Finite Group Databasebeta

← #27280784

Subgroups — #27280784

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic
2#20275539C211yesmaximal, characteristic
3#20273384C211yesmaximal, characteristic
4#20273279C211yesmaximal, characteristic, complemented
5#20272575C211yesmaximal, characteristic, complemented
6#20272546C211yesmaximal, characteristic, complemented
7C53×#2961C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9#16810997C412yescomplemented
10#16810971C412yescomplemented
11#16810823—22no—
12#16810773—22no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15C2×#13133139—21no—
16C2×#13133139—21no—
17#16810489C2211yescharacteristic
18#16810366—22no—
19#16810360—22no—
20C53×#533C2211yescharacteristic
21C53×#446—22no—
22C53×#446—22no—
23C53×#424—21no—
24C53×#423C2211yescharacteristic
25C53×#405—21no—
26C2×#97×C53C2211yescharacteristic
27C22×D212—21no—
28C22×D212—21no—
29C22×D212—21no—
30C2×#11719924—24no—
31#13133214—21no—
32#13133214—21no—
33#13133214—21no—
34#13133214C2×C412yes—
35C2×C4⋊Dic53—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38#13133139—42no—
39#13133139—42no—
40#13133137—24no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43C4×#88322—21no—
44C4×#88322C2×C412yes—
45#13133124—21no—
46#13133124—21no—
47C2×M16×C53—21no—
48C22×C8×C53D411yescharacteristic
49C22×C8×C53D411yescharacteristic
50C4⋊Q8×C53C2×C411yescharacteristic
51C4⋊Q8×C53C2×C411yescharacteristic
52#125×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53D412yes—
55C4×Q8×C53—21no—
56C4×Q8×C53D412yes—
57#117×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—24no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53D412yes—
62#97×C53—22no—
63#97×C53—21no—
64#97×C53—21no—
65C4×C8×C53—22no—
66C23×D53—41no—
67C2×Q8×C53—21no—
68C2×Q8×C53C22⋊C412yes—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53#4111yescharacteristic
72C22×Dic53—41no—
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—82no—
84C2×D212—82no—
85M16×C53—41no—
86M16×C53—41no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—22no—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53SD1612yescomplemented
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53C22⋊C412yes—
100C22⋊C4×C53—22no—
101C42×C53—22no—
102C42×C53C22×C411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106#11719924—84no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109C2×#88322—22no—
110C2×#88322—41no—
111C23×C53#12111yescharacteristic
112C22×D53—42no—
113C22×D53—82no—
114C22×D53—82no—
115C22×D53—81no—
116C22×D53—81no—
117Q8×C53—42no—
118Q8×C53—81no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53C2×C22⋊C411yescharacteristic
124C2×C4×C53C4×D411yescharacteristic
125C2×C4×C53C4×D411yescharacteristic
126C2×C4×C53#12011yescharacteristic
127C2×C4×C53#12111yescharacteristic
128C2×C4×C53#12311yescharacteristic
129C2×C4×C53C2×SD1611yescharacteristic
130C2×C4×C53#13511yescharacteristic
131C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C53#13711yescharacteristic
133C2×Dic53—81no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136D212—44no—
137D212—42no—
138D212—42no—
139D212—82no—
140D212—82no—
141D212—82no—
142D212—81no—
143D212—81no—
144C8×C53—22no—
145C8×C53—41no—
146C8×C53—42no—
147#88322—42no—
148#88322—81no—
149#9294—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53#38611yescharacteristic
153C22×C53#46011yescharacteristic
154C22×C53#46411yescharacteristic
155C2×D53—82no—
156C2×D53—82no—
157C2×D53—82no—
158C2×D53—162no—
159C2×D53—162no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—42no—
165C4×C53—42no—
166C4×C53C4×SD1611yescharacteristic
167C4×C53#44011yescharacteristic
168C4×C53#44811yescharacteristic
169C4×C53#44911yescharacteristic
170Dic53—161no—
171#3208—531no—
172#2961D5311yescharacteristic, complemented
173#1961—531no—
174#1545—531no—
175#1518—531no—
176#1286—531no—
177#1275—531no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53#188011yescharacteristic
181C2×C53#190311yescharacteristic
182C2×C53#191011yescharacteristic
183D53—162no—
184#533C2×D5311yescharacteristic
185C2×#127—531no—
186#492—532no—
187#486—532no—
188#460—1062no—
189#450—1062no—
190#446—22no—
191#446—22no—
192#431—531no—
193#431—531no—
194#424—21no—
195#423C2×D5311yescharacteristic
196C2×#102—1061no—
197C2×#102—1061no—
198#405—21no—
199C2×#97C2×D5311yescharacteristic
200#382—531no—
201#331—1062no—
202#328—1062no—
203C53#929411yescharacteristic, complemented, Sylow
204C22×D4—1061no—
205C22×D4—1061no—
206C22×D4—1061no—
207C2×SD16—1064no—
208C2×M16—21no—
209C22×C8C22⋊D5311yescharacteristic
210C22×C8C22⋊D5311yescharacteristic
211C4⋊Q8C4×D5311yescharacteristic
212C4⋊Q8C4×D5311yescharacteristic
213#127—532no—
214#127—1061no—
215#127—1061no—
216#127—1061no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D21212yes—
220C4×Q8—21no—
221C4×Q8C22⋊D5312yes—
222#117—21no—
223C2×C4⋊C4—1061no—
224C2×C42C22×D5311yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—1061no—
228C4⋊C8—1061no—
229C4⋊C8C22⋊D5312yes—
230#102—1064no—
231#102—2122no—
232#102—2122no—
233C22⋊C8—1061no—
234C22⋊C8—1061no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—532no—
240C4×C8—1061no—
241#95—1061no—
242#95—1061no—
243C24—2121no—
244C2×Q8—21no—
245C2×Q8#1171992112yes—
246C2×D4—1062no—
247C2×D4—1062no—
248C2×D4—1062no—
249C2×D4—4242no—
250C2×D4—4242no—
251C2×D4—2121no—
252C2×D4—2121no—
253C2×D4—2121no—
254C2×D4—2121no—
255C2×D4—2121no—
256C2×D4—2121no—
257C2×D4—2121no—
258C22×C4—2121no—
259C22×C4C2×D21211yescharacteristic
260C22×C4#1171994511yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262SD16—4244no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—1062no—
272C2×C8—2121no—
273C2×C8#1171992412yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C4⋊C4#1171992112yes—
282C22⋊C4—22no—
283C42—22no—
284C42C2×C4×D5311yescharacteristic
285C42#1171994511yescharacteristic
286C42C2×C22⋊D5311yescharacteristic
287C42C2×C22⋊D5311yescharacteristic
288C23—4242no—
289C23—4242no—
290C23—4241no—
291C23—4241no—
292C23—2122no—
293C23#1313319111yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—4242no—
297D4—4242no—
298D4—4242no—
299D4—4241no—
300D4—4241no—
301D4—2124no—
302D4—2122no—
303D4—2122no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—4241no—
309C2×C4—4241no—
310C2×C4—4241no—
311C2×C4C2×#1171992111yescharacteristic
312C2×C4C4×C22⋊D5311yescharacteristic
313C2×C4C4×C22⋊D5311yescharacteristic
314C2×C4#1313318611yescharacteristic
315C2×C4#1313319111yescharacteristic
316C2×C4#1313320011yescharacteristic
317C2×C4C2×#1171992411yescharacteristic
318C2×C4#1313323711yescharacteristic
319C2×C4#1313324211yescharacteristic
320C2×C4#1313324811yescharacteristic
321C8—22no—
322C8—41no—
323C8—42no—
324C8—4241no—
325C8—2122no—
326C22—21no—
327C22—21no—
328C22—8482no—
329C22—8482no—
330C22—4242no—
331C22—4242no—
332C22—4242no—
333C22#1681050211yescharacteristic
334C22#1681082311yescharacteristic
335C22#1681084511yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—8481no—
343C4C4×#1171992411yescharacteristic
344C4#1681072111yescharacteristic
345C4#1681076211yescharacteristic
346C4#1681076811yescharacteristic
347C2—21no—
348C2—21no—
349C2—8482no—
350C2#2027507011yescharacteristic
351C2#2027519511yescharacteristic
352C2#2027524211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2961C2
socleC22×C53#386
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#2961D53
hypocenterC53#9294
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
4. #20273279C2semidirect
5. #20272575C2semidirect
6. #20272546C2semidirect
7. C53×#2961C2semidirect
9. #16810997C4semidirect
10. #16810971C4semidirect
93. C2×C8×C53SD16semidirect
172. #2961D53semidirect
203. C53#9294semidirect
273. C2×C8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202755392
#202733842
#202732792
#202725752
#202725462
C53×#29612
#929453