Finite Group Databasebeta

← #27280639

Subgroups — #27280639

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#20283459C211yesmaximal, characteristic, complemented
2#20275556C211yesmaximal, characteristic, complemented
3#20273392C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5#20272611C211yesmaximal, characteristic, complemented
6#20272564C211yesmaximal, characteristic, complemented
7C53×#3205C211yesmaximal, characteristic
8C2×#13133224—21no—
9#16810701—21no—
10#16810695—22no—
11D4×#88322—21no—
12#16810647C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133134—21no—
16C2×#13133134—21no—
17#16810590—21no—
18C4×#11719917C2211yescharacteristic
19#16810369—21no—
20#16810367—21no—
21#16810367—21no—
22#16810367—21no—
23C53×#532C2211yescharacteristic
24C2×#127×C53C2211yescharacteristic
25C53×#493—21no—
26C53×#486—21no—
27C53×#459—22no—
28C53×#447C414yes—
29C53×#431C2211yescharacteristic
30#13133224—42no—
31#13133224—42no—
32C2×#11719917—22no—
33C2×#11719917—21no—
34C2×#11719917D411yescharacteristic
35C2×#11719917D411yescharacteristic
36C22×#88322—22no—
37C22×#88322—21no—
38C22×#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322D411yescharacteristic
45C4⋊#88322D411yescharacteristic
46#13133134—42no—
47#13133134—42no—
48C22⋊#88322—22no—
49C22⋊#88322—22no—
50#13133128—21no—
51C4×#88322—21no—
52C2×#41×C53—21no—
53C22×D4×C53—21no—
54C22×D4×C53—21no—
55C22×D4×C53C2×C412yes—
56C2×D8×C53—24no—
57C4⋊Q8×C53D411yescharacteristic
58#127×C53—21no—
59#127×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53D411yescharacteristic
63#124×C53—21no—
64#121×C53—21no—
65#121×C53C2×C412yes—
66C4×D4×C53—21no—
67C4×D4×C53D412yes—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70#102×C53—24no—
71C22⋊C8×C53C2×C412yes—
72C4×C8×C53—21no—
73C24×C53—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—22no—
81C2×D4×C53—22no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—81no—
93C2×D4×C53—81no—
94C2×D4×C53—82no—
95C2×D4×C53C22⋊C412yes—
96C22×C4×C53—22no—
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100D8×C53—84no—
101C2×C8×C53—22no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53C22⋊C412yes—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53#4111yescharacteristic
110#11719917—21no—
111#11719917—21no—
112#11719917—21no—
113#11719917—44no—
114#11719917—41no—
115#11719917—41no—
116#11719917D812yescomplemented
117C2×#88322—21no—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—44no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—81no—
127C23×C53—22no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—82no—
132C23×C53—82no—
133C23×C53C4×D411yescharacteristic
134Q8×C53—41no—
135Q8×C53—41no—
136D4×C53—42no—
137D4×C53—44no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—82no—
145D4×C53—82no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—81no—
149D4×C53—81no—
150D4×C53—81no—
151D4×C53—81no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—22no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53C2×C22⋊C411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12311yescharacteristic
164C2×C4×C53C2×D811yescharacteristic
165C2×C4×C53#13511yescharacteristic
166C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C8×C53—42no—
169#88322—22no—
170#88322—41no—
171#88322—41no—
172#88322—41no—
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—82no—
182C22×C53—82no—
183C22×C53—82no—
184C22×C53—162no—
185C22×C53—162no—
186C22×C53#38611yescharacteristic
187C22×C53C4×D811yescharacteristic
188C22×C53#44211yescharacteristic
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#3205D5311yescharacteristic
200#2935—531no—
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—162no—
210C2×C53#187911yescharacteristic
211C2×C53#190711yescharacteristic
212C2×C53#191411yescharacteristic
213C2×#131—1061no—
214#532C2×D5311yescharacteristic
215C2×#127C2×D5311yescharacteristic
216#493—21no—
217#486—21no—
218#459—22no—
219#447Dic5314yes—
220#436—1061no—
221#435—1062no—
222C8×D4—1061no—
223#431C2×D5311yescharacteristic
224#423—531no—
225C2×C4⋊C8—531no—
226C2×C4⋊C8—531no—
227C2×#100—1061no—
228C2×#100—1061no—
229#408—1061no—
230C4×M16—531no—
231#332—1061no—
232#331—1061no—
233#331—1061no—
234#331—1061no—
235C53#927311yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4C2×Dic5312yes—
240C2×D8—24no—
241#131—2122no—
242#131—2122no—
243C2×M16—531no—
244C2×M16—531no—
245C2×M16—1062no—
246C2×M16—1061no—
247C22×C8—1062no—
248C22×C8—1061no—
249C22×C8—1061no—
250C4⋊Q8C22⋊D5311yescharacteristic
251#127—21no—
252#127—21no—
253#127—21no—
254#127—21no—
255#127C22⋊D5311yescharacteristic
256#124—21no—
257#121—21no—
258#121C2×Dic5312yes—
259C4×D4—21no—
260C4×D4C22⋊D5312yes—
261C2×C42C22×D5311yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—531no—
264C4⋊C8—531no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270#102—24no—
271#100—2122no—
272#100—2122no—
273C22⋊C8—1062no—
274C22⋊C8—1062no—
275C22⋊C8C2×Dic5312yes—
276#97—1061no—
277C4×C8—21no—
278C4×C8—1061no—
279C24—22no—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—82no—
301C2×D4C22⋊Dic5312yes—
302C22×C4—22no—
303C22×C4D4×D5311yescharacteristic
304C22×C4#1171994711yescharacteristic
305C22×C4C22×Dic5311yescharacteristic
306D8—84no—
307M16—532no—
308M16—1061no—
309M16—1061no—
310M16—1061no—
311M16—2124no—
312M16—2121no—
313M16—2121no—
314C2×C8—22no—
315C2×C8—1061no—
316C2×C8—1061no—
317C2×C8—1061no—
318C2×C8—1061no—
319C2×C8—4241no—
320C2×C8—2124no—
321C2×C8—2121no—
322C2×C8—2121no—
323C2×C8—2121no—
324C2×C8—2121no—
325C4⋊C4—21no—
326C4⋊C4C22⋊Dic5312yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C42D4×D5311yescharacteristic
330C42#1171994711yescharacteristic
331C42C2×C22⋊D5311yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C23—22no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—82no—
338C23—82no—
339C23D4×Dic5311yescharacteristic
340Q8—41no—
341Q8—41no—
342D4—42no—
343D4—44no—
344D4—42no—
345D4—42no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—82no—
351D4—82no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4C2×C22⋊Dic5311yescharacteristic
365C2×C4D4×Dic5311yescharacteristic
366C2×C4#1313318511yescharacteristic
367C2×C4#1313318911yescharacteristic
368C2×C4#1313318911yescharacteristic
369C2×C4#1313319811yescharacteristic
370C2×C4D8×D5311yescharacteristic
371C2×C4#1313323611yescharacteristic
372C2×C4#1313324111yescharacteristic
373C2×C4#1313324111yescharacteristic
374C8—42no—
375C8—1062no—
376C8—4241no—
377C8—4241no—
378C8—2121no—
379C8—2121no—
380C8—2121no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—162no—
390C22—162no—
391C22#1681050311yescharacteristic
392C22D8×Dic5311yescharacteristic
393C22#1681073511yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4#1681075511yescharacteristic
399C4#1681076511yescharacteristic
400C4#1681081611yescharacteristic
401C4#1681083611yescharacteristic
402C4#1681088411yescharacteristic
403C4#1681089111yescharacteristic
404C2—21no—
405C2—21no—
406C2—82no—
407C2—162no—
408C2#2027506511yescharacteristic
409C2#2027522511yescharacteristic
410C2#2027526311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3205C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3205D53
hypocenterC53#9273
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #20283459C2semidirect
2. #20275556C2semidirect
3. #20273392C2semidirect
4. C2×#16810367C2semidirect
5. #20272611C2semidirect
6. #20272564C2semidirect
116. #11719917D8semidirect
235. C53#9273semidirect

Maximal subgroups

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

subgroupindex
#202834592
#202755562
#202733922
C2×#168103672
#202726112
#202725642
C53×#32052
#927353