Finite Group Databasebeta

← #27287446

Subgroups — #27287446

Order 13568.

576 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 113 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#20280829C211yesmaximal, characteristic
2#20275020C212yesmaximal
3C2×#16810361C212yesmaximal
4#20272971C211yesmaximal, characteristic
5C53×#1280C211yesmaximal, characteristic
6C2×#13133196C2212yes—
7C22×C4⋊Dic53C2211yescharacteristic
8C2×#13133142—22no—
9#16810555—22no—
10#16810544—22no—
11#16810483—21no—
12#16810483—21no—
13C2×#13133124—21no—
14C2×#13133124—21no—
15C2×#13133124C2211yescharacteristic
16#16810386—22no—
17#16810361—24no—
18C2×C22⋊C8×C53C2212yes—
19C2×#95×C53C2211yescharacteristic
20C22×Dic106D412yes—
21C23×Dic53—21no—
22C23×Dic53—21no—
23#13133196—22no—
24#13133196—22no—
25#13133196—22no—
26#13133196—22no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—42no—
33C2×C4⋊Dic53D412yes—
34C2×#11719919—22no—
35C2×C22⋊Dic53—22no—
36#13133142—44no—
37#13133124—22no—
38#13133124—22no—
39#13133124—21no—
40#13133124—42no—
41#13133124—42no—
42#13133124D412yes—
43C23×C4×C53D411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53C2311yescharacteristic
46C22×C8×C53—22no—
47C22⋊C8×C53—22no—
48C22⋊C8×C53—22no—
49#95×C53—22no—
50C24×C53C2×D411yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53SD1612yes—
62C22×C4×C53Q1612yes—
63C22×C4×C53C2×D412yes—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53#4112yes—
66C22×C4×C53#4111yescharacteristic
67C22×Dic53—22no—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53—41no—
74C22×Dic53—41no—
75C22×Dic53—41no—
76C22×Dic53—41no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C2×Dic106—22no—
84C2×Dic106—22no—
85C2×Dic106—42no—
86C2×Dic106—42no—
87C2×C8×C53—42no—
88C2×C8×C53—42no—
89C2×C8×C53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—24no—
93C4⋊Dic53—22no—
94C4⋊Dic53—22no—
95C4⋊Dic53—42no—
96C4⋊Dic53—81no—
97C4⋊Dic53—81no—
98C4⋊Dic53—81no—
99C4⋊Dic53—81no—
100#11719919—42no—
101#11719919—42no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53#12011yescharacteristic
107C23×C53#12111yescharacteristic
108C23×C53#12411yescharacteristic
109C23×C53C2×SD1611yescharacteristic
110C23×C53C2×Q1611yescharacteristic
111C23×C53#13711yescharacteristic
112C23×C53#13711yescharacteristic
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53#12112yes—
129C2×C4×C53#12312yes—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150Dic106—44no—
151Dic106—82no—
152C8×C53—82no—
153#10407—531nomaximal, Sylow
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53#39411yescharacteristic
167C22×C53#44811yescharacteristic
168C22×C53#45111yescharacteristic
169C22×C53#45411yescharacteristic
170C22×C53#45511yescharacteristic
171C22×C53B(4,𝔽2)11yescharacteristic
172C22×C53#45811yescharacteristic
173C22×C53#46111yescharacteristic
174C22×C53#46211yescharacteristic
175C22×C53#48711yescharacteristic
176C22×C53#49111yescharacteristic
177C4×C53—22no—
178C4×C53—22no—
179C4×C53—42no—
180C4×C53—42no—
181Dic53—82no—
182Dic53—82no—
183Dic53—82no—
184Dic53—82no—
185#2659—531no—
186#1872—532no—
187C2×#328—532no—
188#1442—531no—
189#1280D5311yescharacteristic
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53#159911yescharacteristic
195C2×C53#160211yescharacteristic
196C2×C53#160811yescharacteristic
197C2×C53#161711yescharacteristic
198C2×C53#204011yescharacteristic
199C2×C53#204311yescharacteristic
200C2×C53#204411yescharacteristic
201C2×C22⋊Q8—532no—
202C22×C4⋊C4—531no—
203C2×#103—1062no—
204C2×C22⋊C8C2×D5312yes—
205#398—1062no—
206#395—1062no—
207#380—1061no—
208#380—1061no—
209C2×#95—531no—
210C2×#95—1061no—
211C2×#95—1061no—
212C2×#95C2×D5311yescharacteristic
213#340—1062no—
214#328—1064no—
215C53#1040711yescharacteristic, complemented, Sylow
216C22×Q8—532no—
217C23×C4—1061no—
218C23×C4—1061no—
219C23×C4D21211yescharacteristic
220C23×C4D21211yescharacteristic
221C23×C4C22×D5311yescharacteristic
222C22×C8—22no—
223C22⋊Q8—1062no—
224C22⋊Q8—1062no—
225C22⋊Q8—1062no—
226C22⋊Q8—1062no—
227C2×C4⋊C4—532no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1062no—
234C2×C4⋊C4—2122no—
235C2×C22⋊C4—1062no—
236#103—2124no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239#95—22no—
240#95—532no—
241#95—1062no—
242#95—1062no—
243#95—1061no—
244#95—2122no—
245#95—2122no—
246C24C2×D21211yescharacteristic
247C2×Q8—1062no—
248C2×Q8—1062no—
249C2×Q8—2122no—
250C2×Q8—2122no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—1062no—
262C22×C4—1062no—
263C22×C4—1062no—
264C22×C4—1062no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4#1171991312yes—
278C22×C4Dic21212yes—
279C22×C4#1171994512yes—
280C22×C4D4×D5312yes—
281C22×C4D4×D5311yescharacteristic
282C22×C4#1171994911yescharacteristic
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—1064no—
287C4⋊C4—1062no—
288C4⋊C4—1062no—
289C4⋊C4—4241no—
290C4⋊C4—4241no—
291C4⋊C4—4241no—
292C4⋊C4—4241no—
293C4⋊C4—2122no—
294C4⋊C4—2122no—
295C4⋊C4—2122no—
296C22⋊C4—2122no—
297C22⋊C4—2122no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#1313318711yescharacteristic
303C23#1313319211yescharacteristic
304C23#1313320511yescharacteristic
305C23C2×#1171991311yescharacteristic
306C23C2×Dic21211yescharacteristic
307C23#1313325011yescharacteristic
308C23#1313325011yescharacteristic
309Q8—4242no—
310Q8—2124no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—4241no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4#1313319012yes—
347C2×C4#1313320212yes—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22#1681054111yescharacteristic
362C22#1681076411yescharacteristic
363C22#1681078111yescharacteristic
364C22#1681079811yescharacteristic
365C22#1681080311yescharacteristic
366C22#1681081111yescharacteristic
367C22#1681081411yescharacteristic
368C22#1681082911yescharacteristic
369C22#1681083411yescharacteristic
370C22#1681097411yescharacteristic
371C22#1681099211yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—4242no—
377C4—4242no—
378C4—4242no—
379C4—4242no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#2027365711yescharacteristic
385C2#2027367411yescharacteristic
386C2#2027370811yescharacteristic
387C2#2027375911yescharacteristic
388C2#2027594511yescharacteristic
389C2#2027596211yescharacteristic
390C2#2027596711yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1280C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#1280D53
hypocenterC53#10407
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
215. C53#10407semidirect

Maximal subgroups

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

subgroupindex
#202808292
#202750202
C2×#168103612
#202729712
C53×#12802
#1040753