Finite Group Databasebeta

← #27356579

Subgroups — #27356579

Order 13568.

1210 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1210 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284651C212yesmaximal, complemented
2#20279002C211yesmaximal, characteristic, complemented
3#20273752C214yesmaximal, complemented
4#20273718C214yesmaximal, complemented
5#20273244C212yesmaximal, complemented
6#20273045C211yesmaximal, characteristic, complemented
7C2×C53×#537C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic
9#16810944—24no—
10#16810939C2214yescomplemented
11#16810934C2214yescomplemented
12#16810663—22no—
13#16810628—22no—
14#16810587C2212yes—
15#16810562C2212yescomplemented
16#16810553C2212yescomplemented
17#16810504—22no—
18#16810496—21no—
19#16810486—21no—
20#16810486C2211yescharacteristic
21#16810357C2218yescomplemented
22#16810349C2214yescomplemented
23C22×#41×C53—21no—
24C53×#537—24no—
25C53×#537—22no—
26C53×#537C2214yes—
27C2×C22⋊Q8×C53—22no—
28C2×#121×C53—22no—
29C2×#121×C53—21no—
30C2×#121×C53C2212yes—
31C22×C4⋊C4×C53C2211yescharacteristic
32C2×#11719917—22no—
33#13133176—22no—
34#13133176—22no—
35C2×C4⋊Dic53C2311yescharacteristic
36C2×C4⋊Dic53C2311yescharacteristic
37C2×#11719919—22no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C4×Dic53—21no—
41#13133150—24no—
42#13133148—24no—
43#13133138—24no—
44#13133138—24no—
45C22⋊#88322—22no—
46C22⋊#88322C2314yes—
47#13133124—22no—
48#13133124—21no—
49#13133124—22no—
50#13133124—22no—
51#13133124C2312yes—
52C2×#41×C53—24no—
53C2×#41×C53—24no—
54C2×#41×C53—48no—
55C22×Q8×C53—21no—
56C22×D4×C53—21no—
57C22×D4×C53—22no—
58C22×D4×C53—21no—
59C22×D4×C53D412yes—
60C23×C4×C53—22no—
61C23×C4×C53D411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53C2311yescharacteristic
64C22⋊Q8×C53—28no—
65C22⋊Q8×C53—28no—
66#121×C53—28no—
67#121×C53—24no—
68#121×C53—28no—
69#121×C53—24no—
70#121×C53—22no—
71#121×C53—22no—
72#121×C53D414yes—
73#121×C53C2314yes—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53D412yes—
78C2×C4⋊C4×C53D412yes—
79C2×C4⋊C4×C53C2312yes—
80C2×C22⋊C4×C53—22no—
81C2×C22⋊C4×C53—22no—
82C24×C53—22no—
83C24×C53C2×D411yescharacteristic
84#41×C53—416no—
85#41×C53—48no—
86#41×C53—48no—
87C2×Q8×C53—24no—
88C2×Q8×C53—44no—
89C2×D4×C53—24no—
90C2×D4×C53—24no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—22no—
94C2×D4×C53—44no—
95C2×D4×C53—44no—
96C2×D4×C53—48no—
97C2×D4×C53—82no—
98C2×D4×C53C2×D414yes—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—24no—
102C22×C4×C53—28no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—42no—
107C22×C4×C53—48no—
108C22×C4×C53—42no—
109C22×C4×C53C2×D412yes—
110C22×C4×C53C2×D412yes—
111C22×C4×C53C2×D412yes—
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D412yes—
114C22×C4×C53C2411yescharacteristic
115C22×Dic53—22no—
116C22×Dic53—21no—
117C22×Dic53—21no—
118C4⋊C4×C53—24no—
119C4⋊C4×C53—24no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—24no—
122C4⋊C4×C53C2×D414yes—
123C22⋊C4×C53—28no—
124C22⋊C4×C53—24no—
125C22⋊C4×C53—24no—
126C22⋊Dic53—42no—
127C22⋊Dic53—41no—
128C22⋊Dic53—41no—
129C4⋊Dic53—22no—
130C4⋊Dic53—22no—
131#11719919—44no—
132C4×Dic53—42no—
133#11719917—44no—
134C2×#88322—24no—
135C23×C53—22no—
136C23×C53—22no—
137C23×C53—24no—
138C23×C53—41no—
139C23×C53—44no—
140C23×C53—44no—
141C23×C53—44no—
142C23×C53#12011yescharacteristic
143C23×C53#12011yescharacteristic
144C23×C53C22×D411yescharacteristic
145Q8×C53—48no—
146D4×C53—44no—
147D4×C53—48no—
148D4×C53—48no—
149D4×C53—48no—
150D4×C53—82no—
151D4×C53—84no—
152C2×C4×C53—24no—
153C2×C4×C53—24no—
154C2×C4×C53—22no—
155C2×C4×C53—22no—
156C2×C4×C53—24no—
157C2×C4×C53—24no—
158C2×C4×C53—22no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—416no—
163C2×C4×C53—44no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—81no—
167C2×C4×C53#12012yes—
168C2×C4×C53AGL(1,ℤ/8)14yes—
169C2×C4×C53C22×D412yes—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177#88322—44no—
178#18147—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—44no—
187C22×C53—44no—
188C22×C53—44no—
189C22×C53—44no—
190C22×C53—88no—
191C22×C53#45812yes—
192C22×C53C2×#12011yescharacteristic
193C22×C53C2×AGL(1,ℤ/8)12yes—
194C4×C53—24no—
195C4×C53—22no—
196C4×C53—22no—
197C4×C53—44no—
198C4×C53—42no—
199Dic53—82no—
200Dic53—81no—
201Dic53—81no—
202C2×#537D5311yescharacteristic
203#3094—532no—
204#2484—531no—
205#1616—534no—
206#1610—534no—
207#1510—532no—
208#1463—531no—
209C2×C53—22no—
210C2×C53—22no—
211C2×C53—41no—
212C2×C53—84no—
213C2×C53#300311yescharacteristic
214C2×C53#302011yescharacteristic
215C2×C53C2×#45811yescharacteristic
216C22×#41—21no—
217#537—24no—
218#537—22no—
219#537C2×D5314yes—
220C2×C22⋊Q8—22no—
221C2×#121—22no—
222C2×#121—21no—
223C2×#121C2×D5312yes—
224C2×#117—531no—
225C22×C4⋊C4C2×D5311yescharacteristic
226#482—1064no—
227#481—534no—
228#480—534no—
229#428—1062no—
230#418—1062no—
231#407—532no—
232#400—532no—
233#398—532no—
234#387—1062no—
235#385—1061no—
236#381—531no—
237#381—1061no—
238#327—538no—
239#324—534no—
240C53#1814711yescharacteristic, complemented, Sylow
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—48no—
244C22×Q8—21no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—21no—
248C22×D4C22⋊D5312yes—
249C23×C4—22no—
250C23×C4C22⋊D5311yescharacteristic
251C23×C4C22⋊D5311yescharacteristic
252C23×C4C22×D5311yescharacteristic
253C2×M16—1062no—
254C22⋊Q8—28no—
255C22⋊Q8—28no—
256#121—24no—
257#121—28no—
258#121—28no—
259#121—24no—
260#121—22no—
261#121—22no—
262#121C22⋊D5314yes—
263#121C22×D5314yes—
264#117—1062no—
265#117—1062no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—531no—
270C2×C4⋊C4—531no—
271C2×C4⋊C4—1062no—
272C2×C4⋊C4C22⋊D5312yes—
273C2×C4⋊C4C22⋊D5312yes—
274C2×C4⋊C4C22×D5312yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—1061no—
278C2×C22⋊C4—1061no—
279C2×C42—1061no—
280#107—1064no—
281#106—1064no—
282#102—1064no—
283#102—1064no—
284C22⋊C8—534no—
285C22⋊C8—1062no—
286#95—532no—
287#95—1061no—
288#95—1062no—
289#95—1062no—
290#95—1062no—
291C24—22no—
292C24C2×C22⋊D5311yescharacteristic
293#41—416no—
294#41—48no—
295#41—48no—
296C2×Q8—24no—
297C2×Q8—44no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—82no—
307C2×D4C2×C22⋊D5314yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—28no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—48no—
317C22×C4—42no—
318C22×C4—1062no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4C2×C22⋊D5312yes—
322C22×C4C2×C22⋊D5312yes—
323C22×C4C2×C22⋊D5312yes—
324C22×C4C2×C22⋊D5311yescharacteristic
325C22×C4C2×C22⋊D5312yes—
326C22×C4C23×D5311yescharacteristic
327M16—2124no—
328C2×C8—1064no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—1062no—
334C4⋊C4—1062no—
335C4⋊C4—2124no—
336C4⋊C4C2×C22⋊D5314yes—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—2122no—
341C22⋊C4—2121no—
342C22⋊C4—2121no—
343C42—2122no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—41no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23#1313318611yescharacteristic
352C23#1313318611yescharacteristic
353C23C22×C22⋊D5311yescharacteristic
354Q8—48no—
355D4—44no—
356D4—48no—
357D4—48no—
358D4—48no—
359D4—82no—
360D4—84no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—416no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—2122no—
377C2×C4—2122no—
378C2×C4—2122no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4#1313318612yes—
384C2×C4#1313324314yes—
385C2×C4C22×C22⋊D5312yes—
386C8—2124no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—44no—
395C22—44no—
396C22—44no—
397C22—44no—
398C22—88no—
399C22#1681081312yes—
400C22C2×#1313318611yescharacteristic
401C22C2×#1313324312yes—
402C4—24no—
403C4—22no—
404C4—22no—
405C4—44no—
406C4—42no—
407C4—4242no—
408C4—4241no—
409C4—4241no—
410C2—22no—
411C2—22no—
412C2—41no—
413C2—84no—
414C2#2028394211yescharacteristic
415C2#2028407911yescharacteristic
416C2C2×#1681081311yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#537C2
socleC22×C53C2×#120
hypercenterC2×#537D53
hypocenterC53#18147
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20284651C2semidirect
2. #20279002C2semidirect
3. #20273752C2semidirect
4. #20273718C2semidirect
5. #20273244C2semidirect
6. #20273045C2semidirect
10. #16810939C22semidirect
11. #16810934C22semidirect
15. #16810562C22semidirect
16. #16810553C22semidirect
21. #16810357C22semidirect
22. #16810349C22semidirect
240. C53#18147semidirect

Maximal subgroups

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

subgroupindex
#202846512
#202790022
#202737522
#202737182
#202732442
#202730452
C2×C53×#5372
#1814753