Finite Group Databasebeta

← #27537509

Subgroups — #27537509

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287849C211yesmaximal, characteristic
2#20283633C211yesmaximal, characteristic, complemented
3#20276599C214yesmaximal, complemented
4#20276594C214yesmaximal, complemented
5#20276509C211yesmaximal, characteristic, complemented
6C2×#16810430C211yesmaximal, characteristic, complemented
7C2×#16810429C211yesmaximal, characteristic, complemented
8#20276312C211yesmaximal, characteristic, complemented
9C53×#2957C211yesmaximal, characteristic, complemented
10C2×#13133239C2211yescharacteristic
11#16811389—21no—
12#16811064C412yescomplemented
13#16811063C412yescomplemented
14#16811059C412yescomplemented
15#16811058C412yescomplemented
16#16811053C2214yes—
17C2×#13133157C2211yescharacteristic
18C22×#11719908C2211yescharacteristic
19#16810746—22no—
20#16810741C2214yescomplemented
21C2×#13133153C2211yescharacteristic
22#16810641—21no—
23#16810452C2214yescomplemented
24#16810430C2212yescomplemented
25#16810430C2212yescomplemented
26#16810429C2212yescomplemented
27#16810429C2212yescomplemented
28#16810399C2212yescomplemented
29C4×#11719908C2212yescomplemented
30C2×#135×C53C2211yescharacteristic
31C53×#568—21no—
32C53×#444—22no—
33C53×#443C2214yescomplemented
34C2×#108×C53C2211yescharacteristic
35C53×#421—21no—
36C53×#405C2211yescharacteristic
37C2×#11719945—21no—
38#13133239—21no—
39#13133239—21no—
40#13133239C2×C412yes—
41#13133239C2312yes—
42C2×Dic212C2×C411yescharacteristic
43C2×#11719913—21no—
44C2×D424C2×C411yescharacteristic
45#13133225—42no—
46#13133223—24no—
47C2×#11719917—21no—
48M16×D53—42no—
49#13133163C2×C414yes—
50#13133161C2×C414yes—
51#13133160C2×C414yes—
52#13133158C2×C414yes—
53#13133157C2×C412yes—
54#13133157C2×C412yes—
55C2×#11719908—21no—
56C2×#11719908—21no—
57C2×#11719908C2311yescharacteristic
58C2×#11719908C2311yescharacteristic
59C2×#11719908C2312yes—
60#13133153—21no—
61#13133153C2312yes—
62#13133145—24no—
63C2×#41×C53—21no—
64#135×C53—21no—
65#135×C53—21no—
66#135×C53C2×C412yes—
67#135×C53C2312yes—
68C2×Q16×C53C2×C411yescharacteristic
69C2×SD16×C53—21no—
70C2×D8×C53C2×C411yescharacteristic
71#131×C53—24no—
72#131×C53—42no—
73C2×M16×C53—21no—
74C2×M16×C53—42no—
75C2×M16×C53D411yescharacteristic
76C22×C8×C53C2311yescharacteristic
77#117×C53D411yescharacteristic
78#108×C53—21no—
79#108×C53C2312yes—
80#104×C53—24no—
81#97×C53D412yes—
82C4×C8×C53C2312yes—
83#41×C53—22no—
84#41×C53—22no—
85#41×C53—41no—
86#41×C53—41no—
87C2×Q8×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C2×C22⋊D53—41no—
93#11719945—22no—
94#11719945—22no—
95#11719945—41no—
96#11719945—41no—
97C2×D212—21no—
98C2×C4×D53—41no—
99C2×Dic106—21no—
100Q16×C53—21no—
101Q16×C53C22×C412yes—
102SD16×C53—22no—
103SD16×C53—22no—
104D8×C53—21no—
105D8×C53C22×C412yes—
106M16×C53—22no—
107M16×C53—41no—
108M16×C53—44no—
109M16×C53D812yes—
110M16×C53D812yes—
111C2×C8×C53—41no—
112C2×C8×C53—44no—
113C2×C8×C53C22×C411yescharacteristic
114C2×C8×C53C22×C411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D412yes—
117C2×C8×C53#4111yescharacteristic
118C2×C8×C53#4111yescharacteristic
119C2×C8×C53C2411yescharacteristic
120C4⋊C4×C53—21no—
121C22⋊C4×C53—21no—
122C42×C53C2×D412yes—
123#11719917—22no—
124#11719917—41no—
125C2×#88322—41no—
126Dic212—21no—
127Dic212C22×C412yes—
128D424—21no—
129D424C22×C412yes—
130#11719913—22no—
131#11719913—22no—
132#11719912—44no—
133C8×D53—44no—
134#11719908—21no—
135#11719908—21no—
136#11719908C22×C414yes—
137C23×C53—41no—
138C23×C53#13511yescharacteristic
139C22×D53—41no—
140Q8×C53—22no—
141Q8×C53—41no—
142D4×C53—22no—
143D4×C53—42no—
144D4×C53—42no—
145D4×C53—41no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53#13511yescharacteristic
155C2×C4×C53C22×D411yescharacteristic
156C22⋊D53—42no—
157C22⋊D53—42no—
158C2×Dic53—41no—
159D212—22no—
160D212—41no—
161C4×D53—42no—
162C4×D53—42no—
163Dic106—22no—
164Dic106—41no—
165C8×C53—42no—
166C8×C53C4×D411yescharacteristic
167C8×C53C4×D411yescharacteristic
168C8×C53C2×D814yes—
169C8×C53C23×C411yescharacteristic
170C8×C53C2×#4111yescharacteristic
171#88322—42no—
172#33464—531nomaximal, Sylow
173C22×C53—41no—
174C22×C53—42no—
175C22×C53—161no—
176C22×C53C4×D811yescharacteristic
177C22×C53C4×D811yescharacteristic
178C22×C53C2×#13511yescharacteristic
179C2×D53—42no—
180C2×D53—161no—
181C4×C53—42no—
182C4×C53—42no—
183C4×C53C4×D811yescharacteristic
184C4×C53C4×D811yescharacteristic
185C4×C53C2×C4×D411yescharacteristic
186C4×C53C22×D811yescharacteristic
187Dic53—42no—
188#3414—531no—
189#2957—531no—
190#2957D5311yescharacteristic, complemented
191#2179—534no—
192#2178—534no—
193#2161—531no—
194C2×#359—531no—
195C2×#359—531no—
196#2108—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—82no—
201C2×C53C2×C4×D811yescharacteristic
202D53—82no—
203C2×#135—531no—
204C2×#135C2×D5311yescharacteristic
205#568—21no—
206#568—1061no—
207#510—532no—
208#510—532no—
209#509—532no—
210#509—532no—
211#508—534no—
212C2×M32—531no—
213C22×C16—531no—
214#444—22no—
215#444—1062no—
216#443—534no—
217#443C2×D5314yescomplemented
218C2×#108—531no—
219C2×#108C2×D5311yescharacteristic
220#421—21no—
221#421—1061no—
222#405C2×D5311yescharacteristic
223#368—534no—
224#359—532no—
225#359—532no—
226#359—532no—
227#359—532no—
228#346—532no—
229C4×C16—532no—
230C53#3346411yescharacteristic, complemented, Sylow
231C2×#41—21no—
232C2×#41—1061no—
233#135—21no—
234#135—21no—
235#135—532no—
236#135—532no—
237#135—1061no—
238#135—1061no—
239#135C4×D5312yes—
240#135C22×D5312yes—
241C2×Q16—531no—
242C2×Q16C4×D5311yescharacteristic
243C2×SD16—21no—
244C2×SD16—1061no—
245C2×D8—531no—
246C2×D8C4×D5311yescharacteristic
247#131—24no—
248#131—42no—
249#131—1064no—
250#131—2122no—
251C2×M16—21no—
252C2×M16—42no—
253C2×M16—1061no—
254C2×M16—2122no—
255C2×M16C22⋊D5311yescharacteristic
256C22×C8C22×D5311yescharacteristic
257#117C22⋊D5311yescharacteristic
258Q32—534no—
259SD32—534no—
260SD32—534no—
261D16—534no—
262M32—532no—
263M32—532no—
264C2×C16—532no—
265C2×C16—531no—
266C2×C16—531no—
267C2×C16—1061no—
268C2×C16—1061no—
269#108—21no—
270#108—532no—
271#108—1061no—
272#108C22×D5312yes—
273#104—24no—
274#104—1064no—
275#97C22⋊D5312yes—
276C4×C8C22×D5312yes—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—1062no—
282#41—1062no—
283#41—2121no—
284#41—2121no—
285C2×Q8—21no—
286C2×Q8—1061no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—1061no—
290C2×D4—2121no—
291C22×C4—41no—
292C22×C4—2121no—
293C22×C4C2×C22⋊D5311yescharacteristic
294Q16—21no—
295Q16—532no—
296Q16—1061no—
297Q16C2×C4×D5312yes—
298SD16—22no—
299SD16—22no—
300SD16—1062no—
301SD16—1062no—
302D8—21no—
303D8—532no—
304D8—1061no—
305D8C2×C4×D5312yes—
306M16—22no—
307M16—41no—
308M16—44no—
309M16—1062no—
310M16—2121no—
311M16—2124no—
312M16#1171992212yes—
313M16#1171992212yes—
314C2×C8—41no—
315C2×C8—44no—
316C2×C8—2121no—
317C2×C8—2124no—
318C2×C8C2×C4×D5311yescharacteristic
319C2×C8C2×C4×D5311yescharacteristic
320C2×C8#1171994511yescharacteristic
321C2×C8#1171994511yescharacteristic
322C2×C8C2×C22⋊D5311yescharacteristic
323C2×C8C2×C22⋊D5312yes—
324C2×C8C23×D5311yescharacteristic
325C4⋊C4—21no—
326C22⋊C4—21no—
327C42C2×C22⋊D5312yes—
328C16—534no—
329C16—1061no—
330C16—1061no—
331C23—41no—
332C23—2121no—
333C23#1313323711yescharacteristic
334Q8—22no—
335Q8—41no—
336Q8—1062no—
337Q8—2121no—
338D4—22no—
339D4—42no—
340D4—42no—
341D4—41no—
342D4—1062no—
343D4—2122no—
344D4—2122no—
345D4—2121no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—2121no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4C4×C22⋊D5311yescharacteristic
354C2×C4C4×C22⋊D5311yescharacteristic
355C2×C4C2×#1171992211yescharacteristic
356C2×C4C2×#1171992211yescharacteristic
357C2×C4#1313323711yescharacteristic
358C2×C4C22×C22⋊D5311yescharacteristic
359C8—42no—
360C8—2122no—
361C8C4×C22⋊D5311yescharacteristic
362C8C4×C22⋊D5311yescharacteristic
363C8C2×#1171992214yes—
364C8C22×C4×D5311yescharacteristic
365C8C2×#1171994511yescharacteristic
366C22—41no—
367C22—42no—
368C22—161no—
369C22—8481no—
370C22—2122no—
371C22C4×#1171992211yescharacteristic
372C22C4×#1171992211yescharacteristic
373C22C2×#1313323711yescharacteristic
374C4—42no—
375C4—42no—
376C4—2122no—
377C4C4×#1171992211yescharacteristic
378C4C4×#1171992211yescharacteristic
379C4C2×C4×C22⋊D5311yescharacteristic
380C4C22×#1171992211yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—82no—
385C2—4242no—
386C2C2×C4×#1171992211yescharacteristic
subgroupG/H
centerC4C22×#11719922
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2957C2
socleC2×C53C2×C4×D8
prime radical#20287849C2
cosocleC2×C8×C53C24
hypercenter#2957D53
hypocenterC53#33464
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×M16 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. #20283633C2semidirect
3. #20276599C2semidirect
4. #20276594C2semidirect
5. #20276509C2semidirect
6. C2×#16810430C2semidirect
7. C2×#16810429C2semidirect
8. #20276312C2semidirect
9. C53×#2957C2semidirect
12. #16811064C4semidirect
13. #16811063C4semidirect
14. #16811059C4semidirect
15. #16811058C4semidirect
20. #16810741C22semidirect
23. #16810452C22semidirect
24. #16810430C22semidirect
25. #16810430C22semidirect
26. #16810429C22semidirect
27. #16810429C22semidirect
28. #16810399C22semidirect
29. C4×#11719908C22semidirect
33. C53×#443C22semidirect
190. #2957D53semidirect
217. #443C2×D53semidirect
230. C53#33464semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202836332
#202765992
#202765942
#202765092
C2×#168104302
C2×#168104292
#202763122
C53×#29572
#3346453