Finite Group Databasebeta

← #27539019

Subgroups — #27539019

Order 13568.

610 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#20285296C211yesmaximal, characteristic, complemented
2C2×D8×Dic53C211yesmaximal, characteristic
3#20276913C211yesmaximal, characteristic, complemented
4#20276910C211yesmaximal, characteristic, complemented
5#20276860C212yesmaximal, complemented
6#20276758C212yesmaximal, complemented
7#20276644C211yesmaximal, characteristic
8#20276570C212yesmaximal, complemented
9SD32×Dic53C212yesmaximal, complemented
10#20276511C211yesmaximal, characteristic, complemented
11C53×#2187C211yesmaximal, characteristic
12#16811175—21no—
13C2×D4×Dic53—21no—
14C2×#13133161C2212yes—
15#16811027C2211yescharacteristic, complemented
16#16811001C2211yescharacteristic, complemented
17#16810972—21no—
18#16810864C2211yescharacteristic
19#16810862—21no—
20Q16×Dic53C2211yescharacteristic, complemented
21#16810717C2211yescharacteristic, complemented
22#16810715—21no—
23D8×Dic53—21no—
24D8×Dic53—21no—
25D8×Dic53C2212yes—
26D8×Dic53C2212yes—
27C2×#13133151C2211yescharacteristic
28C2×#13133138—21no—
29C2×C8×Dic53C2211yescharacteristic
30#16810451C2212yescomplemented
31#16810450C2212yescomplemented
32#16810441C2212yescomplemented
33#16810428C2212yescomplemented
34#16810427C2212yescomplemented
35#16810424C2212yes—
36C22⋊#11719908C2211yescharacteristic
37C16×Dic53C2212yescomplemented
38C22×D8×C53C2211yescharacteristic
39C2×SD32×C53C2212yes—
40C53×#467C2211yescharacteristic
41C53×#357C2212yes—
42C53×C22⋊C16C2211yescharacteristic
43C23×Dic53—41no—
44C2×Dic212C2311yescharacteristic
45C2×#11719925—21no—
46C22×#88322D411yescharacteristic
47#13133216—21no—
48#13133215—21no—
49#13133207—41no—
50#13133196—21no—
51#13133194—41no—
52Q8×Dic53—21no—
53#13133182—21no—
54#13133182—41no—
55D4×Dic53—22no—
56D4×Dic53—22no—
57D4×Dic53—41no—
58D4×Dic53—41no—
59#13133173—41no—
60C2×C4⋊Dic53—21no—
61C2×C22⋊Dic53—41no—
62C2×C4×Dic53D411yescharacteristic
63#13133161—24no—
64C2×#11719908C2312yes—
65#13133151—21no—
66#13133151C2311yescharacteristic
67#13133151C2312yes—
68#13133150—21no—
69#13133142—21no—
70#13133141—21no—
71#13133140—21no—
72#13133138—22no—
73#13133138—22no—
74C8×Dic53D411yescharacteristic
75C8×Dic53D411yescharacteristic
76C8×Dic53C2311yescharacteristic
77C8×Dic53C2311yescharacteristic
78C22×D4×C53—21no—
79C2×Q16×C53C2311yescharacteristic
80C2×D8×C53—24no—
81C2×D8×C53—21no—
82C2×D8×C53C2312yes—
83C22×C8×C53C2311yescharacteristic
84C22⋊Q8×C53—21no—
85SD32×C53—24no—
86C2×C16×C53C2312yes—
87#107×C53C2311yescharacteristic
88#103×C53—21no—
89C24×C53—41no—
90C2×Q8×C53—21no—
91C2×D4×C53—24no—
92C2×D4×C53—22no—
93C2×D4×C53—42no—
94C2×D4×C53—41no—
95C22×C4×C53C2×D411yescharacteristic
96C22×Dic53—41no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—84no—
100C22×Dic53D811yescharacteristic
101C22×Dic53D811yescharacteristic
102C2×Dic106—21no—
103C2×Dic106—41no—
104Q16×C53—22no—
105D8×C53—24no—
106D8×C53—22no—
107D8×C53#4114yes—
108C2×C8×C53—22no—
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2411yescharacteristic
111C4⋊C4×C53—21no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—42no—
116C22⋊Dic53—42no—
117#11719925—42no—
118C4⋊Dic53—22no—
119C4⋊Dic53—21no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122#11719919—41no—
123#11719919—41no—
124#11719919—41no—
125#11719919—41no—
126#11719919—41no—
127C4×Dic53—41no—
128C4×Dic53—41no—
129C4×Dic53D811yescharacteristic
130C4×Dic53D811yescharacteristic
131C4×Dic53C2×D411yescharacteristic
132C4×Dic53C2×D411yescharacteristic
133C2×#88322—41no—
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136Dic212—22no—
137C16×C53—22no—
138#11719908—22no—
139C23×C53—44no—
140C23×C53—42no—
141C23×C53—82no—
142C23×C53—82no—
143C23×C53C2×D811yescharacteristic
144Q8×C53—41no—
145Q8×C53—41no—
146D4×C53—24no—
147D4×C53—44no—
148D4×C53—42no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53C22×D411yescharacteristic
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—42no—
159C2×Dic53—82no—
160C2×Dic53—84no—
161C2×Dic53—84no—
162C2×Dic53SD3214yescomplemented
163C2×Dic53C2×D811yescharacteristic
164C2×Dic53C2×D811yescharacteristic
165C2×Dic53C2×D811yescharacteristic
166C2×Dic53C2×D811yescharacteristic
167Dic106—41no—
168Dic106—41no—
169Dic106—82no—
170C8×C53—21no—
171C8×C53C2×#4112yes—
172#88322—21no—
173#88322—21no—
174#33661—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—44no—
178C22×C53—88no—
179C22×C53—82no—
180C22×C53—82no—
181C22×C53C2×SD3212yes—
182C22×C53C22×D811yescharacteristic
183C4×C53—21no—
184C4×C53—81no—
185C4×C53—81no—
186C4×C53#52511yescharacteristic
187C4×C53#57511yescharacteristic
188Dic53—21no—
189Dic53—21no—
190Dic53—81no—
191Dic53—82no—
192Dic53—81no—
193Dic53C2×SD3212yes—
194Dic53C2×SD3212yes—
195#3170—531no—
196C2×C4×D8—531no—
197#2235—531no—
198#2235—531no—
199#2226—532no—
200#2208—532no—
201#2187—531no—
202#2187D5311yescharacteristic
203C4×SD32—532no—
204C4×SD32—532no—
205#2162—531no—
206C2×C53—22no—
207C2×C53—84no—
208C2×C53#310011yescharacteristic
209C2×C53C22×SD3211yescharacteristic
210C2×C53#341411yescharacteristic
211C22×D8C2×D5311yescharacteristic
212#533—1061no—
213C2×C4×D4—1061no—
214C2×SD32—532no—
215C2×SD32C2×D5312yes—
216#500—531no—
217#494—531no—
218#487—1061no—
219#467—531no—
220#467—1061no—
221#467C2×D5311yescharacteristic
222C4×Q16—531no—
223C4×Q16—531no—
224C4×Q16—1061no—
225C4×D8—532no—
226C4×D8—532no—
227C4×D8—1061no—
228C4×D8—1061no—
229C2×#107—531no—
230C2×#102—1061no—
231C2×C4×C8—531no—
232#367—532no—
233#367—532no—
234C4⋊C16—532no—
235#358—532no—
236#358—532no—
237#357—532no—
238#357C2×D5312yes—
239C22⋊C16—531no—
240C22⋊C16C2×D5311yescharacteristic
241C4×C16—532no—
242C53#3366111yescharacteristic, complemented, Sylow
243C22×D4—21no—
244C23×C4—2121no—
245C2×Q16—531no—
246C2×Q16—1061no—
247C2×Q16C22×D5311yescharacteristic
248C2×D8—24no—
249C2×D8—21no—
250C2×D8C22×D5312yes—
251C22×C8—531no—
252C22×C8C22×D5311yescharacteristic
253C4⋊Q8—1061no—
254C4⋊Q8—1061no—
255#125—2121no—
256C22⋊Q8—21no—
257C22⋊Q8—1061no—
258C22⋊Q8—2121no—
259C4×Q8—1061no—
260C4×Q8—1061no—
261C4×Q8—2121no—
262C4×D4—1062no—
263C4×D4—1062no—
264C4×D4—2121no—
265C4×D4—2121no—
266#117—2121no—
267C2×C4⋊C4—1061no—
268C2×C22⋊C4—2121no—
269C2×C42—531no—
270SD32—24no—
271SD32—1064no—
272C2×C16—532no—
273C2×C16C22×D5312yes—
274#107—531no—
275#107—532no—
276#107—1061no—
277#107—1061no—
278#107C22×D5311yescharacteristic
279#103—21no—
280#103—1061no—
281#103—1061no—
282#103—1061no—
283#102—1062no—
284#102—1062no—
285C4×C8—531no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—531no—
289C24—41no—
290C2×Q8—21no—
291C2×Q8—1061no—
292C2×Q8—2121no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—42no—
296C2×D4—41no—
297C22×C4—531no—
298C22×C4—531no—
299C22×C4—4244no—
300C22×C4—2122no—
301C22×C4—2122no—
302C22×C4—2121no—
303C22×C4D4×D5311yescharacteristic
304Q16—22no—
305Q16—1062no—
306Q16—2122no—
307D8—24no—
308D8—22no—
309D8#1171994714yes—
310C2×C8—22no—
311C2×C8—531no—
312C2×C8—531no—
313C2×C8—2121no—
314C2×C8D4×D5311yescharacteristic
315C2×C8C23×D5311yescharacteristic
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—1062no—
319C4⋊C4—1061no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C4⋊C4—2121no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C22⋊C4—41no—
328C22⋊C4—2122no—
329C22⋊C4—2122no—
330C22⋊C4—2121no—
331C42—531no—
332C42—531no—
333C42—531no—
334C42—531no—
335C42—2121no—
336C42—2121no—
337C16—22no—
338C16—1062no—
339C23—44no—
340C23—42no—
341C23—82no—
342C23—82no—
343C23D8×D5311yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—4242no—
347Q8—2121no—
348Q8—2121no—
349D4—24no—
350D4—44no—
351D4—42no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—534no—
357C2×C4—531no—
358C2×C4—531no—
359C2×C4—531no—
360C2×C4—531no—
361C2×C4—4244no—
362C2×C4—4244no—
363C2×C4—4242no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2122no—
368C2×C4D8×D5311yescharacteristic
369C2×C4C2×D4×D5311yescharacteristic
370C8—21no—
371C8—1061no—
372C8—1061no—
373C8C2×#1171994712yes—
374C22—21no—
375C22—21no—
376C22—44no—
377C22—88no—
378C22—82no—
379C22—82no—
380C22SD32×D5312yes—
381C22C2×D8×D5311yescharacteristic
382C4—21no—
383C4—81no—
384C4—81no—
385C4—532no—
386C4—532no—
387C4—1061no—
388C4—1061no—
389C4—4241no—
390C4—4242no—
391C4—4241no—
392C4#1681112911yescharacteristic
393C4#1681141911yescharacteristic
394C2—22no—
395C2—84no—
396C2#2028471011yescharacteristic
397C2C2×SD32×D5311yescharacteristic
398C2#2028784711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2187C2
socleC22×C53C22×D8
prime radicalC22×D8×C53C22
hypercenter#2187D53
hypocenterC53#33661
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #20285296C2semidirect
3. #20276913C2semidirect
4. #20276910C2semidirect
5. #20276860C2semidirect
6. #20276758C2semidirect
8. #20276570C2semidirect
9. SD32×Dic53C2semidirect
10. #20276511C2semidirect
15. #16811027C22semidirect
16. #16811001C22semidirect
20. Q16×Dic53C22semidirect
21. #16810717C22semidirect
30. #16810451C22semidirect
31. #16810450C22semidirect
32. #16810441C22semidirect
33. #16810428C22semidirect
34. #16810427C22semidirect
37. C16×Dic53C22semidirect
162. C2×Dic53SD32semidirect
242. C53#33661semidirect

Maximal subgroups

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

subgroupindex
#202852962
C2×D8×Dic532
#202769132
#202769102
#202768602
#202767582
#202766442
#202765702
SD32×Dic532
#202765112
C53×#21872
#3366153