Finite Group Databasebeta

← #27358287

Subgroups — #27358287

Order 13568.

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

normal subgroups only

This group has 950 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#20285536C212yesmaximal
2#20280095C211yesmaximal, characteristic
3#20273723C214yesmaximal, complemented
4#20273689C214yesmaximal, complemented
5#20273235C212yesmaximal
6#20273045C211yesmaximal, characteristic
7C2×C53×#544C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810953—22no—
10#16810951C2214yes—
11#16810879—24no—
12#16810779—22no—
13#16810778C2214yes—
14#16810587C2212yes—
15#16810553C2212yes—
16#16810543C2212yes—
17#16810518—22no—
18#16810496—21no—
19#16810496—21no—
20C4×C22⋊Dic53C2211yescharacteristic
21#16810360C2218yes—
22#16810349C2214yes—
23C53×#544—24no—
24C53×#544—22no—
25C53×#544C2214yes—
26C2×C4⋊Q8×C53—22no—
27C2×#125×C53—22no—
28C2×C22⋊Q8×C53—21no—
29C2×C22⋊Q8×C53C2212yes—
30C2×#117×C53—21no—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic
33C2×#11719925—24no—
34C2×#11719917—22no—
35#13133196—22no—
36#13133196—22no—
37C2×C4⋊Dic53C2311yescharacteristic
38C2×#11719919—22no—
39C2×#11719919—21no—
40C2×#11719919—22no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—21no—
44#13133148—24no—
45#13133141—24no—
46#13133140—24no—
47C22⋊#88322—22no—
48C22⋊#88322C2314yes—
49#13133124—22no—
50#13133124—22no—
51#13133124C2312yes—
52C22×Q8×C53—21no—
53C22×Q8×C53D412yes—
54C23×C4×C53D411yescharacteristic
55C23×C4×C53D411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C4⋊Q8×C53—28no—
58C4⋊Q8×C53—28no—
59#125×C53—28no—
60#125×C53—28no—
61C22⋊Q8×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—22no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53D414yes—
66C22⋊Q8×C53C2314yes—
67#117×C53—24no—
68#117×C53—24no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—22no—
73C2×C4⋊C4×C53—22no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53D412yes—
77C2×C4⋊C4×C53D412yes—
78C2×C4⋊C4×C53C2312yes—
79C2×C22⋊C4×C53—22no—
80C2×C42×C53—22no—
81C24×C53C2×D411yescharacteristic
82C2×Q8×C53—24no—
83C2×Q8×C53—22no—
84C2×Q8×C53—82no—
85C2×Q8×C53—82no—
86C2×Q8×C53C2×D414yes—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—24no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—81no—
95C22×C4×C53—81no—
96C22×C4×C53C2×D412yes—
97C22×C4×C53C2×D412yes—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2411yescharacteristic
102C22×Dic53—22no—
103C22×Dic53—21no—
104C22×Dic53—21no—
105C2×Dic106—22no—
106C2×Dic106—81no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—28no—
111C4⋊C4×C53—24no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53C2×D414yes—
117C22⋊C4×C53—24no—
118C22⋊C4×C53—24no—
119C42×C53—28no—
120C22⋊Dic53—24no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123#11719925—84no—
124C4⋊Dic53—22no—
125#11719919—42no—
126#11719919—44no—
127#11719919—82no—
128C4×Dic53—81no—
129#11719917—82no—
130C2×#88322—24no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53—41no—
134C23×C53#12011yescharacteristic
135C23×C53#12011yescharacteristic
136C23×C53C22×D411yescharacteristic
137Q8×C53—44no—
138Q8×C53—82no—
139Q8×C53—84no—
140C2×C4×C53—24no—
141C2×C4×C53—24no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—24no—
146C2×C4×C53—24no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—44no—
153C2×C4×C53—44no—
154C2×C4×C53—42no—
155C2×C4×C53—81no—
156C2×C4×C53#12012yes—
157C2×C4×C53#13714yes—
158C2×C4×C53C22×D412yes—
159C2×Dic53—24no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—42no—
166Dic106—44no—
167#88322—44no—
168#18327—531nomaximal, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—44no—
177C22×C53#45612yes—
178C22×C53C2×#12011yescharacteristic
179C22×C53C2×#13712yes—
180C4×C53—24no—
181C4×C53—22no—
182C4×C53—22no—
183C4×C53—44no—
184C4×C53—42no—
185C4×C53—44no—
186Dic53—44no—
187Dic53—81no—
188Dic53—81no—
189C2×#544D5311yescharacteristic
190#3195—532no—
191#2583—531no—
192#1611—534no—
193#1605—534no—
194#1508—532no—
195#1463—531no—
196C2×C53—22no—
197C2×C53—22no—
198C2×C53—41no—
199C2×C53#300411yescharacteristic
200C2×C53C2×#45611yescharacteristic
201C2×C53#302611yescharacteristic
202#544—24no—
203#544—22no—
204#544C2×D5314yes—
205C2×C4⋊Q8—22no—
206C2×#125—22no—
207C2×C22⋊Q8—21no—
208C2×C22⋊Q8—531no—
209C2×C22⋊Q8C2×D5312yes—
210C2×#117—21no—
211C22×C4⋊C4C2×D5311yescharacteristic
212#483—534no—
213#483—1062no—
214#470—1064no—
215#451—534no—
216#451—1062no—
217#407—532no—
218#398—532no—
219#395—532no—
220#389—1062no—
221#385—1061no—
222#385—1061no—
223C4×C22⋊C4—531no—
224#328—538no—
225#324—534no—
226C53#1832711yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—531no—
229C22×Q8C22⋊D5312yes—
230C23×C4C22⋊D5311yescharacteristic
231C23×C4C22⋊D5311yescharacteristic
232C23×C4C22×D5311yescharacteristic
233C2×Q16—1064no—
234C2×M16—1062no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237#125—28no—
238#125—28no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—1062no—
244C22⋊Q8—1062no—
245C22⋊Q8C22⋊D5314yes—
246C22⋊Q8C22×D5314yes—
247#117—24no—
248#117—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—1062no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1062no—
260C2×C4⋊C4C22⋊D5312yes—
261C2×C4⋊C4C22⋊D5312yes—
262C2×C4⋊C4C22×D5312yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4—1061no—
266C2×C42—22no—
267C2×C42—1061no—
268#106—1064no—
269#103—1064no—
270#103—1064no—
271C22⋊C8—534no—
272C22⋊C8—1062no—
273#95—532no—
274#95—1062no—
275#95—1062no—
276C24C2×C22⋊D5311yescharacteristic
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—1062no—
282C2×Q8—4241no—
283C2×Q8C2×C22⋊D5314yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—1062no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4C2×C22⋊D5312yes—
297C22×C4C2×C22⋊D5312yes—
298C22×C4C2×C22⋊D5311yescharacteristic
299C22×C4C2×C22⋊D5312yes—
300C22×C4C2×C22⋊D5312yes—
301C22×C4C23×D5311yescharacteristic
302Q16—4244no—
303M16—4242no—
304C2×C8—1064no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—1062no—
315C4⋊C4—4242no—
316C4⋊C4—2122no—
317C4⋊C4—2124no—
318C4⋊C4C2×C22⋊D5314yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—1064no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C42—28no—
325C42—4241no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23#1313318611yescharacteristic
330C23#1313318611yescharacteristic
331C23C22×C22⋊D5311yescharacteristic
332Q8—44no—
333Q8—82no—
334Q8—84no—
335Q8—2124no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—1064no—
353C2×C4—2122no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4#1313318612yes—
360C2×C4#1313324814yes—
361C2×C4C22×C22⋊D5312yes—
362C8—2124no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22#1681080612yes—
372C22C2×#1313318611yescharacteristic
373C22C2×#1313324812yes—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—42no—
379C4—44no—
380C4—4241no—
381C4—4241no—
382C4—2124no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2#2028394911yescharacteristic
387C2C2×#1681080611yescharacteristic
388C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#544C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenterC2×#544D53
hypocenterC53#18327
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20273723C2semidirect
4. #20273689C2semidirect
226. C53#18327semidirect

Maximal subgroups

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

subgroupindex
#202855362
#202800952
#202737232
#202736892
#202732352
#202730452
C2×C53×#5442
#1832753