Finite Group Databasebeta

← #27281264

Subgroups — #27281264

Order 13568.

500 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 93 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#20284659C211yesmaximal, characteristic
2#20277598C211yesmaximal, characteristic
3#20275701C211yesmaximal, characteristic
4#20274812C211yesmaximal, characteristic
5#20272622C211yesmaximal, characteristic
6#20272601C211yesmaximal, characteristic
7C53×#1508C211yesmaximal, characteristic
8C2×#13133176C2211yescharacteristic
9#16810961—22no—
10#16810956—21no—
11#16810951—21no—
12#16810663C412yescomplemented
13#16810632C412yescomplemented
14#16810587C2211yescharacteristic
15#16810521—21no—
16#16810511—22no—
17#16810487—21no—
18#16810483C2211yescharacteristic
19C4×C4⋊Dic53—21no—
20C4×C22⋊Dic53C2211yescharacteristic
21#16810468—21no—
22#16810390—22no—
23#16810384—22no—
24C2×C22⋊Q8×C53C2211yescharacteristic
25C53×#407C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C53×#328—21no—
28C53×#328—21no—
29C2×#11719917C2×C412yes—
30#13133176—21no—
31#13133176—21no—
32#13133176C2×C412yes—
33#13133176D412yes—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719919—22no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53D412yes—
41C2×C4×Dic53—22no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44#13133150—22no—
45#13133148—22no—
46#13133141—22no—
47#13133141—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50#13133124—22no—
51#13133124—22no—
52#13133124—21no—
53#13133124—21no—
54C22×Q8×C53C2×C411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C2×M16×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C42×C53—21no—
66C22⋊C8×C53—21no—
67C22⋊C8×C53—21no—
68#95×C53D411yescharacteristic
69#95×C53D411yescharacteristic
70C24×C53C2×D411yescharacteristic
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—42no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—42no—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—22no—
88C22×Dic53—41no—
89M16×C53—42no—
90C2×C8×C53—41no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—42no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C42×C53—42no—
99C22⋊Dic53—22no—
100C22⋊Dic53—22no—
101C22⋊Dic53—22no—
102C22⋊Dic53—41no—
103C22⋊Dic53—42no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53—42no—
107C4⋊Dic53C22⋊C412yes—
108C4⋊Dic53C22⋊C412yes—
109#11719919—82no—
110C4×Dic53—22no—
111C4×Dic53—22no—
112C4×Dic53—44no—
113C4×Dic53—42no—
114#11719917—42no—
115C2×#88322—22no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53C4×D411yescharacteristic
121C23×C53#12011yescharacteristic
122C23×C53#12111yescharacteristic
123Q8×C53—42no—
124Q8×C53—81no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53C2×C22⋊C411yescharacteristic
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12311yescharacteristic
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—42no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—81no—
155C2×Dic53—81no—
156C8×C53—81no—
157#88322—42no—
158#9363—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53#38611yescharacteristic
170C22×C53#45511yescharacteristic
171C22×C53#45611yescharacteristic
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—42no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53—81no—
179C4×C53#41012yes—
180Dic53—42no—
181Dic53—42no—
182Dic53—82no—
183Dic53—82no—
184#3095—531no—
185#2346—531no—
186#1996—531no—
187#1826—531no—
188#1508D5311yescharacteristic
189#1305—531no—
190#1297—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53#188211yescharacteristic
196C2×C53#189911yescharacteristic
197C2×C53#190611yescharacteristic
198C2×C22⋊Q8C2×D5311yescharacteristic
199C2×#117—531no—
200#485—1062no—
201#484—1061no—
202#483—1061no—
203#428—532no—
204#419—532no—
205#407—531no—
206#407C2×D5311yescharacteristic
207#389—1061no—
208#388—1062no—
209#382—1061no—
210#380—531no—
211C4×C4⋊C4—1061no—
212C4×C22⋊C4—531no—
213C4×C22⋊C4C2×D5311yescharacteristic
214#376—1061no—
215#342—1062no—
216#339—1062no—
217#328—21no—
218#328—21no—
219C53#936311yescharacteristic, complemented, Sylow
220C22×Q8C4×D5311yescharacteristic
221C23×C4C22×D5311yescharacteristic
222C2×M16—21no—
223C2×M16—532no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228#117—532no—
229#117—532no—
230#117—1061no—
231#117—1061no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4C4×D5311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—532no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C42—21no—
244C2×C42—1061no—
245C2×C42—1062no—
246C2×C42—1061no—
247#107—1062no—
248#106—1062no—
249#103—1062no—
250#103—1062no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—1061no—
254C22⋊C8—1061no—
255#95—1062no—
256#95—1062no—
257#95—1061no—
258#95—1061no—
259#95C22⋊D5311yescharacteristic
260#95C22⋊D5311yescharacteristic
261C24D4×D5311yescharacteristic
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1062no—
273C22×C4—2121no—
274C22×C4C2×C4×D5311yescharacteristic
275C22×C4#1171994511yescharacteristic
276C22×C4D4×D5311yescharacteristic
277C22×C4D4×D5311yescharacteristic
278C22×C4#1171994711yescharacteristic
279C22×C4C2×C22⋊D5311yescharacteristic
280M16—42no—
281M16—2122no—
282C2×C8—41no—
283C2×C8—1062no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—532no—
289C4⋊C4—532no—
290C4⋊C4—1061no—
291C4⋊C4—1061no—
292C4⋊C4—4242no—
293C4⋊C4—2122no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—1062no—
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C22⋊C4—2121no—
301C22⋊C4—2122no—
302C42—42no—
303C42—1062no—
304C42—1062no—
305C42—2122no—
306C42—2124no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23#1313317711yescharacteristic
312C23#1313318511yescharacteristic
313C23#1313319011yescharacteristic
314Q8—42no—
315Q8—81no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—1062no—
333C2×C4—1062no—
334C2×C4—4241no—
335C2×C4—4241no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4C22⋊C4×D5311yescharacteristic
344C2×C4C4×C22⋊D5311yescharacteristic
345C2×C4#1313318811yescharacteristic
346C2×C4#1313319911yescharacteristic
347C8—81no—
348C8—2122no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22#1681049911yescharacteristic
360C22#1681079911yescharacteristic
361C22#1681080411yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—4242no—
370C4—4242no—
371C4—2122no—
372C4—2122no—
373C4#1681059512yes—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#2027508111yescharacteristic
379C2#2027517011yescharacteristic
380C2#2027521911yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1508C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1508D53
hypocenterC53#9363
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
12. #16810663C4semidirect
13. #16810632C4semidirect
219. C53#9363semidirect

Maximal subgroups

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

subgroupindex
#202846592
#202775982
#202757012
#202748122
#202726222
#202726012
C53×#15082
#936353