Finite Group Databasebeta

← #27279208

Subgroups — #27279208

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#20284624C211yesmaximal, characteristic
2#20274812C211yesmaximal, characteristic, complemented
3#20274784C211yesmaximal, characteristic, complemented
4#20273206C211yesmaximal, characteristic
5#20272602C211yesmaximal, characteristic, complemented
6#20272562C211yesmaximal, characteristic, complemented
7C53×#2976C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic
9#16810961—21no—
10#16810956C411yescharacteristic, complemented
11#16810951C411yescharacteristic, complemented
12#16810944—21no—
13#16810939C411yescharacteristic, complemented
14#16810934C411yescharacteristic, complemented
15#16810659—21no—
16#16810619—21no—
17C2×C22⋊#88322C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19C4×C22⋊Dic53C2211yescharacteristic
20#16810384—21no—
21#16810384—21no—
22#16810377—21no—
23#16810377—21no—
24#16810360—21no—
25#16810357—21no—
26C53×#525C2211yescharacteristic
27C53×#436—21no—
28C53×#435—21no—
29C53×#435—21no—
30C8×D4×C53—21no—
31C53×#433—21no—
32C53×#408—21no—
33C53×#407C2211yescharacteristic
34C2×C22⋊C8×C53C2211yescharacteristic
35C22×#88322—21no—
36C22×#88322—21no—
37#13133176—21no—
38#13133176—21no—
39#13133176—21no—
40#13133176—21no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53—21no—
47#13133150—21no—
48#13133150—21no—
49#13133148—21no—
50#13133148—21no—
51#13133141—21no—
52#13133141—21no—
53#13133138—21no—
54#13133138—21no—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57C22⋊#88322—21no—
58C22⋊#88322C2×C411yescharacteristic
59C22⋊#88322C2×C411yescharacteristic
60#13133124—21no—
61C2×#41×C53D411yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C2×M16×C53—21no—
64C2×M16×C53D411yescharacteristic
65C2×M16×C53D411yescharacteristic
66C22×C8×C53—21no—
67C22×C8×C53D411yescharacteristic
68C22×C8×C53D411yescharacteristic
69#123×C53—21no—
70C22⋊Q8×C53C2×C411yescharacteristic
71C22⋊Q8×C53D411yescharacteristic
72#121×C53C2×C411yescharacteristic
73#121×C53D411yescharacteristic
74#120×C53—21no—
75C4×D4×C53—21no—
76C4×D4×C53—21no—
77#117×C53D411yescharacteristic
78C4⋊C8×C53—21no—
79C4⋊C8×C53—21no—
80C22⋊C8×C53—21no—
81C22⋊C8×C53—21no—
82C22⋊C8×C53—21no—
83C22⋊C8×C53—21no—
84C22⋊C8×C53—21no—
85C22⋊C8×C53—21no—
86#97×C53—21no—
87C4×C8×C53—21no—
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—41no—
98C22×C4×C53—41no—
99C22×C4×C53C22×C411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—41no—
108M16×C53—41no—
109M16×C53—41no—
110M16×C53—81no—
111C2×C8×C53—21no—
112C2×C8×C53—21no—
113C2×C8×C53—21no—
114C2×C8×C53—21no—
115C2×C8×C53—41no—
116C2×C8×C53—41no—
117C2×C8×C53—81no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53C22⋊C411yescharacteristic
121C4⋊C4×C53C22⋊C411yescharacteristic
122C22⋊C4×C53—21no—
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C22⋊C4×C53—21no—
127C42×C53—21no—
128C22⋊Dic53—22no—
129C22⋊Dic53—41no—
130C22⋊Dic53—41no—
131C22⋊Dic53—41no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4×Dic53—41no—
137C4×Dic53—41no—
138C4×Dic53—41no—
139C4×Dic53—41no—
140C2×#88322—21no—
141C2×#88322—21no—
142C2×#88322—41no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—41no—
146C2×#88322—41no—
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×C53C2×C22⋊C411yescharacteristic
176C2×C4×C53C4×D411yescharacteristic
177C2×C4×C53#12011yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C8×C53—41no—
188C8×C53—41no—
189C8×C53—41no—
190C8×C53—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#45811yescharacteristic
218C4×C53#45811yescharacteristic
219Dic53—82no—
220Dic53—81no—
221Dic53—81no—
222#3091—531no—
223#2976D5311yescharacteristic
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×#117—531no—
239#485—1061no—
240#484—531no—
241#483—531no—
242#482—1061no—
243#481—531no—
244#480—531no—
245#436—21no—
246#435—21no—
247#435—21no—
248C8×D4—21no—
249#433—21no—
250#427—1061no—
251#416—1061no—
252#408—21no—
253#407C2×D5311yescharacteristic
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—21no—
268C2×M16C22⋊D5311yescharacteristic
269C2×M16C22⋊D5311yescharacteristic
270C22×C8—21no—
271C22×C8—1061no—
272C22×C8—1061no—
273C22×C8C22⋊D5311yescharacteristic
274C22×C8C22⋊D5311yescharacteristic
275#123—21no—
276C22⋊Q8C4×D5311yescharacteristic
277C22⋊Q8D21211yescharacteristic
278#121C4×D5311yescharacteristic
279#121D21211yescharacteristic
280#120—21no—
281C4×D4—21no—
282C4×D4—21no—
283#117—1061no—
284#117—1061no—
285#117—1061no—
286#117—1061no—
287#117C22⋊D5311yescharacteristic
288C2×C4⋊C4—531no—
289C2×C4⋊C4—531no—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4—1061no—
292C2×C42—1061no—
293C2×C42—1061no—
294#107—1061no—
295#107—1061no—
296#106—1061no—
297#106—1061no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300#103—1061no—
301#103—1061no—
302#102—1061no—
303#102—1061no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—531no—
311C22⋊C8—531no—
312C22⋊C8—1061no—
313C22⋊C8—1061no—
314C22⋊C8—1061no—
315#97—21no—
316C4×C8—21no—
317#95—1061no—
318C24#1171994511yescharacteristic
319#41—41no—
320#41—41no—
321C2×Q8#1171992111yescharacteristic
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4#1171992111yescharacteristic
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—2121no—
332C22×C4C2×C4×D5311yescharacteristic
333C22×C4C2×D21211yescharacteristic
334C22×C4#1171994511yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338M16—41no—
339M16—41no—
340M16—81no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—81no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—2121no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8—2121no—
354C2×C8—2121no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—1061no—
358C4⋊C4—1061no—
359C4⋊C4—1061no—
360C4⋊C4—1061no—
361C4⋊C4#1171992111yescharacteristic
362C4⋊C4#1171992111yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—1062no—
369C22⋊C4—2121no—
370C22⋊C4—2121no—
371C22⋊C4—2121no—
372C42—21no—
373C42—2121no—
374C42—2121no—
375C42—2121no—
376C42—2121no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—41no—
381C23—41no—
382C23C4×C22⋊D5311yescharacteristic
383C23#1313319111yescharacteristic
384C23#1313320011yescharacteristic
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—4241no—
406C2×C4—4241no—
407C2×C4—2122no—
408C2×C4—2122no—
409C2×C4—2121no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4C2×#1171992111yescharacteristic
414C2×C4C4×C22⋊D5311yescharacteristic
415C2×C4#1313318611yescharacteristic
416C2×C4#1313319111yescharacteristic
417C8—41no—
418C8—41no—
419C8—41no—
420C8—41no—
421C8—4241no—
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#1681050211yescharacteristic
438C22#1681073811yescharacteristic
439C22#1681074311yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—4242no—
447C4—4241no—
448C4—4241no—
449C4#1681081311yescharacteristic
450C4#1681081311yescharacteristic
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2#2027504211yescharacteristic
457C2#2027507711yescharacteristic
458C2#2027512811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2976C2
socleC22×C53#386
prime radicalC53×#525C22
cosocleC23×C4×C53C23
hypercenter#2976D53
hypocenterC53#9038
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2976

Complements

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

NH ≅ G/Ntype
2. #20274812C2semidirect
3. #20274784C2semidirect
5. #20272602C2semidirect
6. #20272562C2semidirect
10. #16810956C4semidirect
11. #16810951C4semidirect
13. #16810939C4semidirect
14. #16810934C4semidirect
264. C53#9038semidirect

Maximal subgroups

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

subgroupindex
#202846242
#202748122
#202747842
#202732062
#202726022
#202725622
C53×#29762
#903853