Finite Group Databasebeta

← #28299714

Subgroups — #28299714

Order 13568.

490 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290607C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20277196C211yesmaximal, characteristic
6C2×#13133274—21no—
7C2×#13133274—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133175C411yescharacteristic
13C2×#13133175C411yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810521C411yescharacteristic
17#16810519C2211yescharacteristic
18#16810497—21no—
19#16810494—21no—
20C4⋊C4×Dic53—21no—
21#16810474—21no—
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133274—42no—
26#13133274—42no—
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30#13133175—22no—
31#13133175—22no—
32#13133175—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×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45C42×D53—42no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53#95×C53—21no—
54C23×D53C22⋊C411yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×C4×D53—22no—
66C2×C4×D53—21no—
67C2×C4×D53—21no—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—22no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53M1612yes—
85C42×C53—41no—
86#11719921—24no—
87#11719921—42no—
88C4⋊Dic53—21no—
89C4⋊Dic53—42no—
90C4⋊Dic53—41no—
91C4⋊Dic53M1612yes—
92#11719919—42no—
93#11719919—42no—
94C4×Dic53—24no—
95C4×Dic53—41no—
96C4×Dic53—41no—
97C4×Dic53—41no—
98C4×Dic53—42no—
99C23×C53C2×C22⋊C411yescharacteristic
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53#9912yes—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53C2×M1611yescharacteristic
122C2×C4×C53C2×M1611yescharacteristic
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—22no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53#10112yes—
136C2×Dic53#13512yes—
137C2×Dic53#13512yes—
138C4×D53—24no—
139C4×D53—24no—
140C4×D53—44no—
141C4×D53—44no—
142#88324—82no—
143#88324—82no—
144#7646—531nomaximal, Sylow
145C22×C53#40711yescharacteristic
146C22×C53C2×#9911yescharacteristic
147C22×C53C2×#10111yescharacteristic
148C22×C53#41611yescharacteristic
149C22×C53#41611yescharacteristic
150C22×C53#42111yescharacteristic
151C22×C53#42111yescharacteristic
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—22no—
155C2×D53—41no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—41no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—42no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—81no—
165Dic53—22no—
166Dic53—22no—
167Dic53—22no—
168Dic53—42no—
169Dic53—81no—
170#2296—531no—
171#1817—531no—
172#1817—531no—
173#1271—532no—
174#1271—532no—
175C2×C53#146311yescharacteristic
176C2×C53#148811yescharacteristic
177C2×C53#148811yescharacteristic
178C2×C53#151611yescharacteristic
179C2×C53#151611yescharacteristic
180C2×C53#153711yescharacteristic
181C2×C53#154411yescharacteristic
182D53—42no—
183C2×#117—531no—
184C2×#117—531no—
185C22×C42—531no—
186C2×C22⋊C8—1061no—
187C2×C22⋊C8—1061no—
188#389—531no—
189#389—531no—
190#389—531no—
191#389#615711yescharacteristic
192#385—1061no—
193#385—1061no—
194C4×C4⋊C4—1061no—
195C4×C4⋊C4—1061no—
196#336—532no—
197#336—532no—
198#336—1061no—
199#336—1061no—
200C53#764611yescharacteristic, complemented, Sylow
201C23×C4—531no—
202C23×C4—531no—
203C23×C4—531no—
204C22×C8—1062no—
205C22×C8—1062no—
206#117—1062no—
207#117—1062no—
208#117—2122no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4C2×#615711yescharacteristic
216C2×C4⋊C4C2×#615711yescharacteristic
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C42—531no—
220C2×C42—531no—
221C2×C42—531no—
222C2×C42—1061no—
223C2×C42—1061no—
224C2×C42—2122no—
225C2×C42C2×#615711yescharacteristic
226C22⋊C8—2122no—
227C22⋊C8—2122no—
228#95—21no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232C24—531no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—1062no—
239C22×C4—1062no—
240C22×C4—1062no—
241C22×C4—1062no—
242C22×C4—1062no—
243C22×C4—1062no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—2121no—
249C22×C4—2121no—
250C22×C4C22⋊#615711yescharacteristic
251C22×C4C22⋊#615711yescharacteristic
252C22×C4C22×#615711yescharacteristic
253C2×C8—2122no—
254C2×C8—2122no—
255C2×C8—2122no—
256C2×C8—2122no—
257C2×C8—2122no—
258C2×C8—2122no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—532no—
263C4⋊C4—1061no—
264C4⋊C4—2121no—
265C4⋊C4—2122no—
266C4⋊C4—2122no—
267C4⋊C4—2122no—
268C4⋊C4#1171993612yes—
269C22⋊C4—1064no—
270C22⋊C4—2122no—
271C42—41no—
272C42—1064no—
273C42—2121no—
274C42—2121no—
275C42—2121no—
276C42—2122no—
277C23—532no—
278C23—1061no—
279C23—1061no—
280C23—1061no—
281C23—1061no—
282C23—1061no—
283C23—1061no—
284C23C2×C22⋊#615711yescharacteristic
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—532no—
300C2×C4—532no—
301C2×C4—532no—
302C2×C4—1064no—
303C2×C4—1064no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1062no—
311C2×C4—2124no—
312C2×C4—2124no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4D53⋊M1611yescharacteristic
319C2×C4C2×#1171993611yescharacteristic
320C8—4242no—
321C8—4242no—
322C22—1062no—
323C22—1062no—
324C22—1062no—
325C22—2121no—
326C22—2121no—
327C22—2121no—
328C22—2121no—
329C22#1681159411yescharacteristic
330C22D53⋊#9911yescharacteristic
331C22D53⋊#10111yescharacteristic
332C22#1681161311yescharacteristic
333C22#1681161311yescharacteristic
334C22#1681162111yescharacteristic
335C22#1681162211yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—81no—
342C4—1062no—
343C4—1062no—
344C4—1062no—
345C4—4241no—
346C4—2122no—
347C2—2122no—
348C2#2028996411yescharacteristic
349C2#2029004711yescharacteristic
350C2#2029004911yescharacteristic
351C2#2029014011yescharacteristic
352C2#2029014211yescharacteristic
353C2#2029020711yescharacteristic
354C2#2029022111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7646
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
200. C53#7646semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202895342
#202895332
#202771962
#764653