Finite Group Databasebeta

← #27369632

Subgroups — #27369632

Order 13568.

714 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#20285658C211yesmaximal, characteristic, complemented
2#20284512C211yesmaximal, characteristic, complemented
3C2×#16810818C211yesmaximal, characteristic
4#20274248C214yesmaximal, complemented
5#20274180C214yesmaximal, complemented
6#20273490C211yesmaximal, characteristic, complemented
7#20273382C211yesmaximal, characteristic, complemented
8#20273370C211yesmaximal, characteristic, complemented
9C53×#2975C211yesmaximal, characteristic, complemented
10C22×#11719922—21no—
11#16811180C2211yescharacteristic
12C2×#13133214C2211yescharacteristic
13#16810996C2212yescomplemented
14#16810970C2212yescomplemented
15#16810923—22no—
16#16810913C2212yescomplemented
17#16810903C2212yescomplemented
18#16810824—22no—
19#16810818—22no—
20#16810818C2214yes—
21#16810776—22no—
22#16810663—21no—
23#16810647C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25#16810628—21no—
26C2×#13133137—21no—
27#16810376C2214yescomplemented
28#16810371C2212yescomplemented
29#16810370C2212yescomplemented
30#16810366C2214yescomplemented
31C2×C4×D4×C53C2211yescharacteristic
32C53×#436—22no—
33C53×#436C2214yescomplemented
34C53×#431C2211yescharacteristic
35C53×#431C2211yescharacteristic
36C53×#407—22no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C2×#11719913—24no—
41C2×#11719922—24no—
42C2×#11719922—44no—
43C2×#11719917—21no—
44C22×#88322—21no—
45#13133217C2311yescharacteristic
46#13133217C2311yescharacteristic
47#13133214—21no—
48#13133214—21no—
49#13133214—21no—
50#13133214C2312yes—
51#13133209—21no—
52#13133196—21no—
53#13133196—21no—
54C4×Dic106—21no—
55C4×Dic106—21no—
56#13133176—21no—
57#13133150—22no—
58#13133148—22no—
59C4⋊#88322—21no—
60C4⋊#88322—21no—
61C4⋊#88322C2312yes—
62C4⋊#88322C2312yes—
63#13133139—24no—
64#13133138—22no—
65#13133138—22no—
66#13133137—24no—
67C22×D4×C53D411yescharacteristic
68C23×C4×C53—21no—
69C2×M16×C53—24no—
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53D412yes—
73C4×D4×C53C2312yes—
74C2×C4⋊C4×C53D411yescharacteristic
75C2×C22⋊C4×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53D412yes—
78C4⋊C8×C53C2312yes—
79C22⋊C8×C53—24no—
80C22⋊C8×C53—21no—
81C22⋊C8×C53D412yes—
82C4×C8×C53—21no—
83C4×C8×C53C2312yes—
84C24×C53—21no—
85C23×D53—41no—
86C2×D4×C53—21no—
87C2×D4×C53—22no—
88C2×D4×C53—22no—
89C2×D4×C53C2×D412yes—
90C22×C4×C53—21no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—82no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C2×D212—22no—
98C2×D212—22no—
99C2×D212—22no—
100C2×D212—42no—
101C2×D212—42no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—82no—
106C2×D212—82no—
107C2×Dic106—21no—
108C2×Dic106—21no—
109M16×C53—84no—
110C2×C8×C53—22no—
111C2×C8×C53C2×D414yes—
112C4⋊C4×C53—21no—
113C4⋊C4×C53C2×D412yes—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—22no—
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119C42×C53C2411yescharacteristic
120C22⋊Dic53—41no—
121#11719922—48no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126#11719919—41no—
127#11719919—41no—
128C4×Dic53—41no—
129#11719917—42no—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322—42no—
133#11719913—84no—
134C23×C53—22no—
135C23×C53—42no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53#12111yescharacteristic
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—82no—
143D4×C53—22no—
144D4×C53—22no—
145D4×C53—42no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—41no—
149C2×C4×C53—44no—
150C2×C4×C53—82no—
151C2×C4×C53—81no—
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)12yes—
156C2×C4×C53#13712yes—
157C2×C4×C53C22×D411yescharacteristic
158C2×C4×C53C22×D411yescharacteristic
159C2×C4×C53C2×#4111yescharacteristic
160C2×Dic53—41no—
161C2×Dic53—41no—
162D212—42no—
163D212—42no—
164D212—44no—
165D212—82no—
166D212—82no—
167D212—82no—
168D212—82no—
169Dic106—42no—
170Dic106—81no—
171C8×C53—24no—
172C8×C53—42no—
173#88322—42no—
174#88322—42no—
175#19420—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—84no—
182C22×C53#46812yes—
183C22×C53C2×#12111yescharacteristic
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—82no—
187C2×D53—164no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—42no—
192C4×C53#46812yes—
193C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
194C4×C53C2×#13711yescharacteristic
195C4×C53#57612yes—
196Dic53—81no—
197Dic53—81no—
198#3208—531no—
199#3077—531no—
200C2×#459—531no—
201#2975D5311yescharacteristic, complemented
202#1697—534no—
203#1685—534no—
204#1571—531no—
205#1545—531no—
206#1542—531no—
207C2×C53—22no—
208C2×C53—82no—
209C2×C53#304111yescharacteristic
210C2×C53C2×#46811yescharacteristic
211C2×C53#306811yescharacteristic
212D53—162no—
213C22×D8—1061no—
214#533—531no—
215C2×#127—531no—
216C2×C4×D4C2×D5311yescharacteristic
217#492—532no—
218#486—532no—
219#478—1062no—
220#476—532no—
221#474—532no—
222#460—1062no—
223#459—534no—
224#459—1062no—
225#450—1062no—
226#436—22no—
227#436C2×D5314yescomplemented
228#431C2×D5311yescharacteristic
229#431C2×D5311yescharacteristic
230#428—1061no—
231#423—531no—
232C2×C4⋊C8—531no—
233#418—1061no—
234C2×#102—1061no—
235#407—22no—
236#335—534no—
237#333—532no—
238#332—532no—
239#331—534no—
240C53#1942011yescharacteristic, complemented, Sylow
241C22×D4—1061no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4D21211yescharacteristic
245C23×C4—21no—
246C2×SD16—1064no—
247C2×D8—1064no—
248C2×D8—2124no—
249C2×M16—24no—
250C2×M16—1061no—
251C22×C8—1061no—
252C4⋊Q8—531no—
253C4⋊Q8—531no—
254#127—532no—
255#127—1061no—
256#127—1061no—
257#127—1061no—
258#125—1061no—
259C22⋊Q8—1061no—
260C22⋊Q8—1061no—
261C4×Q8—1061no—
262C4×Q8—1061no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4D21212yes—
266C4×D4C22×D5312yes—
267#117—1061no—
268C2×C4⋊C4D21211yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C22×D5311yescharacteristic
271#107—1062no—
272#106—1062no—
273C4⋊C8—532no—
274C4⋊C8—532no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8C22⋊D5312yes—
278C4⋊C8C22×D5312yes—
279#102—1062no—
280#102—1062no—
281#102—1064no—
282#102—1064no—
283C22⋊C8—21no—
284C22⋊C8—24no—
285C22⋊C8C22⋊D5312yes—
286C4×C8—21no—
287C4×C8C22×D5312yes—
288C24—21no—
289C24—2121no—
290C2×Q8—1061no—
291C2×Q8—1061no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—4242no—
299C2×D4—4242no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2122no—
304C2×D4—2122no—
305C2×D4C2×D21212yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4C2×D21211yescharacteristic
311C22×C4#1171994511yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313SD16—4244no—
314D8—2128no—
315M16—84no—
316M16—2122no—
317C2×C8—22no—
318C2×C8—1062no—
319C2×C8—1062no—
320C2×C8—2122no—
321C2×C8C2×C22⋊D5314yes—
322C4⋊C4—21no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4—1061no—
327C4⋊C4—2121no—
328C4⋊C4—2121no—
329C4⋊C4C2×D21212yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—2121no—
333C42—2121no—
334C42C2×D21211yescharacteristic
335C42#1171994511yescharacteristic
336C42C2×C22⋊D5311yescharacteristic
337C42C23×D5311yescharacteristic
338C23—22no—
339C23—42no—
340C23—81no—
341C23—81no—
342C23—4242no—
343C23—4242no—
344C23—4242no—
345C23—2122no—
346C23#1313319111yescharacteristic
347Q8—4241no—
348Q8—2122no—
349D4—22no—
350D4—22no—
351D4—42no—
352D4—4242no—
353D4—4242no—
354D4—4242no—
355D4—4242no—
356D4—2122no—
357D4—2122no—
358D4—2124no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4#1313319111yescharacteristic
368C2×C4#1313319111yescharacteristic
369C2×C4#1313319111yescharacteristic
370C2×C4#1313324512yes—
371C2×C4#1313325012yes—
372C2×C4C22×C22⋊D5311yescharacteristic
373C2×C4C22×D21211yescharacteristic
374C2×C4C2×#1171994511yescharacteristic
375C8—24no—
376C8—42no—
377C8—2122no—
378C8—2122no—
379C22—21no—
380C22—21no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—84no—
385C22—8484no—
386C22—4242no—
387C22—4242no—
388C22—4242no—
389C22#1681086912yes—
390C22C2×#1313319111yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—4241no—
396C4—4241no—
397C4#1681086912yes—
398C4C2×#1313324511yescharacteristic
399C4C2×#1313325011yescharacteristic
400C4#1681142612yes—
401C2—22no—
402C2—82no—
403C2—8482no—
404C2#2028421811yescharacteristic
405C2C2×#1681086911yescharacteristic
406C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2975C2
socleC22×C53C2×#121
prime radicalC2×#16810818C2
hypercenter#2975D53
hypocenterC53#19420
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
1. #20285658C2semidirect
2. #20284512C2semidirect
4. #20274248C2semidirect
5. #20274180C2semidirect
6. #20273490C2semidirect
7. #20273382C2semidirect
8. #20273370C2semidirect
9. C53×#2975C2semidirect
13. #16810996C22semidirect
14. #16810970C22semidirect
16. #16810913C22semidirect
17. #16810903C22semidirect
27. #16810376C22semidirect
28. #16810371C22semidirect
29. #16810370C22semidirect
30. #16810366C22semidirect
33. C53×#436C22semidirect
201. #2975D53semidirect
227. #436C2×D53semidirect
240. C53#19420semidirect

Maximal subgroups

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

subgroupindex
#202856582
#202845122
C2×#168108182
#202742482
#202741802
#202734902
#202733822
#202733702
C53×#29752
#1942053