Finite Group Databasebeta

← #28303342

Subgroups — #28303342

Order 13568.

588 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 110 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#20289891C211yesmaximal, characteristic
3#20289588C214yesmaximal
4#20277120C211yesmaximal, characteristic
5C2×#13133274C2212yes—
6C2×(C2×#6157)⋊C4—21no—
7C2×(C2×#6157)⋊C4—21no—
8C2×(C2×#6157)⋊C4C2212yes—
9#16811489—21no—
10#16811489—21no—
11C2×#13133175—21no—
12C2×C42×D53C2211yescharacteristic
13#16810493C411yescharacteristic, complemented
14#16810488—21no—
15#16810480—21no—
16#16810480C2212yes—
17C4⋊C4×Dic53—21no—
18C4⋊C4×Dic53C412yescomplemented
19C53×#384C411yescharacteristic, complemented
20C23×#6157—22no—
21C22×#88324—22no—
22#13133274—24no—
23(C2×#6157)⋊C4—24no—
24(C2×#6157)⋊C4—24no—
25(C2×#6157)⋊C4—42no—
26C22×C4×D53D411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×C4×D53C2311yescharacteristic
29#13133175—42no—
30#13133175—42no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C412yes—
34C2×#11719921—22no—
35C2×C4×Dic53—22no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C42×D53—24no—
40#13133124—22no—
41#13133124—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53C2×C412yes—
45C2×C42×C53C2×C411yescharacteristic
46C23×D53C2×D411yescharacteristic
47C22×#6157—24no—
48C22×#6157—42no—
49C22×#6157—42no—
50C22×#6157—42no—
51C22×#6157—42no—
52C22×#6157—42no—
53C22×#6157—42no—
54C22×C4×C53—22no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C2×C4×D53—22no—
63C2×C4×D53—22no—
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—41no—
67C2×C4×D53—41no—
68C2×C4×D53—41no—
69C2×C4×D53D812yes—
70C2×C4×D53SD1612yes—
71C2×C4×D53#4114yes—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C42×C53—22no—
79#11719921—42no—
80#11719921—42no—
81C4⋊Dic53—42no—
82C4⋊Dic53—42no—
83C4⋊Dic53—42no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—44no—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—21no—
93C22×D53#12312yes—
94C22×D53C2×D811yescharacteristic
95C22×D53C2×SD1611yescharacteristic
96C22×D53#13712yes—
97C2×#6157—44no—
98C2×#6157—44no—
99C2×#6157—44no—
100C2×#6157—82no—
101C2×#6157—82no—
102C2×#6157—82no—
103C2×#6157—82no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53#10212yes—
114C2×C4×C53#10412yes—
115C2×C4×C53#11712yes—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53#10212yes—
127C2×Dic53#12312yes—
128C4×D53—24no—
129C4×D53—24no—
130C4×D53—44no—
131C4×D53—44no—
132#88324—82no—
133#8716—531nomaximal, Sylow
134C22×C53#38111yescharacteristic
135C22×C53#41011yescharacteristic
136C22×C53#41011yescharacteristic
137C22×C53C2×#10211yescharacteristic
138C22×C53#41911yescharacteristic
139C22×C53C2×#10411yescharacteristic
140C22×C53#42111yescharacteristic
141C2×D53—22no—
142C2×D53—22no—
143C2×D53—21no—
144C2×D53—21no—
145C2×D53—22no—
146C2×D53—22no—
147C2×D53—21no—
148C2×D53—21no—
149C2×D53#48012yes—
150C2×D53#48312yes—
151#6157—84no—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—82no—
156Dic53—22no—
157Dic53—22no—
158Dic53—41no—
159Dic53—82no—
160#2285—531no—
161#1817—531no—
162#1434—531no—
163#1294—534no—
164C2×C53#150211yescharacteristic
165C2×C53#150611yescharacteristic
166C2×C53#151511yescharacteristic
167C2×C53#151711yescharacteristic
168C2×C53#182011yescharacteristic
169C2×C53#182311yescharacteristic
170C2×C53#182611yescharacteristic
171D53—22no—
172D53—22no—
173C2×#117—1061no—
174C22×C42—531no—
175C2×C22⋊C8—532no—
176#384—531no—
177#384#615711yescharacteristic, complemented
178#382—1061no—
179#380—532no—
180#380—1061no—
181C4×C4⋊C4—532no—
182C4×C4⋊C4—1061no—
183C2×#95—532no—
184C2×#95—1061no—
185C2×#95—1061no—
186#336—1061no—
187#336—1061no—
188C53#871611yescharacteristic, complemented, Sylow
189C23×C4—531no—
190C23×C4—531no—
191C23×C4—531no—
192C23×C4—1062no—
193C22×C8—1062no—
194#117—2122no—
195#117—2122no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—532no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4C2×#615712yes—
202C2×C22⋊C4—1062no—
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—1064no—
207C2×C42—1062no—
208C2×C42C2×#615711yescharacteristic
209C22⋊C8—1064no—
210#95—1062no—
211#95—1062no—
212#95—1064no—
213#95—1064no—
214#95—2122no—
215C24—531no—
216C22×C4—22no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—532no—
220C22×C4—532no—
221C22×C4—534no—
222C22×C4—531no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—1062no—
226C22×C4—1062no—
227C22×C4—1064no—
228C22×C4—1062no—
229C22×C4—2121no—
230C22×C4—2121no—
231C22×C4—2121no—
232C22×C4—2122no—
233C22×C4—2122no—
234C22×C4—2122no—
235C22×C4—2122no—
236C22×C4—2122no—
237C22×C4—2122no—
238C22×C4C22⋊#615711yescharacteristic
239C22×C4C22⋊#615711yescharacteristic
240C22×C4C22×#615711yescharacteristic
241C2×C8—2122no—
242C2×C8—2122no—
243C2×C8—2122no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—2122no—
248C4⋊C4—2122no—
249C4⋊C4—2122no—
250C22⋊C4—2122no—
251C22⋊C4—2122no—
252C42—22no—
253C42—1062no—
254C42—1062no—
255C42—1062no—
256C42—2124no—
257C23—532no—
258C23—531no—
259C23—531no—
260C23—532no—
261C23—1061no—
262C23—1061no—
263C23—1061no—
264C23—1061no—
265C23C2×C22⋊#615711yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—532no—
276C2×C4—532no—
277C2×C4—1064no—
278C2×C4—1064no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—4242no—
287C2×C4—4242no—
288C2×C4—4242no—
289C2×C4—4242no—
290C2×C4—2124no—
291C2×C4—2124no—
292C2×C4—2124no—
293C2×C4—2124no—
294C2×C4—2124no—
295C2×C4—2122no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4#1313327912yes—
299C2×C4#1313328212yes—
300C2×C4#1313329612yes—
301C8—4242no—
302C22—532no—
303C22—532no—
304C22—1061no—
305C22—1061no—
306C22—1062no—
307C22—1062no—
308C22—1062no—
309C22—1062no—
310C22—1061no—
311C22—1061no—
312C22#1681156611yescharacteristic
313C22#1681160311yescharacteristic
314C22#1681160411yescharacteristic
315C22C2×#1313327911yescharacteristic
316C22#1681161611yescharacteristic
317C22C2×#1313328211yescharacteristic
318C22#1681162111yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—82no—
323C4—1062no—
324C4—1062no—
325C4—4244no—
326C4—4242no—
327C4—2121no—
328C2—1062no—
329C2—1062no—
330C2#2029009611yescharacteristic
331C2#2029010911yescharacteristic
332C2#2029013811yescharacteristic
333C2#2029014411yescharacteristic
334C2#2029061311yescharacteristic
335C2#2029061811yescharacteristic
336C2#2029062211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenter#384#6157
hypocenterC53#8716
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
13. #16810493C4semidirect
18. C4⋊C4×Dic53C4semidirect
19. C53×#384C4semidirect
177. #384#6157semidirect
188. C53#8716semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202898912
#202895882
#202771202
#871653