Finite Group Databasebeta

← #27302342

Subgroups — #27302342

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285139C211yesmaximal, characteristic
2C2×#16810434C211yesmaximal, characteristic, complemented
3C2×#16810433C211yesmaximal, characteristic, complemented
4C2×#16810432C211yesmaximal, characteristic, complemented
5M32×Dic53C211yesmaximal, characteristic, complemented
6#20274876C211yesmaximal, characteristic, complemented
7C2×C53×#360C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9C2×#13133237—21no—
10#16811169—21no—
11C2×#13133157C2211yescharacteristic
12#16811008C411yescharacteristic, complemented
13#16811006C411yescharacteristic, complemented
14#16811004—21no—
15#16811001C411yescharacteristic, complemented
16#16810998C411yescharacteristic, complemented
17#16810994—21no—
18C2×#13133150—21no—
19C2×#13133149C2211yescharacteristic
20C2×C8×Dic53C2211yescharacteristic
21#16810488—21no—
22#16810434—22no—
23#16810433—22no—
24#16810432—22no—
25#16810398—21no—
26C16×Dic53—21no—
27C2×#135×C53C2211yescharacteristic
28C2×M32×C53C2211yescharacteristic
29C2×#106×C53C2211yescharacteristic
30C53×#360—22no—
31C2×#11719945—21no—
32#13133239—21no—
33#13133239—21no—
34#13133239—21no—
35#13133239—21no—
36#13133237—42no—
37#13133237—42no—
38C2×Dic212C2×C411yescharacteristic
39C2×#11719925—21no—
40C2×#11719913—21no—
41C2×#11719923—21no—
42C2×#11719924—21no—
43C2×D424C2×C411yescharacteristic
44C2×#11719922—21no—
45C22×#88322D411yescharacteristic
46#13133215—21no—
47#13133213—21no—
48#13133206—21no—
49#13133204—21no—
50#13133189—42no—
51C4×C22⋊D53—42no—
52C2×C4⋊Dic53—21no—
53C2×C4×Dic53D411yescharacteristic
54#13133157—21no—
55#13133157D412yescomplemented
56C2×#11719908—21no—
57#13133150—42no—
58#13133149—21no—
59#13133149—21no—
60C8×Dic53C2×C411yescharacteristic
61C8×Dic53C2×C411yescharacteristic
62C8×Dic53D411yescharacteristic
63C8×Dic53D411yescharacteristic
64#13133124—42no—
65C2×#41×C53—21no—
66#135×C53—21no—
67#135×C53—21no—
68#135×C53—21no—
69#135×C53—21no—
70C2×Q16×C53C2×C411yescharacteristic
71C2×SD16×C53—21no—
72C2×D8×C53C2×C411yescharacteristic
73C22×C8×C53C2311yescharacteristic
74C2×C4⋊C4×C53—21no—
75M32×C53—21no—
76M32×C53D412yescomplemented
77C2×C16×C53—21no—
78#106×C53—21no—
79#106×C53—21no—
80#41×C53—41no—
81#41×C53—41no—
82#41×C53—41no—
83#41×C53—41no—
84C2×Q8×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90C2×C22⋊D53—41no—
91C2×C22⋊D53—42no—
92C22×Dic53—41no—
93C22×Dic53D812yes—
94#11719945—41no—
95#11719945—41no—
96#11719945—41no—
97#11719945—41no—
98C2×D212—21no—
99C2×C4×D53—41no—
100C2×Dic106—21no—
101Q16×C53—21no—
102Q16×C53—21no—
103SD16×C53—41no—
104SD16×C53—41no—
105D8×C53—21no—
106D8×C53—21no—
107C2×C8×C53C22×C411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C22⋊Dic53—42no—
116#11719925—42no—
117#11719924—42no—
118#11719923—42no—
119#11719922—42no—
120#11719921—42no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123#11719919—42no—
124C4×Dic53C22⋊C411yescharacteristic
125C4×Dic53C22⋊C411yescharacteristic
126C4×Dic53SD1612yes—
127C2×#88322C22⋊C411yescharacteristic
128C2×#88322C22⋊C411yescharacteristic
129C2×#88322D812yes—
130C2×#88322SD1612yes—
131Dic212—21no—
132Dic212—21no—
133D424—21no—
134D424—21no—
135#11719913—41no—
136#11719913—41no—
137C16×C53—22no—
138#11719908—22no—
139C23×C53—41no—
140C23×C53C2×D811yescharacteristic
141C22×D53—41no—
142Q8×C53—41no—
143Q8×C53—41no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—81no—
147D4×C53—81no—
148C2×C4×C53—41no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53—82no—
152C2×C4×C53—81no—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53#12411yescharacteristic
155C2×C4×C53#12711yescharacteristic
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C2×SD1611yescharacteristic
158C2×C4×C53C2×SD1611yescharacteristic
159C22⋊D53—84no—
160C22⋊D53—81no—
161C22⋊D53—81no—
162C2×Dic53—42no—
163C2×Dic53—41no—
164C2×Dic53—82no—
165C2×Dic53—81no—
166C2×Dic53#10212yes—
167C2×Dic53#10212yes—
168D212—41no—
169D212—41no—
170C4×D53—81no—
171C4×D53—81no—
172Dic106—41no—
173Dic106—41no—
174C8×C53C4×D411yescharacteristic
175C8×C53C4×D411yescharacteristic
176C8×C53#12111yescharacteristic
177C8×C53#12111yescharacteristic
178#88322#10212yes—
179#88322#10212yes—
180#13211—531nomaximal, Sylow
181C22×C53—22no—
182C22×C53—82no—
183C22×C53—81no—
184C22×C53C2×#10211yescharacteristic
185C22×C53#48611yescharacteristic
186C22×C53#49311yescharacteristic
187C2×D53—82no—
188C2×D53—81no—
189C4×C53—81no—
190C4×C53—161no—
191C4×C53#39011yescharacteristic
192C4×C53C2×#10211yescharacteristic
193C4×C53#48611yescharacteristic
194C4×C53#49211yescharacteristic
195Dic53—22no—
196Dic53—22no—
197Dic53—81no—
198Dic53—161no—
199#3147—531no—
200C2×#360—531no—
201C2×#360—531no—
202C2×#360—531no—
203C2×#360D5311yescharacteristic, complemented
204C4×M32—531no—
205#1846—531no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—161no—
209C2×C53#195711yescharacteristic
210C2×C53#214512yes—
211D53—161no—
212C2×#135—531no—
213C2×#135—1061no—
214C2×#135C2×D5311yescharacteristic
215#532—1061no—
216C2×M32—531no—
217C2×M32C2×D5311yescharacteristic
218#495—531no—
219#495—531no—
220#495—1061no—
221#494—531no—
222#493—531no—
223#492—1061no—
224C2×#107—1061no—
225C2×#106—531no—
226C2×#106C2×D5311yescharacteristic
227C2×C4×C8—531no—
228#382—1061no—
229#360—22no—
230#360—1062no—
231#360—1062no—
232#360—1062no—
233#346—1061no—
234C4×C16—1061no—
235C53#1321111yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—1061no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135—21no—
242#135—1061no—
243#135—1061no—
244#135—1061no—
245#135—1061no—
246#135—2122no—
247#135—2122no—
248C2×Q16—531no—
249C2×Q16—1061no—
250C2×Q16C4×D5311yescharacteristic
251C2×SD16—21no—
252C2×SD16—1061no—
253C2×SD16—1061no—
254C2×SD16—1061no—
255C2×D8—531no—
256C2×D8—1061no—
257C2×D8C4×D5311yescharacteristic
258C22×C8—531no—
259C22×C8C22×D5311yescharacteristic
260C4⋊Q8—1061no—
261#127—1061no—
262#124—1061no—
263#124—1061no—
264#121—2122no—
265C4×D4—2122no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—1061no—
268C2×C42—531no—
269M32—21no—
270M32—532no—
271M32—1061no—
272M32C22⋊D5312yescomplemented
273C2×C16—21no—
274C2×C16—1061no—
275#107—2122no—
276#106—21no—
277#106—21no—
278#106—1061no—
279#106—1061no—
280C4×C8—531no—
281C4×C8—531no—
282C4×C8—531no—
283C4×C8—531no—
284#95—2122no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—2121no—
290#41—2121no—
291#41—2121no—
292#41—2121no—
293C2×Q8—21no—
294C2×Q8—1061no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1061no—
298C2×D4—2122no—
299C2×D4—2121no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—532no—
303C22×C4—2121no—
304C22×C4—2121no—
305C22×C4D4×D5311yescharacteristic
306Q16—21no—
307Q16—21no—
308Q16—1061no—
309Q16—1061no—
310Q16—2122no—
311SD16—41no—
312SD16—41no—
313SD16—2122no—
314SD16—2122no—
315SD16—2121no—
316SD16—2121no—
317D8—21no—
318D8—21no—
319D8—1061no—
320D8—1061no—
321D8—2122no—
322C2×C8—531no—
323C2×C8—531no—
324C2×C8—532no—
325C2×C8—532no—
326C2×C8C2×C4×D5311yescharacteristic
327C2×C8#1171994511yescharacteristic
328C2×C8D4×D5311yescharacteristic
329C2×C8D4×D5311yescharacteristic
330C2×C8#1171994711yescharacteristic
331C2×C8C2×C22⋊D5311yescharacteristic
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—2122no—
335C4⋊C4—2121no—
336C4⋊C4—2121no—
337C22⋊C4—2122no—
338C22⋊C4—2122no—
339C42—531no—
340C42—531no—
341C42—532no—
342C16—22no—
343C16—1062no—
344C23—41no—
345C23—2121no—
346C23D8×D5311yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—2121no—
350Q8—2121no—
351D4—41no—
352D4—41no—
353D4—81no—
354D4—81no—
355D4—4244no—
356D4—4241no—
357D4—4241no—
358D4—2121no—
359D4—2121no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—532no—
366C2×C4—532no—
367C2×C4—4242no—
368C2×C4—4241no—
369C2×C4—4241no—
370C2×C4—4241no—
371C2×C4—2122no—
372C2×C4—2121no—
373C2×C4C22⋊C4×D5311yescharacteristic
374C2×C4#1313320311yescharacteristic
375C2×C4#1313321311yescharacteristic
376C2×C4D8×D5311yescharacteristic
377C2×C4SD16×D5311yescharacteristic
378C2×C4SD16×D5311yescharacteristic
379C8—532no—
380C8—532no—
381C8#1313317711yescharacteristic
382C8C4×C22⋊D5311yescharacteristic
383C8#1313318911yescharacteristic
384C8#1313319011yescharacteristic
385C22—22no—
386C22—81no—
387C22—82no—
388C22—4241no—
389C22—4242no—
390C22#102×D5311yescharacteristic
391C22#1681096811yescharacteristic
392C22#1681099811yescharacteristic
393C4—81no—
394C4—161no—
395C4—8481no—
396C4—1062no—
397C4—1062no—
398C4—4241no—
399C4#1681052311yescharacteristic
400C4#102×D5311yescharacteristic
401C4#1681096811yescharacteristic
402C4#1681099411yescharacteristic
403C2—21no—
404C2—21no—
405C2—161no—
406C2—8481no—
407C2#2027551911yescharacteristic
408C2#2027644912yes—
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#360C2
socleC22×C53C2×#102
prime radical#20285139C2
cosocleC22×C8×C53C23
hypercenterC2×#360D53
hypocenterC53#13211
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
2. C2×#16810434C2semidirect
3. C2×#16810433C2semidirect
4. C2×#16810432C2semidirect
5. M32×Dic53C2semidirect
6. #20274876C2semidirect
7. C2×C53×#360C2semidirect
12. #16811008C4semidirect
13. #16811006C4semidirect
15. #16811001C4semidirect
16. #16810998C4semidirect
55. #13133157D4semidirect
76. M32×C53D4semidirect
203. C2×#360D53semidirect
235. C53#13211semidirect
272. M32C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202851392
C2×#168104342
C2×#168104332
C2×#168104322
M32×Dic532
#202748762
C2×C53×#3602
#1321153