Finite Group Databasebeta

← #27280642

Subgroups — #27280642

Order 13568.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic
2#20283664C211yesmaximal, characteristic, complemented
3#20275554C211yesmaximal, characteristic
4C2×#16810366C211yesmaximal, characteristic
5#20272610C211yesmaximal, characteristic, complemented
6#20272565C211yesmaximal, characteristic, complemented
7C53×#1550C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9#16810997—21no—
10#16810971—21no—
11#16810823—22no—
12#16810773C414yes—
13#16810754—21no—
14#16810754—22no—
15Q8×#88322—21no—
16#16810676C2211yescharacteristic
17#16810650—21no—
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133133—21no—
20C2×#13133133—21no—
21#16810579—21no—
22C4×#11719917C2211yescharacteristic
23#16810366—21no—
24#16810366—21no—
25C53×#533C2211yescharacteristic
26C53×#423C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#333—21no—
29C53×#332—21no—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212C2×C412yes—
33C2×#11719924—24no—
34C2×#11719917D411yescharacteristic
35C2×#11719917D411yescharacteristic
36C22×#88322—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41C4⋊#88322—22no—
42C4⋊#88322—22no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D411yescharacteristic
46C4⋊#88322D411yescharacteristic
47#13133137—24no—
48#13133133—42no—
49#13133133—42no—
50C22⋊#88322C2×C412yes—
51#13133128—22no—
52#13133128—21no—
53C4×#88322—22no—
54C4×#88322—21no—
55C4×#88322—21no—
56C2×M16×C53—21no—
57C22×C8×C53—21no—
58C4⋊Q8×C53D411yescharacteristic
59C4⋊Q8×C53D411yescharacteristic
60#125×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53C2×C412yes—
63C4×Q8×C53—21no—
64C4×Q8×C53D412yes—
65#117×C53—21no—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C23×D53—22no—
72C2×Q8×C53—21no—
73C2×Q8×C53C22⋊C412yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C2×D212—22no—
78C2×D212—22no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—82no—
88C2×D212—81no—
89C2×D212—81no—
90M16×C53—41no—
91M16×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—81no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53C22⋊C412yes—
101C22⋊C4×C53—22no—
102C42×C53—22no—
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53#4111yescharacteristic
107#11719924—84no—
108#11719917—21no—
109#11719917—21no—
110#11719917—21no—
111#11719917SD1612yescomplemented
112C2×#88322—21no—
113C2×#88322—22no—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322—41no—
118C2×#88322—41no—
119C23×C53C4×D411yescharacteristic
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—82no—
124C22×D53—82no—
125Q8×C53—42no—
126Q8×C53—41no—
127Q8×C53—41no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—41no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53#12011yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12311yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53#13711yescharacteristic
143D212—44no—
144D212—82no—
145D212—82no—
146D212—81no—
147D212—81no—
148D212—81no—
149D212—81no—
150D212—81no—
151D212—81no—
152C8×C53—81no—
153C8×C53—81no—
154#88322—22no—
155#88322—42no—
156#88322—41no—
157#88322—41no—
158#88322—41no—
159#9274—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53#38611yescharacteristic
163C22×C53C4×SD1611yescharacteristic
164C22×C53#44011yescharacteristic
165C2×D53—82no—
166C2×D53—82no—
167C2×D53—82no—
168C2×D53—162no—
169C2×D53—162no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—42no—
174C4×C53—42no—
175C4×C53#44811yescharacteristic
176C4×C53#44911yescharacteristic
177C4×C53#46011yescharacteristic
178C4×C53#46411yescharacteristic
179C4×C53#47111yescharacteristic
180C4×C53#47211yescharacteristic
181#3208—531no—
182#2962—531no—
183#1965—531no—
184#1550D5311yescharacteristic, complemented
185C2×#331—531no—
186#1301—531no—
187#1283—531no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53#188011yescharacteristic
191C2×C53#190711yescharacteristic
192C2×C53#191411yescharacteristic
193D53—162no—
194#533C2×D5311yescharacteristic
195C2×#127—531no—
196#492—1061no—
197#486—1061no—
198#460—1062no—
199#450—534no—
200#446—1061no—
201#446—1062no—
202C8×Q8—1061no—
203#431—531no—
204#424—1061no—
205#423C2×D5311yescharacteristic
206C2×C4⋊C8—531no—
207C2×C4⋊C8C2×D5311yescharacteristic
208C2×#100—1061no—
209C2×#100—1061no—
210#405—1061no—
211C4×M16—531no—
212#333—21no—
213#332—21no—
214#331—1061no—
215#331—1061no—
216C53#927411yescharacteristic, complemented, Sylow
217C22×D4—532no—
218C22×D4—1061no—
219C22×D4—1061no—
220C2×SD16—1064no—
221C2×M16—21no—
222C2×M16—531no—
223C2×M16—531no—
224C22×C8—21no—
225C22×C8—1061no—
226C4⋊Q8C22⋊D5311yescharacteristic
227C4⋊Q8C22⋊D5311yescharacteristic
228#127—1061no—
229#127—1061no—
230#127—1061no—
231#127—1061no—
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8C4×D5312yes—
235C4×Q8—21no—
236C4×Q8D21212yes—
237#117—21no—
238C2×C42C22×D5311yescharacteristic
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—531no—
244C4⋊C8—531no—
245C4⋊C8—1062no—
246C4⋊C8—1062no—
247C4⋊C8—1061no—
248C4⋊C8—1061no—
249#102—1064no—
250#100—2122no—
251#100—2122no—
252C22⋊C8—532no—
253#97—1062no—
254#97—1061no—
255C4×C8—1062no—
256C4×C8—1061no—
257C4×C8—1061no—
258C24—1062no—
259C2×Q8—21no—
260C2×Q8#1171992112yes—
261C2×D4—1062no—
262C2×D4—1062no—
263C2×D4—4241no—
264C2×D4—4241no—
265C2×D4—4242no—
266C2×D4—2121no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4—2121no—
271C2×D4—2121no—
272C2×D4—2121no—
273C2×D4—2121no—
274C22×C4C2×C4×D5311yescharacteristic
275C22×C4D4×D5311yescharacteristic
276C22×C4#1171994711yescharacteristic
277SD16—4244no—
278M16—41no—
279M16—41no—
280M16—532no—
281M16—1061no—
282M16—1061no—
283M16—1061no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—81no—
287C2×C8—1061no—
288C2×C8—1062no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—1061no—
292C2×C8—2121no—
293C2×C8—2121no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4#1171992112yes—
300C22⋊C4—22no—
301C42—22no—
302C42C2×D21211yescharacteristic
303C42D4×D5311yescharacteristic
304C42#1171994711yescharacteristic
305C42C2×C22⋊D5311yescharacteristic
306C23—4242no—
307C23—4242no—
308C23—2122no—
309C23—2122no—
310C23—2122no—
311C23#1313317711yescharacteristic
312Q8—42no—
313Q8—41no—
314Q8—41no—
315D4—4242no—
316D4—4242no—
317D4—4241no—
318D4—4241no—
319D4—4241no—
320D4—4241no—
321D4—4241no—
322D4—4241no—
323D4—2124no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4C2×#1171992111yescharacteristic
330C2×C4#1313317711yescharacteristic
331C2×C4#1313318711yescharacteristic
332C2×C4#1313318911yescharacteristic
333C2×C4#1313318911yescharacteristic
334C2×C4#1313320111yescharacteristic
335C2×C4SD16×D5311yescharacteristic
336C2×C4#1313324011yescharacteristic
337C2×C4#1313324411yescharacteristic
338C2×C4#1313324911yescharacteristic
339C8—81no—
340C8—81no—
341C8—1062no—
342C8—2122no—
343C8—2121no—
344C8—2121no—
345C8—2121no—
346C22—21no—
347C22—21no—
348C22—8482no—
349C22—8482no—
350C22—4242no—
351C22—4242no—
352C22—4242no—
353C22#1681050011yescharacteristic
354C22#1681070811yescharacteristic
355C22#1681072611yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—42no—
361C4#1681076311yescharacteristic
362C4#1681077111yescharacteristic
363C4#1681082111yescharacteristic
364C4#1681084711yescharacteristic
365C4#1681088811yescharacteristic
366C4#1681089111yescharacteristic
367C2—21no—
368C2—21no—
369C2—8482no—
370C2#2027506811yescharacteristic
371C2#2027522111yescharacteristic
372C2#2027526011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1550C2
socleC22×C53#386
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1550D53
hypocenterC53#9274
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
2. #20283664C2semidirect
5. #20272610C2semidirect
6. #20272565C2semidirect
7. C53×#1550C2semidirect
111. #11719917SD16semidirect
184. #1550D53semidirect
216. C53#9274semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202836642
#202755542
C2×#168103662
#202726102
#202725652
C53×#15502
#927453