Finite Group Databasebeta

← #27280637

Subgroups — #27280637

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic
2#20275555C211yesmaximal, characteristic, complemented
3#20273392C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5#20272612C211yesmaximal, characteristic, complemented
6#20272566C211yesmaximal, characteristic, complemented
7C53×#2935C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9#16810999—21no—
10#16810971—21no—
11#16810818—22no—
12#16810756C414yes—
13#16810676C2211yescharacteristic
14#16810647C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133133—21no—
17C2×#13133133—21no—
18#16810369—21no—
19#16810367—21no—
20#16810366—21no—
21#16810366—21no—
22C2×#131×C53—21no—
23C53×#532C2211yescharacteristic
24C53×#436—21no—
25C53×#435—22no—
26C8×D4×C53—21no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#408—21no—
29C4×M16×C53C2211yescharacteristic
30C22×D212—21no—
31C22×D212—21no—
32C22×D212C2×C412yes—
33C2×#11719922—24no—
34C2×#11719917—21no—
35C22×#88322—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45#13133137—24no—
46#13133133—42no—
47#13133133—42no—
48C22⋊#88322C2×C412yes—
49C4×#88322—21no—
50C2×#41×C53—21no—
51#131×C53—42no—
52#131×C53—42no—
53C2×M16×C53—22no—
54C2×M16×C53D411yescharacteristic
55C2×M16×C53D411yescharacteristic
56C22×C8×C53—22no—
57C22×C8×C53—21no—
58C4⋊Q8×C53D411yescharacteristic
59#127×C53D411yescharacteristic
60#124×C53—21no—
61#121×C53—21no—
62#121×C53C2×C412yes—
63C4×D4×C53—21no—
64C4×D4×C53D412yes—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53D411yescharacteristic
68C4⋊C8×C53D411yescharacteristic
69C22⋊C8×C53—22no—
70C22⋊C8×C53—22no—
71#97×C53—21no—
72C4×C8×C53—21no—
73C23×D53—22no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78C2×Q8×C53—21no—
79C2×D4×C53—22no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53C22⋊C412yes—
83C22×C4×C53—22no—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—82no—
98C2×D212—81no—
99C2×D212—81no—
100M16×C53—21no—
101M16×C53—21no—
102M16×C53—21no—
103M16×C53—44no—
104M16×C53D812yescomplemented
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53—44no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53C22⋊C412yes—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—22no—
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120#11719922—84no—
121#11719917—41no—
122#11719917—41no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—81no—
127C23×C53—22no—
128C23×C53C4×D411yescharacteristic
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—82no—
133C22×D53—82no—
134Q8×C53—41no—
135Q8×C53—41no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—42no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—41no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12311yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159D212—44no—
160D212—82no—
161D212—82no—
162D212—81no—
163D212—81no—
164D212—81no—
165D212—81no—
166D212—81no—
167D212—81no—
168C8×C53—22no—
169C8×C53—41no—
170C8×C53—41no—
171C8×C53—41no—
172#88322—42no—
173#88322—81no—
174#88322—81no—
175#9273—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53#38611yescharacteristic
182C22×C53C4×D811yescharacteristic
183C22×C53#44211yescharacteristic
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—82no—
187C2×D53—162no—
188C2×D53—162no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—42no—
193C4×C53#44711yescharacteristic
194C4×C53#44911yescharacteristic
195C4×C53#45911yescharacteristic
196C4×C53#46311yescharacteristic
197C4×C53#47111yescharacteristic
198C4×C53#47211yescharacteristic
199#3205—531no—
200#2935D5311yescharacteristic, complemented
201#1965—531no—
202#1547—531no—
203C2×#331—531no—
204#1301—531no—
205#1283—531no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—82no—
209C2×C53#187911yescharacteristic
210C2×C53#190711yescharacteristic
211C2×C53#191411yescharacteristic
212D53—162no—
213C2×#131—21no—
214#532C2×D5311yescharacteristic
215C2×#127—531no—
216#493—1061no—
217#486—1061no—
218#459—1062no—
219#447—534no—
220#436—21no—
221#435—22no—
222C8×D4—21no—
223#431—531no—
224#423—531no—
225C2×C4⋊C8—531no—
226C2×C4⋊C8C2×D5311yescharacteristic
227C2×#100—1061no—
228C2×#100—1061no—
229#408—21no—
230C4×M16C2×D5311yescharacteristic
231#332—1061no—
232#331—1061no—
233#331—1061no—
234#331—1061no—
235C53#927311yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C22×D4—532no—
238C22×D4—1061no—
239C22×D4—1061no—
240C2×D8—1064no—
241#131—42no—
242#131—42no—
243C2×M16—22no—
244C2×M16—1061no—
245C2×M16C22⋊D5311yescharacteristic
246C2×M16C22⋊D5311yescharacteristic
247C22×C8—22no—
248C22×C8—21no—
249C22×C8—1061no—
250C4⋊Q8D21211yescharacteristic
251#127—1061no—
252#127—1061no—
253#127—1061no—
254#127—1061no—
255#127D21211yescharacteristic
256#124—21no—
257#121—21no—
258#121C4×D5312yes—
259C4×D4—21no—
260C4×D4C22⋊D5312yes—
261C2×C42C22×D5311yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8C22⋊D5311yescharacteristic
269C4⋊C8C22⋊D5311yescharacteristic
270#102—1064no—
271#100—2122no—
272#100—2122no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—532no—
276#97—21no—
277C4×C8—21no—
278C4×C8—1061no—
279C24—1062no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—1062no—
289C2×D4—1062no—
290C2×D4—4241no—
291C2×D4—4241no—
292C2×D4—4242no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4#1171992112yes—
302C22×C4—22no—
303C22×C4C2×C4×D5311yescharacteristic
304C22×C4#1171994511yescharacteristic
305C22×C4C2×C22⋊D5311yescharacteristic
306D8—4244no—
307M16—21no—
308M16—21no—
309M16—21no—
310M16—44no—
311M16—2121no—
312M16—2121no—
313M16#1171992212yescomplemented
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—44no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—1062no—
322C2×C8—4241no—
323C2×C8—2121no—
324C2×C8—2121no—
325C4⋊C4—21no—
326C4⋊C4#1171992112yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C42C2×D21211yescharacteristic
330C42#1171994511yescharacteristic
331C42C2×C22⋊D5311yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C23—22no—
334C23—4242no—
335C23—4242no—
336C23—2122no—
337C23—2122no—
338C23—2122no—
339C23C4×C22⋊D5311yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—4242no—
350D4—4242no—
351D4—4241no—
352D4—4241no—
353D4—4241no—
354D4—4241no—
355D4—4241no—
356D4—4241no—
357D4—2124no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4C2×#1171992111yescharacteristic
365C2×C4C4×C22⋊D5311yescharacteristic
366C2×C4#1313318611yescharacteristic
367C2×C4#1313319111yescharacteristic
368C2×C4#1313319111yescharacteristic
369C2×C4#1313320011yescharacteristic
370C2×C4C2×#1171992211yescharacteristic
371C2×C4#1313323711yescharacteristic
372C2×C4#1313324211yescharacteristic
373C2×C4#1313324311yescharacteristic
374C8—22no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—4241no—
379C8—4241no—
380C8—2122no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—8482no—
387C22—8482no—
388C22—4242no—
389C22—4242no—
390C22—4242no—
391C22#1681050211yescharacteristic
392C22C4×#1171992211yescharacteristic
393C22#1681073311yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4#1681075711yescharacteristic
399C4#1681076811yescharacteristic
400C4#1681081811yescharacteristic
401C4#1681083811yescharacteristic
402C4#1681088611yescharacteristic
403C4#1681089311yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—8482no—
408C2#2027506311yescharacteristic
409C2#2027522311yescharacteristic
410C2#2027526211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2935C2
socleC22×C53#386
prime radical#20285630C2
cosocleC2×C42×C53C23
hypercenter#2935D53
hypocenterC53#9273
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #20275555C2semidirect
3. #20273392C2semidirect
4. C2×#16810366C2semidirect
5. #20272612C2semidirect
6. #20272566C2semidirect
7. C53×#2935C2semidirect
104. M16×C53D8semidirect
200. #2935D53semidirect
235. C53#9273semidirect
313. M16#11719922semidirect

Maximal subgroups

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

subgroupindex
#202856302
#202755552
#202733922
C2×#168103662
#202726122
#202725662
C53×#29352
#927353