Finite Group Databasebeta

← #27280826

Subgroups — #27280826

Order 13568.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285836C211yesmaximal, characteristic
2#20275547C211yesmaximal, characteristic
3#20273387C211yesmaximal, characteristic, complemented
4#20273274C211yesmaximal, characteristic
5#20272573C211yesmaximal, characteristic, complemented
6#20272544C211yesmaximal, characteristic, complemented
7C53×#2327C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16811026C412yescomplemented
10#16810971C412yescomplemented
11#16810936—22no—
12#16810900—22no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×#13133145—21no—
17C2×#13133145—21no—
18#16810369—22no—
19#16810356—22no—
20C2×C4⋊Q8×C53C2211yescharacteristic
21C2×C4×Q8×C53—21no—
22C53×#391—21no—
23C53×#387—22no—
24C53×#387—22no—
25C53×#382C2211yescharacteristic
26C53×#376C2211yescharacteristic
27C2×#11719945—21no—
28C2×#11719917—21no—
29#13133217C2×C411yescharacteristic
30#13133214C2×C411yescharacteristic
31#13133205—21no—
32#13133191—21no—
33#13133191—21no—
34C4×D212—21no—
35C4×D212—21no—
36#13133150—24no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41#13133145—42no—
42#13133145—42no—
43#13133137—24no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46C4×#88322—21no—
47C4×#88322C2×C412yes—
48C22×Q8×C53—21no—
49C4⋊Q8×C53—21no—
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53C2×C412yes—
52C4⋊Q8×C53D412yes—
53C4×Q8×C53—42no—
54C4×Q8×C53—42no—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53D412yes—
59C2×C42×C53—22no—
60C2×C42×C53D411yescharacteristic
61C2×C42×C53D411yescharacteristic
62C2×C42×C53C2311yescharacteristic
63#95×C53—24no—
64#95×C53—21no—
65#95×C53—21no—
66#95×C53—21no—
67#95×C53D412yes—
68C2×Q8×C53—22no—
69C2×Q8×C53—22no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53—41no—
74C22×C4×C53—21no—
75C22×C4×C53—22no—
76C22×C4×C53—21no—
77C22×C4×C53—22no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C2×C22⋊D53—41no—
82#11719945—41no—
83#11719945—41no—
84#11719945—41no—
85#11719945—41no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×C4×D53—41no—
90C2×Dic106—21no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—44no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53C22⋊C414yes—
98C42×C53—21no—
99C42×C53—21no—
100C42×C53—21no—
101C42×C53—44no—
102C42×C53D812yes—
103C42×C53C22×C411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C42×C53#4111yescharacteristic
107#11719921—41no—
108#11719921—41no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111#11719917—41no—
112#11719917—41no—
113C2×#88322—22no—
114C2×#88322—41no—
115C2×#88322—41no—
116C23×C53#12011yescharacteristic
117C22×D53—41no—
118Q8×C53—42no—
119Q8×C53—42no—
120Q8×C53—42no—
121Q8×C53—42no—
122C2×C4×C53—24no—
123C2×C4×C53—22no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12311yescharacteristic
142C2×C4×C53C2×D811yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C22⋊D53—81no—
147C22⋊D53—81no—
148C2×Dic53—41no—
149D212—42no—
150D212—41no—
151D212—41no—
152D212—81no—
153C4×D53—81no—
154C4×D53—81no—
155Dic106—41no—
156Dic106—41no—
157#88322—42no—
158#88322—81no—
159#88322—81no—
160#9300—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53#38611yescharacteristic
164C22×C53#44711yescharacteristic
165C22×C53#44911yescharacteristic
166C2×D53—81no—
167C2×D53—81no—
168C2×D53—81no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—42no—
174C4×C53—44no—
175C4×C53—82no—
176C4×C53—81no—
177C4×C53C4×D811yescharacteristic
178C4×C53#44211yescharacteristic
179C4×C53#45911yescharacteristic
180C4×C53#46311yescharacteristic
181Dic53—81no—
182#3226—531no—
183#2327D5311yescharacteristic, complemented
184#1963—531no—
185#1546—531no—
186#1517—531no—
187#1286—531no—
188#1274—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53#187911yescharacteristic
192C2×C53#190511yescharacteristic
193C2×C53#191211yescharacteristic
194D53—161no—
195#532—531no—
196C2×C4⋊Q8C2×D5311yescharacteristic
197C2×C4×Q8—21no—
198#500—532no—
199#486—532no—
200#480—1062no—
201#474—1062no—
202#431—531no—
203#431—531no—
204#423—531no—
205C2×#104—1061no—
206C2×#104—1061no—
207#391—21no—
208#387—22no—
209#387—22no—
210#382C2×D5311yescharacteristic
211#376C2×D5311yescharacteristic
212#332—1062no—
213#327—1062no—
214C53#930011yescharacteristic, complemented, Sylow
215C2×#41—1061no—
216C22×Q8—21no—
217C2×M16—1061no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—531no—
221C4⋊Q8C4×D5312yes—
222C4⋊Q8D21212yes—
223#127—531no—
224#124—1061no—
225#121—1061no—
226#121—1061no—
227C4×Q8—42no—
228C4×Q8—42no—
229C4×D4—1061no—
230C4×D4—1061no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22⋊D5312yes—
235C2×C42—22no—
236C2×C42C22⋊D5311yescharacteristic
237C2×C42C22⋊D5311yescharacteristic
238C2×C42C22×D5311yescharacteristic
239#107—1064no—
240C4⋊C8—1061no—
241C4⋊C8—1061no—
242C4⋊C8—1061no—
243C4⋊C8—1061no—
244#104—2122no—
245#104—2122no—
246#102—1064no—
247C22⋊C8—1061no—
248C22⋊C8—1061no—
249C4×C8—532no—
250C4×C8—1061no—
251#95—24no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95C22⋊D5312yes—
256#41—2121no—
257#41—2121no—
258#41—2121no—
259#41—2121no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—2121no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—2121no—
276C22×C4C2×C22⋊D5311yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278C22×C4C2×C22⋊D5311yescharacteristic
279M16—2121no—
280M16—2121no—
281C2×C8—1062no—
282C2×C8—2121no—
283C2×C8—2121no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4#1171992114yes—
293C22⋊C4—2121no—
294C22⋊C4—2121no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—44no—
299C42#1171992212yes—
300C42C2×C4×D5311yescharacteristic
301C42C2×D21211yescharacteristic
302C42#1171994511yescharacteristic
303C42#1171994511yescharacteristic
304C23—2121no—
305C23#1313318611yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—42no—
310Q8—2121no—
311Q8—2121no—
312D4—4241no—
313D4—4241no—
314D4—4241no—
315D4—2122no—
316D4—2121no—
317D4—2121no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—2121no—
335C2×C4C2×#1171992111yescharacteristic
336C2×C4C4×C22⋊D5311yescharacteristic
337C2×C4C4×C22⋊D5311yescharacteristic
338C2×C4#1313319111yescharacteristic
339C2×C4#1313319111yescharacteristic
340C2×C4#1313320011yescharacteristic
341C2×C4C2×#1171992211yescharacteristic
342C2×C4#1313323711yescharacteristic
343C2×C4#1313324211yescharacteristic
344C2×C4#1313324311yescharacteristic
345C8—4241no—
346C8—4241no—
347C8—2122no—
348C22—21no—
349C22—21no—
350C22—4241no—
351C22—4241no—
352C22—4241no—
353C22#1681050211yescharacteristic
354C22#1681075711yescharacteristic
355C22#1681076811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—44no—
362C4—82no—
363C4—81no—
364C4—4241no—
365C4C4×#1171992211yescharacteristic
366C4#1681073311yescharacteristic
367C4#1681081811yescharacteristic
368C4#1681083811yescharacteristic
369C2—21no—
370C2—21no—
371C2—8481no—
372C2#2027506311yescharacteristic
373C2#2027520911yescharacteristic
374C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2327C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2327D53
hypocenterC53#9300
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
3. #20273387C2semidirect
5. #20272573C2semidirect
6. #20272544C2semidirect
7. C53×#2327C2semidirect
9. #16811026C4semidirect
10. #16810971C4semidirect
183. #2327D53semidirect
214. C53#9300semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202755472
#202733872
#202732742
#202725732
#202725442
C53×#23272
#930053