Finite Group Databasebeta

← #27280136

Subgroups — #27280136

Order 13568.

684 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 135 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#20277329C211yesmaximal, characteristic
2#20273287C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272582C212yesmaximal, complemented
5C53×#2626C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7C2×C22⋊#88322C2212yes—
8#16810498—24no—
9#16810483—21no—
10#16810483C2212yes—
11#16810360—24no—
12#16810347—22no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#399C412yescomplemented
16C53×#396—22no—
17C53×#393C412yescomplemented
18C53×#385—21no—
19C53×#380C2211yescharacteristic
20C53×#379—21no—
21C23×Dic53—22no—
22C22×#88322—22no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C2×C22⋊Dic53—42no—
28C2×C22⋊Dic53—42no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133124—22no—
32#13133124D414yes—
33C25×C53D411yescharacteristic
34C22×Q8×C53C2×C411yescharacteristic, complemented
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37C22⋊Q8×C53—22no—
38C22⋊Q8×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53C2×C412yes—
50C2×C22⋊C4×C53D412yes—
51#95×C53—22no—
52#95×C53C2×C412yescomplemented
53C24×C53—21no—
54C24×C53—21no—
55C24×C53—21no—
56C24×C53—21no—
57C24×C53—81no—
58C24×C53—81no—
59C24×C53C22⋊C412yes—
60C24×C53SD1612yes—
61C24×C53Q1612yes—
62C24×C53C2×D411yescharacteristic
63C2×Q8×C53—22no—
64C2×Q8×C53—81no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—81no—
71C22×C4×C53—81no—
72C22×C4×C53C22⋊C412yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D412yes—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4112yes—
77C22×Dic53—24no—
78C22×Dic53—44no—
79C22×Dic53—82no—
80C22×Dic53—82no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—81no—
84C4⋊C4×C53—82no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—24no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—82no—
92C22⋊C4×C53—82no—
93C22⋊Dic53—44no—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C4⋊Dic53—82no—
97C4⋊Dic53—82no—
98C2×#88322—42no—
99C2×#88322—82no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—81no—
113C23×C53—81no—
114C23×C53—82no—
115C23×C53—82no—
116C23×C53—82no—
117C23×C53—82no—
118C23×C53#10314yes—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53#12011yescharacteristic
121C23×C53#12311yescharacteristic
122C23×C53C2×SD1611yescharacteristic
123C23×C53C2×Q1611yescharacteristic
124C23×C53AGL(1,ℤ/8)11yescharacteristic
125C23×C53#13711yescharacteristic
126Q8×C53—82no—
127C2×C4×C53—22no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—81no—
142C2×C4×C53C4×D412yes—
143C2×C4×C53#12112yes—
144C2×Dic53—44no—
145C2×Dic53—44no—
146C2×Dic53—44no—
147C2×Dic53—88no—
148#88322—82no—
149#9199—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—24no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—44no—
162C22×C53—44no—
163C22×C53—44no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—82no—
167C22×C53—82no—
168C22×C53—82no—
169C22×C53#38611yescharacteristic
170C22×C53C2×#10311yescharacteristic
171C22×C53#41711yescharacteristic
172C22×C53#45011yescharacteristic
173C22×C53#45111yescharacteristic
174C22×C53#48311yescharacteristic
175C22×C53#48411yescharacteristic
176C4×C53—22no—
177C4×C53—22no—
178C4×C53—42no—
179C4×C53—82no—
180C4×C53—81no—
181C4×C53—81no—
182Dic53—84no—
183#2626D5311yescharacteristic
184#2315—531no—
185#1520—532no—
186C2×#323—531no—
187#1289—532no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—44no—
195C2×C53#187211yescharacteristic
196C2×C53#191111yescharacteristic
197C2×C53#191311yescharacteristic
198C2×C22⋊Q8C2×D5311yescharacteristic
199C22×C22⋊C4—1061no—
200C22×C22⋊C4C2×D5311yescharacteristic
201C2×C22⋊C8—532no—
202#399Dic5312yescomplemented
203#396—22no—
204#393Dic5312yescomplemented
205#385—21no—
206#385—1064no—
207#380—532no—
208#380—1061no—
209#380C2×D5311yescharacteristic
210#379—21no—
211#328—1064no—
212#323—1062no—
213C53#919911yescharacteristic, complemented, Sylow
214C25C22⋊D5311yescharacteristic
215C22×Q8C2×Dic5311yescharacteristic, complemented
216C23×C4—1062no—
217C23×C4C22⋊D5311yescharacteristic
218C23×C4C22×D5311yescharacteristic
219C22×C8—1062no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—1062no—
226C2×C4⋊C4—1062no—
227C2×C4⋊C4C2×Dic5311yescharacteristic, complemented
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—1062no—
235C2×C22⋊C4—1062no—
236C2×C22⋊C4—2122no—
237C2×C22⋊C4—2122no—
238C2×C22⋊C4C2×Dic5312yes—
239C2×C22⋊C4C22⋊D5312yes—
240C22⋊C8—1062no—
241C22⋊C8—1062no—
242#95—22no—
243#95—534no—
244#95—1062no—
245#95C2×Dic5312yescomplemented
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24#1171992412yes—
253C24#1171992512yes—
254C24C22⋊Dic5312yes—
255C24C2×C22⋊D5311yescharacteristic
256C2×Q8—22no—
257C2×Q8—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—1064no—
266C22×C4—4242no—
267C22×C4—4242no—
268C22×C4—2124no—
269C22×C4C22⋊Dic5312yes—
270C22×C4D4×D5312yes—
271C22×C4#1171994712yes—
272C22×C4C22×Dic5311yescharacteristic
273C22×C4C2×C22⋊D5311yescharacteristic
274C2×C8—4242no—
275C2×C8—2122no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4—82no—
280C4⋊C4—4242no—
281C4⋊C4—4242no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C22⋊C4—2124no—
291C22⋊C4—2122no—
292C22⋊C4—2122no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1313314114yes—
312C23C2×C22⋊Dic5311yescharacteristic
313C23#1313318511yescharacteristic
314C23#1313319811yescharacteristic
315C23C2×#1171992411yescharacteristic
316C23C2×#1171992511yescharacteristic
317C23#1313324211yescharacteristic
318C23#1313324711yescharacteristic
319Q8—82no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—4248no—
336C2×C4—2124no—
337C2×C4—2124no—
338C2×C4—2124no—
339C2×C4D4×Dic5312yes—
340C2×C4#1313318912yes—
341C8—4242no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1681050311yescharacteristic
362C22C2×#1313314111yescharacteristic
363C22#1681062411yescharacteristic
364C22#1681077311yescharacteristic
365C22#1681077811yescharacteristic
366C22#1681095111yescharacteristic
367C22#1681095611yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—82no—
372C4—81no—
373C4—81no—
374C4—4244no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2027502311yescharacteristic
383C2#2027524811yescharacteristic
384C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2626C2
socleC22×C53#386
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#2626D53
hypocenterC53#9199
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #20273287C2semidirect
4. #20272582C2semidirect
15. C53×#399C4semidirect
17. C53×#393C4semidirect
34. C22×Q8×C53C2×C4semidirect
42. C2×C4⋊C4×C53C2×C4semidirect
52. #95×C53C2×C4semidirect
202. #399Dic53semidirect
204. #393Dic53semidirect
213. C53#9199semidirect
215. C22×Q8C2×Dic53semidirect
227. C2×C4⋊C4C2×Dic53semidirect
245. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202773292
#202732872
C2×#168103472
#202725822
C53×#26262
#919953