Finite Group Databasebeta

← #27538758

Subgroups — #27538758

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D53C211yesmaximal, characteristic, complemented
2C4×#13133236C211yesmaximal, characteristic
3#20276866C212yesmaximal, complemented
4#20276860C212yesmaximal, complemented
5#20276858C211yesmaximal, characteristic, complemented
6#20276854C211yesmaximal, characteristic, complemented
7#20276852C211yesmaximal, characteristic, complemented
8#20276500C212yesmaximal, complemented
9#20276469C211yesmaximal, characteristic, complemented
10#20276424C212yesmaximal
11C53×#2224C211yesmaximal, characteristic, complemented
12C2×#13133236C2211yescharacteristic
13C4⋊Q8×D53—21no—
14C4×#11719947—21no—
15C2×#13133156C2212yes—
16#16811028C2211yescharacteristic
17#16811027—21no—
18#16811027C2212yes—
19#16811026—21no—
20C4×Dic212C2211yescharacteristic
21#16810717C2211yescharacteristic
22#16810714—21no—
23C4×#11719923—21no—
24D8×Dic53C2211yescharacteristic
25#16810660C2211yescharacteristic
26#107×D53—21no—
27#107×D53C2212yes—
28#16810616—21no—
29C4×C8×D53C2211yescharacteristic
30#16810451C2212yes—
31#16810450C2212yes—
32#16810448C2212yes—
33C4⋊#11719908C2211yescharacteristic
34#16810441C2212yes—
35#16810428C2212yes—
36#16810426C2212yes—
37#16810424C2212yes—
38C53×#500C2211yescharacteristic
39C4×D8×C53C2211yescharacteristic
40C53×#366C2212yes—
41C53×C4⋊C16C2211yescharacteristic
42C53×#357C2212yes—
43C2×#11719947—21no—
44C2×Q8×D53—41no—
45#13133236—22no—
46#13133236Q814yescomplemented
47C2×Dic212C2311yescharacteristic
48C2×#11719923—21no—
49C2×C8×D53D411yescharacteristic
50C2×C8×D53C2311yescharacteristic
51#13133217—21no—
52#13133216—22no—
53#13133216—41no—
54#13133215—41no—
55C4×Dic106—21no—
56#13133182—21no—
57D4×Dic53—21no—
58C4×C22⋊D53—41no—
59#13133177—41no—
60#13133175—21no—
61#13133173—41no—
62C4⋊C4×D53—22no—
63C4⋊C4×D53—41no—
64C2×C4×Dic53—41no—
65C42×D53D411yescharacteristic
66#13133156—24no—
67C2×#11719908C2312yes—
68#13133151—21no—
69#13133151C2312yes—
70#13133151C2311yescharacteristic
71#13133150—22no—
72#13133150—22no—
73#13133148—21no—
74#13133142—21no—
75#13133140—21no—
76#13133138—21no—
77C4×#88322D411yescharacteristic
78C8×Dic53D411yescharacteristic
79C8×Dic53C2311yescharacteristic
80C2×D8×C53C2311yescharacteristic
81C4⋊Q8×C53—21no—
82C4×D4×C53—21no—
83C2×C16×C53C2312yes—
84#107×C53—21no—
85#107×C53C2312yes—
86#107×C53C2311yescharacteristic
87#102×C53—21no—
88C4×C8×C53C2311yescharacteristic
89C2×Q8×C53—41no—
90C2×D4×C53—21no—
91C22×C4×C53—41no—
92C2×C22⋊D53—41no—
93C22×Dic53—41no—
94C22×Dic53—41no—
95Q8×D53—84no—
96#11719947—24no—
97#11719947—42no—
98C2×C4×D53—42no—
99C2×C4×D53D811yescharacteristic
100C2×C4×D53D811yescharacteristic
101C2×C4×D53C2×D411yescharacteristic
102C2×Dic106—21no—
103C2×Dic106—41no—
104C2×Dic106—42no—
105D8×C53—21no—
106D8×C53C2×Q812yescomplemented
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2411yescharacteristic
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C42×C53C2×D411yescharacteristic
114C22⋊Dic53—41no—
115#11719923—24no—
116#11719921—41no—
117C4⋊Dic53—21no—
118C4⋊Dic53—22no—
119C4⋊Dic53—41no—
120#11719919—41no—
121#11719919—42no—
122#11719919—42no—
123#11719919—41no—
124C4×Dic53—41no—
125C4×Dic53—41no—
126C4×Dic53—41no—
127C4×Dic53D811yescharacteristic
128C4×Dic53D811yescharacteristic
129C4×Dic53C2×D411yescharacteristic
130C2×#88322C2×D411yescharacteristic
131C2×#88322C2×D411yescharacteristic
132Dic212—21no—
133Dic212C2×Q812yescomplemented
134C8×D53—22no—
135C8×D53C2×Q812yes—
136C8×D53#4112yes—
137C16×C53—22no—
138#11719908—22no—
139C23×C53—41no—
140C22×D53C2×D811yescharacteristic
141Q8×C53—82no—
142D4×C53—22no—
143D4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—82no—
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53C22×D411yescharacteristic
150C22⋊D53—44no—
151C2×Dic53—41no—
152C2×Dic53—44no—
153C2×Dic53—42no—
154C2×Dic53—41no—
155C2×Dic53—82no—
156C2×Dic53C2×D811yescharacteristic
157C2×Dic53C2×D811yescharacteristic
158C2×Dic53C2×D811yescharacteristic
159C4×D53—22no—
160C4×D53—22no—
161C4×D53—84no—
162C4×D53C22⋊Q812yes—
163C4×D53#13712yes—
164Dic106—22no—
165Dic106—41no—
166Dic106—84no—
167Dic106—82no—
168C8×C53—21no—
169C8×C53C22×Q811yescharacteristic
170C8×C53C2×#4111yescharacteristic
171#88322—21no—
172#88322C22⋊Q812yes—
173#33629—531nomaximal, Sylow
174C22×C53—42no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53C22×D811yescharacteristic
178C2×D53—21no—
179C2×D53—21no—
180C2×D53#47412yes—
181C4×C53—21no—
182C4×C53—82no—
183C4×C53—81no—
184C4×C53#50912yes—
185C4×C53C2×C22⋊Q811yescharacteristic
186C4×C53C2×#13711yescharacteristic
187Dic53—21no—
188Dic53—42no—
189Dic53—82no—
190Dic53—81no—
191Dic53#47412yes—
192Dic53#51012yes—
193C2×#500—531no—
194C4×#135—531no—
195#2227—532no—
196#2226—532no—
197#2225—531no—
198#2225—531no—
199#2224—531no—
200#2224D5311yescharacteristic, complemented
201#2159—532no—
202#2150—531no—
203#2139—532no—
204C2×C53—82no—
205C2×C53C2×#47411yescharacteristic
206C2×C53C2×#50911yescharacteristic
207C2×C53C2×#51011yescharacteristic
208D53—22no—
209C2×#135—531no—
210C2×C4⋊Q8—1061no—
211C4×#41—1061no—
212C2×M32—532no—
213#500—532no—
214#500—531no—
215#500—1061no—
216#500—1061no—
217#500C2×D5311yescharacteristic
218C4×Q16—531no—
219C4×Q16—531no—
220C4×SD16—1061no—
221C4×SD16—1061no—
222C4×D8—531no—
223C4×D8C2×D5311yescharacteristic
224#427—531no—
225C2×#107—532no—
226C2×#107—1061no—
227#416—1061no—
228C2×C4×C8—531no—
229#367—532no—
230#367—532no—
231#366—532no—
232#366C2×D5312yes—
233C4⋊C16—532no—
234C4⋊C16—531no—
235C4⋊C16C2×D5311yescharacteristic
236#358—532no—
237#358—532no—
238#357—532no—
239#357C2×D5312yes—
240C53#3362911yescharacteristic, complemented, Sylow
241C2×#41—1061no—
242C22×Q8—2121no—
243#135—534no—
244#135—1062no—
245C2×Q16—531no—
246C2×SD16—1061no—
247C2×D8C22×D5311yescharacteristic
248C22×C8—531no—
249C22×C8—531no—
250C4⋊Q8—21no—
251C4⋊Q8—1062no—
252C4⋊Q8—1061no—
253C4⋊Q8—2121no—
254C4⋊Q8—2121no—
255C4×Q8—1061no—
256C4×Q8—1061no—
257C4×D4—21no—
258C4×D4—1061no—
259C4×D4—2121no—
260C4×D4—2121no—
261#117—1061no—
262#117—2121no—
263C2×C4⋊C4—1062no—
264C2×C4⋊C4—2121no—
265C2×C42—531no—
266C2×C42—2121no—
267M32—1064no—
268C2×C16—532no—
269C2×C16C22×D5312yes—
270#107—21no—
271#107—532no—
272#107—531no—
273#107—1062no—
274#107—1062no—
275#107—1061no—
276#107C22×D5312yes—
277#107C22×D5311yescharacteristic
278#106—1061no—
279#103—1061no—
280#103—1061no—
281#102—21no—
282#102—1061no—
283C4×C8—531no—
284C4×C8—531no—
285C4×C8—531no—
286C4×C8C22×D5311yescharacteristic
287#41—1064no—
288#41—2122no—
289C2×Q8—41no—
290C2×Q8—1061no—
291C2×Q8—4244no—
292C2×Q8—2122no—
293C2×Q8—2121no—
294C2×D4—21no—
295C2×D4—2121no—
296C22×C4—41no—
297C22×C4—531no—
298C22×C4—531no—
299C22×C4—531no—
300C22×C4—2122no—
301C22×C4—2121no—
302C22×C4—2121no—
303Q16—532no—
304Q16—1061no—
305SD16—1064no—
306D8—21no—
307D8Q8×D5312yescomplemented
308C2×C8—531no—
309C2×C8—532no—
310C2×C8—532no—
311C2×C8—531no—
312C2×C8—1062no—
313C2×C8D4×D5311yescharacteristic
314C2×C8C23×D5311yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—1061no—
319C4⋊C4—1062no—
320C4⋊C4—2122no—
321C4⋊C4—2122no—
322C4⋊C4—2121no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C22⋊C4—41no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C42—531no—
329C42—531no—
330C42—531no—
331C42—2121no—
332C42—2121no—
333C42—2121no—
334C42D4×D5311yescharacteristic
335C16—22no—
336C16—1062no—
337C23—41no—
338C23—531no—
339Q8—82no—
340Q8—1062no—
341Q8—4244no—
342Q8—4242no—
343Q8—2121no—
344D4—22no—
345D4—41no—
346D4—2124no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—531no—
351C2×C4—531no—
352C2×C4—532no—
353C2×C4—532no—
354C2×C4—531no—
355C2×C4—1062no—
356C2×C4—1062no—
357C2×C4—4244no—
358C2×C4—4242no—
359C2×C4—2124no—
360C2×C4—2121no—
361C2×C4—2122no—
362C2×C4—2121no—
363C2×C4D8×D5311yescharacteristic
364C2×C4D8×D5311yescharacteristic
365C2×C4C2×D4×D5311yescharacteristic
366C8—21no—
367C8—532no—
368C8—1061no—
369C8C2×Q8×D5311yescharacteristic
370C8#41×D5311yescharacteristic
371C22—42no—
372C22—81no—
373C22—81no—
374C22—532no—
375C22—1061no—
376C22—1061no—
377C22C2×D8×D5311yescharacteristic
378C4—21no—
379C4—81no—
380C4—82no—
381C4—532no—
382C4—532no—
383C4—1061no—
384C4—4242no—
385C4—4241no—
386C4—2122no—
387C4#1681105712yes—
388C4C22⋊Q8×D5311yescharacteristic
389C4#137×D5311yescharacteristic
390C2—82no—
391C2—1062no—
392C2#474×D5311yescharacteristic
393C2C2×#1681105711yescharacteristic
394C2#510×D5311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2224C2
socleC22×C53C22×D8
prime radicalC2×#13133236C22
hypercenter#2224D53
hypocenterC53#33629
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2224

Complements

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

NH ≅ G/Ntype
1. #500×D53C2semidirect
3. #20276866C2semidirect
4. #20276860C2semidirect
5. #20276858C2semidirect
6. #20276854C2semidirect
7. #20276852C2semidirect
8. #20276500C2semidirect
9. #20276469C2semidirect
11. C53×#2224C2semidirect
46. #13133236Q8semidirect
106. D8×C53C2×Q8semidirect
133. Dic212C2×Q8semidirect
200. #2224D53semidirect
240. C53#33629semidirect
307. D8Q8×D53semidirect

Maximal subgroups

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

subgroupindex
#500×D532
C4×#131332362
#202768662
#202768602
#202768582
#202768542
#202768522
#202765002
#202764692
#202764242
C53×#22242
#3362953