Finite Group Databasebeta

← #27293654

Subgroups — #27293654

Order 13568.

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

normal subgroups only

This group has 408 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#20279173C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic, complemented
3#20272658C212yesmaximal, complemented
4#20272646C212yesmaximal, complemented
5C53×#1464C211yesmaximal, characteristic, complemented
6C2×#13133191C411yescharacteristic
7#16810676—21no—
8#16810538—21no—
9#16810538—21no—
10#16810524C411yescharacteristic
11#16810524C2211yescharacteristic
12#16810514—21no—
13#16810514—21no—
14#16810492C411yescharacteristic
15#16810488C2211yescharacteristic
16C4×C22⋊Dic53C411yescharacteristic
17#16810349C2212yes—
18#16810347—21no—
19C2×#117×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324C2212yes—
22C53×#323—21no—
23C22×D212C2×C411yescharacteristic
24C22×C22⋊D53—21no—
25#13133191—41no—
26#13133191—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—21no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53C2×C411yescharacteristic
38C4⋊#88322—41no—
39C22⋊#88322—22no—
40C4×#88322—41no—
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
57C23×D53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—41no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C2×C22⋊D53—42no—
69C2×C22⋊D53—81no—
70C2×C22⋊D53—81no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C2×D212—22no—
75C2×D212—41no—
76C2×D212—41no—
77C2×C8×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C22⋊Dic53—42no—
87#11719921—42no—
88#11719921—42no—
89#11719921—41no—
90#11719921—41no—
91#11719921—81no—
92#11719921—81no—
93C4⋊Dic53—41no—
94#11719919—81no—
95#11719919—81no—
96C4×Dic53—22no—
97C4×Dic53—41no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C22×D53—41no—
106C22×D53—41no—
107C22×D53—41no—
108C22×D53—41no—
109C22×D53—41no—
110C22×D53—41no—
111C22×D53—41no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53#13112yes—
129C22⋊D53—84no—
130C2×Dic53—22no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139D212—82no—
140C8×C53—82no—
141#88322—82no—
142#11550—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53#40811yescharacteristic
151C22×C53C2×#9911yescharacteristic
152C22×C53#41311yescharacteristic
153C2×D53—81no—
154C2×D53—81no—
155C2×D53—81no—
156C2×D53—81no—
157C2×D53—81no—
158C2×D53—81no—
159C2×D53—81no—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53—41no—
167Dic53—42no—
168Dic53—81no—
169Dic53—81no—
170#2500—531no—
171#1464D5311yescharacteristic, complemented
172#1458—531no—
173#1324—532no—
174#1318—532no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53—81no—
178C2×C53#146811yescharacteristic
179C2×C53#212611yescharacteristic
180C2×C53#213411yescharacteristic
181D53—161no—
182C2×#121—531no—
183C2×#117C2×D5311yescharacteristic
184#432—21no—
185#431—1061no—
186#394—1061no—
187#394—1061no—
188#390—531no—
189#390—531no—
190#388—1061no—
191#388—1061no—
192#384—531no—
193#382—531no—
194C4×C22⋊C4—531no—
195#324—532no—
196#324C2×D5312yes—
197#323—21no—
198#323—1061no—
199C53#1155011yescharacteristic, complemented, Sylow
200C22×D4—531no—
201C22×D4—1061no—
202C23×C4C4×D5311yescharacteristic
203#121—2121no—
204#121—2121no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4C4×D5311yescharacteristic
211C2×C4⋊C4C22×D5311yescharacteristic
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4D21211yescharacteristic
219C2×C22⋊C4C22⋊D5311yescharacteristic
220C2×C42—531no—
221C2×C42—531no—
222C2×C42D21211yescharacteristic
223C2×C42C22⋊D5311yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—2121no—
226C22⋊C8—22no—
227C22⋊C8—1062no—
228#97—41no—
229C4×C8—2121no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233C24—1061no—
234C24#1171992111yescharacteristic
235C2×D4—1062no—
236C2×D4—4241no—
237C2×D4—4241no—
238C2×D4—2121no—
239C2×D4—2121no—
240C2×D4—2122no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4#1171992111yescharacteristic
249C22×C4#1171992111yescharacteristic
250C22×C4#1171992111yescharacteristic
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4C2×D21211yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254C2×C8—42no—
255C2×C8—2122no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—4241no—
259C4⋊C4—4241no—
260C4⋊C4—2121no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—4241no—
264C22⋊C4—4241no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C22⋊C4—2122no—
268C22⋊C4—2121no—
269C22⋊C4—2121no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—1062no—
275C42—2121no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—2121no—
282C23—2121no—
283C23—2121no—
284C23—2121no—
285C23—2121no—
286C23—2121no—
287C23—2121no—
288C23C2×#1171992111yescharacteristic
289D4—4242no—
290D4—4244no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—1062no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4#1313313111yescharacteristic
315C2×C4#1313313111yescharacteristic
316C2×C4#1313322512yes—
317C8—82no—
318C8—4242no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—4241no—
327C22—4241no—
328C22—4241no—
329C22—4241no—
330C22—4241no—
331C22—4241no—
332C22—4241no—
333C22#1681059111yescharacteristic
334C22C2×#1313313111yescharacteristic
335C22#1681060511yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—4241no—
344C4—4241no—
345C4—2122no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—8481no—
350C2#2027306411yescharacteristic
351C2#2027637811yescharacteristic
352C2#2027640611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1464C2
socleC22×C53C2×#99
prime radicalC2×#13133191C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1464D53
hypocenterC53#11550
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
2. #20273029C2semidirect
3. #20272658C2semidirect
4. #20272646C2semidirect
5. C53×#1464C2semidirect
171. #1464D53semidirect
199. C53#11550semidirect

Maximal subgroups

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

subgroupindex
#202791732
#202730292
#202726582
#202726462
C53×#14642
#1155053