Finite Group Databasebeta

← #27299287

Subgroups — #27299287

Order 13568.

460 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283425C211yesmaximal, characteristic
2#20276398C211yesmaximal, characteristic
3#20276378C211yesmaximal, characteristic
4#20274515C211yesmaximal, characteristic, complemented
5#20272845C211yesmaximal, characteristic, complemented
6#20272843C211yesmaximal, characteristic, complemented
7C53×#1757C211yesmaximal, characteristic, complemented
8C2×#13133223C411yescharacteristic, complemented
9#16811169C2211yescharacteristic
10#16810699—21no—
11#16810699—21no—
12C8×C22⋊D53—21no—
13C8×C22⋊D53—21no—
14#16810646C411yescharacteristic, complemented
15#16810607C2211yescharacteristic
16#16810597C2211yescharacteristic
17#16810591C411yescharacteristic, complemented
18#16810590C2211yescharacteristic
19#16810578—21no—
20C2×C8×Dic53C411yescharacteristic, complemented
21#16810421C412yescomplemented
22#16810419C412yescomplemented
23#16810417C412yes—
24#16810415C412yes—
25#16810394—21no—
26C2×#131×C53C2211yescharacteristic
27C53×#413C2211yescharacteristic
28C53×#410C2211yescharacteristic
29C53×#405—21no—
30C53×#344—21no—
31C2×#11719945C2×C411yescharacteristic
32#13133223—41no—
33#13133223—41no—
34C2×#11719917—21no—
35C2×#11719917C2×C411yescharacteristic
36C2×#11719912—21no—
37C22×#88322C2×C411yescharacteristic
38C2×C8×D53—21no—
39#13133215C2×C411yescharacteristic
40#13133213C2×C411yescharacteristic
41#13133206C2×C411yescharacteristic
42#13133204C2×C411yescharacteristic
43#13133189—21no—
44#13133189—21no—
45C4×C22⋊D53—21no—
46C4×C22⋊D53—21no—
47#13133176—21no—
48C2×C4×Dic53C2×C411yescharacteristic
49#13133146—21no—
50#13133146—21no—
51#13133136C2×C412yes—
52#13133134—21no—
53#13133132—21no—
54#13133132C2×C412yes—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57#13133129—21no—
58#13133129—21no—
59#13133127—41no—
60C8×Dic53—21no—
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×C53C2×C412yes—
75#100×C53—21no—
76#99×C53—21no—
77#99×C53C2×C412yes—
78#97×C53—41no—
79C4×C8×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82C2×Q8×C53C22×C411yescharacteristic
83C2×D4×C53C22×C411yescharacteristic
84C2×D4×C53#4111yescharacteristic
85C2×D4×C53#4111yescharacteristic
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C2×C22⋊D53—21no—
90C2×C22⋊D53—21no—
91C2×C22⋊D53—21no—
92C22×Dic53—21no—
93#11719945—41no—
94#11719945—41no—
95C2×D212C4211yescharacteristic
96C2×C4×D53—21no—
97C2×Dic106C4211yescharacteristic
98M16×C53—22no—
99M16×C53—41no—
100M16×C53—41no—
101M16×C53—41no—
102M16×C53—41no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53—41no—
110C2×C8×C53C22⋊C411yescharacteristic
111C2×C8×C53C22⋊C411yescharacteristic
112C2×C8×C53C22⋊C411yescharacteristic
113C2×C8×C53C22⋊C411yescharacteristic
114C4⋊C4×C53—41no—
115C22⋊C4×C53—22no—
116C42×C53—41no—
117C22⋊Dic53—21no—
118C22⋊Dic53—21no—
119C22⋊Dic53—22no—
120#11719921—21no—
121#11719921—21no—
122C4⋊Dic53—41no—
123#11719919—21no—
124#11719919—21no—
125C4×Dic53—21no—
126C4×Dic53—41no—
127C4×Dic53C4211yescharacteristic
128C4×Dic53C4211yescharacteristic
129#11719917—22no—
130#11719917—41no—
131#11719917—41no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—41no—
135C2×#88322—41no—
136#11719912—42no—
137C8×D53—41no—
138C8×D53—41no—
139C23×C53#11711yescharacteristic
140C23×C53C4×D411yescharacteristic
141C23×C53C4×D411yescharacteristic
142C22×D53—21no—
143Q8×C53—41no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—41no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—42no—
154C2×C4×C53C2×C4211yescharacteristic
155C2×C4×C53C2×C22⋊C411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53C4×D411yescharacteristic
158C22⋊D53—42no—
159C22⋊D53—42no—
160C22⋊D53—42no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—42no—
165C2×Dic53—42no—
166D212—41no—
167C4×D53—41no—
168C4×D53—41no—
169Dic106—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#12633—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—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—82no—
194C4×C53C2×#9911yescharacteristic
195C4×C53#41011yescharacteristic
196Dic53—22no—
197Dic53—41no—
198Dic53—41no—
199Dic53—82no—
200#2931—531no—
201#2132—531no—
202#2126—531no—
203#1757—531no—
204#1757D5311yescharacteristic, complemented
205#1395—531no—
206#1395—531no—
207C2×C53—21no—
208C2×C53—41no—
209C2×C53—41no—
210C2×C53—41no—
211C2×C53C4×#9911yescharacteristic
212D53—81no—
213C2×#131—531no—
214C2×#131C2×D5311yescharacteristic
215#532—531no—
216#436—1061no—
217#436—1061no—
218C8×D4—1061no—
219C8×D4—1061no—
220#423—531no—
221#413—531no—
222#413C2×D5311yescharacteristic
223#410—531no—
224#410C2×D5311yescharacteristic
225#408—531no—
226#408—531no—
227#405—21no—
228#405—1061no—
229C2×C4×C8—531no—
230#356—532no—
231#355—532no—
232#354—532no—
233#353—532no—
234#344—21no—
235#344—1061no—
236C53#1263311yescharacteristic, complemented, Sylow
237C2×#41—531no—
238C2×#41C22×D5311yescharacteristic
239#131—41no—
240#131—41no—
241#131—2121no—
242#131—2121no—
243C2×M16—21no—
244C2×M16—531no—
245C2×M16—1061no—
246C2×M16—1061no—
247C2×M16C4×D5311yescharacteristic
248C2×M16D21211yescharacteristic
249C2×M16C22⋊D5311yescharacteristic
250C22×C8—531no—
251C22×C8—1061no—
252C22×C8C4×D5311yescharacteristic
253C22×C8D21211yescharacteristic
254C22×C8C22⋊D5311yescharacteristic
255C4⋊Q8—531no—
256#127—531no—
257#124—531no—
258#124—531no—
259#121—1061no—
260#121—1061no—
261C4×D4—1061no—
262C4×D4—1061no—
263#117—21no—
264#117—1061no—
265C2×C42—531no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268#101—532no—
269#101C4×D5312yes—
270#100—21no—
271#100—1061no—
272#99—21no—
273#99—532no—
274#99—1061no—
275#99C4×D5312yes—
276C22⋊C8—1061no—
277C22⋊C8—1061no—
278C22⋊C8—1061no—
279C22⋊C8—1061no—
280#97—41no—
281#97—2121no—
282C4×C8—41no—
283C4×C8—1061no—
284C4×C8—1061no—
285C4×C8—2121no—
286#41—41no—
287#41—41no—
288#41—2121no—
289#41—2121no—
290C2×Q8—531no—
291C2×Q8C2×C4×D5311yescharacteristic
292C2×D4—531no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4C2×C4×D5311yescharacteristic
297C2×D4#1171994511yescharacteristic
298C2×D4#1171994511yescharacteristic
299C22×C4—1061no—
300C22×C4—1061no—
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—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—41no—
320C2×C8—1061no—
321C2×C8—1061no—
322C2×C8—2121no—
323C2×C8—2121no—
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—2121no—
334C22⋊C4—22no—
335C22⋊C4—1061no—
336C22⋊C4—1061no—
337C22⋊C4—1061no—
338C22⋊C4—1061no—
339C22⋊C4—1062no—
340C42—41no—
341C42—531no—
342C42—531no—
343C42—1061no—
344C42—2121no—
345C23—1061no—
346C23#1313317411yescharacteristic
347C23C4×C22⋊D5311yescharacteristic
348C23C4×D21211yescharacteristic
349Q8—41no—
350Q8—2121no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—2121no—
355D4—2122no—
356D4—2122no—
357D4—2122no—
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×C4C42×D5311yescharacteristic
373C2×C4C2×#1171992111yescharacteristic
374C2×C4C4×C22⋊D5311yescharacteristic
375C2×C4C4×D21211yescharacteristic
376C8—21no—
377C8—21no—
378C8—21no—
379C8—42no—
380C8—2122no—
381C8—2121no—
382C8—2121no—
383C8#1313313111yescharacteristic
384C8#1313313111yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22C4×#1171992111yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—82no—
399C4—1062no—
400C4—4242no—
401C4—2121no—
402C4—2121no—
403C4C2×#1313313111yescharacteristic
404C4#1681059611yescharacteristic
405C2—21no—
406C2—41no—
407C2—41no—
408C2—41no—
409C2—4241no—
410C2C4×#1313313111yescharacteristic
subgroupG/H
centerC8#13133131
commutatorC2×C4×C53C2×C42
FrattiniC2×#41C22×D53
FittingC53×#1757C2
socleC2×C53C4×#99
prime radical#16811169C22
cosocleC2×#41×C53C23
hypercenter#1757D53
hypocenterC53#12633
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ #1757

Complements

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

NH ≅ G/Ntype
4. #20274515C2semidirect
5. #20272845C2semidirect
6. #20272843C2semidirect
7. C53×#1757C2semidirect
8. C2×#13133223C4semidirect
14. #16810646C4semidirect
17. #16810591C4semidirect
20. C2×C8×Dic53C4semidirect
21. #16810421C4semidirect
22. #16810419C4semidirect
204. #1757D53semidirect
236. C53#12633semidirect

Maximal subgroups

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

subgroupindex
#202834252
#202763982
#202763782
#202745152
#202728452
#202728432
C53×#17572
#1263353