Finite Group Databasebeta

← #27538744

Subgroups — #27538744

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285227C211yesmaximal, characteristic, complemented
2C2×D8×Dic53C211yesmaximal, characteristic
3#20276874C212yesmaximal, complemented
4#20276870C212yesmaximal, complemented
5#20276860C212yesmaximal, complemented
6#20276850C212yesmaximal, complemented
7#20276504C211yesmaximal, characteristic, complemented
8#20276503C211yesmaximal, characteristic, complemented
9#20276470C211yesmaximal, characteristic, complemented
10#20276431C211yesmaximal, characteristic
11C53×#2140C211yesmaximal, characteristic
12#16811176—21no—
13C2×D4×Dic53—21no—
14C2×#13133157C2211yescharacteristic
15#16811027C2211yescharacteristic, complemented
16#16811027C2211yescharacteristic, complemented
17#16811024C2211yescharacteristic, complemented
18#16811024C2211yescharacteristic, complemented
19#16811023—21no—
20#16811020—21no—
21D8×Dic53—21no—
22D8×Dic53—21no—
23D8×Dic53C2212yes—
24D8×Dic53C2212yes—
25#16810661C2211yescharacteristic
26#16810659—21no—
27C2×#13133151C2211yescharacteristic
28C2×#13133138—21no—
29C2×C8×Dic53C2211yescharacteristic
30#16810451C2212yescomplemented
31#16810450C2212yescomplemented
32#16810449C2212yescomplemented
33#16810448C2212yescomplemented
34#16810441C2212yescomplemented
35#16810441C2212yescomplemented
36#16810424C2212yes—
37#16810424C2212yes—
38C22×D8×C53C2211yescharacteristic
39C2×M32×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C53×#357C2212yes—
42C53×#357C2212yes—
43C23×Dic53—41no—
44C22×#88322D411yescharacteristic
45#13133216—21no—
46#13133216—21no—
47#13133208—21no—
48#13133208—21no—
49#13133194—41no—
50#13133194—41no—
51C4×Dic106—41no—
52C4×Dic106—41no—
53D4×Dic53—22no—
54D4×Dic53—22no—
55D4×Dic53—41no—
56D4×Dic53—41no—
57#13133176—21no—
58C2×C4⋊Dic53—21no—
59C2×C22⋊Dic53—41no—
60C2×C4×Dic53D411yescharacteristic
61#13133157—22no—
62C2×#11719908C2312yes—
63#13133151—21no—
64#13133151C2311yescharacteristic
65#13133151C2311yescharacteristic
66#13133151C2312yes—
67#13133150—21no—
68#13133150—21no—
69#13133149—21no—
70#13133148—21no—
71#13133148—21no—
72#13133138—22no—
73#13133138—22no—
74C8×Dic53D411yescharacteristic
75C8×Dic53D411yescharacteristic
76C8×Dic53C2311yescharacteristic
77C8×Dic53C2311yescharacteristic
78C22×D4×C53—21no—
79C2×D8×C53—21no—
80C2×D8×C53—22no—
81C2×D8×C53Q814yescomplemented
82C2×D8×C53C2312yes—
83C22×C8×C53C2311yescharacteristic
84#117×C53—21no—
85M32×C53—22no—
86C2×C16×C53C2312yes—
87#107×C53C2311yescharacteristic
88#107×C53C2311yescharacteristic
89#106×C53—21no—
90C24×C53—41no—
91C2×D4×C53—24no—
92C2×D4×C53—22no—
93C2×D4×C53—41no—
94C2×D4×C53—42no—
95C22×C4×C53C2×D411yescharacteristic
96C22×Dic53—41no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—84no—
100C22×Dic53D811yescharacteristic
101C22×Dic53D811yescharacteristic
102C2×Dic106—41no—
103C2×Dic106—41no—
104D8×C53—24no—
105D8×C53—22no—
106D8×C53C2×Q814yes—
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×Q812yes—
109C2×C8×C53#4112yes—
110C2×C8×C53C2411yescharacteristic
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C22⋊C4×C53—41no—
114C42×C53—41no—
115C22⋊Dic53—41no—
116C22⋊Dic53—42no—
117C22⋊Dic53—42no—
118C4⋊Dic53—22no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C4⋊Dic53—41no—
124#11719919—41no—
125#11719919—41no—
126#11719919—41no—
127#11719919—41no—
128C4×Dic53—41no—
129C4×Dic53D811yescharacteristic
130C4×Dic53D811yescharacteristic
131C4×Dic53C2×D411yescharacteristic
132C4×Dic53C2×D411yescharacteristic
133C2×#88322—22no—
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136C16×C53—22no—
137#11719908—22no—
138C23×C53—42no—
139C23×C53—44no—
140C23×C53—82no—
141C23×C53—82no—
142C23×C53C2×D811yescharacteristic
143D4×C53—24no—
144D4×C53—42no—
145D4×C53—44no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53C22⋊Q812yes—
149C2×C4×C53C2×D811yescharacteristic
150C2×C4×C53#13712yes—
151C2×C4×C53C22×D411yescharacteristic
152C2×Dic53—22no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—42no—
157C2×Dic53—84no—
158C2×Dic53—82no—
159C2×Dic53—84no—
160C2×Dic53C2×D811yescharacteristic
161C2×Dic53C2×D811yescharacteristic
162C2×Dic53C2×D811yescharacteristic
163C2×Dic53C2×D811yescharacteristic
164Dic106—82no—
165Dic106—82no—
166C8×C53C22⋊Q812yes—
167C8×C53C22×Q811yescharacteristic
168C8×C53C2×#4111yescharacteristic
169#88322—21no—
170#88322—21no—
171#33627—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—44no—
175C22×C53—88no—
176C22×C53—82no—
177C22×C53—82no—
178C22×C53#47412yes—
179C22×C53C22×D811yescharacteristic
180C4×C53—81no—
181C4×C53—81no—
182C4×C53#47412yes—
183C4×C53C2×C22⋊Q811yescharacteristic
184C4×C53C2×#13711yescharacteristic
185Dic53—21no—
186Dic53—21no—
187Dic53—21no—
188Dic53—81no—
189Dic53—82no—
190Dic53—81no—
191Dic53#50912yes—
192#3160—531no—
193C2×C4×D8—531no—
194#2228—532no—
195#2228—532no—
196#2226—532no—
197#2224—532no—
198#2159—531no—
199#2159—531no—
200#2150—531no—
201#2140—531no—
202#2140D5311yescharacteristic
203C2×C53—22no—
204C2×C53—84no—
205C2×C53C2×#47411yescharacteristic
206C2×C53C2×#50911yescharacteristic
207C2×C53#341411yescharacteristic
208C22×D8C2×D5311yescharacteristic
209#533—1061no—
210C2×C4×D4—1061no—
211C2×M32—531no—
212C2×M32C2×D5311yescharacteristic
213#500—531no—
214#500—531no—
215#499—531no—
216#499—531no—
217#499—1061no—
218#498—1061no—
219C4×D8—532no—
220C4×D8—532no—
221C4×D8—1061no—
222C4×D8—1061no—
223#427—531no—
224#427—1061no—
225#427C2×D5311yescharacteristic
226C2×#107—531no—
227C2×#102—1061no—
228C2×C4×C8—531no—
229#367—532no—
230#367—532no—
231#366—532no—
232#366—532no—
233C4⋊C16—532no—
234C4⋊C16—532no—
235#357—532no—
236#357—532no—
237#357C2×D5312yes—
238#357C2×D5312yes—
239C53#3362711yescharacteristic, complemented, Sylow
240C22×D4—21no—
241C23×C4—2121no—
242C2×D8—21no—
243C2×D8—22no—
244C2×D8Dic10614yescomplemented
245C2×D8C22×D5312yes—
246C22×C8—531no—
247C22×C8C22×D5311yescharacteristic
248C4⋊Q8—1061no—
249C4⋊Q8—1061no—
250#125—1061no—
251#125—1061no—
252C22⋊Q8—2121no—
253C22⋊Q8—2121no—
254C4×Q8—2121no—
255C4×Q8—2121no—
256C4×D4—1062no—
257C4×D4—1062no—
258C4×D4—2121no—
259C4×D4—2121no—
260#117—21no—
261#117—1061no—
262C2×C4⋊C4—1061no—
263C2×C22⋊C4—2121no—
264C2×C42—531no—
265M32—22no—
266M32—1062no—
267C2×C16—532no—
268C2×C16C22×D5312yes—
269#107—531no—
270#107—531no—
271#107—532no—
272#107—1061no—
273#107—1061no—
274#107—1061no—
275#107C22×D5311yescharacteristic
276#107C22×D5311yescharacteristic
277#106—21no—
278#106—1061no—
279#106—1061no—
280#106—1061no—
281#102—1062no—
282#102—1062no—
283C4×C8—531no—
284C4×C8—531no—
285C4×C8—531no—
286C4×C8—531no—
287C24—41no—
288C2×Q8—2121no—
289C2×Q8—2121no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—42no—
294C22×C4—531no—
295C22×C4—531no—
296C22×C4—4244no—
297C22×C4—2122no—
298C22×C4—2122no—
299C22×C4—2121no—
300C22×C4D4×D5311yescharacteristic
301D8—24no—
302D8—22no—
303D8C2×Dic10614yes—
304C2×C8—531no—
305C2×C8—531no—
306C2×C8—1062no—
307C2×C8C2×Dic10612yes—
308C2×C8D4×D5311yescharacteristic
309C2×C8#1171994712yes—
310C2×C8C23×D5311yescharacteristic
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—1062no—
314C4⋊C4—1061no—
315C4⋊C4—1061no—
316C4⋊C4—2121no—
317C4⋊C4—2121no—
318C4⋊C4—2121no—
319C4⋊C4—2121no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C4⋊C4—2121no—
323C22⋊C4—41no—
324C22⋊C4—2122no—
325C22⋊C4—2122no—
326C22⋊C4—2121no—
327C42—41no—
328C42—531no—
329C42—531no—
330C42—531no—
331C42—531no—
332C42—2121no—
333C16—22no—
334C16—1062no—
335C23—42no—
336C23—44no—
337C23—82no—
338C23—82no—
339C23D8×D5311yescharacteristic
340Q8—4242no—
341Q8—4242no—
342D4—24no—
343D4—44no—
344D4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—531no—
348C2×C4—531no—
349C2×C4—531no—
350C2×C4—531no—
351C2×C4—1062no—
352C2×C4—4244no—
353C2×C4—4244no—
354C2×C4—4242no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2122no—
359C2×C4#1313319412yes—
360C2×C4D8×D5311yescharacteristic
361C2×C4#1313324612yes—
362C2×C4C2×D4×D5311yescharacteristic
363C8—1061no—
364C8—1061no—
365C8#1313319412yes—
366C8C22×Dic10611yescharacteristic
367C8C2×#1171994711yescharacteristic
368C22—21no—
369C22—21no—
370C22—44no—
371C22—88no—
372C22—82no—
373C22—82no—
374C22#1681090112yes—
375C22C2×D8×D5311yescharacteristic
376C4—81no—
377C4—81no—
378C4—532no—
379C4—1061no—
380C4—1061no—
381C4—1061no—
382C4—4242no—
383C4—4241no—
384C4—4241no—
385C4#1681090112yes—
386C4C2×#1313319411yescharacteristic
387C4C2×#1313324611yescharacteristic
388C2—22no—
389C2—84no—
390C2C2×#1681090111yescharacteristic
391C2#509×D5311yescharacteristic
392C2#2028784711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2140C2
socleC22×C53C22×D8
prime radicalC22×D8×C53C22
hypercenter#2140D53
hypocenterC53#33627
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2140

Complements

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

NH ≅ G/Ntype
1. #20285227C2semidirect
3. #20276874C2semidirect
4. #20276870C2semidirect
5. #20276860C2semidirect
6. #20276850C2semidirect
7. #20276504C2semidirect
8. #20276503C2semidirect
9. #20276470C2semidirect
15. #16811027C22semidirect
16. #16811027C22semidirect
17. #16811024C22semidirect
18. #16811024C22semidirect
30. #16810451C22semidirect
31. #16810450C22semidirect
32. #16810449C22semidirect
33. #16810448C22semidirect
34. #16810441C22semidirect
35. #16810441C22semidirect
81. C2×D8×C53Q8semidirect
239. C53#33627semidirect
244. C2×D8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202852272
C2×D8×Dic532
#202768742
#202768702
#202768602
#202768502
#202765042
#202765032
#202764702
#202764312
C53×#21402
#3362753