Finite Group Databasebeta

← #27279210

Subgroups — #27279210

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283763C211yesmaximal, characteristic
2#20274814C211yesmaximal, characteristic, complemented
3#20274786C211yesmaximal, characteristic, complemented
4#20273206C211yesmaximal, characteristic
5#20272603C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#3091C211yesmaximal, characteristic
8#16810701—21no—
9#16810695—21no—
10#16810695—21no—
11D4×#88322—21no—
12#16810684—21no—
13#16810590—21no—
14#16810587C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16C2×C22⋊#88322C2211yescharacteristic
17C4×C22⋊Dic53C2211yescharacteristic
18#16810382—21no—
19#16810382—21no—
20#16810378—21no—
21#16810378—21no—
22#16810360—21no—
23#16810357—21no—
24C53×#525C2211yescharacteristic
25C2×#117×C53C2211yescharacteristic
26C53×#485—21no—
27C53×#484C411yescharacteristic, complemented
28C53×#483C411yescharacteristic, complemented
29C53×#482—21no—
30C53×#481C411yescharacteristic, complemented
31C53×#480C411yescharacteristic, complemented
32C53×#427—21no—
33C53×#416—21no—
34C2×C22⋊C8×C53C2211yescharacteristic
35C2×#11719917—21no—
36C2×#11719917D411yescharacteristic
37C2×#11719917D411yescharacteristic
38C22×#88322—21no—
39C22×#88322—21no—
40C22×#88322D411yescharacteristic
41C22×#88322D411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133128—21no—
55C4×#88322—21no—
56#13133124—21no—
57C2×#41×C53D411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C22×C8×C53—21no—
60#123×C53—21no—
61C22⋊Q8×C53C2×C411yescharacteristic
62C22⋊Q8×C53D411yescharacteristic
63#121×C53C2×C411yescharacteristic
64#121×C53D411yescharacteristic
65#120×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68#117×C53—21no—
69#117×C53—21no—
70#117×C53—21no—
71#117×C53—21no—
72#117×C53D411yescharacteristic
73C2×C4⋊C4×C53C2×C411yescharacteristic
74C2×C4⋊C4×C53C2×C411yescharacteristic
75C2×C22⋊C4×C53—21no—
76C2×C42×C53—21no—
77#107×C53—21no—
78#107×C53—21no—
79#106×C53—21no—
80#106×C53—21no—
81#103×C53—21no—
82#103×C53—21no—
83#102×C53—21no—
84#102×C53—21no—
85C22⋊C8×C53—21no—
86C22⋊C8×C53C2×C411yescharacteristic
87C22⋊C8×C53C2×C411yescharacteristic
88C24×C53#4111yescharacteristic
89#41×C53—41no—
90#41×C53—41no—
91C2×Q8×C53C22⋊C411yescharacteristic
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53C22⋊C411yescharacteristic
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×Dic53—41no—
108C2×C8×C53—21no—
109C2×C8×C53—21no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53C22⋊C411yescharacteristic
119C4⋊C4×C53C22⋊C411yescharacteristic
120C22⋊C4×C53—21no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C42×C53—21no—
128C42×C53—41no—
129C42×C53—41no—
130C22⋊Dic53—22no—
131C22⋊Dic53—41no—
132C4×Dic53—41no—
133C4×Dic53—41no—
134#11719917—41no—
135#11719917—41no—
136#11719917—81no—
137C2×#88322—21no—
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—21no—
141C2×#88322—41no—
142C2×#88322—41no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—41no—
146C2×#88322—81no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53C4×D411yescharacteristic
153C23×C53#12111yescharacteristic
154C23×C53#12311yescharacteristic
155Q8×C53—41no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53—41no—
159D4×C53—81no—
160D4×C53—81no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—42no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53C2×C22⋊C411yescharacteristic
181C2×C4×C53C4×D411yescharacteristic
182C2×C4×C53#12011yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×Dic53—42no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C8×C53—41no—
188C8×C53—41no—
189#88322—41no—
190#88322—41no—
191#88322—41no—
192#88322—41no—
193#88322—81no—
194#9038—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—81no—
207C22×C53—81no—
208C22×C53#38611yescharacteristic
209C22×C53#44311yescharacteristic
210C22×C53#44411yescharacteristic
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—41no—
215C4×C53—41no—
216C4×C53—41no—
217C4×C53—81no—
218C4×C53—81no—
219C4×C53#45811yescharacteristic
220C4×C53#45811yescharacteristic
221Dic53—82no—
222#3091D5311yescharacteristic
223#2976—531no—
224#1826—531no—
225#1818—531no—
226#1501—531no—
227#1298—531no—
228#1282—531no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—41no—
232C2×C53—41no—
233C2×C53—81no—
234C2×C53#187611yescharacteristic
235C2×C53#188111yescharacteristic
236C2×C53#189011yescharacteristic
237#525C2×D5311yescharacteristic
238C2×#117C2×D5311yescharacteristic
239#485—21no—
240#484Dic5311yescharacteristic, complemented
241#483Dic5311yescharacteristic, complemented
242#482—21no—
243#481Dic5311yescharacteristic, complemented
244#480Dic5311yescharacteristic, complemented
245#436—1061no—
246#435—1061no—
247#435—1061no—
248C8×D4—1061no—
249#433—1061no—
250#427—21no—
251#416—21no—
252#408—1061no—
253#407—531no—
254C2×C22⋊C8—531no—
255C2×C22⋊C8—531no—
256C2×C22⋊C8C2×D5311yescharacteristic
257C4×C22⋊C4—531no—
258#339—1061no—
259#339—1061no—
260#336—1061no—
261#336—1061no—
262#328—1061no—
263#327—1061no—
264C53#903811yescharacteristic, complemented, Sylow
265C2×#41C22⋊D5311yescharacteristic
266C23×C4C22×D5311yescharacteristic
267C2×M16—531no—
268C2×M16—531no—
269C2×M16—1061no—
270C22×C8—21no—
271C22×C8—531no—
272C22×C8—531no—
273C22×C8—1061no—
274C22×C8—1061no—
275#123—21no—
276C22⋊Q8C2×Dic5311yescharacteristic
277C22⋊Q8C22⋊D5311yescharacteristic
278#121C2×Dic5311yescharacteristic
279#121C22⋊D5311yescharacteristic
280#120—21no—
281C4×D4—21no—
282C4×D4—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117C22⋊D5311yescharacteristic
288C2×C4⋊C4C2×Dic5311yescharacteristic
289C2×C4⋊C4C2×Dic5311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—1061no—
292C2×C42—21no—
293C2×C42—1061no—
294#107—21no—
295#107—21no—
296#106—21no—
297#106—21no—
298C4⋊C8—1061no—
299C4⋊C8—1061no—
300#103—21no—
301#103—21no—
302#102—21no—
303#102—21no—
304C22⋊C8—21no—
305C22⋊C8—1061no—
306C22⋊C8—1061no—
307C22⋊C8—1061no—
308C22⋊C8—1061no—
309C22⋊C8—1061no—
310C22⋊C8—1061no—
311C22⋊C8—1061no—
312C22⋊C8—1061no—
313C22⋊C8C2×Dic5311yescharacteristic
314C22⋊C8C2×Dic5311yescharacteristic
315#97—1061no—
316C4×C8—1061no—
317#95—1061no—
318C24#1171994711yescharacteristic
319#41—41no—
320#41—41no—
321C2×Q8C22⋊Dic5311yescharacteristic
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4C22⋊Dic5311yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—2121no—
332C22×C4D4×D5311yescharacteristic
333C22×C4D4×D5311yescharacteristic
334C22×C4#1171994711yescharacteristic
335C22×C4C22×Dic5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338M16—4241no—
339M16—2121no—
340M16—2121no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—1061no—
346C2×C8—1061no—
347C2×C8—1061no—
348C2×C8—1061no—
349C2×C8—4241no—
350C2×C8—2121no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8—2121no—
354C2×C8—2121no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4C22⋊Dic5311yescharacteristic
362C4⋊C4C22⋊Dic5311yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—1062no—
371C22⋊C4—2121no—
372C42—21no—
373C42—41no—
374C42—41no—
375C42—2121no—
376C42—2121no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—41no—
381C23—41no—
382C23D4×Dic5311yescharacteristic
383C23#1313318911yescharacteristic
384C23#1313319811yescharacteristic
385Q8—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—81no—
390D4—81no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
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×C4#1313318511yescharacteristic
416C2×C4#1313318911yescharacteristic
417C8—41no—
418C8—41no—
419C8—4241no—
420C8—2121no—
421C8—2121no—
422C8—2121no—
423C8—2121no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22#1681050311yescharacteristic
438C22#1681074011yescharacteristic
439C22#1681074511yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—81no—
447C4—81no—
448C4—4242no—
449C4#1681081211yescharacteristic
450C4#1681081211yescharacteristic
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2#2027504411yescharacteristic
457C2#2027507911yescharacteristic
458C2#2027512911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3091C2
socleC22×C53#386
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#3091D53
hypocenterC53#9038
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3091

Complements

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

NH ≅ G/Ntype
2. #20274814C2semidirect
3. #20274786C2semidirect
5. #20272603C2semidirect
6. #20272561C2semidirect
27. C53×#484C4semidirect
28. C53×#483C4semidirect
30. C53×#481C4semidirect
31. C53×#480C4semidirect
240. #484Dic53semidirect
241. #483Dic53semidirect
243. #481Dic53semidirect
244. #480Dic53semidirect
264. C53#9038semidirect

Maximal subgroups

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

subgroupindex
#202837632
#202748142
#202747862
#202732062
#202726032
#202725612
C53×#30912
#903853