Finite Group Databasebeta

← #27538924

Subgroups — #27538924

Order 13568.

580 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810861C211yesmaximal, characteristic
2#427×D53C211yesmaximal, characteristic
3#20276922C212yesmaximal, complemented
4#20276921C212yesmaximal, complemented
5#20276916C212yesmaximal, complemented
6#20276911C212yesmaximal, complemented
7#20276504C211yesmaximal, characteristic, complemented
8#20276503C211yesmaximal, characteristic, complemented
9#20276435C211yesmaximal, characteristic, complemented
10#20276325C211yesmaximal, characteristic
11C53×#2141C211yesmaximal, characteristic, complemented
12C22×C8×D53C2211yescharacteristic
13C2×#13133193—21no—
14#117×D53—21no—
15C2×#13133157C2211yescharacteristic
16#16810861—21no—
17#16810861—21no—
18#16810861C2212yes—
19#16810861C2212yes—
20#16810661C2211yescharacteristic
21#16810660C2211yescharacteristic
22#16810660C2211yescharacteristic
23#16810659—21no—
24#16810658—21no—
25C2×#13133151C2211yescharacteristic
26#107×D53C2211yescharacteristic
27#107×D53C2211yescharacteristic
28#106×D53—21no—
29C2×#13133141—21no—
30#16810451C2212yescomplemented
31#16810450C2212yescomplemented
32#16810449C2212yescomplemented
33#16810448C2212yescomplemented
34#16810427C2212yescomplemented
35#16810427C2212yescomplemented
36#16810402C2212yes—
37#16810402C2212yes—
38C22×Q16×C53C2211yescharacteristic
39C2×M32×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C53×#358C2212yescomplemented
42C53×#358C2212yescomplemented
43C22×C4×D53D411yescharacteristic
44C22×#88322D411yescharacteristic
45C2×C8×D53—22no—
46C2×C8×D53—22no—
47C2×C8×D53D411yescharacteristic
48C2×C8×D53D411yescharacteristic
49C2×C8×D53C2311yescharacteristic
50C2×C8×D53C2311yescharacteristic
51#13133193—22no—
52#13133193—22no—
53#13133193—41no—
54#13133193—41no—
55#13133176—21no—
56#13133175—21no—
57#13133175—21no—
58#13133174—41no—
59#13133173—41no—
60C2×C4⋊Dic53—21no—
61C2×#11719919—41no—
62C4⋊C4×D53—21no—
63C4⋊C4×D53—21no—
64C2×#11719921—41no—
65C22⋊C4×D53—41no—
66C42×D53—41no—
67#13133157—22no—
68C2×#11719908C2312yes—
69#13133151—21no—
70#13133151C2312yes—
71#13133151C2311yescharacteristic
72#13133151C2311yescharacteristic
73#13133150—21no—
74#13133150—21no—
75#13133149—21no—
76#13133148—21no—
77#13133148—21no—
78#13133141—22no—
79#13133141—22no—
80C22×Q8×C53—21no—
81C2×Q16×C53—21no—
82C2×Q16×C53—22no—
83C2×Q16×C53—22no—
84C2×Q16×C53C2312yes—
85C22×C8×C53C2311yescharacteristic
86#117×C53—21no—
87M32×C53—22no—
88C2×C16×C53C2312yes—
89#107×C53C2311yescharacteristic
90#107×C53C2311yescharacteristic
91#106×C53—21no—
92C23×D53D811yescharacteristic
93C2×Q8×C53—22no—
94C2×Q8×C53—41no—
95C2×Q8×C53—42no—
96C2×Q8×C53—42no—
97C22×C4×C53—41no—
98C22×C4×C53C2×D411yescharacteristic
99C22×Dic53—41no—
100C22×Dic53D811yescharacteristic
101C2×C4×D53—22no—
102C2×C4×D53—22no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C4×D53D811yescharacteristic
106C2×C4×D53D811yescharacteristic
107C2×C4×D53C2×D411yescharacteristic
108C2×C4×D53C2×D411yescharacteristic
109Q16×C53—22no—
110Q16×C53—22no—
111Q16×C53—42no—
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53#4112yes—
114C2×C8×C53#4112yes—
115C2×C8×C53C2411yescharacteristic
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C22⋊C4×C53—41no—
119C42×C53—41no—
120C22⋊Dic53—41no—
121#11719921—41no—
122#11719921—41no—
123#11719921—42no—
124#11719921—42no—
125C4⋊Dic53—22no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128C4⋊Dic53—41no—
129#11719919—41no—
130#11719919—41no—
131#11719919—42no—
132#11719919—42no—
133C4×Dic53—41no—
134C4×Dic53—41no—
135C4×Dic53—41no—
136C2×#88322—22no—
137C2×#88322C2×D411yescharacteristic
138C2×#88322C2×D411yescharacteristic
139C8×D53—22no—
140C8×D53—22no—
141C8×D53—21no—
142C8×D53—21no—
143C8×D53—21no—
144C8×D53—21no—
145C16×C53—22no—
146#11719908—22no—
147C23×C53C2×D811yescharacteristic
148C22×D53—22no—
149C22×D53—22no—
150C22×D53—41no—
151C22×D53C2×D811yescharacteristic
152C22×D53C2×D811yescharacteristic
153Q8×C53—42no—
154Q8×C53—42no—
155Q8×C53—82no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—82no—
160C2×C4×C53#12312yes—
161C2×C4×C53C2×D811yescharacteristic
162C2×C4×C53#13712yes—
163C2×C4×C53C22×D411yescharacteristic
164C2×Dic53—22no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—42no—
168C2×Dic53—82no—
169C2×Dic53C2×D811yescharacteristic
170C2×Dic53C2×D811yescharacteristic
171C4×D53—21no—
172C4×D53—21no—
173C4×D53—21no—
174C4×D53—21no—
175C4×D53—22no—
176C4×D53—22no—
177C4×D53—82no—
178C4×D53—82no—
179C8×C53#12312yes—
180C8×C53C2×#4111yescharacteristic
181C8×C53C2×#4111yescharacteristic
182#88322—21no—
183#88322—21no—
184#33649—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53#48012yes—
188C22×C53C22×D811yescharacteristic
189C2×D53—21no—
190C2×D53—21no—
191C2×D53—21no—
192C2×D53—21no—
193C2×D53—21no—
194C2×D53—44no—
195C2×D53#51012yes—
196C4×C53—82no—
197C4×C53—81no—
198C4×C53—81no—
199C4×C53#48012yes—
200C4×C53C2×#12311yescharacteristic
201C4×C53C2×#13711yescharacteristic
202Dic53—21no—
203Dic53—21no—
204Dic53—81no—
205Dic53—81no—
206Dic53—82no—
207C2×#467—531no—
208C2×#427—531no—
209#2237—532no—
210#2237—532no—
211#2236—532no—
212#2235—532no—
213#2159—531no—
214#2159—531no—
215#2141—531no—
216#2141D5311yescharacteristic, complemented
217#2112—531no—
218C2×C53—22no—
219C2×C53C2×#48011yescharacteristic
220C2×C53C2×#51011yescharacteristic
221C2×C53#341411yescharacteristic
222D53—22no—
223D53—41no—
224C22×Q16C2×D5311yescharacteristic
225C23×C8—531no—
226C2×C22⋊Q8—1061no—
227C2×#117—1061no—
228C2×M32—531no—
229C2×M32C2×D5311yescharacteristic
230#467—532no—
231#467—532no—
232#467—1061no—
233#467—1061no—
234#427—531no—
235#427—531no—
236#427—531no—
237#427—1061no—
238#427—1061no—
239#427C2×D5311yescharacteristic
240C2×#107—531no—
241C2×#107—531no—
242C2×#107—531no—
243C2×#106—1061no—
244C2×#103—1061no—
245#367—532no—
246#367—532no—
247#366—532no—
248#366—532no—
249#358—532no—
250#358—532no—
251#358C2×D5312yescomplemented
252#358C2×D5312yescomplemented
253C22⋊C16—532no—
254C22⋊C16—532no—
255C53#3364911yescharacteristic, complemented, Sylow
256C22×Q8—21no—
257C23×C4—531no—
258C2×Q16—22no—
259C2×Q16—21no—
260C2×Q16—22no—
261C2×Q16C22×D5312yes—
262C22×C8—531no—
263C22×C8—531no—
264C22×C8—531no—
265C22×C8—531no—
266C22×C8—531no—
267C22×C8—1062no—
268C22×C8—1062no—
269C22×C8C22×D5311yescharacteristic
270C22⋊Q8—1062no—
271C22⋊Q8—1062no—
272C22⋊Q8—2121no—
273C22⋊Q8—2121no—
274#117—21no—
275#117—1061no—
276#117—1061no—
277#117—1061no—
278#117—2121no—
279#117—2121no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1061no—
283C2×C4⋊C4—2121no—
284C2×C22⋊C4—2121no—
285C2×C22⋊C4—2121no—
286C2×C42—2121no—
287M32—22no—
288M32—1062no—
289C2×C16—532no—
290C2×C16C22×D5312yes—
291#107—531no—
292#107—531no—
293#107—532no—
294#107—1061no—
295#107—1061no—
296#107—1061no—
297#107C22×D5311yescharacteristic
298#107C22×D5311yescharacteristic
299#106—21no—
300#106—1061no—
301#106—1061no—
302#106—1061no—
303#103—1062no—
304#103—1062no—
305C24—531no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C22×C4—41no—
311C22×C4—531no—
312C22×C4—531no—
313C22×C4—531no—
314C22×C4—531no—
315C22×C4—531no—
316C22×C4—1062no—
317C22×C4—1062no—
318C22×C4—2121no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4D4×D5311yescharacteristic
322Q16—22no—
323Q16—22no—
324Q16—42no—
325C2×C8—531no—
326C2×C8—531no—
327C2×C8—1062no—
328C2×C8—1062no—
329C2×C8—1062no—
330C2×C8—1061no—
331C2×C8—1061no—
332C2×C8—1061no—
333C2×C8—1061no—
334C2×C8#1171994512yes—
335C2×C8D4×D5311yescharacteristic
336C2×C8#1171994712yes—
337C2×C8C23×D5311yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—1062no—
341C4⋊C4—1061no—
342C4⋊C4—1061no—
343C4⋊C4—2121no—
344C4⋊C4—2121no—
345C4⋊C4—2122no—
346C4⋊C4—2122no—
347C4⋊C4—2121no—
348C22⋊C4—41no—
349C22⋊C4—2121no—
350C22⋊C4—2121no—
351C22⋊C4—2122no—
352C22⋊C4—2122no—
353C22⋊C4—2121no—
354C42—41no—
355C42—2121no—
356C42—2121no—
357C42—2121no—
358C16—22no—
359C16—1062no—
360C23—531no—
361C23—531no—
362C23—1062no—
363C23—1062no—
364C23—2121no—
365C23D8×D5311yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—82no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—531no—
374C2×C4—531no—
375C2×C4—1062no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1062no—
381C2×C4—1062no—
382C2×C4—4242no—
383C2×C4—4242no—
384C2×C4—4242no—
385C2×C4—2121no—
386C2×C4—2121no—
387C2×C4—2122no—
388C2×C4#1313319912yes—
389C2×C4D8×D5311yescharacteristic
390C2×C4#1313324612yes—
391C2×C4C2×D4×D5311yescharacteristic
392C8—1061no—
393C8—1061no—
394C8#1313319912yes—
395C8C2×#1171994711yescharacteristic
396C8C2×#1171994511yescharacteristic
397C22—21no—
398C22—21no—
399C22—532no—
400C22—1061no—
401C22—1061no—
402C22—1061no—
403C22—1061no—
404C22—1061no—
405C22—2124no—
406C22#1681093512yes—
407C22C2×D8×D5311yescharacteristic
408C4—81no—
409C4—81no—
410C4—82no—
411C4—1061no—
412C4—1061no—
413C4—4241no—
414C4—4241no—
415C4—4242no—
416C4#1681093512yes—
417C4C2×#1313319911yescharacteristic
418C4C2×#1313324611yescharacteristic
419C2—22no—
420C2—1062no—
421C2—2121no—
422C2C2×#1681093511yescharacteristic
423C2#510×D5311yescharacteristic
424C2#2028785011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2141C2
socleC22×C53C22×D8
prime radicalC23×D53D8
hypercenter#2141D53
hypocenterC53#33649
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
3. #20276922C2semidirect
4. #20276921C2semidirect
5. #20276916C2semidirect
6. #20276911C2semidirect
7. #20276504C2semidirect
8. #20276503C2semidirect
9. #20276435C2semidirect
11. C53×#2141C2semidirect
30. #16810451C22semidirect
31. #16810450C22semidirect
32. #16810449C22semidirect
33. #16810448C22semidirect
34. #16810427C22semidirect
35. #16810427C22semidirect
41. C53×#358C22semidirect
42. C53×#358C22semidirect
216. #2141D53semidirect
251. #358C2×D53semidirect
252. #358C2×D53semidirect
255. C53#33649semidirect

Maximal subgroups

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

subgroupindex
C2×#168108612
#427×D532
#202769222
#202769212
#202769162
#202769112
#202765042
#202765032
#202764352
#202763252
C53×#21412
#3364953