Finite Group Databasebeta

← #27539670

Subgroups — #27539670

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285285C211yesmaximal, characteristic
2#20276998C211yesmaximal, characteristic, complemented
3#20276984C211yesmaximal, characteristic, complemented
4#20276960C211yesmaximal, characteristic, complemented
5#20276954C211yesmaximal, characteristic, complemented
6#20276830C212yesmaximal
7#20276821C212yesmaximal, complemented
8C4×#13133163C212yesmaximal, complemented
9C4×#13133160C212yesmaximal, complemented
10C4×#13133157C211yesmaximal, characteristic, complemented
11C53×#3170C211yesmaximal, characteristic, complemented
12#16811157—21no—
13C2×#13133163C2212yes—
14C2×#13133160C2212yes—
15C2×#13133157C2212yes—
16#16811025C2211yescharacteristic
17#16811007C2211yescharacteristic
18#16810978—21no—
19#16810864C2211yescharacteristic
20#16810859C2211yescharacteristic
21#16810852—21no—
22C4×Dic212C2211yescharacteristic
23C4×#11719913—21no—
24C4×D424C2211yescharacteristic
25#16810450C2212yes—
26#16810448C2212yes—
27#16810427C2212yes—
28#16810427C2212yes—
29#16810426C2212yes—
30#16810423C2212yes—
31#16810399C2211yescharacteristic
32#16810399C2211yescharacteristic
33C4×#11719908C2212yes—
34C53×#533—21no—
35C53×#500C2211yescharacteristic
36C53×#494C2211yescharacteristic
37C53×#487—21no—
38C53×#467—21no—
39C53×#467C2212yes—
40C4×Q16×C53—21no—
41C4×Q16×C53C2212yes—
42C2×C4×C8×C53C2211yescharacteristic
43C2×Dic212C2311yescharacteristic
44C2×#11719913—21no—
45C2×D424C2311yescharacteristic
46#13133211—41no—
47#13133209—21no—
48#13133205—21no—
49#13133200—41no—
50#13133196—21no—
51#13133191—21no—
52C4×Dic106—21no—
53C4×D212—21no—
54C4×C22⋊D53—41no—
55#13133174—41no—
56#13133163—24no—
57#13133160—24no—
58#13133157—24no—
59C2×#11719908C2312yes—
60C2×#11719908C2312yes—
61#13133151C2311yescharacteristic
62#13133151C2311yescharacteristic
63#13133149—21no—
64#13133142—21no—
65#13133142—21no—
66#13133139—21no—
67#13133139—21no—
68C2×Q16×C53—21no—
69C2×Q16×C53C2312yes—
70C22×C8×C53D411yescharacteristic
71C22×C8×C53C2311yescharacteristic
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74#125×C53—41no—
75C22⋊Q8×C53—22no—
76C22⋊Q8×C53—41no—
77C4×Q8×C53—22no—
78C4×Q8×C53—41no—
79#117×C53—41no—
80C2×C42×C53D411yescharacteristic
81#107×C53—21no—
82#107×C53C2312yes—
83#103×C53—22no—
84#103×C53—22no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C2×Q8×C53—22no—
90C2×Q8×C53—41no—
91C22×C4×C53D811yescharacteristic
92C22×C4×C53D811yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C2×C22⋊D53—41no—
95C2×D212—21no—
96C2×C4×D53—41no—
97C2×Dic106—21no—
98Q16×C53—22no—
99Q16×C53—22no—
100Q16×C53—42no—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2411yescharacteristic
107C4⋊C4×C53—22no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—42no—
111C4⋊C4×C53—42no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—42no—
114C42×C53—42no—
115C42×C53D811yescharacteristic
116C42×C53D811yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C22⋊Dic53—41no—
120#11719921—41no—
121#11719921—41no—
122#11719921—41no—
123#11719921—41no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126#11719919—41no—
127#11719919—41no—
128#11719919—41no—
129#11719919—41no—
130C4×Dic53—41no—
131Dic212—22no—
132D424—22no—
133#11719913—42no—
134#11719908—22no—
135#11719908#4114yes—
136C23×C53C2×D811yescharacteristic
137C22×D53—41no—
138Q8×C53—42no—
139Q8×C53—42no—
140Q8×C53—82no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53C2×D811yescharacteristic
150C2×C4×C53C2×D811yescharacteristic
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53C22×D411yescharacteristic
153C22⋊D53—82no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157D212—41no—
158D212—41no—
159C4×D53—82no—
160Dic106—41no—
161Dic106—41no—
162C8×C53—21no—
163C8×C53—21no—
164C8×C53#13512yes—
165C8×C53C2×#4112yes—
166#33738—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53C22×D811yescharacteristic
171C2×D53—81no—
172C2×D53—81no—
173C2×D53—81no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—82no—
179C4×C53—82no—
180C4×C53#51012yes—
181C4×C53#53211yescharacteristic
182C4×C53C2×#13511yescharacteristic
183Dic53—81no—
184Dic53—81no—
185Dic53—81no—
186#3170D5311yescharacteristic, complemented
187#3168—531no—
188#2255—531no—
189#2251—531no—
190#2245—531no—
191#2243—531no—
192#2220—532no—
193#2219—532no—
194C4×Q32—532no—
195C4×SD32—532no—
196C4×M32—531no—
197C2×C53—41no—
198C2×C53#316911yescharacteristic
199C2×C53C2×#51011yescharacteristic
200C2×C53#341411yescharacteristic
201D53—161no—
202#533—21no—
203#529—1061no—
204C2×Q32—532no—
205C2×SD32—532no—
206C2×M32—532no—
207#500C2×D5311yescharacteristic
208#499—531no—
209#495—531no—
210#494C2×D5311yescharacteristic
211#488—1061no—
212#487—21no—
213#467—21no—
214#467—531no—
215#467C2×D5312yes—
216#466—531no—
217#465—1061no—
218C4×Q16—21no—
219C4×Q16—531no—
220C4×Q16C2×D5312yes—
221C4×SD16—1061no—
222C4×D8—531no—
223C2×C4×C8C2×D5311yescharacteristic
224#367—532no—
225#366—532no—
226#358—532no—
227#358—532no—
228#358—532no—
229#357—532no—
230#346—531no—
231#346—531no—
232C4×C16—532no—
233C53#3373811yescharacteristic, complemented, Sylow
234C2×Q16—21no—
235C2×Q16—531no—
236C2×Q16C22×D5312yes—
237C2×SD16—1061no—
238C2×D8—531no—
239C22×C8C22⋊D5311yescharacteristic
240C22×C8C22×D5311yescharacteristic
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243#126—2121no—
244#125—41no—
245#125—1061no—
246#124—1061no—
247#123—2121no—
248C22⋊Q8—22no—
249C22⋊Q8—41no—
250C22⋊Q8—1061no—
251#121—1061no—
252C4×Q8—22no—
253C4×Q8—41no—
254C4×Q8—1061no—
255C4×D4—1061no—
256C4×D4—2121no—
257#117—41no—
258#117—2121no—
259C2×C42C22⋊D5311yescharacteristic
260Q32—1064no—
261SD32—1064no—
262M32—1064no—
263C2×C16—532no—
264C2×C16—532no—
265#107—21no—
266#107—531no—
267#107—531no—
268#107C22×D5312yes—
269#106—1061no—
270#103—22no—
271#103—22no—
272#103—1061no—
273#103—1061no—
274#102—1061no—
275#102—1061no—
276C4×C8C22⋊D5311yescharacteristic
277C4×C8C22⋊D5311yescharacteristic
278C4×C8C22×D5311yescharacteristic
279C4×C8C22×D5311yescharacteristic
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—1061no—
283C2×D4—1061no—
284C2×D4—2121no—
285C22×C4—2121no—
286C22×C4#1171992211yescharacteristic
287C22×C4#1171992211yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—42no—
292Q16—1062no—
293SD16—2122no—
294D8—1062no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C2×C8C23×D5311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C4⋊C4—2121no—
312C4⋊C4—2121no—
313C22⋊C4—42no—
314C22⋊C4—2121no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C22⋊C4—2121no—
319C42—42no—
320C42—2121no—
321C42#1171992211yescharacteristic
322C42#1171992211yescharacteristic
323C42C2×C22⋊D5311yescharacteristic
324C42C2×C22⋊D5311yescharacteristic
325C16—534no—
326C16—1062no—
327C23—2121no—
328C23C2×#1171992211yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—82no—
332Q8—2121no—
333Q8—2121no—
334D4—4242no—
335D4—2121no—
336D4—2121no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—4242no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4C2×#1171992211yescharacteristic
348C2×C4C2×#1171992211yescharacteristic
349C2×C4C2×#1171992211yescharacteristic
350C2×C4C2×#1171992211yescharacteristic
351C2×C4C2×#1171992211yescharacteristic
352C2×C4C22×C22⋊D5311yescharacteristic
353C8—21no—
354C8—21no—
355C8#1313323712yes—
356C8#41×D5312yes—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22C22×#1171992211yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—82no—
370C4—4241no—
371C4—4241no—
372C4—4241no—
373C4#1681106412yes—
374C4#1681116911yescharacteristic
375C4C2×#1313323711yescharacteristic
376C2—41no—
377C2—8481no—
378C2#2028529011yescharacteristic
379C2C2×#1681106411yescharacteristic
380C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3170C2
socleC22×C53C22×D8
prime radical#16810859C22
hypercenter#3170D53
hypocenterC53#33738
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3170

Complements

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

NH ≅ G/Ntype
2. #20276998C2semidirect
3. #20276984C2semidirect
4. #20276960C2semidirect
5. #20276954C2semidirect
7. #20276821C2semidirect
8. C4×#13133163C2semidirect
9. C4×#13133160C2semidirect
10. C4×#13133157C2semidirect
11. C53×#3170C2semidirect
186. #3170D53semidirect
233. C53#33738semidirect

Maximal subgroups

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

subgroupindex
#202852852
#202769982
#202769842
#202769602
#202769542
#202768302
#202768212
C4×#131331632
C4×#131331602
C4×#131331572
C53×#31702
#3373853