Finite Group Databasebeta

← #27302756

Subgroups — #27302756

Order 13568.

488 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 99 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
2C4×#13133132C211yesmaximal, characteristic
3#20272834C212yesmaximal
4#20272834C212yesmaximal
5C53×#1800C211yesmaximal, 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#16810483C411yescharacteristic, complemented
17C4×C22⋊Dic53—21no—
18#16810390—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#423—21no—
21C2×#100×C53C2212yes—
22C53×#343—21no—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53—21no—
26C2×#11719919—21no—
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—41no—
31C2×#11719919—41no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53C2×C411yescharacteristic
36C2×C22⋊Dic53C2×C411yescharacteristic
37C2×C4×Dic53—41no—
38#13133132—22no—
39#13133132Q814yes—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—41no—
46C22×D4×C53C2311yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C2×M16×C53—22no—
50C4×D4×C53—41no—
51C4×D4×C53—41no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C4⋊C8×C53—41no—
57C4⋊C8×C53—41no—
58#100×C53—24no—
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—42no—
83C22×Dic53—41no—
84C22×Dic53—81no—
85M16×C53—42no—
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—24no—
94C22⋊Dic53—42no—
95C22⋊Dic53—42no—
96C22⋊Dic53—41no—
97C22⋊Dic53—41no—
98C22⋊Dic53—81no—
99C4⋊Dic53—81no—
100#11719919—42no—
101#11719919—42no—
102#11719919—41no—
103#11719919—41no—
104#11719919—81no—
105C4×Dic53—82no—
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×C53C22⋊Q812yes—
117D4×C53—42no—
118D4×C53—42no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—81no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#9911yescharacteristic
134C2×C4×C53#12411yescharacteristic
135C2×C4×C53C4⋊Q811yescharacteristic
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—44no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—82no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154C8×C53—82no—
155#13297—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—22no—
159C22×C53—22no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—81no—
168C22×C53#39111yescharacteristic
169C22×C53C2×#9911yescharacteristic
170C22×C53#41011yescharacteristic
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178Dic53—42no—
179Dic53—81no—
180Dic53—81no—
181Dic53—84no—
182Dic53—81no—
183#2579—531no—
184#1800D5311yescharacteristic
185C4×#99—531no—
186#1390—532no—
187#1390—532no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53—42no—
191C2×C53—42no—
192C2×C53#199611yescharacteristic
193C2×C53#212011yescharacteristic
194C2×C53#212111yescharacteristic
195C2×C4×D4C2×D5311yescharacteristic
196C22×C4⋊C4—531no—
197#423—21no—
198C2×#100C2×D5312yes—
199C2×#99—532no—
200#398—1061no—
201#396—1061no—
202#387—1061no—
203#386—531no—
204#386—531no—
205#385—531no—
206#385—531no—
207#385—1061no—
208#380—531no—
209C4×C22⋊C4—1061no—
210#343—21no—
211#342—1061no—
212C53#1329711yescharacteristic, complemented, Sylow
213C22×D4C22×D5311yescharacteristic
214C23×C4—531no—
215C23×C4—531no—
216C23×C4C4×D5311yescharacteristic
217C23×C4D21211yescharacteristic
218C2×M16—22no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—2121no—
227C2×C4⋊C4—2121no—
228C2×C4⋊C4C22⋊D5311yescharacteristic
229C2×C22⋊C4—531no—
230C2×C22⋊C4—531no—
231C2×C22⋊C4—1062no—
232C2×C22⋊C4—1061no—
233C2×C22⋊C4—1061no—
234C2×C22⋊C4C4×D5311yescharacteristic
235C2×C22⋊C4D21211yescharacteristic
236C2×C42—2121no—
237C2×C42C22⋊D5311yescharacteristic
238C4⋊C8—41no—
239C4⋊C8—41no—
240#100—24no—
241#99—534no—
242#99—1062no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246#95—1061no—
247#95—1061no—
248#95—2121no—
249C24C2×C4×D5311yescharacteristic
250C24C2×D21211yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4#1171994712yes—
254C2×D4Q8×D5312yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—4241no—
266C22×C4—2122no—
267C22×C4—2122no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4#1171992111yescharacteristic
271C22×C4#1171992111yescharacteristic
272C22×C4#1171992111yescharacteristic
273C22×C4#1171992111yescharacteristic
274C22×C4C2×C22⋊D5311yescharacteristic
275M16—42no—
276M16—42no—
277C2×C8—42no—
278C4⋊C4—41no—
279C4⋊C4—4241no—
280C4⋊C4—4241no—
281C4⋊C4—2122no—
282C4⋊C4—2122no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—1064no—
288C22⋊C4—4241no—
289C22⋊C4—2122no—
290C22⋊C4—2122no—
291C22⋊C4—2121no—
292C22⋊C4—2121no—
293C42—21no—
294C42—21no—
295C42—4242no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C2×#1171992111yescharacteristic
305C23#1313318212yes—
306C23#1313319712yes—
307D4—42no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—1062no—
323C2×C4—4242no—
324C2×C4—4242no—
325C2×C4—4242no—
326C2×C4—2124no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2122no—
338C2×C4—2121no—
339C2×C4—2122no—
340C2×C4#1313313111yescharacteristic
341C2×C4#1313313111yescharacteristic
342C2×C4#1313320411yescharacteristic
343C2×C4#1313321511yescharacteristic
344C8—82no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22#1681052811yescharacteristic
358C22C2×#1313313111yescharacteristic
359C22#1681059611yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—4241no—
368C4—4241no—
369C4—4244no—
370C4—4241no—
371C4—2122no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#2027569911yescharacteristic
377C2#2027635711yescharacteristic
378C2#2027636211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1800C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1800D53
hypocenterC53#13297
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
6. C22×#11719919C4semidirect
14. #16810495C4semidirect
15. #16810495C4semidirect
16. #16810483C4semidirect
212. C53#13297semidirect

Maximal subgroups

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

subgroupindex
#202800582
C4×#131331322
#202728342
C53×#18002
#1329753