Finite Group Databasebeta

← #27294379

Subgroups — #27294379

Order 13568.

448 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 83 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#20280535C211yesmaximal, characteristic
2#20274446C211yesmaximal, characteristic
3#20273026C211yesmaximal, characteristic
4#20272737C211yesmaximal, characteristic
5#20272736C211yesmaximal, characteristic
6#20272735C211yesmaximal, characteristic
7C53×#1351C211yesmaximal, characteristic
8C2×#13133215C411yescharacteristic, complemented
9C2×#13133173—21no—
10C2×#13133173C2211yescharacteristic
11#16810673—21no—
12#16810564—21no—
13#16810559—21no—
14#16810544C411yescharacteristic, complemented
15#16810542C411yescharacteristic, complemented
16#16810535C2211yescharacteristic
17#16810520—21no—
18#16810487C411yescharacteristic, complemented
19#16810481—21no—
20#16810390C2211yescharacteristic
21#16810389—21no—
22#16810348—21no—
23#16810347C2211yescharacteristic
24C53×#393C2211yescharacteristic
25C53×#342C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C22×Dic106C2×C411yescharacteristic
28C23×Dic53D411yescharacteristic
29#13133215—41no—
30#13133215—41no—
31#13133173—24no—
32#13133173—41no—
33#13133173—41no—
34#13133173—42no—
35C2×C4⋊Dic53—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—41no—
41C2×#11719919C2×C411yescharacteristic
42C2×#11719919D411yescharacteristic
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53D411yescharacteristic
46C2×C4×Dic53—41no—
47C2×C4×Dic53C2×C411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49#13133146—41no—
50C22⋊#88322—21no—
51C8×Dic53—41no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56#13133124C2×C411yescharacteristic
57C22×Q8×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53C2311yescharacteristic
61C22⋊C8×C53—21no—
62#95×C53—21no—
63#95×C53C2×C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×Q8×C53—41no—
66C2×Q8×C53—81no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—22no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—41no—
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53SD1612yes—
83C22×Dic53Q1612yes—
84C2×Dic106—41no—
85C2×Dic106—81no—
86C2×C8×C53—41no—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—81no—
91C22⋊Dic53—22no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94C22⋊Dic53—81no—
95C4⋊Dic53—81no—
96#11719919—41no—
97#11719919—41no—
98#11719919—42no—
99#11719919—44no—
100#11719919—41no—
101#11719919—41no—
102#11719919—42no—
103#11719919—81no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53—44no—
109C2×#88322—41no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53C2×SD1611yescharacteristic
116C23×C53C2×Q1611yescharacteristic
117Q8×C53—82no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—42no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—81no—
149C2×Dic53—81no—
150C2×Dic53#9911yescharacteristic
151C2×Dic53#9911yescharacteristic
152C2×Dic53#10312yes—
153C2×Dic53#10312yes—
154C2×Dic53#10412yes—
155Dic106—82no—
156C8×C53—81no—
157#88322—81no—
158#11717—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—81no—
168C22×C53C2×#9911yescharacteristic
169C22×C53C2×#10311yescharacteristic
170C22×C53C2×#10411yescharacteristic
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—81no—
174C4×C53—82no—
175C4×C53—81no—
176Dic53—22no—
177Dic53—41no—
178Dic53—41no—
179Dic53—81no—
180Dic53—82no—
181Dic53—81no—
182Dic53#32812yes—
183Dic53#32812yes—
184#2628—531no—
185#1740—531no—
186#1458—531no—
187#1351—531no—
188#1351—531no—
189#1351—531no—
190#1351D5311yescharacteristic
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—41no—
194C2×C53—41no—
195C2×C53C2×#32811yescharacteristic
196C2×C53#212211yescharacteristic
197C2×C53#212911yescharacteristic
198C2×C4⋊Q8—531no—
199C2×#117—531no—
200C2×#117—1061no—
201#431—1061no—
202#400—1061no—
203#399—1061no—
204#395—531no—
205#395—531no—
206#393—531no—
207#393C2×D5311yescharacteristic
208#389—1061no—
209#382—531no—
210#380—1061no—
211#342—531no—
212#342—1061no—
213#342C2×D5311yescharacteristic
214#324—1061no—
215#323—531no—
216#323C2×D5311yescharacteristic
217C53#1171711yescharacteristic, complemented, Sylow
218C22×Q8—531no—
219C22×Q8C4×D5311yescharacteristic
220C23×C4—531no—
221C4⋊Q8—2121no—
222C4⋊Q8—2121no—
223#117—1064no—
224#117—2121no—
225#117—2121no—
226#117—2122no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—2121no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—531no—
238C2×C22⋊C4—1061no—
239C2×C22⋊C4—1061no—
240C2×C22⋊C4C22×D5311yescharacteristic
241C2×C42—531no—
242C2×C42—531no—
243C2×C42—2121no—
244C4⋊C8—2121no—
245C22⋊C8—21no—
246C22⋊C8—1061no—
247C4×C8—2121no—
248#95—21no—
249#95—531no—
250#95—1061no—
251#95—1061no—
252#95—1061no—
253#95—1061no—
254#95C4×D5311yescharacteristic
255C24D4×D5311yescharacteristic
256C2×Q8—41no—
257C2×Q8—81no—
258C2×Q8—4241no—
259C2×Q8—2121no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—532no—
266C22×C4—532no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1062no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—2121no—
275C22×C4C2×C4×D5311yescharacteristic
276C22×C4D4×D5311yescharacteristic
277C2×C8—41no—
278C2×C8—2121no—
279C4⋊C4—4241no—
280C4⋊C4—4241no—
281C4⋊C4—2121no—
282C4⋊C4—2121no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C4⋊C4—2122no—
286C4⋊C4—2122no—
287C4⋊C4—2124no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—81no—
292C22⋊C4—1062no—
293C22⋊C4—4241no—
294C22⋊C4—2121no—
295C22⋊C4—2121no—
296C42—1061no—
297C42—1061no—
298C42—1061no—
299C42—1061no—
300C42—2124no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23C22⋊C4×D5311yescharacteristic
306C23SD16×D5311yescharacteristic
307C23Q16×D5311yescharacteristic
308Q8—82no—
309Q8—4242no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—531no—
324C2×C4—531no—
325C2×C4—532no—
326C2×C4—532no—
327C2×C4—532no—
328C2×C4—1062no—
329C2×C4—1062no—
330C2×C4—1062no—
331C2×C4—1062no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—4241no—
337C2×C4—4241no—
338C2×C4—2122no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C8—81no—
348C8—4241no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22#99×D5311yescharacteristic
359C22#103×D5311yescharacteristic
360C22#104×D5311yescharacteristic
361C4—41no—
362C4—41no—
363C4—81no—
364C4—82no—
365C4—81no—
366C4—532no—
367C4—532no—
368C4—1062no—
369C4—4241no—
370C4—4242no—
371C4—4241no—
372C4—2121no—
373C4—2121no—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#328×D5311yescharacteristic
379C2#2027636311yescharacteristic
380C2#2027638811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1351C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1351D53
hypocenterC53#11717
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
8. C2×#13133215C4semidirect
14. #16810544C4semidirect
15. #16810542C4semidirect
18. #16810487C4semidirect
217. C53#11717semidirect

Maximal subgroups

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

subgroupindex
#202805352
#202744462
#202730262
#202727372
#202727362
#202727352
C53×#13512
#1171753