Finite Group Databasebeta

← #27302745

Subgroups — #27302745

Order 13568.

484 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 103 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#20280058C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20272834C212yesmaximal
4#20272834C212yesmaximal
5C4×#100×C53C211yesmaximal, characteristic
6C22×#11719919C411yescharacteristic, complemented
7C2×#13133132C2212yes—
8#16810556—21no—
9#16810546—21no—
10#16810507—21no—
11#16810503C2211yescharacteristic
12#16810503C2211yescharacteristic
13#16810496—21no—
14#16810495C411yescharacteristic, complemented
15#16810495C411yescharacteristic, complemented
16#16810483—21no—
17#16810483C411yescharacteristic, complemented
18#16810390—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#100×C53C2212yes—
21C4×M16×C53—21no—
22C53×#343—21no—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—41no—
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—41no—
32C2×#11719919—41no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53C2×C411yescharacteristic
37C2×C22⋊Dic53C2×C411yescharacteristic
38#13133132—24no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—41no—
45C22×D4×C53C2311yescharacteristic
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C2×M16×C53—22no—
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55#100×C53—22no—
56#100×C53Q814yes—
57#97×C53—41no—
58C4×C8×C53—41no—
59C24×C53C22×C411yescharacteristic
60C24×C53C2×D411yescharacteristic
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53C2×Q812yes—
64C2×D4×C53#4112yes—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—81no—
70C22×C4×C53—81no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—42no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—42no—
84C22×Dic53—81no—
85M16×C53—24no—
86M16×C53—42no—
87C2×C8×C53—42no—
88C4⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C42×C53—21no—
92C42×C53—21no—
93C22⋊Dic53—42no—
94C22⋊Dic53—41no—
95C22⋊Dic53—41no—
96C22⋊Dic53—42no—
97C22⋊Dic53—42no—
98C22⋊Dic53—81no—
99C4⋊Dic53—81no—
100C4⋊Dic53—82no—
101#11719919—42no—
102#11719919—42no—
103#11719919—41no—
104#11719919—41no—
105#11719919—81no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53C4×Q812yes—
116C23×C53#12411yescharacteristic
117C23×C53C4⋊Q811yescharacteristic
118D4×C53—42no—
119D4×C53—42no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53#9911yescharacteristic
134C2×C4×C53#9911yescharacteristic
135C2×C4×C53C22⋊Q812yes—
136C2×Dic53—22no—
137C2×Dic53—41no—
138C2×Dic53—42no—
139C2×Dic53—41no—
140C2×Dic53—42no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—82no—
151C2×Dic53—82no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154C8×C53—44no—
155#13294—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—22no—
160C22×C53—21no—
161C22×C53—42no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—42no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—81no—
170C22×C53#39111yescharacteristic
171C22×C53C2×#9911yescharacteristic
172C22×C53#41011yescharacteristic
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—22no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—41no—
179Dic53—42no—
180Dic53—81no—
181Dic53—81no—
182Dic53—81no—
183Dic53—162no—
184#2579—531no—
185#1799—531no—
186C4×#100D5311yescharacteristic
187#1390—532no—
188#1390—532no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—41no—
192C2×C53—42no—
193C2×C53—41no—
194C2×C53#199611yescharacteristic
195C2×C53#212011yescharacteristic
196C2×C53#212111yescharacteristic
197C2×C4×D4C2×D5311yescharacteristic
198C22×C4⋊C4—531no—
199C2×#100C2×D5312yes—
200C2×#99—532no—
201C4×M16—21no—
202#398—1061no—
203#396—1061no—
204#387—1061no—
205#386—531no—
206#386—531no—
207#385—531no—
208#385—531no—
209#385—1061no—
210#380—531no—
211#380—1061no—
212#343—21no—
213#342—1061no—
214C53#1329411yescharacteristic, complemented, Sylow
215C22×D4C22×D5311yescharacteristic
216C23×C4—531no—
217C23×C4—531no—
218C23×C4C4×D5311yescharacteristic
219C23×C4D21211yescharacteristic
220C2×M16—22no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—2121no—
229C2×C4⋊C4—2121no—
230C2×C4⋊C4—2121no—
231C2×C4⋊C4C22⋊D5311yescharacteristic
232C2×C22⋊C4—531no—
233C2×C22⋊C4—531no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C22⋊C4—1062no—
237C2×C22⋊C4C4×D5311yescharacteristic
238C2×C22⋊C4D21211yescharacteristic
239C2×C42C22⋊D5311yescharacteristic
240#100—22no—
241#100Dic10614yes—
242#99—1064no—
243#97—41no—
244C4×C8—41no—
245#95—1061no—
246#95—1061no—
247#95—1061no—
248#95—1061no—
249#95—1061no—
250#95—2121no—
251C24C2×C4×D5311yescharacteristic
252C24C2×D21211yescharacteristic
253C2×D4—41no—
254C2×D4—41no—
255C2×D4C2×Dic10612yes—
256C2×D4#1171994512yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—4241no—
268C22×C4—2122no—
269C22×C4—2122no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4#1171992111yescharacteristic
273C22×C4#1171992111yescharacteristic
274C22×C4#1171992111yescharacteristic
275C22×C4#1171992111yescharacteristic
276C22×C4C2×C22⋊D5311yescharacteristic
277M16—24no—
278M16—42no—
279C2×C8—42no—
280C4⋊C4—41no—
281C4⋊C4—4242no—
282C4⋊C4—4241no—
283C4⋊C4—4241no—
284C4⋊C4—2122no—
285C4⋊C4—2122no—
286C4⋊C4—2121no—
287C4⋊C4—2121no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—4241no—
291C22⋊C4—2122no—
292C22⋊C4—2122no—
293C22⋊C4—2122no—
294C22⋊C4—2121no—
295C22⋊C4—2121no—
296C42—21no—
297C42—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23C2×#1171992111yescharacteristic
307C23C4×Dic10612yes—
308C23#1313320511yescharacteristic
309C23#1313321711yescharacteristic
310D4—42no—
311D4—42no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—1062no—
326C2×C4—4242no—
327C2×C4—4242no—
328C2×C4—4242no—
329C2×C4—4242no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2122no—
341C2×C4—2121no—
342C2×C4—2122no—
343C2×C4#1313313111yescharacteristic
344C2×C4#1313313111yescharacteristic
345C2×C4#1313319612yes—
346C8—44no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22#1681052911yescharacteristic
362C22C2×#1313313111yescharacteristic
363C22#1681059611yescharacteristic
364C4—21no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—8482no—
371C4—4241no—
372C4—4241no—
373C4—4241no—
374C4—2122no—
375C2—21no—
376C2—21no—
377C2—41no—
378C2—42no—
379C2—41no—
380C2#2027570011yescharacteristic
381C2#2027635711yescharacteristic
382C2#2027636211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC4×#100×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#100D53
hypocenterC53#13294
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
6. C22×#11719919C4semidirect
14. #16810495C4semidirect
15. #16810495C4semidirect
17. #16810483C4semidirect
214. C53#13294semidirect

Maximal subgroups

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

subgroupindex
#202800582
#202747032
#202728342
C4×#100×C532
#1329453