Finite Group Databasebeta

← #27293664

Subgroups — #27293664

Order 13568.

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

normal subgroups only

This group has 366 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#20279164C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272660C212yesmaximal, complemented
4#20272648C212yesmaximal, complemented
5C53×#1464C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7#16810676—21no—
8#16810551—21no—
9#16810551—21no—
10#16810528C411yescharacteristic
11#16810528C2211yescharacteristic
12#16810507—21no—
13#16810507—21no—
14#16810492C2211yescharacteristic
15#16810488C411yescharacteristic
16C4×C22⋊Dic53C411yescharacteristic
17#16810349C2212yes—
18#16810347—21no—
19C2×#117×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324C2212yes—
22C53×#323—21no—
23C22×Dic106C2×C411yescharacteristic
24#13133196—41no—
25#13133196—41no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C4⋊#88322—41no—
35C22⋊#88322—22no—
36C4×#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45#117×C53—41no—
46#117×C53—41no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2311yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C4⋊C8×C53—41no—
54C22⋊C8×C53—22no—
55#97×C53—41no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C2×Dic106—22no—
72C2×Dic106—41no—
73C2×Dic106—41no—
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—
85C4⋊Dic53—41no—
86#11719919—42no—
87#11719919—42no—
88#11719919—41no—
89#11719919—41no—
90C4×Dic53—22no—
91C4×Dic53—41no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53C2×C22⋊C411yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
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—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53#9911yescharacteristic
114C2×C4×C53#9911yescharacteristic
115C2×C4×C53#13112yes—
116C2×Dic53—22no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128Dic106—82no—
129C8×C53—82no—
130#88322—82no—
131#11552—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—81no—
138C22×C53—81no—
139C22×C53#40811yescharacteristic
140C22×C53C2×#9911yescharacteristic
141C22×C53#41311yescharacteristic
142C4×C53—22no—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—41no—
149Dic53—42no—
150Dic53—81no—
151Dic53—81no—
152Dic53—81no—
153#2499—531no—
154#1464D5311yescharacteristic
155#1458—531no—
156#1325—532no—
157#1319—532no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—81no—
161C2×C53#146811yescharacteristic
162C2×C53#212611yescharacteristic
163C2×C53#213511yescharacteristic
164C2×C22⋊Q8—531no—
165C2×#117C2×D5311yescharacteristic
166#432—21no—
167#431—1061no—
168#397—1061no—
169#397—1061no—
170#391—531no—
171#391—531no—
172#387—1061no—
173#387—1061no—
174#384—531no—
175#382—531no—
176C4×C22⋊C4—531no—
177#324—532no—
178#324C2×D5312yes—
179#323—21no—
180#323—1061no—
181C53#1155211yescharacteristic, complemented, Sylow
182C22×Q8—531no—
183C23×C4C4×D5311yescharacteristic
184C22⋊Q8—2121no—
185C22⋊Q8—2121no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4C4×D5311yescharacteristic
193C2×C4⋊C4C22×D5311yescharacteristic
194C2×C22⋊C4—1061no—
195C2×C22⋊C4—1061no—
196C2×C22⋊C4D21211yescharacteristic
197C2×C22⋊C4C22⋊D5311yescharacteristic
198C2×C42—531no—
199C2×C42—531no—
200C2×C42D21211yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—2121no—
204C22⋊C8—22no—
205C22⋊C8—1062no—
206#97—41no—
207C4×C8—2121no—
208#95—1061no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215C24#1171992111yescharacteristic
216C2×Q8—1062no—
217C2×Q8—2121no—
218C2×Q8—2121no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4—1061no—
226C22×C4—1061no—
227C22×C4#1171992111yescharacteristic
228C22×C4#1171992111yescharacteristic
229C22×C4#1171992111yescharacteristic
230C22×C4C2×C4×D5311yescharacteristic
231C22×C4C2×D21211yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233C2×C8—42no—
234C2×C8—2122no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—2121no—
238C4⋊C4—2122no—
239C4⋊C4—2122no—
240C4⋊C4—2121no—
241C4⋊C4—2121no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—1062no—
251C42—2121no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×#1171992111yescharacteristic
258Q8—4242no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—1062no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4#1313313111yescharacteristic
286C2×C4#1313313111yescharacteristic
287C2×C4#1313322512yes—
288C8—82no—
289C8—4242no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1681059111yescharacteristic
298C22C2×#1313313111yescharacteristic
299C22#1681060511yescharacteristic
300C4—21no—
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—4241no—
308C4—4241no—
309C4—4241no—
310C4—2122no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#2027306411yescharacteristic
315C2#2027637811yescharacteristic
316C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11552
LCSG ▷ C22×C4×C53 ▷ C22×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
3. #20272660C2semidirect
4. #20272648C2semidirect
181. C53#11552semidirect

Maximal subgroups

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

subgroupindex
#202791642
#202730292
#202726602
#202726482
C53×#14642
#1155253