Finite Group Databasebeta

← #27539150

Subgroups — #27539150

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D424C211yesmaximal, characteristic, complemented
2#20276910C211yesmaximal, characteristic, complemented
3#20276893C211yesmaximal, characteristic, complemented
4#20276869C212yesmaximal, complemented
5#20276767C212yesmaximal, complemented
6#20276643C211yesmaximal, characteristic, complemented
7#20276626C211yesmaximal, characteristic
8#20276589C212yesmaximal, complemented
9#20276577C212yesmaximal, complemented
10#20276518C211yesmaximal, characteristic, complemented
11C53×#3168C211yesmaximal, characteristic, complemented
12C22×D424C2211yescharacteristic
13C2×C4×D212—21no—
14C2×#13133160C2212yes—
15C2×#13133158C2212yes—
16C4×D424—21no—
17C4×D424—21no—
18C4×D424C2212yes—
19C4×D424C2212yes—
20C2×#13133151C2211yescharacteristic
21C2×#13133139—21no—
22#16810450C2212yescomplemented
23#16810448C2212yescomplemented
24C4⋊#11719908C2212yescomplemented
25#16810427C2212yescomplemented
26#16810424C2212yescomplemented
27#16810423C2212yes—
28#16810423C2212yes—
29C22⋊#11719908C2211yescharacteristic
30C22⋊#11719908C2211yescharacteristic
31#16810399C2212yescomplemented
32C53×#529—21no—
33C53×#499C2211yescharacteristic, complemented
34C53×#495C2211yescharacteristic, complemented
35C53×#488—21no—
36C53×#467C2211yescharacteristic
37C53×#466C2211yescharacteristic
38C53×#465—21no—
39C4×Q16×C53C2211yescharacteristic, complemented
40C4×SD16×C53—21no—
41C4×D8×C53C2211yescharacteristic, complemented
42C2×C4×C8×C53C2211yescharacteristic
43C22×D212—21no—
44C22×C4×D53—41no—
45C2×D424—21no—
46C2×D424—24no—
47C2×D424C2312yes—
48C4×D212—22no—
49C4×D212—22no—
50C4×D212—41no—
51C4×D212—41no—
52C2×C4⋊Dic53—21no—
53C2×#11719921—41no—
54#13133160—24no—
55#13133158—24no—
56C2×#11719908C2312yes—
57C2×#11719908C2312yes—
58#13133151—21no—
59#13133151C2312yes—
60#13133139—22no—
61#13133139—22no—
62C2×Q16×C53C2311yescharacteristic
63C2×SD16×C53—21no—
64C2×D8×C53C2311yescharacteristic
65C22×C8×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67#126×C53—41no—
68#125×C53—21no—
69#124×C53—21no—
70#123×C53—41no—
71C22⋊Q8×C53—21no—
72#121×C53—21no—
73C4×Q8×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—41no—
76#117×C53—41no—
77C2×C42×C53D411yescharacteristic
78#107×C53C2311yescharacteristic
79#107×C53C2311yescharacteristic
80#106×C53—21no—
81#103×C53—21no—
82#103×C53—21no—
83#102×C53—21no—
84#102×C53—21no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C23×D53—41no—
90C2×Q8×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53D811yescharacteristic
95C22×C4×C53D811yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×Dic53—41no—
98C2×D212—24no—
99C2×D212—22no—
100C2×D212—42no—
101C2×D212—41no—
102C2×C4×D53—42no—
103C2×C4×D53—42no—
104C2×C4×D53—84no—
105Q16×C53—22no—
106SD16×C53—42no—
107D8×C53—22no—
108C2×C8×C53—22no—
109C2×C8×C53—41no—
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2411yescharacteristic
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C42×C53—41no—
126C42×C53D811yescharacteristic
127C42×C53D811yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130#11719921—42no—
131#11719921—42no—
132C4⋊Dic53—22no—
133C4⋊Dic53—41no—
134D424—24no—
135D424—22no—
136D424#4114yes—
137#11719908—22no—
138#11719908—22no—
139C23×C53—41no—
140C23×C53C2×D811yescharacteristic
141C22×D53—44no—
142C22×D53—42no—
143C22×D53—82no—
144C22×D53—82no—
145Q8×C53—41no—
146Q8×C53—41no—
147D4×C53—41no—
148D4×C53—41no—
149D4×C53—82no—
150C2×C4×C53—22no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—82no—
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53C2×D811yescharacteristic
160C2×C4×C53C2×D811yescharacteristic
161C2×C4×C53C2×D811yescharacteristic
162C2×C4×C53C2×D811yescharacteristic
163C2×C4×C53C22×D411yescharacteristic
164C2×Dic53—42no—
165C2×Dic53—82no—
166D212—24no—
167D212—44no—
168D212—42no—
169C4×D53—84no—
170C4×D53—84no—
171C8×C53—21no—
172C8×C53—21no—
173C8×C53—21no—
174C8×C53C2×#4112yes—
175#33674—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53#50912yes—
182C22×C53C22×D811yescharacteristic
183C2×D53—44no—
184C2×D53—88no—
185C2×D53—82no—
186C2×D53—82no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53—81no—
195C4×C53#52511yescharacteristic
196C4×C53#57511yescharacteristic
197Dic53—82no—
198#3168D5311yescharacteristic, complemented
199C2×C4×D8—531no—
200#2235—531no—
201#2232—531no—
202#2228—532no—
203#2210—532no—
204#2187—531no—
205#2184—531no—
206#2177—532no—
207#2175—532no—
208#2163—531no—
209C2×C53—22no—
210C2×C53—161no—
211C2×C53#310011yescharacteristic
212C2×C53C2×#50911yescharacteristic
213C2×C53#341411yescharacteristic
214D53—84no—
215C22×D8—531no—
216#529—21no—
217C2×C4×D4—1061no—
218C2×SD32—532no—
219C2×D16—532no—
220#499C2×D5311yescharacteristic, complemented
221#495C2×D5311yescharacteristic, complemented
222#488—21no—
223#467C2×D5311yescharacteristic
224#466C2×D5311yescharacteristic
225#465—21no—
226C4×Q16C2×D5311yescharacteristic, complemented
227C4×SD16—21no—
228C4×D8—532no—
229C4×D8—532no—
230C4×D8—1061no—
231C4×D8—1061no—
232C4×D8C2×D5311yescharacteristic, complemented
233C2×#107—531no—
234C2×#102—1061no—
235C2×C4×C8C2×D5311yescharacteristic
236#367—532no—
237#366—532no—
238C4⋊C16—532no—
239#358—532no—
240#357—532no—
241#357—532no—
242#357—532no—
243C22⋊C16—531no—
244C22⋊C16—531no—
245#346—532no—
246C53#3367411yescharacteristic, complemented, Sylow
247C22×D4—1061no—
248C23×C4—2121no—
249C2×Q16C22×D5311yescharacteristic
250C2×SD16—21no—
251C2×D8—532no—
252C2×D8—1064no—
253C2×D8—1061no—
254C2×D8C22×D5311yescharacteristic
255C22×C8C22⋊D5311yescharacteristic
256C22×C8C22×D5311yescharacteristic
257#126—41no—
258#125—21no—
259#124—21no—
260#123—41no—
261C22⋊Q8—21no—
262#121—21no—
263C4×Q8—21no—
264C4×D4—21no—
265C4×D4—41no—
266C4×D4—1062no—
267C4×D4—1062no—
268C4×D4—2121no—
269C4×D4—2121no—
270#117—41no—
271C2×C4⋊C4—1061no—
272C2×C22⋊C4—2121no—
273C2×C42C22⋊D5311yescharacteristic
274SD32—1064no—
275D16—1064no—
276C2×C16—532no—
277C2×C16—532no—
278#107—532no—
279#107—1061no—
280#107C22×D5311yescharacteristic
281#107C22×D5311yescharacteristic
282#106—21no—
283#103—21no—
284#103—21no—
285#102—21no—
286#102—21no—
287#102—1062no—
288#102—1062no—
289C4×C8C22⋊D5311yescharacteristic
290C4×C8C22⋊D5311yescharacteristic
291C4×C8C22×D5311yescharacteristic
292C4×C8C22×D5311yescharacteristic
293C24—2121no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1064no—
298C2×D4—1062no—
299C2×D4—2122no—
300C2×D4—2121no—
301C22×C4—41no—
302C22×C4—4244no—
303C22×C4—2121no—
304C22×C4—2122no—
305C22×C4—2122no—
306C22×C4#1171992211yescharacteristic
307C22×C4#1171992211yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309Q16—22no—
310SD16—42no—
311D8—22no—
312D8—534no—
313D8—1064no—
314D8—1062no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8C2×C22⋊D5311yescharacteristic
318C2×C8C2×C22⋊D5311yescharacteristic
319C2×C8C2×C22⋊D5311yescharacteristic
320C2×C8C23×D5311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—1062no—
328C4⋊C4—2121no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—2122no—
335C22⋊C4—2122no—
336C42—41no—
337C42#1171992211yescharacteristic
338C42#1171992211yescharacteristic
339C42C2×C22⋊D5311yescharacteristic
340C42C2×C22⋊D5311yescharacteristic
341C16—1062no—
342C16—1062no—
343C23—41no—
344C23—4242no—
345C23—4242no—
346C23—2124no—
347C23—2122no—
348C23C2×#1171992211yescharacteristic
349Q8—41no—
350Q8—41no—
351D4—41no—
352D4—41no—
353D4—82no—
354D4—1064no—
355D4—2124no—
356D4—2122no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—4244no—
366C2×C4—4244no—
367C2×C4—4242no—
368C2×C4—2122no—
369C2×C4C2×#1171992211yescharacteristic
370C2×C4C2×#1171992211yescharacteristic
371C2×C4C2×#1171992211yescharacteristic
372C2×C4C2×#1171992211yescharacteristic
373C2×C4C2×#1171992211yescharacteristic
374C2×C4C22×C22⋊D5311yescharacteristic
375C8—21no—
376C8—21no—
377C8—21no—
378C8#41×D5312yes—
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—4248no—
385C22—4242no—
386C22—4242no—
387C22—2124no—
388C22#1681105812yes—
389C22C22×#1171992211yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—4242no—
399C4#1681113011yescharacteristic
400C4#1681142111yescharacteristic
401C2—22no—
402C2—161no—
403C2—4244no—
404C2#2028471111yescharacteristic
405C2C2×#1681105811yescharacteristic
406C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3168C2
socleC22×C53C22×D8
prime radical#20276626C2
hypercenter#3168D53
hypocenterC53#33674
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. C2×C4×D424C2semidirect
2. #20276910C2semidirect
3. #20276893C2semidirect
4. #20276869C2semidirect
5. #20276767C2semidirect
6. #20276643C2semidirect
8. #20276589C2semidirect
9. #20276577C2semidirect
10. #20276518C2semidirect
11. C53×#3168C2semidirect
22. #16810450C22semidirect
23. #16810448C22semidirect
24. C4⋊#11719908C22semidirect
25. #16810427C22semidirect
26. #16810424C22semidirect
31. #16810399C22semidirect
33. C53×#499C22semidirect
34. C53×#495C22semidirect
39. C4×Q16×C53C22semidirect
41. C4×D8×C53C22semidirect
198. #3168D53semidirect
220. #499C2×D53semidirect
221. #495C2×D53semidirect
226. C4×Q16C2×D53semidirect
232. C4×D8C2×D53semidirect
246. C53#33674semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4242
#202769102
#202768932
#202768692
#202767672
#202766432
#202766262
#202765892
#202765772
#202765182
C53×#31682
#3367453