Finite Group Databasebeta

← #27279639

Subgroups — #27279639

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283741C211yesmaximal, characteristic
2#20277384C211yesmaximal, characteristic
3#20273031C211yesmaximal, characteristic
4C2×#16810349C211yesmaximal, characteristic, complemented
5#20272562C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C2×C53×#324C211yesmaximal, characteristic, complemented
8C2×#13133200C411yescharacteristic
9C2×C4×C22⋊D53C2211yescharacteristic
10#16810694—22no—
11#16810691—22no—
12#16810677—21no—
13#16810585—21no—
14C2×C22⋊#88322C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16#16810518—21no—
17#16810513—21no—
18#16810504—21no—
19#16810502C411yescharacteristic
20#16810496C411yescharacteristic
21#16810494—21no—
22#16810486C411yescharacteristic
23C4×C22⋊Dic53C2211yescharacteristic
24#16810349—22no—
25#16810348—21no—
26#16810346—21no—
27C22×C4⋊C4×C53C2211yescharacteristic
28C2×C22⋊C8×C53C2211yescharacteristic
29C2×C22⋊C8×C53C2211yescharacteristic
30C53×#324—22no—
31C22×C22⋊D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C2×#11719912—22no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133200—22no—
37#13133200—22no—
38C4×C22⋊D53—21no—
39C4×C22⋊D53—21no—
40C4×C22⋊D53—21no—
41C4×C22⋊D53—21no—
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719919D411yescharacteristic
45C2×C22⋊Dic53C2×C411yescharacteristic
46C2×C22⋊Dic53C2×C411yescharacteristic
47C2×C22⋊Dic53C2×C411yescharacteristic
48C2×#11719921—21no—
49C2×#11719921—21no—
50C2×#11719921D411yescharacteristic
51C2×C4×Dic53—21no—
52C2×C4×Dic53D411yescharacteristic
53#13133146—22no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57C22⋊#88322D412yes—
58#13133129—22no—
59#13133127—22no—
60#13133124—21no—
61#13133124—21no—
62#13133124—21no—
63#13133124—21no—
64#13133124—21no—
65C23×C4×C53C2×C411yescharacteristic
66C23×C4×C53C2×C411yescharacteristic
67C23×C4×C53C2311yescharacteristic
68C22×C8×C53—21no—
69C22×C8×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—21no—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—21no—
76C22⋊C8×C53—21no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53—21no—
79C22⋊C8×C53D412yes—
80C24×C53C22×C411yescharacteristic
81C23×D53C22⋊C411yescharacteristic
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—81no—
92C22×C4×C53—81no—
93C22×C4×C53M1612yescomplemented
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C2×C22⋊D53—21no—
101C2×C22⋊D53—21no—
102C2×C22⋊D53—21no—
103C2×C22⋊D53—21no—
104C2×C22⋊D53—21no—
105C2×C22⋊D53—21no—
106C22×Dic53—21no—
107C22×Dic53—21no—
108C22×Dic53C22⋊C411yescharacteristic
109C22×Dic53C22⋊C411yescharacteristic
110C22×Dic53C22⋊C411yescharacteristic
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×C4×D53—81no—
114C2×C8×C53—22no—
115C2×C8×C53—41no—
116C2×C8×C53—81no—
117C2×C8×C53—81no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C4⋊C4×C53—41no—
122C22⋊Dic53—21no—
123C22⋊Dic53—21no—
124C22⋊Dic53—21no—
125C22⋊Dic53—21no—
126C22⋊Dic53—21no—
127C22⋊Dic53M1612yescomplemented
128#11719921—24no—
129#11719921—21no—
130#11719921—21no—
131#11719921—81no—
132#11719919—24no—
133#11719919—21no—
134#11719919—21no—
135#11719919—81no—
136C4×Dic53—21no—
137C4×Dic53—21no—
138C4×Dic53—81no—
139C2×#88322—22no—
140C2×#88322—41no—
141C2×#88322—81no—
142C2×#88322—81no—
143#11719912—82no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53C2×C22⋊C411yescharacteristic
147C23×C53C4×D411yescharacteristic
148C23×C53C4×D411yescharacteristic
149C23×C53C2×M1611yescharacteristic
150C23×C53C2×M1611yescharacteristic
151C23×C53#13111yescharacteristic
152C23×C53#13111yescharacteristic
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—41no—
157C22×D53—41no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
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—81no—
175C2×C4×C53—81no—
176C2×C4×C53#12011yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12311yescharacteristic
180C22⋊D53—41no—
181C22⋊D53—41no—
182C22⋊D53—41no—
183C22⋊D53—41no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—22no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C4×D53—42no—
198C4×D53—81no—
199C8×C53—42no—
200C8×C53—81no—
201#88322—42no—
202#88322—81no—
203#9110—531nomaximal, Sylow
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53#38611yescharacteristic
215C22×C53#40711yescharacteristic
216C22×C53#40811yescharacteristic
217C22×C53#43511yescharacteristic
218C22×C53#43511yescharacteristic
219C22×C53#43611yescharacteristic
220C22×C53#43611yescharacteristic
221C2×D53—42no—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—81no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—42no—
231C4×C53—81no—
232Dic53—42no—
233Dic53—41no—
234Dic53—41no—
235Dic53—41no—
236Dic53—81no—
237#2973—531no—
238#2322—531no—
239#1459—531no—
240C2×#324—531no—
241C2×#324D5311yescharacteristic, complemented
242#1282—531no—
243#1282—531no—
244C2×C53—21no—
245C2×C53—21no—
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53#187011yescharacteristic
251C2×C53#191311yescharacteristic
252C2×C53#191511yescharacteristic
253D53—81no—
254C2×#123—531no—
255C2×C4×D4—531no—
256C22×C4⋊C4C2×D5311yescharacteristic
257#435—1062no—
258#435—1062no—
259#432—1061no—
260#407—1061no—
261C2×C22⋊C8—531no—
262C2×C22⋊C8—531no—
263C2×C22⋊C8C2×D5311yescharacteristic
264C2×C22⋊C8C2×D5311yescharacteristic
265#389—1061no—
266#388—1061no—
267#387—1061no—
268#386—531no—
269#385—531no—
270#385—1061no—
271#381—531no—
272C4×C22⋊C4—531no—
273#324—22no—
274#324—1061no—
275#324—1062no—
276#323—1061no—
277C53#911011yescharacteristic, complemented, Sylow
278C22×D4—531no—
279C23×C4—531no—
280C23×C4C4×D5311yescharacteristic
281C23×C4C4×D5311yescharacteristic
282C23×C4C22×D5311yescharacteristic
283C2×M16—1062no—
284C22×C8—21no—
285C22×C8—21no—
286C22×C8—1061no—
287C22×C8—1061no—
288#123—1062no—
289#123—1062no—
290C4×D4—1061no—
291C4×D4—1061no—
292C4×D4—1061no—
293C4×D4—1061no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—1061no—
302C2×C4⋊C4—1061no—
303C2×C22⋊C4—531no—
304C2×C22⋊C4—531no—
305C2×C22⋊C4—531no—
306C2×C22⋊C4—531no—
307C2×C22⋊C4—1061no—
308C2×C22⋊C4—1061no—
309C2×C42—531no—
310C2×C42—1061no—
311C4⋊C8—1062no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C22⋊C8—21no—
315C22⋊C8—532no—
316C22⋊C8—1062no—
317C22⋊C8—1061no—
318C22⋊C8—1061no—
319C22⋊C8—1061no—
320C22⋊C8C22⋊D5312yes—
321#97—1062no—
322#95—1061no—
323#95—1061no—
324#95—1061no—
325#95—1061no—
326#95—1061no—
327C24—531no—
328C24C2×C4×D5311yescharacteristic
329C2×D4—1061no—
330C2×D4—1061no—
331C2×D4—1061no—
332C2×D4—1061no—
333C2×D4—1061no—
334C2×D4—1061no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—531no—
347C22×C4—531no—
348C22×C4—531no—
349C22×C4—1061no—
350C22×C4—1061no—
351C22×C4—1061no—
352C22×C4—1061no—
353C22×C4—4241no—
354C22×C4#1171991212yescomplemented
355C22×C4#1171994511yescharacteristic
356C22×C4D4×D5311yescharacteristic
357C22×C4D4×D5311yescharacteristic
358C22×C4D4×D5311yescharacteristic
359C22×C4#1171994711yescharacteristic
360C22×C4C2×C22⋊D5311yescharacteristic
361M16—4242no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—81no—
365C2×C8—81no—
366C2×C8—1062no—
367C2×C8—4241no—
368C2×C8—4241no—
369C2×C8—2121no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—1064no—
375C4⋊C4—1061no—
376C4⋊C4—1061no—
377C4⋊C4—4241no—
378C22⋊C4—532no—
379C22⋊C4—1064no—
380C22⋊C4—1061no—
381C22⋊C4—1061no—
382C22⋊C4—1061no—
383C22⋊C4—1061no—
384C22⋊C4—1061no—
385C22⋊C4—1061no—
386C22⋊C4—1061no—
387C22⋊C4—4241no—
388C42—1061no—
389C42—1061no—
390C42—4241no—
391C23—41no—
392C23—41no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—2121no—
397C23—2121no—
398C23C22⋊C4×D5311yescharacteristic
399C23#1313317711yescharacteristic
400C23C4×C22⋊D5311yescharacteristic
401C23M16×D5311yescharacteristic
402C23C2×#1171991211yescharacteristic
403C23#1313322311yescharacteristic
404C23#1313322511yescharacteristic
405D4—2121no—
406D4—2121no—
407D4—2121no—
408D4—2121no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1062no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—4241no—
434C2×C4—2122no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4#1313318511yescharacteristic
443C2×C4#1313318811yescharacteristic
444C2×C4#1313319011yescharacteristic
445C2×C4#1313319911yescharacteristic
446C8—42no—
447C8—81no—
448C8—4241no—
449C8—2122no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—4241no—
461C22—2122no—
462C22—2121no—
463C22—2121no—
464C22—2121no—
465C22#1681049911yescharacteristic
466C22#1681058511yescharacteristic
467C22#1681058811yescharacteristic
468C22#1681069111yescharacteristic
469C22#1681069411yescharacteristic
470C22#1681069811yescharacteristic
471C22#1681069911yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—81no—
478C4—4241no—
479C4—2122no—
480C4—2121no—
481C4—2121no—
482C4—2121no—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—4241no—
490C2#2027500911yescharacteristic
491C2#2027525411yescharacteristic
492C2#2027526411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC22×C53#386
prime radicalC22×C22⋊D53C2×C4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#9110
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
4. C2×#16810349C2semidirect
5. #20272562C2semidirect
6. #20272561C2semidirect
7. C2×C53×#324C2semidirect
93. C22×C4×C53M16semidirect
127. C22⋊Dic53M16semidirect
241. C2×#324D53semidirect
277. C53#9110semidirect
354. C22×C4#11719912semidirect

Maximal subgroups

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

subgroupindex
#202837412
#202773842
#202730312
C2×#168103492
#202725622
#202725612
C2×C53×#3242
#911053