8×C53—42no—
216C8×C53—81no—
217#88322—42no—
218#88322—81no—
219#9141—531nomaximal, Sylow
220C22×C53#38611yescharacteristic
221C22×C53C2×#10011yescharacteristic
222C22×C53#41311yescharacteristic
223C22×C53C2×#10211yescharacteristic
224C22×C53#41711yescharacteristic
225C22×C53#41711yescharacteristic
226C22×C53#41811yescharacteristic
227C2×D53—22no—
228C2×D53—22no—
229C2×D53—22no—
230C2×D53—41no—
231C2×D53—41no—
232C2×D53—41no—
233C2×D53—41no—
234C2×D53—82no—
235C2×D53—82no—
236C2×D53—82no—
237C2×D53—82no—
238C2×D53—82no—
239C2×D53—82no—
240C2×D53—82no—
241C4×C53—21no—
242C4×C53—21no—
243C4×C53—82no—
244C4×C53#44711yescharacteristic
245C4×C53#44811yescharacteristic
246C4×C53#46811yescharacteristic
247C4×C53#46811yescharacteristic
248C4×C53#46911yescharacteristic
249C4×C53#46911yescharacteristic
250C4×C53#48011yescharacteristic
251C4×C53#48111yescharacteristic
252Dic53—22no—
253Dic53—22no—
254Dic53—41no—
255C2×#423—531no—
256#2856—531no—
257#1797—531no—
258C2×#331—531no—
259C2×#331—531no—
260#1296—531no—
261#1296D5311yescharacteristic, complemented
262C2×C53#154511yescharacteristic
263C2×C53#154711yescharacteristic
264C2×C53#155711yescharacteristic
265C2×C53#156011yescharacteristic
266C2×C53#187411yescharacteristic
267C2×C53#189511yescharacteristic
268C2×C53#190011yescharacteristic
269D53—42no—
270D53—162no—
271C22×M16—1061no—
272C2×#127—531no—
273C2×#121—532no—
274C2×#121—1061no—
275C2×#121—1061no—
276C22×C42—531no—
277#423—1062no—
278#423—1062no—
279C2×C4⋊C8—531no—
280C2×C4⋊C8—531no—
281C2×C4⋊C8—1061no—
282C2×C4⋊C8C2×D5311yescharacteristic
283C2×C4⋊C8C2×D5311yescharacteristic
284C2×C22⋊C8—1061no—
285#390—532no—
286#390—1061no—
287#390—1061no—
288#384C2×D5311yescharacteristic
289#336—1061no—
290#331—1062no—
291#331—1062no—
292C53#914111yescharacteristic, complemented, Sylow
293C22×D4—532no—
294C22×D4—1062no—
295C22×D4—1061no—
296C22×D4—1061no—
297C23×C4—531no—
298C23×C4—531no—
299C23×C4—531no—
300C2×M16—1064no—
301C2×M16—2122no—
302C2×M16—2122no—
303C22×C8—21no—
304C22×C8—21no—
305C22×C8—1061no—
306C22×C8—1061no—
307#127—1061no—
308#127—1061no—
309#127—1061no—
310#127—1061no—
311#121—2122no—
312#121—2122no—
313#121—2122no—
314#121—2122no—
315#121—2124no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C4×D5312yes—
319C2×C22⋊C4—1062no—
320C2×C22⋊C4—1062no—
321C2×C22⋊C4—1062no—
322C2×C42—531no—
323C2×C42—531no—
324C2×C42—531no—
325C2×C42—1061no—
326C2×C42—1061no—
327C2×C42—1061no—
328C2×C42—1061no—
329C2×C42C22×D5311yescharacteristic
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—532no—
334C4⋊C8—1064no—
335C4⋊C8—1061no—
336C4⋊C8—1061no—
337C4⋊C8—1061no—
338C4⋊C8D21212yescomplemented
339C22⋊C8—2122no—
340C24—531no—
341C24—1062no—
342C2×D4—1062no—
343C2×D4—1062no—
344C2×D4—2122no—
345C2×D4—2122no—
346C2×D4—2124no—
347C2×D4—2124no—
348C2×D4—2124no—
349C2×D4—2121no—
350C2×D4—2121no—
351C2×D4—2121no—
352C2×D4—2121no—
353C2×D4—2121no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4—2121no—
358C2×D4—2121no—
359C2×D4—2121no—
360C2×D4—2121no—
361C22×C4—22no—
362C22×C4—531no—
363C22×C4—531no—
364C22×C4—532no—
365C22×C4—532no—
366C22×C4—531no—
367C22×C4—1062no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—1061no—
371C22×C4—1061no—
372C22×C4—1061no—
373C22×C4—1061no—
374C22×C4—1061no—
375C22×C4—1061no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4—1061no—
380C22×C4—2121no—
381C22×C4C2×C4×D5311yescharacteristic
382C22×C4D4×D5311yescharacteristic
383C22×C4D4×D5311yescharacteristic
384M16—2128no—
385C2×C8—22no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—1062no—
392C2×C8—2121no—
393C2×C8—2121no—
394C2×C8—2121no—
395C2×C8—2121no—
396C2×C8—2121no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C22⋊C4—2124no—
403C22⋊C4—2124no—
404C22⋊C4—2124no—
405C42—1061no—
406C42—1061no—
407C42—1061no—
408C42—1061no—
409C42—1061no—
410C42—1061no—
411C42C2×D21211yescharacteristic
412C42#1171994511yescharacteristic
413C42D4×D5311yescharacteristic
414C42#1171994911yescharacteristic
415C23—532no—
416C23—1061no—
417C23—1061no—
418C23—1061no—
419C23—1061no—
420C23—1061no—
421C23—1061no—
422C23—2122no—
423C23—2122no—
424C23—2122no—
425C23—2122no—
426C23—2122no—
427C23—2122no—
428C23—2122no—
429C23C22⋊C4×D5311yescharacteristic
430D4—4242no—
431D4—4242no—
432D4—4248no—
433D4—4241no—
434D4—4241no—
435D4—4241no—
436D4—4241no—
437D4—4241no—
438D4—4241no—
439D4—4241no—
440D4—4241no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—532no—
447C2×C4—532no—
448C2×C4—1061no—
449C2×C4—1061no—
450C2×C4—1061no—
451C2×C4—1061no—
452C2×C4—1062no—
453C2×C4—1062no—
454C2×C4—1062no—
455C2×C4—1062no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—2122no—
460C2×C4—2122no—
461C2×C4—2121no—
462C2×C4—2121no—
463C2×C4—2121no—
464C2×C4—2121no—
465C2×C4#1313317911yescharacteristic
466C2×C4C4×D21211yescharacteristic
467C2×C4#1313318711yescharacteristic
468C2×C4#1313319011yescharacteristic
469C2×C4#1313319211yescharacteristic
470C2×C4#1313320211yescharacteristic
471C2×C4D8×D5311yescharacteristic
472C2×C4SD16×D5311yescharacteristic
473C2×C4#1313324111yescharacteristic
474C2×C4#1313324411yescharacteristic
475C2×C4#1313324411yescharacteristic
476C2×C4#1313324411yescharacteristic
477C2×C4#1313324911yescharacteristic
478C2×C4#1313324911yescharacteristic
479C8—42no—
480C8—81no—
481C8—4241no—
482C8—2122no—
483C22—1062no—
484C22—1062no—
485C22—1062no—
486C22—4242no—
487C22—4242no—
488C22—4242no—
489C22—4242no—
490C22—4242no—
491C22—4242no—
492C22—4242no—
493C22—2121no—
494C22—2121no—
495C22—2121no—
496C22—2121no—
497C22#1681050111yescharacteristic
498C22#100×D5311yescharacteristic
499C22#1681060611yescharacteristic
500C22#102×D5311yescharacteristic
501C22#1681062311yescharacteristic
502C22#1681062311yescharacteristic
503C22#1681062611yescharacteristic
504C4—21no—
505C4—21no—
506C4—82no—
507C4—1062no—
508C4—1062no—
509C4—2121no—
510C4#1681075811yescharacteristic
511C4#1681076311yescharacteristic
512C4#1681087011yescharacteristic
513C4#1681087211yescharacteristic
514C4#1681087511yescharacteristic
515C4#1681087711yescharacteristic
516C4#1681093811yescharacteristic
517C4#1681094311yescharacteristic
518C2—8482no—
519C2—2122no—
520C2#2027338111yescharacteristic
521C2#2027339111yescharacteristic
522C2#2027343611yescharacteristic
523C2#2027344911yescharacteristic
524C2#2027503111yescharacteristic
525C2#2027515211yescharacteristic
526C2#2027517711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53C2×C22⋊C4
prime radical#20283061C2
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#9141
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
1. C2×#16810645C2semidirect
3. #20274692C2semidirect
4. C2×#16810366C2semidirect
5. C2×#16810366C2semidirect
6. #20272598C2semidirect
7. C53×#1296C2semidirect
63. C4⋊#88322D4semidirect
75. C4⋊C8×C53D4semidirect
261. #1296D53semidirect
292. C53#9141semidirect
338. C4⋊C8D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168106452
#202830612
#202746922
C2×#168103662
#202725982
C53×#12962
#914153