Finite Group Databasebeta

← #28299911

Subgroups — #28299911

Order 13568.

470 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 88 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#20290572C211yesmaximal, characteristic
2#20290442C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20283056C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×#13133274—21no—
8C2×#13133274—21no—
9C2×#13133272—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133193C411yescharacteristic
13C2×C42×D53C2211yescharacteristic
14#16810530C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810521C411yescharacteristic
17#16810506—21no—
18#16810505—21no—
19#16810497—21no—
20#16810485—21no—
21#16810480C411yescharacteristic
22C53×#391C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26#13133274—24no—
27#13133274—42no—
28#13133272—42no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D411yescharacteristic
32#13133193—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43C42×D53—42no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×Q8×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54C23×D53C22⋊C411yescharacteristic
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C2×Q8×C53—41no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—21no—
72C2×C4×D53—22no—
73C2×C4×D53—21no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×#88324—24no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C4⋊C4×C53—22no—
85C42×C53—41no—
86#11719921—42no—
87#11719921—42no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92#11719919—42no—
93#11719919—42no—
94C4×Dic53—22no—
95C4×Dic53—22no—
96C4×Dic53—41no—
97C23×C53C2×C22⋊C411yescharacteristic
98C22×D53—21no—
99C22×D53—21no—
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53#10412yes—
105Q8×C53—82no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53#13112yes—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53#9912yes—
134C2×Dic53AGL(1,ℤ/8)11yescharacteristic
135C2×Dic53#13711yescharacteristic
136C2×Dic53#13712yes—
137C4×D53—44no—
138C4×D53—44no—
139C4×D53—42no—
140C4×D53—42no—
141#88324—44no—
142#88324—82no—
143#7701—531nomaximal, Sylow
144C22×C53#40811yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53#41311yescharacteristic
147C22×C53#41711yescharacteristic
148C22×C53#41911yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C22×C53#42111yescharacteristic
151C2×D53—22no—
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—41no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—81no—
162C4×C53—81no—
163Dic53—22no—
164Dic53—22no—
165Dic53—21no—
166Dic53—21no—
167Dic53—81no—
168Dic53—81no—
169#2855—531no—
170#1797—531no—
171#1749—531no—
172#1272—532no—
173#1271—532no—
174C2×C53#146811yescharacteristic
175C2×C53#148311yescharacteristic
176C2×C53#149711yescharacteristic
177C2×C53#150811yescharacteristic
178C2×C53#151311yescharacteristic
179C2×C53#153211yescharacteristic
180C2×C53#155911yescharacteristic
181D53—42no—
182C2×C22⋊Q8—531no—
183C22×C42—531no—
184C2×C4⋊C8—1061no—
185C2×C22⋊C8—1061no—
186C2×C22⋊C8—1061no—
187C2×#97—1061no—
188#391—531no—
189#391#615711yescharacteristic
190#389—531no—
191#389—531no—
192#387—1061no—
193#387—1061no—
194#385—1061no—
195#381—1061no—
196#380—531no—
197#336—532no—
198#336—532no—
199C53#770111yescharacteristic, complemented, Sylow
200C22×Q8C2×#615711yescharacteristic
201C23×C4—531no—
202C23×C4—531no—
203C23×C4—531no—
204C22×C8—1062no—
205C22×C8—1062no—
206C22⋊Q8—2122no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4C2×#615711yescharacteristic
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—2122no—
219C2×C42C2×#615711yescharacteristic
220C4⋊C8—2122no—
221C22⋊C8—1064no—
222C22⋊C8—2122no—
223#97—2122no—
224#95—21no—
225#95—21no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231C24—531no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—1061no—
241C22×C4—1062no—
242C22×C4—1061no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—1062no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—2121no—
249C22×C4—2121no—
250C22×C4—2121no—
251C22×C4—2121no—
252C22×C4C22⋊#615711yescharacteristic
253C22×C4C22⋊#615711yescharacteristic
254C22×C4C22×#615711yescharacteristic
255C2×C8—1064no—
256C2×C8—2122no—
257C2×C8—2122no—
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C4⋊C4—22no—
262C4⋊C4—2121no—
263C4⋊C4—2121no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C22⋊C4—2122no—
269C22⋊C4—2122no—
270C42—41no—
271C42—1062no—
272C42—1062no—
273C42—2121no—
274C23—532no—
275C23—1061no—
276C23—1061no—
277C23—1061no—
278C23—1061no—
279C23—1061no—
280C23—1061no—
281C23C2×C22⋊#615711yescharacteristic
282Q8—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
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—41no—
297C2×C4—532no—
298C2×C4—531no—
299C2×C4—531no—
300C2×C4—532no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—2124no—
308C2×C4—2124no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4#1313330412yes—
318C8—4242no—
319C8—2124no—
320C22—1062no—
321C22—1062no—
322C22—1062no—
323C22—2121no—
324C22—2121no—
325C22—2121no—
326C22—2121no—
327C22#1681159711yescharacteristic
328C22D53⋊#9911yescharacteristic
329C22#1681160911yescharacteristic
330C22#1681161411yescharacteristic
331C22#1681161611yescharacteristic
332C22D53⋊#10411yescharacteristic
333C22#1681162211yescharacteristic
334C4—22no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—1062no—
340C4—1061no—
341C4—1061no—
342C4—1062no—
343C4—4241no—
344C4—4241no—
345C2—2122no—
346C2#2028998111yescharacteristic
347C2#2029003111yescharacteristic
348C2#2029007811yescharacteristic
349C2#2029011311yescharacteristic
350C2#2029013311yescharacteristic
351C2#2029019211yescharacteristic
352C2#2029025011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#7701
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #20289535C2semidirect
4. #20289533C2semidirect
199. C53#7701semidirect

Maximal subgroups

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

subgroupindex
#202905722
#202904422
#202895352
#202895332
#202830562
#770153