Finite Group Databasebeta

← #28299634

Subgroups — #28299634

Order 13568.

506 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 96 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#20290607C211yesmaximal, characteristic
2#20290442C211yesmaximal, characteristic
3#20289535C212yesmaximal, complemented
4#20289534C212yesmaximal, complemented
5C4×#13133193C211yesmaximal, characteristic
6C2×#13133274—21no—
7C2×#13133274—21no—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133193C411yescharacteristic
13C2×C42×D53C2211yescharacteristic
14#16810530C2211yescharacteristic
15#16810519C2211yescharacteristic
16#16810504—21no—
17#16810494—21no—
18#16810480C411yescharacteristic
19C4×#11719919—21no—
20C4×C4⋊Dic53C411yescharacteristic
21C4×#11719921—21no—
22C2×C4×Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133274—24no—
26#13133274—42no—
27#13133272—42no—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31#13133193—22no—
32#13133193—22no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53—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
50C4×Q8×C53—42no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53C2×C411yescharacteristic
55C23×D53C22⋊C411yescharacteristic
56C2×Q8×C53—21no—
57C2×Q8×C53—41no—
58C2×Q8×C53—41no—
59C2×Q8×C53M1612yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53—22no—
75C2×C4×D53—22no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×#88324—24no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C2×#88324—42no—
85C2×#88324—42no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—41no—
88C42×C53—22no—
89C42×C53—42no—
90#11719921—24no—
91#11719921—42no—
92C4⋊Dic53—21no—
93C4⋊Dic53—41no—
94C4⋊Dic53M1612yes—
95#11719919—24no—
96#11719919—42no—
97C4×Dic53—22no—
98C4×Dic53—41no—
99C4×Dic53—41no—
100C4×Dic53—42no—
101C23×C53C2×C22⋊C411yescharacteristic
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53#10412yes—
109Q8×C53—42no—
110Q8×C53—42no—
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#9912yes—
125C2×C4×C53C2×M1611yescharacteristic
126C2×C4×C53C2×M1611yescharacteristic
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—22no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53#13512yes—
141C2×Dic53#13712yes—
142C4×D53—44no—
143C4×D53—44no—
144C4×D53—42no—
145C4×D53—42no—
146#88324—44no—
147#88324—82no—
148#7623—531nomaximal, Sylow
149C22×C53#40711yescharacteristic
150C22×C53C2×#9911yescharacteristic
151C22×C53#41311yescharacteristic
152C22×C53#41611yescharacteristic
153C22×C53#41911yescharacteristic
154C22×C53C2×#10411yescharacteristic
155C22×C53#42111yescharacteristic
156C2×D53—22no—
157C2×D53—22no—
158C2×D53—22no—
159C2×D53—41no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—42no—
167C4×C53—41no—
168C4×C53—81no—
169Dic53—22no—
170Dic53—22no—
171Dic53—41no—
172Dic53—42no—
173Dic53—81no—
174C4×C22⋊Q8—531no—
175#1817—531no—
176#1749—531no—
177#1272—532no—
178#1271—532no—
179C2×C53#146511yescharacteristic
180C2×C53#148811yescharacteristic
181C2×C53#149111yescharacteristic
182C2×C53#150111yescharacteristic
183C2×C53#151111yescharacteristic
184C2×C53#153211yescharacteristic
185C2×C53#155411yescharacteristic
186D53—42no—
187C2×C22⋊Q8—531no—
188C2×C4×Q8#615711yescharacteristic
189C22×C42—531no—
190C2×C22⋊C8—1061no—
191C2×C22⋊C8—1061no—
192C2×#97—1061no—
193#391—531no—
194#389—531no—
195#387—1061no—
196#385—1061no—
197#380—531no—
198C4×C4⋊C4—531no—
199C4×C4⋊C4—1061no—
200C4×C22⋊C4—1061no—
201#336—532no—
202#336—532no—
203#336—1061no—
204C53#762311yescharacteristic, complemented, Sylow
205C22×Q8C2×#615711yescharacteristic
206C23×C4—531no—
207C23×C4—531no—
208C23×C4—531no—
209C22×C8—1062no—
210C22×C8—1062no—
211C22⋊Q8—1062no—
212C22⋊Q8—1062no—
213C4×Q8—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4C2×#615711yescharacteristic
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1061no—
222C2×C42—21no—
223C2×C42—531no—
224C2×C42—531no—
225C2×C42—531no—
226C2×C42—1061no—
227C2×C42—2122no—
228C2×C42C2×#615711yescharacteristic
229C22⋊C8—1064no—
230C22⋊C8—2122no—
231#97—2122no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237C24—531no—
238C2×Q8—21no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8#1171993612yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—1062no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1061no—
251C22×C4—1062no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—2121no—
256C22×C4—2121no—
257C22×C4—2121no—
258C22×C4—2121no—
259C22×C4C22⋊#615711yescharacteristic
260C22×C4C22⋊#615711yescharacteristic
261C22×C4C22×#615711yescharacteristic
262C2×C8—1064no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—532no—
271C4⋊C4—1064no—
272C4⋊C4—1061no—
273C4⋊C4—2121no—
274C4⋊C4—2122no—
275C22⋊C4—1064no—
276C22⋊C4—2122no—
277C42—22no—
278C42—42no—
279C42—1062no—
280C42—2121no—
281C42—2121no—
282C42—2122no—
283C23—532no—
284C23—1061no—
285C23—1061no—
286C23—1061no—
287C23—1061no—
288C23—1061no—
289C23—1061no—
290C23C2×C22⋊#615711yescharacteristic
291Q8—42no—
292Q8—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—532no—
307C2×C4—532no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1062no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—2124no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2124no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4#1313327512yes—
326C2×C4D53⋊M1611yescharacteristic
327C2×C4C2×#1171993611yescharacteristic
328C8—4242no—
329C8—2124no—
330C22—1062no—
331C22—1062no—
332C22—1062no—
333C22—2121no—
334C22—2121no—
335C22—2121no—
336C22—2121no—
337C22#1681159411yescharacteristic
338C22C2×#1313327511yescharacteristic
339C22#1681160911yescharacteristic
340C22#1681161311yescharacteristic
341C22#1681161611yescharacteristic
342C22D53⋊#10411yescharacteristic
343C22#1681162211yescharacteristic
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—42no—
349C4—81no—
350C4—1062no—
351C4—1062no—
352C4—4241no—
353C4—2122no—
354C4—2121no—
355C2—2122no—
356C2#2028997011yescharacteristic
357C2#2029004911yescharacteristic
358C2#2029005811yescharacteristic
359C2#2029009011yescharacteristic
360C2#2029012511yescharacteristic
361C2#2029019211yescharacteristic
362C2#2029024011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#7623
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20289535C2semidirect
4. #20289534C2semidirect
204. C53#7623semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202904422
#202895352
#202895342
C4×#131331932
#762353