Finite Group Databasebeta

← #27299251

Subgroups — #27299251

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283416C211yesmaximal, characteristic
2#20276357C211yesmaximal, characteristic
3#20276354C211yesmaximal, characteristic
4#20274515C211yesmaximal, characteristic, complemented
5#20272837C211yesmaximal, characteristic, complemented
6#20272835C211yesmaximal, characteristic, complemented
7C53×#1757C211yesmaximal, characteristic, complemented
8C22×#11719912C411yescharacteristic, complemented
9#16811130C2211yescharacteristic
10#16810699—21no—
11#16810699—21no—
12#16810694—21no—
13#16810694—21no—
14#16810651C411yescharacteristic, complemented
15#16810607C2211yescharacteristic
16#16810597C2211yescharacteristic
17#16810590C2211yescharacteristic
18#16810586C411yescharacteristic, complemented
19#16810578—21no—
20#16810578C411yescharacteristic, complemented
21#16810414C412yescomplemented
22#16810412C412yescomplemented
23#16810411C412yescomplemented
24#16810409C412yescomplemented
25#16810394—21no—
26C2×#131×C53C2211yescharacteristic
27C53×#413C2211yescharacteristic
28C53×#410C2211yescharacteristic
29C53×#405—21no—
30C53×#344—21no—
31C22×C4×D53C2×C411yescharacteristic
32C2×#11719917—21no—
33C2×#11719917C2×C411yescharacteristic
34C2×#11719912—21no—
35C2×#11719912—21no—
36C2×#11719912—41no—
37C2×#11719912—41no—
38C22×#88322C2×C411yescharacteristic
39#13133198C2×C411yescharacteristic
40#13133198C2×C411yescharacteristic
41#13133193—21no—
42#13133188—21no—
43#13133185C2×C411yescharacteristic
44#13133185C2×C411yescharacteristic
45C4×C22⋊D53—21no—
46C4×C22⋊D53—21no—
47#13133176—21no—
48#13133176C2×C411yescharacteristic
49#13133146—21no—
50#13133146—21no—
51#13133136—21no—
52#13133134C2×C412yes—
53#13133132—21no—
54#13133132C2×C412yes—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57#13133129—21no—
58#13133129—21no—
59#13133127—21no—
60#13133127—41no—
61C8×Dic53—21no—
62C8×Dic53—41no—
63C2×#41×C53C2311yescharacteristic
64#131×C53—41no—
65#131×C53—41no—
66C2×M16×C53—21no—
67C2×M16×C53C2×C411yescharacteristic
68C2×M16×C53D411yescharacteristic
69C2×M16×C53D411yescharacteristic
70C22×C8×C53C2×C411yescharacteristic
71C22×C8×C53D411yescharacteristic
72C22×C8×C53D411yescharacteristic
73#117×C53—21no—
74#101×C53—21no—
75#100×C53C2×C412yes—
76#99×C53—21no—
77#99×C53C2×C412yes—
78#97×C53—41no—
79C4×C8×C53—41no—
80C23×D53C4211yescharacteristic
81#41×C53—41no—
82#41×C53—41no—
83C2×Q8×C53#4111yescharacteristic
84C2×D4×C53C22×C411yescharacteristic
85C2×D4×C53C22×C411yescharacteristic
86C2×D4×C53#4111yescharacteristic
87C22×C4×C53C22×C411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C2×C22⋊D53—21no—
91C2×C22⋊D53—21no—
92C22×Dic53C4211yescharacteristic
93C2×C4×D53—21no—
94C2×C4×D53—21no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97M16×C53—22no—
98M16×C53—41no—
99M16×C53—41no—
100M16×C53—41no—
101M16×C53—41no—
102C2×C8×C53—21no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—41no—
109C2×C8×C53C22⋊C411yescharacteristic
110C2×C8×C53C22⋊C411yescharacteristic
111C2×C8×C53C22⋊C411yescharacteristic
112C2×C8×C53C22⋊C411yescharacteristic
113C4⋊C4×C53—41no—
114C22⋊C4×C53—22no—
115C42×C53—41no—
116C22⋊Dic53—21no—
117C22⋊Dic53—21no—
118C22⋊Dic53—22no—
119C22⋊Dic53C4211yescharacteristic
120C22⋊Dic53C4211yescharacteristic
121#11719921—21no—
122#11719921—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—41no—
125#11719919—21no—
126#11719919—21no—
127C4×Dic53—21no—
128C4×Dic53—41no—
129#11719917—22no—
130#11719917—41no—
131#11719917—41no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—41no—
135C2×#88322—41no—
136#11719912—42no—
137#11719912—41no—
138#11719912—41no—
139C23×C53C2×C4211yescharacteristic
140C23×C53C4×D411yescharacteristic
141C23×C53C4×D411yescharacteristic
142C22×D53—21no—
143C22×D53—41no—
144C22×D53—81no—
145Q8×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—41no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—42no—
156C2×C4×C53C2×C22⋊C411yescharacteristic
157C2×C4×C53#11711yescharacteristic
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C22⋊D53—81no—
161C22⋊D53—81no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—41no—
166C2×Dic53—42no—
167C4×D53—42no—
168C4×D53—41no—
169C4×D53—41no—
170C8×C53—21no—
171C8×C53—21no—
172C8×C53—21no—
173C8×C53—42no—
174C8×C53#9911yescharacteristic
175C8×C53#9911yescharacteristic
176#88322—42no—
177#88322—41no—
178#88322—41no—
179#12624—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53C4×C22⋊C411yescharacteristic
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—82no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—82no—
195C4×C53C2×#9911yescharacteristic
196C4×C53#41011yescharacteristic
197Dic53—41no—
198Dic53—41no—
199Dic53—41no—
200Dic53—82no—
201#2930—531no—
202#2120—531no—
203#2120—531no—
204#1757—531no—
205#1757D5311yescharacteristic, complemented
206#1392—531no—
207#1391—531no—
208C2×C53—21no—
209C2×C53—41no—
210C2×C53—41no—
211C2×C53—41no—
212C2×C53C4×#9911yescharacteristic
213D53—81no—
214C2×#131C2×D5311yescharacteristic
215C22×M16—531no—
216#525—531no—
217#436—1061no—
218#436—1061no—
219#435—1061no—
220#435—1061no—
221#424—531no—
222#413—531no—
223#413C2×D5311yescharacteristic
224#410—531no—
225#410C2×D5311yescharacteristic
226#408—531no—
227#407—531no—
228#405—21no—
229#405—531no—
230#405—1061no—
231#352—532no—
232#352—532no—
233AΣL(1,𝔽16)—532no—
234AΣL(1,𝔽16)—532no—
235#344—21no—
236#344—1061no—
237C53#1262411yescharacteristic, complemented, Sylow
238C2×#41C22×D5311yescharacteristic
239C23×C4—531no—
240#131—41no—
241#131—41no—
242C2×M16—21no—
243C2×M16—531no—
244C2×M16—1061no—
245C2×M16—1061no—
246C2×M16—1061no—
247C2×M16—2121no—
248C2×M16—2121no—
249C2×M16C4×D5311yescharacteristic
250C2×M16D21211yescharacteristic
251C2×M16C22⋊D5311yescharacteristic
252C22×C8—531no—
253C22×C8C4×D5311yescharacteristic
254C22×C8D21211yescharacteristic
255C22×C8C22⋊D5311yescharacteristic
256#123—531no—
257#123—531no—
258C22⋊Q8—1061no—
259#121—1061no—
260#120—531no—
261#120—531no—
262C4×D4—1061no—
263C4×D4—1061no—
264#117—21no—
265#117—531no—
266#117—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269#101—21no—
270#101—1061no—
271#100—532no—
272#100C4×D5312yes—
273#99—21no—
274#99—532no—
275#99—1061no—
276#99C4×D5312yes—
277C22⋊C8—1061no—
278C22⋊C8—1061no—
279C22⋊C8—1061no—
280C22⋊C8—1061no—
281#97—41no—
282#97—1061no—
283#97—2121no—
284C4×C8—41no—
285C4×C8—1061no—
286C4×C8—2121no—
287C24—531no—
288#41—41no—
289#41—41no—
290C2×Q8#1171994511yescharacteristic
291C2×D4—1061no—
292C2×D4—1061no—
293C2×D4C2×C4×D5311yescharacteristic
294C2×D4C2×C4×D5311yescharacteristic
295C2×D4#1171994511yescharacteristic
296C22×C4—531no—
297C22×C4—1061no—
298C22×C4—1061no—
299C22×C4—2121no—
300C22×C4—2121no—
301C22×C4C2×C4×D5311yescharacteristic
302C22×C4C2×D21211yescharacteristic
303C22×C4C2×C22⋊D5311yescharacteristic
304M16—22no—
305M16—41no—
306M16—41no—
307M16—41no—
308M16—41no—
309M16—1062no—
310M16—2122no—
311M16—2121no—
312M16—2121no—
313M16—2121no—
314M16—2121no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—1061no—
323C2×C8—1061no—
324C2×C8—2121no—
325C2×C8—2121no—
326C2×C8#1171992111yescharacteristic
327C2×C8#1171992111yescharacteristic
328C2×C8#1171992111yescharacteristic
329C2×C8#1171992111yescharacteristic
330C4⋊C4—41no—
331C4⋊C4—1061no—
332C4⋊C4—1061no—
333C4⋊C4—1061no—
334C4⋊C4—2121no—
335C22⋊C4—22no—
336C22⋊C4—531no—
337C22⋊C4—531no—
338C22⋊C4—1061no—
339C22⋊C4—1061no—
340C22⋊C4—1061no—
341C22⋊C4—1061no—
342C22⋊C4—1062no—
343C42—41no—
344C42—1061no—
345C42—2121no—
346C23—1061no—
347C23—4241no—
348C23—2121no—
349C23C42×D5311yescharacteristic
350C23C4×C22⋊D5311yescharacteristic
351C23C4×D21211yescharacteristic
352Q8—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—4241no—
357D4—4241no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4C2×#1171992111yescharacteristic
374C2×C4#1313317411yescharacteristic
375C2×C4C4×C22⋊D5311yescharacteristic
376C2×C4C4×D21211yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8—42no—
381C8—2122no—
382C8—2121no—
383C8—2121no—
384C8#1313313111yescharacteristic
385C8#1313313111yescharacteristic
386C22—21no—
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—4242no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22C4×#1171992111yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—4242no—
402C4—2121no—
403C4—2121no—
404C4—2121no—
405C4C2×#1313313111yescharacteristic
406C4#1681059611yescharacteristic
407C2—21no—
408C2—41no—
409C2—41no—
410C2—41no—
411C2—4241no—
412C2C4×#1313313111yescharacteristic
subgroupG/H
centerC4C2×#13133131
commutatorC23×C53C2×C42
FrattiniC2×#41C22×D53
FittingC53×#1757C2
socleC2×C53C4×#99
prime radical#16811130C22
cosocleC2×#41×C53C23
hypercenter#1757D53
hypocenterC53#12624
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #1757

Complements

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

NH ≅ G/Ntype
4. #20274515C2semidirect
5. #20272837C2semidirect
6. #20272835C2semidirect
7. C53×#1757C2semidirect
8. C22×#11719912C4semidirect
14. #16810651C4semidirect
18. #16810586C4semidirect
20. #16810578C4semidirect
21. #16810414C4semidirect
22. #16810412C4semidirect
23. #16810411C4semidirect
24. #16810409C4semidirect
205. #1757D53semidirect
237. C53#12624semidirect

Maximal subgroups

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

subgroupindex
#202834162
#202763572
#202763542
#202745152
#202728372
#202728352
C53×#17572
#1262453