Finite Group Databasebeta

← #28299878

Subgroups — #28299878

Order 13568.

444 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 90 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#20290432C211yesmaximal, characteristic, complemented
2#20290430C211yesmaximal, characteristic, complemented
3#20289534C212yesmaximal, complemented
4#20289532C212yesmaximal
5#20283083C211yesmaximal, characteristic
6C2×#13133272—21no—
7C2×C4×#88324—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133212—21no—
13#16810525C411yescharacteristic
14#16810524C411yescharacteristic
15#16810524C2211yescharacteristic
16#16810522C411yescharacteristic
17#16810519C2211yescharacteristic
18#16810515—21no—
19#16810513—21no—
20#16810512—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133272—24no—
26C4×#88324—24no—
27C22×D212C2×C411yescharacteristic
28#13133212—44no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53C2311yescharacteristic
44#13133124—21no—
45#13133124—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—21no—
51C23×D53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—21no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C2×D212—22no—
63C2×D212—41no—
64C2×D212—41no—
65C2×#88324—24no—
66C2×#88324—24no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C42×C53—41no—
75#11719921—42no—
76#11719921—42no—
77#11719921—42no—
78#11719921—42no—
79#11719921—42no—
80#11719921—42no—
81C4⋊Dic53—22no—
82C4⋊Dic53—42no—
83#11719919—42no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—22no—
88C4×Dic53—22no—
89C4×Dic53—41no—
90C23×C53C2×C22⋊C411yescharacteristic
91C22×D53—41no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53—41no—
96C22×D53—41no—
97C22×D53—41no—
98C2×C4×C53—21no—
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—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53#10412yes—
112C2×C4×C53#13112yes—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53#10412yes—
126C2×Dic53#13512yes—
127C2×Dic53AGL(1,ℤ/8)12yes—
128D212—82no—
129#88324—44no—
130#88324—44no—
131#7692—531nomaximal, Sylow
132C22×C53#40811yescharacteristic
133C22×C53#41011yescharacteristic
134C22×C53#41311yescharacteristic
135C22×C53#41611yescharacteristic
136C22×C53#41811yescharacteristic
137C22×C53C2×#10411yescharacteristic
138C22×C53C2×#10411yescharacteristic
139C2×D53—81no—
140C2×D53—81no—
141C2×D53—81no—
142C2×D53—81no—
143C2×D53—81no—
144C2×D53—81no—
145C2×D53—81no—
146C4×C53—22no—
147C4×C53—22no—
148C4×C53—41no—
149C4×C53—81no—
150C4×C53—81no—
151Dic53—22no—
152Dic53—22no—
153Dic53—22no—
154Dic53—41no—
155Dic53—81no—
156#2860—531no—
157#1745—531no—
158#1744—531no—
159#1271—532no—
160#1270—532no—
161C2×C53#147311yescharacteristic
162C2×C53#148411yescharacteristic
163C2×C53#149411yescharacteristic
164C2×C53#150011yescharacteristic
165C2×C53#150511yescharacteristic
166C2×C53#152411yescharacteristic
167C2×C53#155311yescharacteristic
168D53—161no—
169C2×#126—1061no—
170C2×#97—1061no—
171C2×C4×C8—1061no—
172#390—531no—
173#390—531no—
174#390—531no—
175#389—531no—
176#389—531no—
177#389#615711yescharacteristic
178#388—1061no—
179#388—1061no—
180#388—1061no—
181C43—531no—
182#336—532no—
183#336—532no—
184#336—1061no—
185#336—1061no—
186C53#769211yescharacteristic, complemented, Sylow
187C22×D4—531no—
188C22×C8—1062no—
189C22×C8—1062no—
190#126—2124no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4C2×#615711yescharacteristic
196C2×C4⋊C4C2×#615711yescharacteristic
197C2×C22⋊C4—1061no—
198C2×C22⋊C4—1061no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1061no—
201C2×C22⋊C4—1061no—
202C2×C22⋊C4—1061no—
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42C2×#615711yescharacteristic
210#97—1064no—
211C4×C8—1064no—
212#95—21no—
213#95—1061no—
214#95—1061no—
215C24—1061no—
216C2×D4—1062no—
217C2×D4—2121no—
218C2×D4—2121no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—1061no—
226C22×C4C22⋊#615711yescharacteristic
227C22×C4C22⋊#615711yescharacteristic
228C22×C4C22×#615711yescharacteristic
229C2×C8—1064no—
230C2×C8—1064no—
231C2×C8—2122no—
232C2×C8—2122no—
233C2×C8—2122no—
234C2×C8—2122no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—1062no—
239C4⋊C4—2122no—
240C4⋊C4—2122no—
241C22⋊C4—2122no—
242C22⋊C4—2122no—
243C22⋊C4—2122no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C22⋊C4—2122no—
247C42—41no—
248C42—1062no—
249C42—1062no—
250C42—1062no—
251C42—1062no—
252C42—1062no—
253C42—2121no—
254C23—2121no—
255C23—2121no—
256C23—2121no—
257C23—2121no—
258C23—2121no—
259C23—2121no—
260C23—2121no—
261C23C2×C22⋊#615711yescharacteristic
262D4—4242no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—532no—
277C2×C4—532no—
278C2×C4—532no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4#1313328212yes—
292C2×C4#1313330512yes—
293C8—2124no—
294C8—2124no—
295C22—4241no—
296C22—4241no—
297C22—4241no—
298C22—4241no—
299C22—4241no—
300C22—4241no—
301C22—4241no—
302C22#1681159811yescharacteristic
303C22#1681160211yescharacteristic
304C22#1681161011yescharacteristic
305C22#1681161311yescharacteristic
306C22#1681161511yescharacteristic
307C22D53⋊#10411yescharacteristic
308C22C2×#1313328211yescharacteristic
309C4—22no—
310C4—22no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—1062no—
315C4—1062no—
316C4—1062no—
317C4—4241no—
318C4—2121no—
319C2—8481no—
320C2#2028999811yescharacteristic
321C2#2029003611yescharacteristic
322C2#2029006911yescharacteristic
323C2#2029008811yescharacteristic
324C2#2029010511yescharacteristic
325C2#2029016811yescharacteristic
326C2#2029023911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7692
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
1. #20290432C2semidirect
2. #20290430C2semidirect
3. #20289534C2semidirect
186. C53#7692semidirect

Maximal subgroups

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

subgroupindex
#202904322
#202904302
#202895342
#202895322
#202830832
#769253