Finite Group Databasebeta

← #28299924

Subgroups — #28299924

Order 13568.

500 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 92 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#20290605C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20283065C211yesmaximal, characteristic
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133216C411yescharacteristic
13C2×#13133194—21no—
14C22×C4×Dic53C2211yescharacteristic
15#16810527—21no—
16#16810522C2211yescharacteristic
17#16810520—21no—
18#16810519C2211yescharacteristic
19#16810492C411yescharacteristic
20#16810483C411yescharacteristic
21#16810481—21no—
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324—42no—
27#13133272—42no—
28C22×Dic106—21no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31#13133216—42no—
32#13133194—44no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—42no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53—21no—
53#95×C53—21no—
54C24×C53C22⋊C411yescharacteristic
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—41no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—41no—
69C22×Dic53—41no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C2×Dic106—42no—
75C2×Dic106—42no—
76C2×Dic106—42no—
77C2×#88324—24no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—41no—
85C22⋊C4×C53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—22no—
88C4⋊Dic53—42no—
89C4⋊Dic53—41no—
90#11719919—42no—
91#11719919—42no—
92#11719919—42no—
93#11719919—42no—
94C4×Dic53—22no—
95C4×Dic53—22no—
96C4×Dic53—41no—
97C4×Dic53—41no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53#10112yes—
105C23×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—44no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53#13112yes—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—44no—
128C2×Dic53—44no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53#9912yes—
136C2×Dic53#10412yes—
137C2×Dic53AGL(1,ℤ/8)12yes—
138C2×Dic53#13712yes—
139Dic106—84no—
140#88324—44no—
141#88324—82no—
142#7705—531nomaximal, Sylow
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53#40811yescharacteristic
151C22×C53C2×#9911yescharacteristic
152C22×C53C2×#10111yescharacteristic
153C22×C53#41811yescharacteristic
154C22×C53#41911yescharacteristic
155C22×C53C2×#10411yescharacteristic
156C22×C53#42111yescharacteristic
157C4×C53—22no—
158C4×C53—41no—
159C4×C53—81no—
160C4×C53—81no—
161Dic53—22no—
162Dic53—22no—
163Dic53—22no—
164Dic53—22no—
165Dic53—81no—
166Dic53—81no—
167#2857—531no—
168#1817—531no—
169#1749—531no—
170#1271—532no—
171#1271—532no—
172C2×C53—42no—
173C2×C53#147211yescharacteristic
174C2×C53#148511yescharacteristic
175C2×C53#149611yescharacteristic
176C2×C53#150711yescharacteristic
177C2×C53#151111yescharacteristic
178C2×C53#153111yescharacteristic
179C2×C53#155611yescharacteristic
180C2×C4⋊Q8—531no—
181C2×C22⋊Q8—1061no—
182C22×C42—531no—
183C2×C22⋊C8—1061no—
184C2×C22⋊C8—1061no—
185C2×#97—1061no—
186#391—1061no—
187#389—531no—
188#389—531no—
189#389—1061no—
190#384—531no—
191#380—531no—
192#380—1061no—
193#380#615711yescharacteristic
194#336—532no—
195#336—532no—
196#336—1061no—
197C53#770511yescharacteristic, complemented, Sylow
198C22×Q8—1061no—
199C23×C4—531no—
200C23×C4—531no—
201C23×C4C2×#615711yescharacteristic
202C22×C8—1062no—
203C22×C8—1062no—
204C4⋊Q8—2122no—
205C22⋊Q8—2124no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4C2×#615711yescharacteristic
214C2×C4⋊C4C2×#615711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—1061no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42—2122no—
222C22⋊C8—1064no—
223C22⋊C8—2122no—
224#97—2122no—
225#95—21no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C24C22⋊#615711yescharacteristic
230C2×Q8—2122no—
231C2×Q8—2122no—
232C2×Q8—2122no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4—1062no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—2121no—
249C22×C4—2121no—
250C22×C4C22⋊#615711yescharacteristic
251C22×C4C22×#615711yescharacteristic
252C2×C8—1064no—
253C2×C8—2122no—
254C2×C8—2122no—
255C2×C8—2122no—
256C2×C8—2122no—
257C2×C8—2122no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—1062no—
261C4⋊C4—2121no—
262C4⋊C4—2122no—
263C4⋊C4—2122no—
264C4⋊C4—2122no—
265C4⋊C4—2122no—
266C4⋊C4—2122no—
267C22⋊C4—42no—
268C22⋊C4—2122no—
269C42—1062no—
270C42—1062no—
271C42—2121no—
272C42—2121no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#1313327812yes—
280C23C2×C22⋊#615711yescharacteristic
281Q8—4244no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—532no—
295C2×C4—532no—
296C2×C4—532no—
297C2×C4—532no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—2124no—
307C2×C4—2124no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4#1313330412yes—
315C8—4242no—
316C8—2124no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22#1681159611yescharacteristic
325C22D53⋊#9911yescharacteristic
326C22C2×#1313327811yescharacteristic
327C22#1681161511yescharacteristic
328C22#1681161611yescharacteristic
329C22D53⋊#10411yescharacteristic
330C22#1681162011yescharacteristic
331C4—22no—
332C4—41no—
333C4—81no—
334C4—81no—
335C4—1062no—
336C4—1062no—
337C4—1062no—
338C4—1062no—
339C4—4241no—
340C4—4241no—
341C2—42no—
342C2#2028999411yescharacteristic
343C2#2029003711yescharacteristic
344C2#2029007411yescharacteristic
345C2#2029011011yescharacteristic
346C2#2029012411yescharacteristic
347C2#2029018711yescharacteristic
348C2#2029024411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7705
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
197. C53#7705semidirect

Maximal subgroups

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

subgroupindex
#202906052
#202904412
#202895342
#202895332
#202830652
#770553