Finite Group Databasebeta

← #28308396

Subgroups — #28308396

Order 13568.

412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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#20289978C211yesmaximal, characteristic, complemented
2#20289958C211yesmaximal, characteristic, complemented
3#20289645C212yesmaximal, complemented
4#20289644C212yesmaximal
5#20279692C211yesmaximal, characteristic
6#16811643—21no—
7#16811639—21no—
8#16811468—21no—
9#16811468—21no—
10#16811468C2212yes—
11#16811468C2212yes—
12C2×#13133206C411yescharacteristic
13C2×#13133206C411yescharacteristic
14C2×Q8×Dic53C2211yescharacteristic
15#16810545—21no—
16#16810545—21no—
17#16810530C2211yescharacteristic
18#16810525C2211yescharacteristic
19#16810513—21no—
20#16810513—21no—
21C4×#11719919C411yescharacteristic
22C2×C4⋊Q8×C53C411yescharacteristic
23#13133284—41no—
24C4⋊#88324—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27#13133272—41no—
28C4×#88324—41no—
29C22×D212C2×C411yescharacteristic
30#13133206—42no—
31#13133206—41no—
32#13133206—41no—
33Q8×Dic53—41no—
34Q8×Dic53—41no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53C2311yescharacteristic
38C2×#11719919—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C22×Q8×C53C2×C411yescharacteristic
53C22×Q8×C53C2×C411yescharacteristic
54C4⋊Q8×C53—42no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C42×C53C2×C411yescharacteristic
58C23×D53—21no—
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53—22no—
62C2×Q8×C53—41no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C2×D212—22no—
76C2×D212—41no—
77C2×D212—41no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C42×C53—41no—
83#11719921—42no—
84#11719921—42no—
85#11719921—42no—
86#11719921—42no—
87#11719921—42no—
88#11719921—41no—
89#11719921—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93#11719919—42no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—41no—
101C23×C53C2×C22⋊C411yescharacteristic
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—41no—
105C22×D53—41no—
106C22×D53—41no—
107C22×D53—41no—
108C22×D53—41no—
109Q8×C53—44no—
110Q8×C53—82no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53#10012yes—
125C2×C4×C53#13112yes—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53#9911yescharacteristic
137C2×Dic53#9911yescharacteristic
138D212—82no—
139#88324—82no—
140#88324—82no—
141#11565—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53#40811yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53C2×#10011yescharacteristic
147C2×D53—81no—
148C2×D53—81no—
149C2×D53—81no—
150C2×D53—81no—
151C2×D53—81no—
152C2×D53—81no—
153C2×D53—81no—
154C4×C53—22no—
155C4×C53—22no—
156C4×C53—42no—
157C4×C53—41no—
158C4×C53—81no—
159Dic53—21no—
160Dic53—21no—
161Dic53—41no—
162Dic53—41no—
163Dic53—41no—
164Dic53—81no—
165#2545—531no—
166#1467—531no—
167#1461—531no—
168#1319—532no—
169#1318—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53#147111yescharacteristic
173C2×C53#212611yescharacteristic
174C2×C53#213311yescharacteristic
175D53—161no—
176C2×C4⋊Q8#615711yescharacteristic
177C2×#124—531no—
178C2×#124—531no—
179C2×C4×Q8—531no—
180#432—1061no—
181#431—1061no—
182#396—1061no—
183#396—1061no—
184#391—531no—
185#390—531no—
186#388—1061no—
187#388—1061no—
188C4×C4⋊C4—531no—
189#324—532no—
190#324—532no—
191#324—1061no—
192#324—1061no—
193C53#1156511yescharacteristic, complemented, Sylow
194C22×Q8C2×#615711yescharacteristic
195C22×Q8C2×#615711yescharacteristic
196C22×D4—531no—
197C4⋊Q8—42no—
198#124—2121no—
199#124—2121no—
200#124—2122no—
201C4×Q8—2121no—
202C4×Q8—2121no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C22⋊C4—1061no—
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42C2×#615711yescharacteristic
220C4⋊C8—2121no—
221C4⋊C8—2121no—
222C22⋊C8—1062no—
223C22⋊C8—1062no—
224#97—2121no—
225C4×C8—2121no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C24—1061no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—1062no—
237C2×D4—2121no—
238C2×D4—2121no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—1061no—
246C22×C4C22⋊#615711yescharacteristic
247C22×C4C22⋊#615711yescharacteristic
248C22×C4C22×#615711yescharacteristic
249C2×C8—2122no—
250C2×C8—2122no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—2121no—
254C4⋊C4—2121no—
255C4⋊C4—2121no—
256C4⋊C4—2122no—
257C22⋊C4—2122no—
258C22⋊C4—2122no—
259C22⋊C4—2122no—
260C22⋊C4—2121no—
261C22⋊C4—2121no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C42—41no—
265C42—1061no—
266C42—1061no—
267C42—1061no—
268C42—1061no—
269C42—1061no—
270C42—1061no—
271C42—2121no—
272C23—2121no—
273C23—2121no—
274C23—2121no—
275C23—2121no—
276C23—2121no—
277C23—2121no—
278C23—2121no—
279C23C2×C22⋊#615711yescharacteristic
280Q8—44no—
281Q8—82no—
282D4—4242no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—531no—
297C2×C4—531no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4#1313327712yes—
309C2×C4#1313330412yes—
310C8—4242no—
311C8—4242no—
312C22—21no—
313C22—21no—
314C22—4241no—
315C22—4241no—
316C22—4241no—
317C22—4241no—
318C22—4241no—
319C22—4241no—
320C22—4241no—
321C22#1681159611yescharacteristic
322C22D53⋊#9911yescharacteristic
323C22C2×#1313327711yescharacteristic
324C4—22no—
325C4—22no—
326C4—42no—
327C4—41no—
328C4—81no—
329C4—1061no—
330C4—1061no—
331C4—4241no—
332C4—2121no—
333C4—2121no—
334C4—2121no—
335C2—21no—
336C2—21no—
337C2—8481no—
338C2#2028999111yescharacteristic
339C2#2029098911yescharacteristic
340C2#2029101311yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11565
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20289978C2semidirect
2. #20289958C2semidirect
3. #20289645C2semidirect
193. C53#11565semidirect

Maximal subgroups

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

subgroupindex
#202899782
#202899582
#202896452
#202896442
#202796922
#1156553