Finite Group Databasebeta

← #27538967

Subgroups — #27538967

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284385C211yesmaximal, characteristic
2#20276923C212yesmaximal, complemented
3#20276912C212yesmaximal, complemented
4#20276906C212yesmaximal, complemented
5#20276900C212yesmaximal, complemented
6#20276504C211yesmaximal, characteristic, complemented
7C2×#16810450C211yesmaximal, characteristic, complemented
8#20276425C211yesmaximal, characteristic, complemented
9#20276420C211yesmaximal, characteristic, complemented
10#20276329C211yesmaximal, characteristic, complemented
11C53×#2913C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13#16811201—21no—
14C2×#13133157C2211yescharacteristic
15C22×#11719908C2211yescharacteristic
16#16810869—22no—
17#16810864C2212yes—
18#16810859C2212yes—
19C2×#13133151C2211yescharacteristic
20#16810625—21no—
21#16810450C2212yes—
22#16810450C2212yes—
23#16810450C2212yes—
24#16810448C2212yes—
25#16810426C2211yescharacteristic
26#16810426C2212yes—
27#16810426C2211yescharacteristic
28#16810423C2211yescharacteristic
29#16810423C2212yes—
30#16810423C2211yescharacteristic
31C22⋊#11719908C2212yes—
32C22⋊#11719908C2212yes—
33C2×#131×C53C2211yescharacteristic
34C53×#520—21no—
35C53×#428—22no—
36C53×#428—21no—
37C53×#427C2211yescharacteristic
38C53×#427C2212yes—
39C2×#107×C53C2211yescharacteristic
40C2×#107×C53C2212yes—
41C2×#11719945—21no—
42#13133239—22no—
43#13133239—22no—
44C2×Dic212C2311yescharacteristic
45C2×#11719913—21no—
46C2×D424C2311yescharacteristic
47#13133196—22no—
48#13133193—41no—
49#13133191—22no—
50#13133189—42no—
51#13133188—41no—
52C2×C4⋊Dic53—21no—
53#13133157—22no—
54C2×#11719908—22no—
55C2×#11719908C2311yescharacteristic
56C2×#11719908C2311yescharacteristic
57C2×#11719908C2312yes—
58#13133151—21no—
59#13133151C2312yes—
60#13133142—22no—
61#13133139—22no—
62C2×#41×C53D411yescharacteristic
63#131×C53—22no—
64#131×C53—22no—
65C2×M16×C53D412yes—
66C2×M16×C53D411yescharacteristic
67C22×C8×C53C2312yes—
68C22×C8×C53C2311yescharacteristic
69C4×Q8×C53—41no—
70C4×D4×C53—41no—
71C4×D4×C53—42no—
72#117×C53—22no—
73#117×C53—21no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—21no—
76#107×C53—22no—
77#107×C53—21no—
78#107×C53C2311yescharacteristic
79#107×C53C2312yes—
80#107×C53C2311yescharacteristic
81#106×C53—22no—
82#106×C53—21no—
83#41×C53—22no—
84#41×C53—22no—
85C2×Q8×C53D811yescharacteristic
86C2×D4×C53D811yescharacteristic
87C2×D4×C53D812yes—
88C22×C4×C53—42no—
89C22×C4×C53—41no—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D411yescharacteristic
92C2×C22⋊D53—41no—
93C2×C22⋊D53—42no—
94C22×Dic53—41no—
95#11719945—42no—
96#11719945—42no—
97C2×D212—21no—
98C2×C4×D53—41no—
99C2×Dic106—21no—
100M16×C53—22no—
101M16×C53—24no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53C2×D412yes—
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4112yes—
107C2×C8×C53#4112yes—
108C2×C8×C53C2411yescharacteristic
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—42no—
113C4⋊C4×C53—41no—
114C22⋊C4×C53—42no—
115C22⋊C4×C53—41no—
116C42×C53—42no—
117C42×C53—41no—
118C22⋊Dic53—42no—
119#11719921—42no—
120C4⋊Dic53—22no—
121C4⋊Dic53—41no—
122#11719919—42no—
123Dic212—21no—
124Dic212—21no—
125D424—21no—
126D424—21no—
127#11719913—42no—
128#11719908—21no—
129#11719908—21no—
130#11719908—22no—
131C23×C53C2×D812yes—
132C23×C53C2×D811yescharacteristic
133C22×D53—41no—
134Q8×C53—22no—
135D4×C53—24no—
136D4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—161no—
143C2×C4×C53—162no—
144C2×C4×C53#12312yes—
145C2×C4×C53C2×D812yes—
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53#13712yes—
148C2×C4×C53C22×D411yescharacteristic
149C22⋊D53—82no—
150C22⋊D53—162no—
151C2×Dic53—41no—
152C2×Dic53—42no—
153C2×Dic53—161no—
154D212—41no—
155D212—41no—
156C4×D53—82no—
157Dic106—41no—
158Dic106—41no—
159C8×C53—22no—
160C8×C53#12312yes—
161C8×C53C2×#4111yescharacteristic
162C8×C53C2×#4111yescharacteristic
163#33653—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53#48012yes—
170C22×C53C22×D811yescharacteristic
171C2×D53—81no—
172C2×D53—81no—
173C2×D53—81no—
174C4×C53—22no—
175C4×C53—81no—
176C4×C53—82no—
177C4×C53—81no—
178C4×C53#48012yes—
179C4×C53C2×#12311yescharacteristic
180C4×C53C2×#13711yescharacteristic
181Dic53—81no—
182Dic53—82no—
183#3061—531no—
184#2913D5311yescharacteristic, complemented
185#2237—532no—
186#2235—532no—
187#2234—532no—
188#2233—532no—
189#2159—531no—
190C2×#367—531no—
191#2139—531no—
192#2138—531no—
193#2113—531no—
194C2×C53—22no—
195C2×C53—42no—
196C2×C53C2×#48011yescharacteristic
197C2×C53#341611yescharacteristic
198C2×C53#341611yescharacteristic
199D53—161no—
200C2×#135—531no—
201C2×#131C2×D5311yescharacteristic
202#537—1061no—
203#520—21no—
204C2×M32—531no—
205C22×C16—531no—
206#468—1062no—
207#467—532no—
208#466—532no—
209#428—21no—
210#428—22no—
211#427C2×D5311yescharacteristic
212#427C2×D5312yes—
213C2×#107—531no—
214C2×#107C2×D5311yescharacteristic
215C2×#107C2×D5312yes—
216#417—1061no—
217#367—532no—
218#367—532no—
219#367—532no—
220#366—532no—
221#358—531no—
222#358—531no—
223#358—532no—
224#357—531no—
225#357—531no—
226#357—532no—
227C22⋊C16—532no—
228C22⋊C16—532no—
229C53#3365311yescharacteristic, complemented, Sylow
230C2×#41—1061no—
231C2×#41C22⋊D5311yescharacteristic
232#135—1062no—
233#135—1062no—
234C2×Q16—531no—
235C2×SD16—1061no—
236C2×D8—531no—
237#131—22no—
238#131—22no—
239C2×M16C22⋊D5311yescharacteristic
240C2×M16C22⋊D5312yes—
241C22×C8C22×D5312yes—
242C22×C8C22×D5311yescharacteristic
243C22⋊Q8—1062no—
244C22⋊Q8—2121no—
245#121—1062no—
246#121—2121no—
247#121—2122no—
248C4×Q8—41no—
249C4×D4—42no—
250C4×D4—41no—
251#117—22no—
252#117—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1061no—
256M32—1062no—
257C2×C16—532no—
258C2×C16—531no—
259C2×C16—531no—
260C2×C16—1062no—
261#107—22no—
262#107—21no—
263#107—532no—
264#107—1061no—
265#107C22×D5311yescharacteristic
266#107C22×D5311yescharacteristic
267#107C22×D5312yes—
268#106—22no—
269#106—21no—
270#103—1062no—
271#102—1062no—
272#41—22no—
273#41—22no—
274#41—2122no—
275#41—2122no—
276C2×Q8—1061no—
277C2×Q8#1171992211yescharacteristic
278C2×D4—1061no—
279C2×D4—2122no—
280C2×D4—2121no—
281C2×D4#1171992212yes—
282C2×D4#1171992211yescharacteristic
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4C2×C22⋊D5312yes—
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—1061no—
290Q16—1061no—
291SD16—2122no—
292D8—1061no—
293D8—1061no—
294M16—22no—
295M16—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8#1171994512yes—
299C2×C8#1171994512yes—
300C2×C8C2×C22⋊D5312yes—
301C2×C8C2×C22⋊D5311yescharacteristic
302C2×C8C23×D5311yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—1062no—
309C4⋊C4—2121no—
310C4⋊C4—2122no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—2122no—
314C22⋊C4—2122no—
315C42—42no—
316C42—41no—
317C16—1061no—
318C16—1061no—
319C16—1062no—
320C23—2121no—
321C23C2×#1171992212yes—
322C23C2×#1171992211yescharacteristic
323Q8—22no—
324Q8—2121no—
325Q8—2121no—
326D4—24no—
327D4—22no—
328D4—8482no—
329D4—4242no—
330D4—2121no—
331D4—2121no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—161no—
338C2×C4—162no—
339C2×C4—8481no—
340C2×C4—4242no—
341C2×C4—2122no—
342C2×C4—2121no—
343C2×C4#1313320012yes—
344C2×C4C2×#1171992212yes—
345C2×C4C2×#1171992211yescharacteristic
346C2×C4#1313324812yes—
347C2×C4C22×C22⋊D5311yescharacteristic
348C8—22no—
349C8#1313320012yes—
350C8C2×#1171994511yescharacteristic
351C8C2×#1171994511yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—4241no—
358C22—4241no—
359C22—4241no—
360C22#1681093612yes—
361C22C22×#1171992211yescharacteristic
362C4—22no—
363C4—81no—
364C4—82no—
365C4—81no—
366C4—4242no—
367C4—4241no—
368C4#1681093612yes—
369C4C2×#1313320011yescharacteristic
370C4C2×#1313324811yescharacteristic
371C2—22no—
372C2—42no—
373C2—8481no—
374C2C2×#1681093611yescharacteristic
375C2#2028785911yescharacteristic
376C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2913C2
socleC22×C53C22×D8
prime radical#20284385C2
hypercenter#2913D53
hypocenterC53#33653
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #2913

Complements

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

NH ≅ G/Ntype
2. #20276923C2semidirect
3. #20276912C2semidirect
4. #20276906C2semidirect
5. #20276900C2semidirect
6. #20276504C2semidirect
7. C2×#16810450C2semidirect
8. #20276425C2semidirect
9. #20276420C2semidirect
10. #20276329C2semidirect
11. C53×#2913C2semidirect
184. #2913D53semidirect
229. C53#33653semidirect

Maximal subgroups

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

subgroupindex
#202843852
#202769232
#202769122
#202769062
#202769002
#202765042
C2×#168104502
#202764252
#202764202
#202763292
C53×#29132
#3365353