Finite Group Databasebeta

← #27294810

Subgroups — #27294810

Order 13568.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#20277983C211yesmaximal, characteristic
2#20274792C211yesmaximal, characteristic
3#20272740C212yesmaximal
4#20272737C212yesmaximal
5C53×#1464C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic, complemented
7#16810562C411yescharacteristic, complemented
8#16810559C2211yescharacteristic
9#16810556—21no—
10#16810546—21no—
11#16810543C411yescharacteristic, complemented
12#16810535C2211yescharacteristic
13#16810486—21no—
14C4×#11719919—21no—
15C4×C4⋊Dic53C411yescharacteristic, complemented
16C4×C22⋊Dic53—21no—
17#16810390—21no—
18#16810390C2212yes—
19C2×#117×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C4×Dic106—41no—
25C4×Dic106—41no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—22no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—41no—
41#13133124—41no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45#117×C53—41no—
46#117×C53—41no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53C2311yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C4⋊C8×C53—41no—
54C22⋊C8×C53—22no—
55#97×C53—41no—
56C24×C53C2×D411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—42no—
72C2×Dic106—41no—
73C2×Dic106—42no—
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C22⋊Dic53—42no—
84C22⋊Dic53—42no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C4⋊Dic53—42no—
88C4⋊Dic53—41no—
89#11719919—42no—
90#11719919—42no—
91#11719919—42no—
92C4×Dic53—42no—
93C4×Dic53—41no—
94C4×Dic53—41no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53#13712yes—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
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—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#9911yescharacteristic
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—82no—
131C2×Dic53—82no—
132C2×Dic53—82no—
133Dic106—42no—
134Dic106—42no—
135C8×C53—82no—
136#11810—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—42no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—81no—
144C22×C53C2×#9911yescharacteristic
145C22×C53#41911yescharacteristic
146C22×C53#42111yescharacteristic
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—41no—
154Dic53—42no—
155Dic53—42no—
156Dic53—81no—
157Dic53—81no—
158Dic53—162no—
159#2396—531no—
160#1820—531no—
161#1464D5311yescharacteristic
162#1352—532no—
163#1351—532no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—42no—
167C2×C53#151111yescharacteristic
168C2×C53#212011yescharacteristic
169C2×C53#212611yescharacteristic
170C2×C4×Q8—531no—
171C2×#117C2×D5311yescharacteristic
172#432—21no—
173#400—531no—
174#399—531no—
175#398—1061no—
176#396—1061no—
177#395—531no—
178#393—531no—
179#381—1061no—
180C4×C4⋊C4—531no—
181C4×C4⋊C4—1061no—
182C4×C22⋊C4—1061no—
183#342—532no—
184#342—1061no—
185#324—21no—
186#323C2×D5312yes—
187C53#1181011yescharacteristic, complemented, Sylow
188C22×Q8—531no—
189C23×C4D21211yescharacteristic
190C4×Q8—2121no—
191C4×Q8—2121no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4C4×D5311yescharacteristic
200C2×C4⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4—1061no—
202C2×C22⋊C4—1061no—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4D21211yescharacteristic
205C2×C22⋊C4C22×D5311yescharacteristic
206C2×C42—1061no—
207C2×C42—1061no—
208C2×C42C4×D5311yescharacteristic
209C2×C42C22⋊D5311yescharacteristic
210C4⋊C8—41no—
211C22⋊C8—22no—
212#97—41no—
213#95—531no—
214#95—531no—
215#95—1061no—
216#95—1061no—
217#95—1061no—
218#95—1061no—
219#95—2121no—
220#95—2121no—
221C24C2×D21211yescharacteristic
222C2×Q8—2121no—
223C2×Q8—2122no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—1061no—
229C22×C4—1061no—
230C22×C4—1061no—
231C22×C4—1061no—
232C22×C4—2122no—
233C22×C4#1171992111yescharacteristic
234C22×C4#1171992111yescharacteristic
235C22×C4#1171992111yescharacteristic
236C22×C4#1171992111yescharacteristic
237C22×C4C2×C4×D5311yescharacteristic
238C22×C4C2×C22⋊D5311yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—2121no—
243C4⋊C4—2122no—
244C4⋊C4—2122no—
245C4⋊C4—2122no—
246C4⋊C4—2122no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—2122no—
250C22⋊C4—2122no—
251C22⋊C4—2122no—
252C22⋊C4—2122no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—2122no—
258C42—2121no—
259C42—2121no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×#1171992111yescharacteristic
265C23#1313325012yes—
266Q8—2122no—
267Q8—2122no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—1062no—
284C2×C4—1062no—
285C2×C4—4242no—
286C2×C4—4242no—
287C2×C4—4242no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4#1313313111yescharacteristic
299C2×C4#1313313111yescharacteristic
300C8—82no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#1313313111yescharacteristic
309C22#1681063311yescharacteristic
310C22#1681063911yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—8482no—
319C4—4241no—
320C4—4241no—
321C4—2122no—
322C4—2122no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2027324911yescharacteristic
327C2#2027635711yescharacteristic
328C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11810
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic106C4semidirect
7. #16810562C4semidirect
11. #16810543C4semidirect
15. C4×C4⋊Dic53C4semidirect
187. C53#11810semidirect

Maximal subgroups

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

subgroupindex
#202779832
#202747922
#202727402
#202727372
C53×#14642
#1181053