Finite Group Databasebeta

← #27280896

Subgroups — #27280896

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284187C211yesmaximal, characteristic
2#20274730C211yesmaximal, characteristic, complemented
3#20273276C211yesmaximal, characteristic
4#20273115C211yesmaximal, characteristic
5#20272611C211yesmaximal, characteristic, complemented
6#20272574C211yesmaximal, characteristic, complemented
7C53×#2983C211yesmaximal, characteristic, complemented
8C2×#13133248—21no—
9#16811173C2211yescharacteristic
10#16810845—22no—
11#16810833C412yescomplemented
12#16810823C412yescomplemented
13#16810647C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15#16810622—21no—
16C2×#13133141—21no—
17#16810489C2211yescharacteristic
18#16810385—21no—
19#16810358—22no—
20#16810352—22no—
21C2×C4×Q8×C53C2211yescharacteristic
22C53×#446—22no—
23C53×#446—22no—
24C53×#432—22no—
25C53×#431C2211yescharacteristic
26C53×#431C2211yescharacteristic
27C2×#11719945—21no—
28#13133248—42no—
29#13133248—42no—
30C2×#11719925—22no—
31C2×#11719924—22no—
32C2×#11719917—21no—
33C22×#88322—21no—
34#13133217C2×C411yescharacteristic
35#13133214C2×C411yescharacteristic
36#13133205—21no—
37#13133191—21no—
38#13133191—21no—
39C4×D212—21no—
40C4×D212—21no—
41C2×C4⋊Dic53—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322C2×C412yes—
46#13133141—42no—
47#13133140—22no—
48#13133137—22no—
49#13133124—21no—
50#13133124—21no—
51C22×Q8×C53D411yescharacteristic
52C4×Q8×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53C2×C412yes—
55C4×Q8×C53D412yes—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C42×C53—21no—
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—24no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C22⋊C8×C53—21no—
64C22⋊C8×C53D412yes—
65#97×C53—24no—
66C4×C8×C53—21no—
67C4×C8×C53D412yes—
68C2×Q8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53C22⋊C412yes—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C2×C22⋊D53—41no—
80C22×Dic53—41no—
81#11719945—41no—
82#11719945—41no—
83#11719945—41no—
84#11719945—41no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×C4×D53—41no—
89C2×Dic106—21no—
90C2×C8×C53—22no—
91C2×C8×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53C22⋊C412yes—
96C42×C53—22no—
97C42×C53—22no—
98C42×C53C22×C411yescharacteristic
99C42×C53C2×D411yescharacteristic
100C42×C53C2×D411yescharacteristic
101C42×C53#4111yescharacteristic
102#11719925—44no—
103#11719924—44no—
104#11719921—41no—
105#11719921—41no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110#11719917—41no—
111#11719917—41no—
112C2×#88322—22no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C23×C53#12111yescharacteristic
117C22×D53—41no—
118Q8×C53—22no—
119Q8×C53—22no—
120Q8×C53—41no—
121Q8×C53—41no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—81no—
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53#12311yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C53#13711yescharacteristic
139C22⋊D53—81no—
140C22⋊D53—81no—
141C2×Dic53—41no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145D212—42no—
146D212—41no—
147D212—41no—
148D212—81no—
149C4×D53—81no—
150C4×D53—81no—
151Dic106—41no—
152Dic106—41no—
153C8×C53—42no—
154C8×C53—42no—
155#88322—42no—
156#88322—81no—
157#9310—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53#38611yescharacteristic
161C22×C53#46811yescharacteristic
162C22×C53#46911yescharacteristic
163C2×D53—81no—
164C2×D53—81no—
165C2×D53—81no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—42no—
171C4×C53—42no—
172C4×C53#41711yescharacteristic
173C4×C53#41811yescharacteristic
174C4×C53#46811yescharacteristic
175C4×C53#46911yescharacteristic
176Dic53—81no—
177Dic53—161no—
178#3037—531no—
179#2983D5311yescharacteristic, complemented
180#1805—531no—
181#1517—531no—
182#1480—531no—
183#1301—531no—
184#1286—531no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#188111yescharacteristic
188C2×C53#189511yescharacteristic
189C2×C53#190511yescharacteristic
190D53—161no—
191C2×#137—1061no—
192#532—531no—
193C2×C4×Q8C2×D5311yescharacteristic
194#464—1062no—
195#462—532no—
196#460—532no—
197#446—22no—
198#446—22no—
199#432—22no—
200#431C2×D5311yescharacteristic
201#431C2×D5311yescharacteristic
202#423—531no—
203C2×C4⋊C8—531no—
204#417—1061no—
205C2×#103—1061no—
206#382—531no—
207#340—1061no—
208#327—1062no—
209D4⋊C8—1062no—
210C53#931011yescharacteristic, complemented, Sylow
211C2×#41—1061no—
212C22×Q8D21211yescharacteristic
213#137—2122no—
214#137—2122no—
215C2×Q16—1062no—
216C2×SD16—1062no—
217C2×M16—1061no—
218C22×C8—1061no—
219C4⋊Q8—531no—
220#127—531no—
221#124—1061no—
222#121—1061no—
223#121—1061no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8C4×D5312yes—
227C4×Q8C22⋊D5312yes—
228C4×D4—1061no—
229C4×D4—1061no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4D21211yescharacteristic
233C2×C42—21no—
234C2×C42C22×D5311yescharacteristic
235C4⋊C8—24no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—532no—
239C4⋊C8—1061no—
240C4⋊C8—1061no—
241C4⋊C8—1061no—
242#103—1062no—
243#103—2122no—
244#102—1062no—
245C22⋊C8—21no—
246C22⋊C8C22⋊D5312yes—
247#97—24no—
248C4×C8—21no—
249C4×C8C22⋊D5312yes—
250#95—1061no—
251#95—1061no—
252#41—2121no—
253#41—2121no—
254#41—2121no—
255#41—2121no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—1061no—
262C2×Q8#1171992112yes—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4—2121no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4C2×D21211yescharacteristic
272C22×C4#1171994511yescharacteristic
273C22×C4C2×C22⋊D5311yescharacteristic
274Q16—2124no—
275SD16—2124no—
276M16—2121no—
277M16—2121no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—1062no—
281C2×C8—2121no—
282C2×C8—2121no—
283C2×C8—2121no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—2121no—
290C4⋊C4—2121no—
291C4⋊C4#1171992112yes—
292C22⋊C4—2121no—
293C22⋊C4—2121no—
294C42—22no—
295C42—22no—
296C42C2×C4×D5311yescharacteristic
297C42#1171994511yescharacteristic
298C42C2×C22⋊D5311yescharacteristic
299C42C2×C22⋊D5311yescharacteristic
300C23—2121no—
301C23#1313319111yescharacteristic
302Q8—22no—
303Q8—22no—
304Q8—41no—
305Q8—41no—
306Q8—2121no—
307Q8—2121no—
308D4—4241no—
309D4—4241no—
310D4—4241no—
311D4—2122no—
312D4—2121no—
313D4—2121no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—4241no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—2121no—
327C2×C4C2×#1171992111yescharacteristic
328C2×C4C4×C22⋊D5311yescharacteristic
329C2×C4C4×C22⋊D5311yescharacteristic
330C2×C4#1313318611yescharacteristic
331C2×C4#1313319111yescharacteristic
332C2×C4#1313320011yescharacteristic
333C2×C4#1313324511yescharacteristic
334C2×C4#1313324511yescharacteristic
335C2×C4#1313324511yescharacteristic
336C2×C4#1313325011yescharacteristic
337C8—42no—
338C8—42no—
339C8—4241no—
340C8—2122no—
341C22—21no—
342C22—21no—
343C22—4241no—
344C22—4241no—
345C22—4241no—
346C22#1681050211yescharacteristic
347C22#1681086911yescharacteristic
348C22#1681087611yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—42no—
355C4—8481no—
356C4—4241no—
357C4#1681062511yescharacteristic
358C4#1681062911yescharacteristic
359C4#1681086911yescharacteristic
360C4#1681087611yescharacteristic
361C2—21no—
362C2—21no—
363C2—8481no—
364C2#2027507711yescharacteristic
365C2#2027515311yescharacteristic
366C2#2027520911yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2983C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2983D53
hypocenterC53#9310
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. #20274730C2semidirect
5. #20272611C2semidirect
6. #20272574C2semidirect
7. C53×#2983C2semidirect
11. #16810833C4semidirect
12. #16810823C4semidirect
179. #2983D53semidirect
210. C53#9310semidirect

Maximal subgroups

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

subgroupindex
#202841872
#202747302
#202732762
#202731152
#202726112
#202725742
C53×#29832
#931053