Finite Group Databasebeta

← #28299926

Subgroups — #28299926

Order 13568.

426 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 86 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#20290631C211yesmaximal, characteristic, complemented
2#20290432C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289531C212yesmaximal
5#20283079C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×#13133272—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133210—21no—
13#16810526C411yescharacteristic
14#16810526C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810521C411yescharacteristic
17#16810519C411yescharacteristic
18#16810517—21no—
19#16810516—21no—
20#16810511—21no—
21C42×Dic53C2211yescharacteristic
22C53×#390C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26#13133272—24no—
27#13133210—44no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×D4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C42×C53C2×C411yescharacteristic
51C24×C53—21no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C22×C4×C53—21no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×Dic53—21no—
60C22×Dic53—21no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×#88324—24no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C4⋊C4×C53—22no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—42no—
74C42×C53—41no—
75C22⋊Dic53—42no—
76C22⋊Dic53—42no—
77C22⋊Dic53—41no—
78C22⋊Dic53—42no—
79C22⋊Dic53—41no—
80C4⋊Dic53—42no—
81C4⋊Dic53—41no—
82C4⋊Dic53—41no—
83#11719919—42no—
84#11719919—42no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—22no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—41no—
91C4×Dic53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100D4×C53—82no—
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—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
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—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53#10412yes—
129C2×Dic53AGL(1,ℤ/8)11yescharacteristic
130C2×Dic53AGL(1,ℤ/8)12yes—
131C2×Dic53#13711yescharacteristic
132#88324—44no—
133#88324—82no—
134#7706—531nomaximal, Sylow
135C22×C53—81no—
136C22×C53—81no—
137C22×C53—81no—
138C22×C53—81no—
139C22×C53—81no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53#40811yescharacteristic
143C22×C53#41011yescharacteristic
144C22×C53#41311yescharacteristic
145C22×C53#41711yescharacteristic
146C22×C53#41811yescharacteristic
147C22×C53C2×#10411yescharacteristic
148C22×C53#42111yescharacteristic
149C4×C53—22no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—81no—
153Dic53—22no—
154Dic53—22no—
155Dic53—21no—
156Dic53—21no—
157Dic53—41no—
158Dic53—81no—
159Dic53—81no—
160#2860—531no—
161#1831—531no—
162#1745—531no—
163#1271—532no—
164#1270—532no—
165C2×C53—161no—
166C2×C53#147311yescharacteristic
167C2×C53#148311yescharacteristic
168C2×C53#149611yescharacteristic
169C2×C53#150511yescharacteristic
170C2×C53#151211yescharacteristic
171C2×C53#152911yescharacteristic
172C2×C53#155711yescharacteristic
173C2×#126—1061no—
174C2×C4⋊C8—1061no—
175C2×#97—1061no—
176#390—531no—
177#390—531no—
178#390#615711yescharacteristic
179#389—531no—
180#389—531no—
181#389—531no—
182#388—1061no—
183#388—1061no—
184#388—1061no—
185C43—531no—
186#336—532no—
187#336—532no—
188#336—1061no—
189#336—1061no—
190C53#770611yescharacteristic, complemented, Sylow
191C22×D4C2×#615711yescharacteristic
192C22×C8—1062no—
193C22×C8—1062no—
194#126—2124no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4C2×#615711yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—1061no—
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1061no—
206C2×C22⋊C4—1061no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42—531no—
211C2×C42—531no—
212C2×C42—531no—
213C2×C42C2×#615711yescharacteristic
214C4⋊C8—2122no—
215#97—1064no—
216#95—1061no—
217#95—1061no—
218#95—1061no—
219C24—21no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—1061no—
229C22×C4—1061no—
230C22×C4C22⋊#615711yescharacteristic
231C22×C4C22⋊#615711yescharacteristic
232C22×C4C22×#615711yescharacteristic
233C2×C8—1064no—
234C2×C8—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C2×C8—2122no—
239C4⋊C4—22no—
240C4⋊C4—2121no—
241C4⋊C4—2121no—
242C4⋊C4—2122no—
243C4⋊C4—2122no—
244C4⋊C4—2122no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—2122no—
248C22⋊C4—2121no—
249C22⋊C4—2121no—
250C22⋊C4—2122no—
251C22⋊C4—2122no—
252C42—41no—
253C42—1062no—
254C42—1061no—
255C42—1062no—
256C42—1061no—
257C42—1062no—
258C42—2121no—
259C42—2121no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23C2×C22⋊#615711yescharacteristic
268D4—82no—
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—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—531no—
281C2×C4—531no—
282C2×C4—532no—
283C2×C4—532no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4#1313330412yes—
300C8—4242no—
301C8—2124no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#1681159711yescharacteristic
310C22#1681160211yescharacteristic
311C22#1681161011yescharacteristic
312C22#1681161411yescharacteristic
313C22#1681161511yescharacteristic
314C22D53⋊#10411yescharacteristic
315C22#1681162111yescharacteristic
316C4—22no—
317C4—41no—
318C4—41no—
319C4—81no—
320C4—1062no—
321C4—1061no—
322C4—1061no—
323C4—1062no—
324C4—4241no—
325C4—4241no—
326C4—2121no—
327C2—161no—
328C2#2028999611yescharacteristic
329C2#2029003111yescharacteristic
330C2#2029007511yescharacteristic
331C2#2029010511yescharacteristic
332C2#2029012611yescharacteristic
333C2#2029018111yescharacteristic
334C2#2029024711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#7706
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20290631C2semidirect
2. #20290432C2semidirect
3. #20289533C2semidirect
190. C53#7706semidirect

Maximal subgroups

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

subgroupindex
#202906312
#202904322
#202895332
#202895312
#202830792
#770653