Finite Group Databasebeta

← #27539167

Subgroups — #27539167

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285294C211yesmaximal, characteristic, complemented
2#20276906C212yesmaximal, complemented
3#20276857C211yesmaximal, characteristic, complemented
4#20276852C211yesmaximal, characteristic, complemented
5#20276755C211yesmaximal, characteristic, complemented
6#20276750C211yesmaximal, characteristic, complemented
7#20276638C212yesmaximal, complemented
8#20276589C212yesmaximal, complemented
9#20276577C212yesmaximal, complemented
10#20276518C211yesmaximal, characteristic, complemented
11C4×#135×C53C211yesmaximal, characteristic, complemented
12#16811173—21no—
13C2×#13133160C2212yes—
14C2×#13133158C2212yes—
15#16811028C2211yescharacteristic
16#16811000C2211yescharacteristic
17#16810970—21no—
18#16810859—21no—
19#16810859C2212yes—
20C4×D424—21no—
21C4×D424C2212yes—
22#16810450C2212yes—
23#16810448C2212yes—
24C4⋊#11719908C2211yescharacteristic
25C4⋊#11719908C2211yescharacteristic
26#16810427C2212yes—
27#16810424C2212yes—
28#16810423C2212yes—
29#16810423C2212yes—
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×#11719945—41no—
44C2×D424—21no—
45C2×D424C2312yes—
46#13133217—21no—
47#13133214—21no—
48#13133205—41no—
49#13133191—22no—
50#13133191—41no—
51C4×D212—22no—
52C4×D212—41no—
53#13133160—24no—
54#13133158—24no—
55C2×#11719908C2312yes—
56C2×#11719908C2312yes—
57#13133151—21no—
58#13133151C2312yes—
59#13133139—22no—
60#13133139—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×C22⋊D53—42no—
102#11719945—84no—
103C2×D212—22no—
104C2×D212—41no—
105C2×D212—41no—
106C2×C4×D53—42no—
107C2×Dic106—41no—
108Q16×C53—21no—
109Q16×C53#4112yescomplemented
110SD16×C53—24no—
111D8×C53—21no—
112D8×C53#4112yescomplemented
113C2×C8×C53—21no—
114C2×C8×C53—21no—
115C2×C8×C53—41no—
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53C2411yescharacteristic
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C42×C53—41no—
128C42×C53—41no—
129C42×C53—41no—
130C42×C53D811yescharacteristic
131C42×C53D811yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134#11719921—42no—
135#11719921—42no—
136C4⋊Dic53—22no—
137C4⋊Dic53—41no—
138D424—22no—
139D424—22no—
140D424—42no—
141#11719908—22no—
142#11719908—22no—
143C23×C53—41no—
144C23×C53C2×D811yescharacteristic
145C22×D53—42no—
146Q8×C53—22no—
147Q8×C53—41no—
148D4×C53—22no—
149D4×C53—44no—
150D4×C53—41no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—22no—
154C2×C4×C53—41no—
155C2×C4×C53—44no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—82no—
160C2×C4×C53—82no—
161C2×C4×C53C2×D811yescharacteristic
162C2×C4×C53C2×D811yescharacteristic
163C2×C4×C53C2×D811yescharacteristic
164C2×C4×C53C2×D811yescharacteristic
165C2×C4×C53C2×D811yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C22⋊D53—84no—
168C2×Dic53—42no—
169D212—42no—
170D212—42no—
171D212—82no—
172D212—82no—
173C4×D53—84no—
174Dic106—82no—
175C8×C53—21no—
176C8×C53—21no—
177C8×C53—21no—
178C8×C53C2×#4111yescharacteristic
179C8×C53C2×#4111yescharacteristic
180#33675—531nomaximal, Sylow
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—42no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53C22×D811yescharacteristic
188C2×D53—82no—
189C2×D53—82no—
190C2×D53—82no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—42no—
196C4×C53—81no—
197C4×C53—81no—
198C4×C53#50912yes—
199C4×C53#52511yescharacteristic
200C4×C53#57511yescharacteristic
201Dic53—82no—
202#3169—531no—
203C4×#135D5311yescharacteristic, complemented
204#2234—532no—
205#2225—531no—
206#2224—531no—
207#2207—531no—
208#2206—531no—
209#2186—532no—
210#2177—532no—
211#2175—532no—
212#2163—531no—
213C2×C53—41no—
214C2×C53—82no—
215C2×C53#310011yescharacteristic
216C2×C53C2×#50911yescharacteristic
217C2×C53#341411yescharacteristic
218D53—162no—
219C2×#135C2×D5311yescharacteristic
220#532—1061no—
221C4×#41—21no—
222C2×SD32—532no—
223C2×D16—532no—
224#500—531no—
225#493—531no—
226#486—1061no—
227#466—532no—
228#466—1061no—
229C4×Q16C2×D5311yescharacteristic
230C4×Q16C2×D5311yescharacteristic
231C4×SD16—21no—
232C4×SD16—21no—
233C4×D8—532no—
234C4×D8—1061no—
235C4×D8C2×D5311yescharacteristic
236C4×D8C2×D5311yescharacteristic
237#427C2×D5311yescharacteristic
238#416—21no—
239C2×C4×C8C2×D5311yescharacteristic
240#367—532no—
241#366—532no—
242C4⋊C16—531no—
243C4⋊C16—531no—
244#358—532no—
245#357—532no—
246#357—532no—
247#357—532no—
248C22⋊C16—532no—
249#346—532no—
250C53#3367511yescharacteristic, complemented, Sylow
251C2×#41—21no—
252C2×#41—2121no—
253#135—22no—
254#135—22no—
255C2×Q16C22×D5311yescharacteristic
256C2×SD16—21no—
257C2×D8—532no—
258C2×D8—1061no—
259C2×D8C22×D5311yescharacteristic
260C22×C8C22⋊D5311yescharacteristic
261C22×C8C22×D5311yescharacteristic
262C4⋊Q8—1061no—
263#127—1061no—
264#124—2121no—
265#121—1062no—
266#121—2121no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—1062no—
274C4×D4—2121no—
275#117—21no—
276#117—41no—
277C2×C42—41no—
278C2×C42C22⋊D5311yescharacteristic
279SD32—1064no—
280D16—1064no—
281C2×C16—532no—
282C2×C16—532no—
283#107—532no—
284#107—1061no—
285#107C22×D5311yescharacteristic
286#107C22×D5311yescharacteristic
287#106—21no—
288#103—21no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—1062no—
293#102—1062no—
294C4×C8C22⋊D5311yescharacteristic
295C4×C8C22⋊D5311yescharacteristic
296C4×C8C22×D5311yescharacteristic
297C4×C8C22×D5311yescharacteristic
298#41—24no—
299#41—42no—
300#41—4244no—
301C2×Q8—21no—
302C2×Q8—2121no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—1062no—
306C2×D4—2122no—
307C2×D4—2121no—
308C2×D4—2121no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—2122no—
313C22×C4#1171992211yescharacteristic
314C22×C4#1171992211yescharacteristic
315C22×C4C2×C22⋊D5311yescharacteristic
316Q16—21no—
317Q16#1171994512yescomplemented
318SD16—24no—
319D8—21no—
320D8—1062no—
321D8—1062no—
322D8—2122no—
323D8#1171994512yescomplemented
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8C2×C22⋊D5311yescharacteristic
328C2×C8C2×C22⋊D5311yescharacteristic
329C2×C8C2×C22⋊D5311yescharacteristic
330C2×C8C23×D5311yescharacteristic
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—1062no—
336C4⋊C4—2121no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—2122no—
341C22⋊C4—2122no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42#1171992211yescharacteristic
346C42#1171992211yescharacteristic
347C42C2×C22⋊D5311yescharacteristic
348C42C2×C22⋊D5311yescharacteristic
349C16—1062no—
350C16—1062no—
351C23—41no—
352C23—2122no—
353C23C2×#1171992211yescharacteristic
354Q8—22no—
355Q8—41no—
356Q8—4242no—
357D4—22no—
358D4—44no—
359D4—41no—
360D4—4242no—
361D4—4244no—
362D4—4242no—
363D4—2122no—
364D4—2122no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—4244no—
376C2×C4—2122no—
377C2×C4C2×#1171992211yescharacteristic
378C2×C4C2×#1171992211yescharacteristic
379C2×C4C2×#1171992211yescharacteristic
380C2×C4C2×#1171992211yescharacteristic
381C2×C4C2×#1171992211yescharacteristic
382C2×C4C22×C22⋊D5311yescharacteristic
383C8—21no—
384C8—21no—
385C8—21no—
386C8C2×#1171994511yescharacteristic
387C8C2×#1171994511yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—81no—
393C22—81no—
394C22—4242no—
395C22—4242no—
396C22—4242no—
397C22C22×#1171992211yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—4242no—
406C4#1681105812yes—
407C4#1681113411yescharacteristic
408C4#1681142111yescharacteristic
409C2—41no—
410C2—82no—
411C2—8482no—
412C2#2028471511yescharacteristic
413C2C2×#1681105811yescharacteristic
414C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC4×#135×C53C2
socleC22×C53C22×D8
hypercenterC4×#135D53
hypocenterC53#33675
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. #20285294C2semidirect
2. #20276906C2semidirect
3. #20276857C2semidirect
4. #20276852C2semidirect
5. #20276755C2semidirect
6. #20276750C2semidirect
7. #20276638C2semidirect
8. #20276589C2semidirect
9. #20276577C2semidirect
10. #20276518C2semidirect
11. C4×#135×C53C2semidirect
109. Q16×C53#41semidirect
112. D8×C53#41semidirect
203. C4×#135D53semidirect
250. C53#33675semidirect
317. Q16#11719945semidirect
323. D8#11719945semidirect

Maximal subgroups

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

subgroupindex
#202852942
#202769062
#202768572
#202768522
#202767552
#202767502
#202766382
#202765892
#202765772
#202765182
C4×#135×C532
#3367553