Finite Group Databasebeta

← #27281246

Subgroups — #27281246

Order 13568.

538 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#20284195C211yesmaximal, characteristic
2#20275554C211yesmaximal, characteristic, complemented
3#20273382C211yesmaximal, characteristic, complemented
4#20273154C211yesmaximal, characteristic, complemented
5#20272628C211yesmaximal, characteristic, complemented
6#20272606C211yesmaximal, characteristic, complemented
7C53×#2991C211yesmaximal, characteristic, complemented
8C2×#13133242C412yes—
9C2×#13133214C2211yescharacteristic
10#16810823—21no—
11#16810823—21no—
12#16810818—21no—
13#16810818—21no—
14#16810647C2211yescharacteristic
15#16810647C2211yescharacteristic
16#16810627C412yes—
17C2×#13133133—22no—
18C4×#11719917C2211yescharacteristic
19#16810366—21no—
20#16810366—21no—
21Q8⋊#88322—21no—
22D4⋊#88322—21no—
23C4×#41×C53C2211yescharacteristic
24C8×Q8×C53—21no—
25C53×#436—22no—
26C53×#436—21no—
27C53×#431—21no—
28C53×#431C2211yescharacteristic
29C53×#423—21no—
30C53×#423C2211yescharacteristic
31C22×D212—21no—
32C22×D212—21no—
33C22×D212C2×C412yes—
34#13133242—44no—
35C2×#11719924—22no—
36C2×#11719922—22no—
37C2×#11719917—21no—
38C2×#11719917—21no—
39C2×#11719917C2×C412yes—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133137—22no—
49#13133137—22no—
50#13133133—44no—
51#13133128—21no—
52C4×#88322—21no—
53C2×#41×C53C2×C411yescharacteristic
54C2×M16×C53—22no—
55C2×M16×C53D412yes—
56C4×Q8×C53D411yescharacteristic
57C4×Q8×C53D411yescharacteristic
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53D411yescharacteristic
61C4×D4×C53D411yescharacteristic
62#117×C53—21no—
63#117×C53C2×C411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—22no—
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53D411yescharacteristic
70C4⋊C8×C53D411yescharacteristic
71C22⋊C8×C53—22no—
72C22⋊C8×C53—21no—
73C4×C8×C53—22no—
74C4×C8×C53—21no—
75C23×D53—22no—
76#41×C53—22no—
77#41×C53—41no—
78C2×Q8×C53C22⋊C411yescharacteristic
79C2×D4×C53—21no—
80C2×D4×C53C22⋊C411yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—42no—
90C2×D212—42no—
91C2×D212—42no—
92C2×D212—42no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—81no—
98C2×D212—81no—
99M16×C53—42no—
100M16×C53—82no—
101C2×C8×C53—21no—
102C2×C8×C53—22no—
103C2×C8×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53C22⋊C411yescharacteristic
107C4⋊C4×C53C22⋊C411yescharacteristic
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C42×C53—21no—
112C42×C53—21no—
113C42×C53—21no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118#11719924—44no—
119#11719922—44no—
120#11719917—22no—
121#11719917—41no—
122#11719917—42no—
123#11719917—81no—
124C2×#88322—22no—
125C2×#88322—41no—
126C2×#88322—41no—
127C23×C53—21no—
128C23×C53C4×D411yescharacteristic
129C22×D53—42no—
130C22×D53—44no—
131C22×D53—82no—
132C22×D53—82no—
133Q8×C53—21no—
134Q8×C53#10012yescomplemented
135D4×C53—21no—
136D4×C53—21no—
137D4×C53—42no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12311yescharacteristic
158D212—42no—
159D212—42no—
160D212—82no—
161D212—82no—
162D212—82no—
163D212—81no—
164D212—81no—
165D212—81no—
166D212—81no—
167C8×C53—42no—
168C8×C53—41no—
169C8×C53—41no—
170#88322—42no—
171#88322—42no—
172#88322—81no—
173#9360—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53#38611yescharacteristic
181C2×D53—82no—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—164no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—21no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53C2×#10011yescharacteristic
192C4×C53#41311yescharacteristic
193C4×C53#45311yescharacteristic
194C4×C53#45411yescharacteristic
195C4×C53#47111yescharacteristic
196C4×C53#47211yescharacteristic
197#3038—531no—
198#2991D5311yescharacteristic, complemented
199#1965—531no—
200#1545—531no—
201#1489—531no—
202#1309—531no—
203#1299—531no—
204C2×C53—41no—
205C2×C53—81no—
206C2×C53#188311yescharacteristic
207C2×C53#190011yescharacteristic
208C2×C53#190711yescharacteristic
209D53—162no—
210C2×AGL(1,ℤ/8)—532no—
211C2×#127—531no—
212C4×#41C2×D5311yescharacteristic
213#460—1061no—
214#460—1061no—
215#459—1061no—
216#459—1061no—
217C8×Q8—21no—
218#436—22no—
219#436—21no—
220#431—21no—
221#431C2×D5311yescharacteristic
222#423—21no—
223#423—531no—
224#423—531no—
225#423C2×D5311yescharacteristic
226#418—532no—
227C2×#100—1062no—
228C4×M16—531no—
229#331—1061no—
230#331—1061no—
231Q8⋊C8—1061no—
232D4⋊C8—1061no—
233C53#936011yescharacteristic, complemented, Sylow
234C2×#41C4×D5311yescharacteristic
235C22×D4—532no—
236C22×D4—1061no—
237C22×D4—1061no—
238AGL(1,ℤ/8)—2124no—
239C2×SD16—1062no—
240C2×D8—1062no—
241C2×M16—22no—
242C2×M16—532no—
243C2×M16—1061no—
244C2×M16—1061no—
245C2×M16C22⋊D5312yes—
246#127—1061no—
247#127—1061no—
248#127—1061no—
249#127—1061no—
250C4×Q8D21211yescharacteristic
251C4×Q8C22⋊D5311yescharacteristic
252C4×D4—21no—
253C4×D4—21no—
254C4×D4D21211yescharacteristic
255C4×D4C22⋊D5311yescharacteristic
256#117—21no—
257#117C4×D5311yescharacteristic
258C2×C42—21no—
259C2×C42C22×D5311yescharacteristic
260C4⋊C8—22no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8C22⋊D5311yescharacteristic
268C4⋊C8C22⋊D5311yescharacteristic
269#102—1062no—
270#102—1062no—
271#100—2124no—
272C22⋊C8—22no—
273C22⋊C8—21no—
274#97—1061no—
275C4×C8—22no—
276C4×C8—21no—
277C4×C8—1061no—
278C24—1062no—
279#41—22no—
280#41—41no—
281C2×Q8#1171992111yescharacteristic
282C2×D4—21no—
283C2×D4—1062no—
284C2×D4—1062no—
285C2×D4—4241no—
286C2×D4—4241no—
287C2×D4—2122no—
288C2×D4—2122no—
289C2×D4—2122no—
290C2×D4—2122no—
291C2×D4—2121no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4#1171992111yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4C2×C4×D5311yescharacteristic
300C22×C4#1171994511yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302SD16—2124no—
303D8—2124no—
304M16—42no—
305M16—82no—
306M16—1062no—
307M16—4241no—
308M16—2121no—
309M16—2122no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—1062no—
314C2×C8—2121no—
315C2×C8—2121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4#1171992111yescharacteristic
319C4⋊C4#1171992111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42C2×D21211yescharacteristic
327C42#1171994511yescharacteristic
328C42C2×C22⋊D5311yescharacteristic
329C42C2×C22⋊D5311yescharacteristic
330C23—21no—
331C23—4242no—
332C23—4242no—
333C23—2124no—
334C23—2122no—
335C23C4×C22⋊D5311yescharacteristic
336Q8—21no—
337Q8#1313313312yescomplemented
338D4—21no—
339D4—21no—
340D4—42no—
341D4—4242no—
342D4—4242no—
343D4—4242no—
344D4—4241no—
345D4—4241no—
346D4—4241no—
347D4—4241no—
348D4—2122no—
349D4—2122no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4C2×#1171992111yescharacteristic
365C2×C4C4×C22⋊D5311yescharacteristic
366C2×C4#1313318611yescharacteristic
367C2×C4#1313319111yescharacteristic
368C2×C4#1313319111yescharacteristic
369C2×C4#1313320011yescharacteristic
370C8—42no—
371C8—41no—
372C8—41no—
373C8—4241no—
374C8—2122no—
375C8—2122no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—8484no—
383C22—4242no—
384C22—4242no—
385C22—4242no—
386C22#1681050211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4C2×#1313313311yescharacteristic
394C4#1681060511yescharacteristic
395C4#1681079111yescharacteristic
396C4#1681079611yescharacteristic
397C4#1681088611yescharacteristic
398C4#1681089311yescharacteristic
399C2—41no—
400C2—81no—
401C2—8482no—
402C2#2027509111yescharacteristic
403C2#2027517811yescharacteristic
404C2#2027522311yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2991C2
socleC22×C53#386
prime radical#20284195C2
cosocleC2×C42×C53C23
hypercenter#2991D53
hypocenterC53#9360
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2991

Complements

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

NH ≅ G/Ntype
2. #20275554C2semidirect
3. #20273382C2semidirect
4. #20273154C2semidirect
5. #20272628C2semidirect
6. #20272606C2semidirect
7. C53×#2991C2semidirect
134. Q8×C53#100semidirect
198. #2991D53semidirect
233. C53#9360semidirect
337. Q8#13133133semidirect

Maximal subgroups

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

subgroupindex
#202841952
#202755542
#202733822
#202731542
#202726282
#202726062
C53×#29912
#936053