Finite Group Databasebeta

← #27539156

Subgroups — #27539156

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
1#20285285C211yesmaximal, characteristic, complemented
2C2×C4×D424C211yesmaximal, characteristic, complemented
3#20276913C211yesmaximal, characteristic, complemented
4#20276896C211yesmaximal, characteristic, complemented
5#20276873C212yesmaximal, complemented
6#20276771C212yesmaximal, complemented
7#20276646C211yesmaximal, characteristic, complemented
8#20276629C211yesmaximal, characteristic
9#20276592C212yesmaximal, complemented
10#20276581C212yesmaximal, complemented
11C53×#2163C211yesmaximal, characteristic, complemented
12C22×D424C2211yescharacteristic
13#16811157—21no—
14C2×C4×D212—21no—
15C2×#13133162C2212yes—
16C2×D848C2212yes—
17#16811025C2211yescharacteristic, complemented
18#16811007C2211yescharacteristic, complemented
19#16810978—21no—
20#16810864C2211yescharacteristic
21#16810859C2211yescharacteristic
22#16810852—21no—
23C4×Dic212C2211yescharacteristic, complemented
24C4×#11719913—21no—
25C4×D424—21no—
26C4×D424—21no—
27C4×D424C2212yes—
28C4×D424C2212yes—
29C4×D424C2211yescharacteristic, complemented
30C2×#13133151C2211yescharacteristic
31C2×#13133139—21no—
32#16810451C2212yescomplemented
33#16810449C2212yescomplemented
34#16810428C2212yescomplemented
35#16810425C2212yes—
36#16810425C2212yes—
37#16810425C2212yescomplemented
38C2×C4×C8×C53C2211yescharacteristic
39C53×C4⋊C16C2212yescomplemented
40C53×C22⋊C16C2211yescharacteristic
41C53×C22⋊C16C2211yescharacteristic
42C53×#346C2212yescomplemented
43C22×D212—21no—
44C22×C4×D53—41no—
45C2×Dic212C2311yescharacteristic
46C2×#11719913—21no—
47C2×D424—21no—
48C2×D424—24no—
49C2×D424C2311yescharacteristic
50C2×D424C2312yes—
51#13133211—41no—
52#13133209—21no—
53#13133205—21no—
54#13133200—41no—
55#13133196—21no—
56#13133191—21no—
57C4×Dic106—21no—
58C4×D212—22no—
59C4×D212—22no—
60C4×D212—21no—
61C4×D212—41no—
62C4×D212—41no—
63C4×C22⋊D53—41no—
64#13133174—41no—
65C2×C4⋊Dic53—21no—
66C2×#11719921—41no—
67#13133162—24no—
68D848—24no—
69#13133151—21no—
70#13133151C2311yescharacteristic
71#13133151C2311yescharacteristic
72#13133151C2312yes—
73#13133149—21no—
74#13133142—21no—
75#13133142—21no—
76#13133139—21no—
77#13133139—21no—
78#13133139—22no—
79#13133139—22no—
80C22×C8×C53D411yescharacteristic
81C22×C8×C53C2311yescharacteristic
82C2×C42×C53D411yescharacteristic
83C2×C16×C53C2312yes—
84C2×C16×C53C2312yes—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89C23×D53—41no—
90C22×C4×C53D811yescharacteristic
91C22×C4×C53D811yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C2×C22⋊D53—41no—
94C22×Dic53—41no—
95C2×D212—24no—
96C2×D212—22no—
97C2×D212—21no—
98C2×D212—42no—
99C2×D212—41no—
100C2×C4×D53—41no—
101C2×C4×D53—42no—
102C2×C4×D53—42no—
103C2×C4×D53—84no—
104C2×Dic106—21no—
105C2×C8×C53—22no—
106C2×C8×C53—41no—
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2411yescharacteristic
111C42×C53D811yescharacteristic
112C42×C53D811yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C22⋊Dic53—41no—
116#11719921—42no—
117#11719921—42no—
118#11719921—41no—
119#11719921—41no—
120#11719921—41no—
121#11719921—41no—
122C4⋊Dic53—22no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—41no—
126#11719919—41no—
127#11719919—41no—
128#11719919—41no—
129#11719919—41no—
130C4×Dic53—41no—
131Dic212—22no—
132D424—24no—
133D424—22no—
134D424—22no—
135D424#4114yes—
136#11719913—42no—
137C16×C53—22no—
138C16×C53—22no—
139C23×C53C2×D811yescharacteristic
140C22×D53—44no—
141C22×D53—42no—
142C22×D53—41no—
143C22×D53—82no—
144C22×D53—82no—
145C2×C4×C53—22no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53C2×D811yescharacteristic
150C2×C4×C53C2×D811yescharacteristic
151C2×C4×C53C2×D811yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×D811yescharacteristic
154C2×C4×C53C22×D411yescharacteristic
155C22⋊D53—82no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—82no—
161D212—24no—
162D212—44no—
163D212—41no—
164D212—42no—
165D212—41no—
166C4×D53—82no—
167C4×D53—84no—
168C4×D53—84no—
169Dic106—41no—
170Dic106—41no—
171C8×C53—21no—
172C8×C53—21no—
173C8×C53—21no—
174C8×C53C2×#4112yes—
175#33674—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53#50912yes—
179C22×C53C22×D811yescharacteristic
180C2×D53—44no—
181C2×D53—88no—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—81no—
185C2×D53—81no—
186C2×D53—81no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53#52511yescharacteristic
193C4×C53#57511yescharacteristic
194Dic53—81no—
195Dic53—82no—
196Dic53—81no—
197Dic53—81no—
198#3168—531no—
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#2163D5311yescharacteristic, complemented
209C2×C53—22no—
210C2×C53#310011yescharacteristic
211C2×C53C2×#50911yescharacteristic
212C2×C53#341411yescharacteristic
213D53—84no—
214D53—161no—
215C22×D8—531no—
216#529—1061no—
217C2×C4×D4—1061no—
218C2×SD32—532no—
219C2×D16—532no—
220#499—531no—
221#495—531no—
222#488—1061no—
223#467—531no—
224#466—531no—
225#465—1061no—
226C4×Q16—531no—
227C4×SD16—1061no—
228C4×D8—531no—
229C4×D8—532no—
230C4×D8—532no—
231C4×D8—1061no—
232C4×D8—1061no—
233C2×#107—531no—
234C2×#102—1061no—
235C2×C4×C8C2×D5311yescharacteristic
236#367—532no—
237#366—532no—
238C4⋊C16C2×D5312yescomplemented
239#358—532no—
240#357—532no—
241#357—532no—
242#357—532no—
243C22⋊C16C2×D5311yescharacteristic
244C22⋊C16C2×D5311yescharacteristic
245#346C2×D5312yescomplemented
246C53#3367411yescharacteristic, complemented, Sylow
247C22×D4—1061no—
248C23×C4—2121no—
249C2×Q16—531no—
250C2×SD16—1061no—
251C2×D8—532no—
252C2×D8—531no—
253C2×D8—1064no—
254C2×D8—1061no—
255C22×C8D21211yescharacteristic
256C22×C8C22×D5311yescharacteristic
257#126—2121no—
258#125—1061no—
259#124—1061no—
260#123—2121no—
261C22⋊Q8—1061no—
262#121—1061no—
263C4×Q8—1061no—
264C4×D4—1062no—
265C4×D4—1062no—
266C4×D4—1061no—
267C4×D4—2121no—
268C4×D4—2121no—
269C4×D4—2121no—
270#117—2121no—
271C2×C4⋊C4—1061no—
272C2×C22⋊C4—2121no—
273C2×C42D21211yescharacteristic
274SD32—1064no—
275D16—1064no—
276C2×C16C22×D5312yes—
277C2×C16C22×D5312yes—
278#107—531no—
279#107—531no—
280#107—532no—
281#107—1061no—
282#106—1061no—
283#103—1061no—
284#103—1061no—
285#102—1062no—
286#102—1062no—
287#102—1061no—
288#102—1061no—
289C4×C8D21211yescharacteristic
290C4×C8D21211yescharacteristic
291C4×C8C22×D5311yescharacteristic
292C4×C8C22×D5311yescharacteristic
293C24—2121no—
294C2×Q8—1061no—
295C2×D4—1064no—
296C2×D4—1062no—
297C2×D4—1061no—
298C2×D4—2122no—
299C2×D4—2121no—
300C2×D4—2121no—
301C22×C4—4244no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4—2122no—
305C22×C4—2122no—
306C22×C4D42411yescharacteristic
307C22×C4D42411yescharacteristic
308C22×C4C2×D21211yescharacteristic
309Q16—1062no—
310SD16—2122no—
311D8—534no—
312D8—1064no—
313D8—1062no—
314D8—1062no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8C2×D21211yescharacteristic
318C2×C8C2×D21211yescharacteristic
319C2×C8C2×D21211yescharacteristic
320C2×C8C23×D5311yescharacteristic
321C4⋊C4—1062no—
322C4⋊C4—1061no—
323C4⋊C4—1061no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C4⋊C4—2121no—
328C4⋊C4—2121no—
329C22⋊C4—2121no—
330C22⋊C4—2122no—
331C22⋊C4—2122no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C22⋊C4—2121no—
335C22⋊C4—2121no—
336C42—2121no—
337C42D42411yescharacteristic
338C42D42411yescharacteristic
339C42C2×D21211yescharacteristic
340C42C2×D21211yescharacteristic
341C16—22no—
342C16—22no—
343C23—4242no—
344C23—4242no—
345C23—2124no—
346C23—2121no—
347C23—2122no—
348C23C2×D42411yescharacteristic
349Q8—2121no—
350Q8—2121no—
351D4—1064no—
352D4—4242no—
353D4—2124no—
354D4—2121no—
355D4—2121no—
356D4—2122no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—4244no—
362C2×C4—4244no—
363C2×C4—4242no—
364C2×C4—4242no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2122no—
369C2×C4C2×D42411yescharacteristic
370C2×C4C2×D42411yescharacteristic
371C2×C4C2×D42411yescharacteristic
372C2×C4C2×D42411yescharacteristic
373C2×C4C2×D42411yescharacteristic
374C2×C4C22×D21211yescharacteristic
375C8—21no—
376C8—21no—
377C8—21no—
378C8#41×D5312yes—
379C22—21no—
380C22—21no—
381C22—4248no—
382C22—4241no—
383C22—4241no—
384C22—4241no—
385C22—4242no—
386C22—4242no—
387C22—2124no—
388C22#1681106112yes—
389C22C22×D42411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—4241no—
396C4—4242no—
397C4—4241no—
398C4—4241no—
399C4#1681113511yescharacteristic
400C4#1681142311yescharacteristic
401C2—22no—
402C2—8481no—
403C2—4244no—
404C2#2028471611yescharacteristic
405C2C2×#1681106111yescharacteristic
406C2#2028785111yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2163C2
socleC22×C53C22×D8
prime radical#20276629C2
hypercenter#2163D53
hypocenterC53#33674
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #2163

Complements

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

NH ≅ G/Ntype
1. #20285285C2semidirect
2. C2×C4×D424C2semidirect
3. #20276913C2semidirect
4. #20276896C2semidirect
5. #20276873C2semidirect
6. #20276771C2semidirect
7. #20276646C2semidirect
9. #20276592C2semidirect
10. #20276581C2semidirect
11. C53×#2163C2semidirect
17. #16811025C22semidirect
18. #16811007C22semidirect
23. C4×Dic212C22semidirect
29. C4×D424C22semidirect
32. #16810451C22semidirect
33. #16810449C22semidirect
34. #16810428C22semidirect
37. #16810425C22semidirect
39. C53×C4⋊C16C22semidirect
42. C53×#346C22semidirect
208. #2163D53semidirect
238. C4⋊C16C2×D53semidirect
245. #346C2×D53semidirect
246. C53#33674semidirect

Maximal subgroups

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

subgroupindex
#202852852
C2×C4×D4242
#202769132
#202768962
#202768732
#202767712
#202766462
#202766292
#202765922
#202765812
C53×#21632
#3367453