Finite Group Databasebeta

← #27274475

Subgroups — #27274475

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283347C211yesmaximal, characteristic
2#20283345C211yesmaximal, characteristic
3#432×D53C211yesmaximal, characteristic
4#20272626C211yesmaximal, characteristic, complemented
5#20272625C211yesmaximal, characteristic, complemented
6#20272625C211yesmaximal, characteristic, complemented
7C53×#1307C211yesmaximal, characteristic, complemented
8C2×C42×D53C2211yescharacteristic
9#16810680C2211yescharacteristic
10#16810680C2211yescharacteristic
11#16810679—21no—
12#16810679—21no—
13#16810679—21no—
14#16810678—21no—
15#16810678—21no—
16#16810677C412yes—
17#16810675—21no—
18#16810675—21no—
19#16810675—21no—
20#16810673C412yes—
21#16810647C2211yescharacteristic
22#16810645—21no—
23C4⋊C8×D53—21no—
24#16810586C412yes—
25#16810585C412yes—
26C22⋊C8×D53C412yes—
27#16810575—21no—
28M16×Dic53C412yes—
29#97×D53—21no—
30C53×#432C2211yescharacteristic
31C53×#432C2211yescharacteristic
32C53×#423C2211yescharacteristic
33C22×C4×D53C2×C411yescharacteristic
34C22×C4×D53C2×C411yescharacteristic
35C22×C4×D53C2×C411yescharacteristic
36C2×#11719917C2×C412yes—
37C2×#11719912—22no—
38C2×C8×D53—22no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42C42×D53D411yescharacteristic
43C42×D53D411yescharacteristic
44C42×D53D411yescharacteristic
45C42×D53D411yescharacteristic
46C42×D53D411yescharacteristic
47C42×D53D411yescharacteristic
48C42×D53Q811yescharacteristic
49C42×D53Q811yescharacteristic
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54#13133146—22no—
55#13133146—22no—
56C22⋊#88322C2×C412yes—
57C22⋊#88322C2×C412yes—
58#13133129—22no—
59#13133129—22no—
60#13133129—22no—
61#13133129—22no—
62#13133128—21no—
63#13133128—21no—
64#13133127—22no—
65#13133127—22no—
66C8×Dic53—22no—
67C8×Dic53—22no—
68C2×M16×C53C2×C412yes—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73C4⋊C8×C53—21no—
74C22⋊C8×C53C2×C412yes—
75C22⋊C8×C53C2×C412yes—
76#97×C53—21no—
77#97×C53—21no—
78C23×D53C4211yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×Dic53C4211yescharacteristic
83C22×Dic53C4211yescharacteristic
84C22×Dic53C4211yescharacteristic
85C2×C4×D53—42no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—42no—
89C2×C4×D53C22⋊C411yescharacteristic
90C2×C4×D53C22⋊C411yescharacteristic
91C2×C4×D53C22⋊C411yescharacteristic
92C2×C4×D53C22⋊C411yescharacteristic
93C2×C4×D53C22⋊C411yescharacteristic
94C2×C4×D53C22⋊C411yescharacteristic
95C2×C4×D53C4⋊C411yescharacteristic
96C2×C4×D53C4⋊C411yescharacteristic
97C2×C4×D53C4⋊C411yescharacteristic
98C2×C4×D53C4⋊C411yescharacteristic
99C2×C4×D53C4⋊C411yescharacteristic
100C2×C4×D53C4⋊C411yescharacteristic
101M16×C53—42no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53C2×Q811yescharacteristic
109C4×Dic53C22⋊C411yescharacteristic
110C4×Dic53C22⋊C411yescharacteristic
111C4×Dic53C22⋊C411yescharacteristic
112C4×Dic53C22⋊C411yescharacteristic
113C4×Dic53C22⋊C411yescharacteristic
114C4×Dic53C22⋊C411yescharacteristic
115C4×Dic53C4⋊C411yescharacteristic
116C4×Dic53C4⋊C411yescharacteristic
117C4×Dic53C4⋊C411yescharacteristic
118C4×Dic53C4⋊C411yescharacteristic
119C4×Dic53C4⋊C411yescharacteristic
120C4×Dic53C4⋊C411yescharacteristic
121#11719917—42no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—22no—
125#11719912—44no—
126C8×D53—44no—
127C23×C53C2×C4211yescharacteristic
128C22×D53—22no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53#9511yescharacteristic
132C22×D53#9511yescharacteristic
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53C2×C22⋊C411yescharacteristic
140C2×C4×C53C2×C4⋊C411yescharacteristic
141C2×C4×C53C2×C4⋊C411yescharacteristic
142C2×C4×C53C2×C4⋊C411yescharacteristic
143C2×Dic53—22no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53#9511yescharacteristic
147C2×Dic53#9511yescharacteristic
148C2×Dic53#9511yescharacteristic
149C2×Dic53#9511yescharacteristic
150C2×Dic53#9511yescharacteristic
151C2×Dic53#9511yescharacteristic
152C4×D53—22no—
153C4×D53—22no—
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—22no—
159C4×D53—22no—
160C4×D53—41no—
161C4×D53—41no—
162C4×D53—41no—
163C4×D53—41no—
164C4×D53—41no—
165C4×D53—41no—
166C8×C53—42no—
167C8×C53—42no—
168C8×C53—42no—
169#88322—42no—
170#88322—42no—
171#88322—42no—
172#8118—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53C2×#9511yescharacteristic
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—21no—
181C2×D53—21no—
182C2×D53—44no—
183C2×D53#34412yes—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53#41311yescharacteristic
190C4×C53#41311yescharacteristic
191Dic53—21no—
192Dic53—21no—
193Dic53—21no—
194Dic53—21no—
195Dic53—21no—
196Dic53#34412yes—
197#2920—531no—
198#2920—531no—
199C2×#432—531no—
200#1307—531no—
201#1307—531no—
202#1307—531no—
203#1307D5311yescharacteristic, complemented
204C2×C53—41no—
205C2×C53#174211yescharacteristic
206C2×C53C2×#34411yescharacteristic
207C2×C53C2×#34411yescharacteristic
208D53—22no—
209D53—41no—
210C22×C42—531no—
211#432—531no—
212#432—531no—
213#432—532no—
214#432—1061no—
215#432—1061no—
216#432—1061no—
217#432—1061no—
218#432—1061no—
219#432C2×D5311yescharacteristic
220#432C2×D5311yescharacteristic
221#431—532no—
222#431—1061no—
223#431—1061no—
224#431—1061no—
225#423—531no—
226#423—1061no—
227#423C2×D5311yescharacteristic
228C2×C4⋊C8—1061no—
229#407—532no—
230#407—532no—
231C2×C22⋊C8—532no—
232C4×M16—532no—
233C4×M16—1061no—
234C2×#97—1061no—
235C53#811811yescharacteristic, complemented, Sylow
236C23×C4—531no—
237C23×C4—531no—
238C23×C4—531no—
239C2×M16—532no—
240C2×M16—1062no—
241C2×M16C4×D5312yes—
242C22×C8—1062no—
243C2×C42—531no—
244C2×C42—531no—
245C2×C42—531no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42C22×D5311yescharacteristic
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—1062no—
260C4⋊C8—1062no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C22⋊C8—532no—
266C22⋊C8—532no—
267C22⋊C8—1062no—
268C22⋊C8—1062no—
269C22⋊C8—1062no—
270C22⋊C8—1062no—
271C22⋊C8C4×D5312yes—
272C22⋊C8C4×D5312yes—
273#97—21no—
274#97—21no—
275#97—1062no—
276#97—1062no—
277#97—1061no—
278#97—1061no—
279C4×C8—1062no—
280C4×C8—1062no—
281C24—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—531no—
291C22×C4—531no—
292C22×C4—531no—
293C22×C4—531no—
294C22×C4—531no—
295C22×C4—531no—
296C22×C4—531no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4—2122no—
300C22×C4—2122no—
301C22×C4C2×C4×D5311yescharacteristic
302C22×C4C2×C4×D5311yescharacteristic
303C22×C4C2×C4×D5311yescharacteristic
304M16—42no—
305M16—2124no—
306M16—2122no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—1062no—
311C2×C8—1062no—
312C2×C8—1062no—
313C2×C8—2124no—
314C42—531no—
315C42—531no—
316C42—531no—
317C42—531no—
318C42—531no—
319C42—531no—
320C42—531no—
321C42—531no—
322C42—531no—
323C42—531no—
324C42—531no—
325C42—531no—
326C42D4×D5311yescharacteristic
327C42D4×D5311yescharacteristic
328C42D4×D5311yescharacteristic
329C42Q8×D5311yescharacteristic
330C23—531no—
331C23—531no—
332C23—1062no—
333C23—2121no—
334C23—2121no—
335C23C42×D5311yescharacteristic
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—531no—
341C2×C4—531no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—531no—
345C2×C4—531no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4C22⋊C4×D5311yescharacteristic
364C2×C4C22⋊C4×D5311yescharacteristic
365C2×C4C22⋊C4×D5311yescharacteristic
366C2×C4C4⋊C4×D5311yescharacteristic
367C2×C4C4⋊C4×D5311yescharacteristic
368C2×C4C4⋊C4×D5311yescharacteristic
369C8—42no—
370C8—42no—
371C8—42no—
372C8—2122no—
373C8—2122no—
374C8—2122no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—532no—
379C22—1061no—
380C22—1061no—
381C22—1061no—
382C22—1061no—
383C22—1061no—
384C22—2124no—
385C22#95×D5311yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—532no—
392C4—1061no—
393C4—1061no—
394C4—1061no—
395C4—1061no—
396C4—1061no—
397C4#1681060411yescharacteristic
398C4#1681060611yescharacteristic
399C2—41no—
400C2—1062no—
401C2—2121no—
402C2#2027445211yescharacteristic
403C2#344×D5311yescharacteristic
404C2#344×D5311yescharacteristic
subgroupG/H
centerC22#95×D53
commutatorC23×C53C2×C42
FrattiniC2×C42C22×D53
FittingC53×#1307C2
socleC22×C53C2×#95
prime radicalC23×D53C42
cosocleC2×C42×C53C23
hypercenter#1307D53
hypocenterC53#8118
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
4. #20272626C2semidirect
5. #20272625C2semidirect
6. #20272625C2semidirect
7. C53×#1307C2semidirect
203. #1307D53semidirect
235. C53#8118semidirect

Maximal subgroups

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

subgroupindex
#202833472
#202833452
#432×D532
#202726262
#202726252
C53×#13072
#811853