Finite Group Databasebeta

← #27281248

Subgroups — #27281248

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#20283847C211yesmaximal, characteristic, complemented
2#20275556C211yesmaximal, characteristic, complemented
3#20273383C211yesmaximal, characteristic, complemented
4#20273154C211yesmaximal, characteristic, complemented
5#20272628C211yesmaximal, characteristic, complemented
6#20272604C211yesmaximal, characteristic, complemented
7C53×#3038C211yesmaximal, characteristic
8Q8×#88322—21no—
9#16810701—22no—
10#16810701—21no—
11#16810676—21no—
12#16810676C2211yescharacteristic
13#16810647—21no—
14#16810647C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×#13133134—22no—
17C4×#11719917C2211yescharacteristic
18#16810367—21no—
19#16810367—21no—
20Q8⋊#88322—21no—
21D4⋊#88322—21no—
22C2×AGL(1,ℤ/8)×C53C412yes—
23C2×#127×C53C2211yescharacteristic
24C4×#41×C53C2211yescharacteristic
25C53×#460—21no—
26C53×#460—21no—
27C53×#459—21no—
28C53×#459—21no—
29C53×#423C2211yescharacteristic
30C53×#418C412yes—
31C2×#11719917—22no—
32C2×#11719917—21no—
33C2×#11719917—21no—
34C2×#11719917D412yes—
35C4⋊#88322—22no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322D411yescharacteristic
41C4⋊#88322D411yescharacteristic
42#13133134—44no—
43C22⋊#88322—22no—
44C22⋊#88322—21no—
45#13133128—21no—
46C4×#88322—22no—
47C4×#88322—21no—
48C4×#88322—21no—
49C2×#41×C53C2×C411yescharacteristic
50C22×D4×C53—21no—
51C22×D4×C53—21no—
52C22×D4×C53C2×C412yes—
53AGL(1,ℤ/8)×C53—44no—
54C2×SD16×C53—22no—
55C2×D8×C53—22no—
56C2×M16×C53C2×C412yes—
57#127×C53—21no—
58#127×C53—21no—
59#127×C53—21no—
60#127×C53—21no—
61C4×Q8×C53D411yescharacteristic
62C4×Q8×C53D411yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53D411yescharacteristic
66C4×D4×C53D411yescharacteristic
67#117×C53—21no—
68#117×C53C2×C411yescharacteristic
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73#102×C53—22no—
74#102×C53—22no—
75C24×C53—22no—
76#41×C53—22no—
77#41×C53—41no—
78C2×Q8×C53C22⋊C411yescharacteristic
79C2×D4×C53—21no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—42no—
83C2×D4×C53—42no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—81no—
91C2×D4×C53—81no—
92C2×D4×C53C22⋊C411yescharacteristic
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99SD16×C53—44no—
100D8×C53—44no—
101M16×C53—42no—
102C2×C8×C53—22no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53C22⋊C411yescharacteristic
106C4⋊C4×C53C22⋊C411yescharacteristic
107C22⋊C4×C53—21no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C42×C53—21no—
111C42×C53—21no—
112C42×C53—21no—
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117#11719917—22no—
118#11719917—41no—
119#11719917—42no—
120#11719917—82no—
121#11719917—81no—
122C2×#88322—21no—
123C2×#88322—22no—
124C2×#88322—21no—
125C2×#88322—41no—
126C2×#88322—41no—
127C23×C53—21no—
128C23×C53—44no—
129C23×C53—42no—
130C23×C53—82no—
131C23×C53—82no—
132C23×C53C4×D411yescharacteristic
133Q8×C53—21no—
134Q8×C53#10012yescomplemented
135D4×C53—21no—
136D4×C53—21no—
137D4×C53—42no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—82no—
141D4×C53—82no—
142D4×C53—82no—
143D4×C53—81no—
144D4×C53—81no—
145D4×C53—81no—
146D4×C53—81no—
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—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53C2×C22⋊C411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12311yescharacteristic
167C8×C53—42no—
168#88322—42no—
169#88322—42no—
170#88322—41no—
171#88322—41no—
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—82no—
181C22×C53—82no—
182C22×C53—82no—
183C22×C53—164no—
184C22×C53#38611yescharacteristic
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#3038D5311yescharacteristic
198#2991—531no—
199#1965—531no—
200#1545—531no—
201#1489—531no—
202#1309—531no—
203#1299—531no—
204C2×C53—41no—
205C2×C53—81no—
206C2×C53—162no—
207C2×C53#188311yescharacteristic
208C2×C53#190011yescharacteristic
209C2×C53#190711yescharacteristic
210C2×AGL(1,ℤ/8)Dic5312yes—
211C2×#127C2×D5311yescharacteristic
212C4×#41C2×D5311yescharacteristic
213#460—21no—
214#460—21no—
215#459—21no—
216#459—21no—
217C8×Q8—1061no—
218#436—1062no—
219#436—1061no—
220#431—531no—
221#431—1061no—
222#423—531no—
223#423—531no—
224#423—1061no—
225#423C2×D5311yescharacteristic
226#418Dic5312yes—
227C2×#100—1062no—
228C4×M16—531no—
229#331—1061no—
230#331—1061no—
231Q8⋊C8—1061no—
232D4⋊C8—1061no—
233C53#936011yescharacteristic, complemented, Sylow
234C2×#41C2×Dic5311yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×Dic5312yes—
238AGL(1,ℤ/8)—44no—
239C2×SD16—22no—
240C2×D8—22no—
241C2×M16—532no—
242C2×M16—1062no—
243C2×M16—1061no—
244C2×M16—1061no—
245C2×M16C2×Dic5312yes—
246#127—21no—
247#127—21no—
248#127—21no—
249#127—21no—
250C4×Q8C22⋊D5311yescharacteristic
251C4×Q8C22⋊D5311yescharacteristic
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C22⋊D5311yescharacteristic
255C4×D4C22⋊D5311yescharacteristic
256#117—21no—
257#117C2×Dic5311yescharacteristic
258C2×C42—21no—
259C2×C42C22×D5311yescharacteristic
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—531no—
263C4⋊C8—531no—
264C4⋊C8—1062no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269#102—22no—
270#102—22no—
271#100—2124no—
272C22⋊C8—1062no—
273C22⋊C8—1061no—
274#97—1061no—
275C4×C8—1062no—
276C4×C8—1061no—
277C4×C8—1061no—
278C24—22no—
279#41—22no—
280#41—41no—
281C2×Q8C22⋊Dic5311yescharacteristic
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4C22⋊Dic5311yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4D4×D5311yescharacteristic
300C22×C4#1171994711yescharacteristic
301C22×C4C22×Dic5311yescharacteristic
302SD16—44no—
303D8—44no—
304M16—42no—
305M16—1062no—
306M16—4241no—
307M16—4242no—
308M16—2121no—
309M16—2122no—
310C2×C8—22no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—1062no—
314C2×C8—2121no—
315C2×C8—2121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4C22⋊Dic5311yescharacteristic
319C4⋊C4C22⋊Dic5311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42D4×D5311yescharacteristic
327C42#1171994711yescharacteristic
328C42C2×C22⋊D5311yescharacteristic
329C42C2×C22⋊D5311yescharacteristic
330C23—21no—
331C23—44no—
332C23—42no—
333C23—82no—
334C23—82no—
335C23D4×Dic5311yescharacteristic
336Q8—21no—
337Q8#1313313412yescomplemented
338D4—21no—
339D4—21no—
340D4—42no—
341D4—42no—
342D4—42no—
343D4—82no—
344D4—82no—
345D4—82no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
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×C22⋊Dic5311yescharacteristic
365C2×C4D4×Dic5311yescharacteristic
366C2×C4#1313318511yescharacteristic
367C2×C4#1313318911yescharacteristic
368C2×C4#1313318911yescharacteristic
369C2×C4#1313319811yescharacteristic
370C8—42no—
371C8—4241no—
372C8—2122no—
373C8—2121no—
374C8—2122no—
375C8—2121no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—164no—
386C22#1681050311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4C2×#1313313411yescharacteristic
394C4#1681060711yescharacteristic
395C4#1681078911yescharacteristic
396C4#1681079411yescharacteristic
397C4#1681088411yescharacteristic
398C4#1681089111yescharacteristic
399C2—41no—
400C2—81no—
401C2—162no—
402C2#2027509311yescharacteristic
403C2#2027517911yescharacteristic
404C2#2027522511yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3038C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3038D53
hypocenterC53#9360
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #20283847C2semidirect
2. #20275556C2semidirect
3. #20273383C2semidirect
4. #20273154C2semidirect
5. #20272628C2semidirect
6. #20272604C2semidirect
134. Q8×C53#100semidirect
233. C53#9360semidirect
337. Q8#13133134semidirect

Maximal subgroups

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

subgroupindex
#202838472
#202755562
#202733832
#202731542
#202726282
#202726042
C53×#30382
#936053