Finite Group Databasebeta

← #27539178

Subgroups — #27539178

Order 13568.

548 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285301C211yesmaximal, characteristic, complemented
2#20276923C212yesmaximal, complemented
3#20276867C211yesmaximal, characteristic, complemented
4#20276862C211yesmaximal, characteristic, complemented
5#20276765C211yesmaximal, characteristic, complemented
6#20276760C211yesmaximal, characteristic, complemented
7#20276655C212yesmaximal
8#20276584C212yesmaximal, complemented
9#20276579C212yesmaximal, complemented
10#20276518C211yesmaximal, characteristic, complemented
11C4×#135×C53C211yesmaximal, characteristic
12#16811180—21no—
13C2×#13133163C2212yes—
14C2×#13133161C2212yes—
15#16811028C2211yescharacteristic
16#16811003C2211yescharacteristic
17#16810974—21no—
18#16810864—21no—
19#16810864C2212yes—
20C4×Dic212—21no—
21C4×Dic212C2212yes—
22#16810450C2212yes—
23#16810448C2212yes—
24C4⋊#11719908C2211yescharacteristic
25C4⋊#11719908C2211yescharacteristic
26#16810427C2212yes—
27#16810426C2212yes—
28#16810426C2212yes—
29#16810424C2212yes—
30C22⋊#11719908C2212yes—
31#16810399C2212yes—
32C2×#135×C53C2211yescharacteristic
33C4×#41×C53—21no—
34C4×Q16×C53C2211yescharacteristic
35C4×Q16×C53C2211yescharacteristic
36C4×SD16×C53—21no—
37C4×SD16×C53—21no—
38C4×D8×C53C2211yescharacteristic
39C4×D8×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C53×#416—21no—
42C2×C4×C8×C53C2211yescharacteristic
43C2×Dic212—21no—
44C2×Dic212C2312yes—
45#13133217—21no—
46#13133217—21no—
47#13133209—41no—
48#13133196—22no—
49#13133196—41no—
50C4×Dic106—22no—
51C4×Dic106—41no—
52#13133176—41no—
53#13133163—24no—
54#13133161—24no—
55C2×#11719908C2312yes—
56C2×#11719908C2312yes—
57#13133151—21no—
58#13133151C2312yes—
59#13133142—22no—
60#13133142—22no—
61C2×#41×C53—21no—
62#135×C53—22no—
63#135×C53—22no—
64C2×Q16×C53C2311yescharacteristic
65C2×SD16×C53—21no—
66C2×D8×C53C2311yescharacteristic
67C22×C8×C53D411yescharacteristic
68C22×C8×C53C2311yescharacteristic
69C4×Q8×C53—21no—
70C4×Q8×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53—41no—
74C4×D4×C53—41no—
75#117×C53—21no—
76#117×C53—41no—
77C2×C42×C53—41no—
78C2×C42×C53D411yescharacteristic
79#107×C53C2311yescharacteristic
80#107×C53C2311yescharacteristic
81#106×C53—21no—
82#103×C53—21no—
83#103×C53—21no—
84#102×C53—21no—
85#102×C53—21no—
86C4×C8×C53D411yescharacteristic
87C4×C8×C53D411yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C4×C8×C53C2311yescharacteristic
90#41×C53—24no—
91#41×C53—42no—
92C2×Q8×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53D811yescharacteristic
99C22×C4×C53D811yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C2×Dic106—22no—
102C2×Dic106—41no—
103Q16×C53—21no—
104Q16×C53#4112yes—
105SD16×C53—24no—
106D8×C53—21no—
107D8×C53#4112yes—
108C2×C8×C53—21no—
109C2×C8×C53—21no—
110C2×C8×C53—41no—
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53C2411yescharacteristic
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53—41no—
125C42×C53D811yescharacteristic
126C42×C53D811yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C22⋊Dic53—42no—
130C4⋊Dic53—22no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133C4⋊Dic53—41no—
134#11719919—42no—
135#11719919—42no—
136C4×Dic53—42no—
137Dic212—22no—
138Dic212—22no—
139Dic212—42no—
140#11719908—22no—
141#11719908—22no—
142C23×C53—41no—
143C23×C53C2×D811yescharacteristic
144Q8×C53—22no—
145Q8×C53—41no—
146D4×C53—22no—
147D4×C53—44no—
148D4×C53—41no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—22no—
152C2×C4×C53—41no—
153C2×C4×C53—44no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53C2×D811yescharacteristic
160C2×C4×C53C2×D811yescharacteristic
161C2×C4×C53C2×D811yescharacteristic
162C2×C4×C53C2×D811yescharacteristic
163C2×C4×C53C2×D811yescharacteristic
164C2×C4×C53C22×D411yescharacteristic
165C2×Dic53—42no—
166C2×Dic53—42no—
167Dic106—42no—
168Dic106—42no—
169Dic106—82no—
170C8×C53—21no—
171C8×C53—21no—
172C8×C53—21no—
173C8×C53C2×#4111yescharacteristic
174C8×C53C2×#4111yescharacteristic
175#33676—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—42no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53C22×D811yescharacteristic
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—42no—
188C4×C53—81no—
189C4×C53—81no—
190C4×C53#51012yes—
191C4×C53#52511yescharacteristic
192C4×C53#57511yescharacteristic
193Dic53—82no—
194Dic53—82no—
195#3170—531no—
196C4×#135D5311yescharacteristic
197#2237—532no—
198#2227—531no—
199#2226—531no—
200#2209—531no—
201#2208—531no—
202#2189—532no—
203#2176—532no—
204#2175—532no—
205#2163—531no—
206C2×C53—41no—
207C2×C53—82no—
208C2×C53#310011yescharacteristic
209C2×C53C2×#51011yescharacteristic
210C2×C53#341411yescharacteristic
211C2×#135C2×D5311yescharacteristic
212#533—1061no—
213C4×#41—21no—
214C2×Q32—532no—
215C2×SD32—532no—
216#500—531no—
217#494—531no—
218#487—1061no—
219#467—532no—
220#467—1061no—
221C4×Q16—532no—
222C4×Q16—1061no—
223C4×Q16C2×D5311yescharacteristic
224C4×Q16C2×D5311yescharacteristic
225C4×SD16—21no—
226C4×SD16—21no—
227C4×D8C2×D5311yescharacteristic
228C4×D8C2×D5311yescharacteristic
229#427C2×D5311yescharacteristic
230#416—21no—
231C2×C4×C8C2×D5311yescharacteristic
232#367—532no—
233#366—532no—
234C4⋊C16—531no—
235C4⋊C16—531no—
236#358—532no—
237#358—532no—
238#358—532no—
239#357—532no—
240C22⋊C16—532no—
241#346—532no—
242C53#3367611yescharacteristic, complemented, Sylow
243C2×#41—21no—
244#135—22no—
245#135—22no—
246C2×Q16—532no—
247C2×Q16—1061no—
248C2×Q16C22×D5311yescharacteristic
249C2×SD16—21no—
250C2×D8C22×D5311yescharacteristic
251C22×C8C22⋊D5311yescharacteristic
252C22×C8C22×D5311yescharacteristic
253C4⋊Q8—1061no—
254C4⋊Q8—1061no—
255#125—2121no—
256C22⋊Q8—1062no—
257C22⋊Q8—2121no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—1062no—
261C4×Q8—2121no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—41no—
265C4×D4—41no—
266#117—21no—
267#117—41no—
268#117—2121no—
269C2×C42—41no—
270C2×C42C22⋊D5311yescharacteristic
271Q32—1064no—
272SD32—1064no—
273C2×C16—532no—
274C2×C16—532no—
275#107—532no—
276#107—1061no—
277#107C22×D5311yescharacteristic
278#107C22×D5311yescharacteristic
279#106—21no—
280#103—21no—
281#103—21no—
282#103—1062no—
283#103—1062no—
284#102—21no—
285#102—21no—
286C4×C8C22⋊D5311yescharacteristic
287C4×C8C22⋊D5311yescharacteristic
288C4×C8C22×D5311yescharacteristic
289C4×C8C22×D5311yescharacteristic
290#41—24no—
291#41—42no—
292C2×Q8—21no—
293C2×Q8—1062no—
294C2×Q8—2121no—
295C2×D4—21no—
296C2×D4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4#1171992211yescharacteristic
301C22×C4#1171992211yescharacteristic
302C22×C4C2×C22⋊D5311yescharacteristic
303Q16—21no—
304Q16—1062no—
305Q16—1062no—
306Q16—2122no—
307Q16#1171994512yes—
308SD16—24no—
309D8—21no—
310D8#1171994512yes—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8C2×C22⋊D5311yescharacteristic
315C2×C8C2×C22⋊D5311yescharacteristic
316C2×C8C2×C22⋊D5311yescharacteristic
317C2×C8C23×D5311yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—1062no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C4⋊C4—2122no—
326C4⋊C4—2122no—
327C4⋊C4—2121no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—2122no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—2122no—
336C42#1171992211yescharacteristic
337C42#1171992211yescharacteristic
338C42C2×C22⋊D5311yescharacteristic
339C42C2×C22⋊D5311yescharacteristic
340C16—1062no—
341C16—1062no—
342C23—41no—
343C23C2×#1171992211yescharacteristic
344Q8—22no—
345Q8—41no—
346Q8—4242no—
347Q8—2122no—
348Q8—2122no—
349D4—22no—
350D4—44no—
351D4—41no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4C2×#1171992211yescharacteristic
365C2×C4C2×#1171992211yescharacteristic
366C2×C4C2×#1171992211yescharacteristic
367C2×C4C2×#1171992211yescharacteristic
368C2×C4C2×#1171992211yescharacteristic
369C2×C4C22×C22⋊D5311yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8C2×#1171994511yescharacteristic
374C8C2×#1171994511yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22C22×#1171992211yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—42no—
387C4—81no—
388C4—81no—
389C4—4242no—
390C4—4242no—
391C4#1681106312yes—
392C4#1681113411yescharacteristic
393C4#1681142111yescharacteristic
394C2—41no—
395C2—82no—
396C2#2028471511yescharacteristic
397C2C2×#1681106311yescharacteristic
398C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC4×#135×C53C2
socleC22×C53C22×D8
prime radicalC2×#135×C53C22
hypercenterC4×#135D53
hypocenterC53#33676
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #20285301C2semidirect
2. #20276923C2semidirect
3. #20276867C2semidirect
4. #20276862C2semidirect
5. #20276765C2semidirect
6. #20276760C2semidirect
8. #20276584C2semidirect
9. #20276579C2semidirect
10. #20276518C2semidirect
242. C53#33676semidirect

Maximal subgroups

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

subgroupindex
#202853012
#202769232
#202768672
#202768622
#202767652
#202767602
#202766552
#202765842
#202765792
#202765182
C4×#135×C532
#3367653