Finite Group Databasebeta

← #27281178

Subgroups — #27281178

Order 13568.

472 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283735C211yesmaximal, characteristic, complemented
2#20273313C211yesmaximal, characteristic, complemented
3#20273149C211yesmaximal, characteristic, complemented
4#20273067C211yesmaximal, characteristic, complemented
5#20272626C211yesmaximal, characteristic, complemented
6#20272604C211yesmaximal, characteristic, complemented
7C53×#3040C211yesmaximal, characteristic
8Q8×#88322—21no—
9#16810701—22no—
10#16810701—21no—
11#16810680—21no—
12#16810680C2211yescharacteristic
13#16810680C2211yescharacteristic
14#16810676C2211yescharacteristic
15C4×#11719917—21no—
16C4×#11719917C2211yescharacteristic
17#16810363—21no—
18#16810363—21no—
19Q8⋊#88322—21no—
20D4⋊#88322—21no—
21#16810347—22no—
22C2×#137×C53C411yescharacteristic
23C2×AGL(1,ℤ/8)×C53C411yescharacteristic
24C2×#124×C53C2211yescharacteristic
25C4×#41×C53C2211yescharacteristic
26C53×#464—21no—
27C53×#464—21no—
28C53×#463—21no—
29C53×#463—21no—
30C53×#423C2211yescharacteristic
31C53×#419C411yescharacteristic
32C53×#418C411yescharacteristic
33C2×#11719917—22no—
34C2×#11719917—21no—
35C4⋊#88322—22no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C22⋊#88322—22no—
40C22⋊#88322—21no—
41C22⋊#88322—21no—
42C22⋊#88322D412yes—
43#13133128—21no—
44#13133128—21no—
45#13133128D411yescharacteristic
46#13133128D411yescharacteristic
47C4×#88322—22no—
48C4×#88322—21no—
49C4×#88322—21no—
50C2×#41×C53C2×C411yescharacteristic
51C22×Q8×C53C2×C411yescharacteristic
52C22×D4×C53C2×C411yescharacteristic
53#137×C53—42no—
54AGL(1,ℤ/8)×C53—42no—
55C2×Q16×C53—21no—
56C2×SD16×C53—21no—
57C2×SD16×C53—21no—
58C2×D8×C53—21no—
59C2×M16×C53C2×C411yescharacteristic
60C2×M16×C53C2×C411yescharacteristic
61#124×C53—21no—
62#124×C53—21no—
63#124×C53—21no—
64#124×C53—21no—
65C4×Q8×C53D411yescharacteristic
66C4×Q8×C53D411yescharacteristic
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53D411yescharacteristic
70C4×D4×C53D411yescharacteristic
71#117×C53—21no—
72#117×C53C2×C411yescharacteristic
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—21no—
75C2×C42×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—21no—
78C4⋊C8×C53—21no—
79#103×C53—21no—
80#103×C53—21no—
81#102×C53—21no—
82#102×C53—21no—
83C24×C53—21no—
84#41×C53—22no—
85#41×C53—41no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53—41no—
89C2×Q8×C53—41no—
90C2×Q8×C53C22⋊C411yescharacteristic
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53C22⋊C411yescharacteristic
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104Q16×C53—42no—
105SD16×C53—42no—
106SD16×C53—42no—
107D8×C53—42no—
108M16×C53—41no—
109M16×C53—41no—
110C2×C8×C53—21no—
111C2×C8×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53C22⋊C411yescharacteristic
115C4⋊C4×C53C22⋊C411yescharacteristic
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C42×C53—21no—
124C42×C53—21no—
125C42×C53—21no—
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53#4111yescharacteristic
130#11719917—42no—
131#11719917—44no—
132C2×#88322—21no—
133C2×#88322—22no—
134C2×#88322—21no—
135C2×#88322—41no—
136C2×#88322—41no—
137C23×C53—21no—
138C23×C53—42no—
139C23×C53—41no—
140C23×C53—81no—
141C23×C53—81no—
142C23×C53C4×D411yescharacteristic
143Q8×C53—21no—
144Q8×C53—41no—
145Q8×C53—41no—
146Q8×C53—81no—
147Q8×C53#10012yes—
148D4×C53—21no—
149D4×C53—21no—
150D4×C53—42no—
151D4×C53—41no—
152D4×C53—41no—
153D4×C53—81no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53C2×C22⋊C411yescharacteristic
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12311yescharacteristic
172C8×C53—41no—
173C8×C53—41no—
174#88322—22no—
175#88322—42no—
176#88322—41no—
177#88322—81no—
178#88322—81no—
179#9350—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53—162no—
190C22×C53#38611yescharacteristic
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—22no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—81no—
200C4×C53C2×#10011yescharacteristic
201C4×C53#41311yescharacteristic
202#3040D5311yescharacteristic
203#2971—531no—
204#1527—531no—
205#1488—531no—
206#1469—531no—
207#1307—531no—
208#1299—531no—
209C2×C53—41no—
210C2×C53—81no—
211C2×C53—161no—
212C2×C53#188111yescharacteristic
213C2×C53#190011yescharacteristic
214C2×C53#190611yescharacteristic
215C2×#137Dic5311yescharacteristic
216C2×AGL(1,ℤ/8)Dic5311yescharacteristic
217C2×#124C2×D5311yescharacteristic
218C4×#41C2×D5311yescharacteristic
219#464—21no—
220#464—21no—
221#463—21no—
222#463—21no—
223C8×Q8—1061no—
224#436—1062no—
225#436—1061no—
226#432—531no—
227#432—531no—
228#432—1061no—
229#431—531no—
230#423C2×D5311yescharacteristic
231#419Dic5311yescharacteristic
232#418Dic5311yescharacteristic
233C4×M16—531no—
234C4×M16—1061no—
235#329—1061no—
236#329—1061no—
237Q8⋊C8—1061no—
238D4⋊C8—1061no—
239#323—1062no—
240C53#935011yescharacteristic, complemented, Sylow
241C2×#41C2×Dic5311yescharacteristic
242C22×Q8C2×Dic5311yescharacteristic
243C22×D4C2×Dic5311yescharacteristic
244#137—42no—
245AGL(1,ℤ/8)—42no—
246C2×Q16—21no—
247C2×SD16—21no—
248C2×SD16—21no—
249C2×D8—21no—
250C2×M16—1062no—
251C2×M16—1061no—
252C2×M16C2×Dic5311yescharacteristic
253C2×M16C2×Dic5311yescharacteristic
254#124—21no—
255#124—21no—
256#124—21no—
257#124—21no—
258C4×Q8C22⋊D5311yescharacteristic
259C4×Q8C22⋊D5311yescharacteristic
260C4×D4—21no—
261C4×D4—21no—
262C4×D4C22⋊D5311yescharacteristic
263C4×D4C22⋊D5311yescharacteristic
264#117—21no—
265#117C2×Dic5311yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—21no—
269C2×C42C22×D5311yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—1062no—
273C4⋊C8—1061no—
274C4⋊C8—1061no—
275C4⋊C8—1061no—
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280C22⋊C8—532no—
281C22⋊C8—1062no—
282C22⋊C8—1061no—
283C22⋊C8—1061no—
284#97—531no—
285#97—531no—
286#97—1061no—
287#97—1061no—
288C4×C8—1062no—
289C4×C8—1061no—
290C4×C8—1061no—
291C24—21no—
292#41—22no—
293#41—41no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8C22⋊Dic5311yescharacteristic
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C22⋊Dic5311yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4D4×D5311yescharacteristic
310C22×C4#1171994711yescharacteristic
311C22×C4C22×Dic5311yescharacteristic
312Q16—42no—
313SD16—42no—
314SD16—42no—
315D8—42no—
316M16—41no—
317M16—41no—
318M16—2122no—
319M16—2124no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—1061no—
323C2×C8—1061no—
324C2×C8—1062no—
325C2×C8—2121no—
326C2×C8—2121no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4C22⋊Dic5311yescharacteristic
330C4⋊C4C22⋊Dic5311yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42D4×D5311yescharacteristic
342C42#1171994711yescharacteristic
343C42C2×C22⋊D5311yescharacteristic
344C42C2×C22⋊D5311yescharacteristic
345C23—21no—
346C23—42no—
347C23—41no—
348C23—81no—
349C23—81no—
350C23D4×Dic5311yescharacteristic
351Q8—21no—
352Q8—41no—
353Q8—41no—
354Q8—81no—
355Q8#1313313412yes—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—41no—
360D4—41no—
361D4—81no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4C2×C22⋊Dic5311yescharacteristic
375C2×C4D4×Dic5311yescharacteristic
376C2×C4#1313318511yescharacteristic
377C2×C4#1313318911yescharacteristic
378C2×C4#1313318911yescharacteristic
379C2×C4#1313319811yescharacteristic
380C8—41no—
381C8—41no—
382C8—1062no—
383C8—4241no—
384C8—4241no—
385C8—2121no—
386C8—2122no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—162no—
397C22#1681050311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—22no—
404C4—41no—
405C4—41no—
406C4—81no—
407C4C2×#1313313411yescharacteristic
408C4#1681060711yescharacteristic
409C2—41no—
410C2—81no—
411C2—161no—
412C2#2027507911yescharacteristic
413C2#2027517911yescharacteristic
414C2#2027521811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3040C2
socleC22×C53#386
prime radicalC2×AGL(1,ℤ/8)×C53C4
cosocleC2×C42×C53C23
hypercenter#3040D53
hypocenterC53#9350
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3040

Complements

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

NH ≅ G/Ntype
1. #20283735C2semidirect
2. #20273313C2semidirect
3. #20273149C2semidirect
4. #20273067C2semidirect
5. #20272626C2semidirect
6. #20272604C2semidirect
240. C53#9350semidirect

Maximal subgroups

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

subgroupindex
#202837352
#202733132
#202731492
#202730672
#202726262
#202726042
C53×#30402
#935053