Finite Group Databasebeta

← #28305708

Subgroups — #28305708

Order 13568.

714 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 142 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#20290647C211yesmaximal, characteristic, complemented
2#20290430C211yesmaximal, characteristic, complemented
3C2×#16811471C212yesmaximal, complemented
4C2×D4⋊#88324C212yesmaximal, complemented
5#20277279C211yesmaximal, characteristic
6C2×C4⋊#88324—22no—
7C2×C4⋊#88324C2212yes—
8C2×C4×#88324C2212yes—
9#16811489—22no—
10#16811471—24no—
11D4⋊#88324—24no—
12C2×#13133213C411yescharacteristic
13C2×D4×Dic53C2211yescharacteristic
14C2×#13133179C2211yescharacteristic
15C2×#13133177—22no—
16#16810525C411yescharacteristic
17#16810523—22no—
18#16810492C411yescharacteristic
19C42×Dic53C2211yescharacteristic
20C53×#390C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23C4⋊#88324—22no—
24C4⋊#88324—22no—
25C4⋊#88324—44no—
26C4×#88324—22no—
27C4×#88324D414yescomplemented
28C22×D212C2×C411yescharacteristic
29C22×C22⋊D53—22no—
30C23×Dic53—21no—
31C22×C4×D53—21no—
32#13133213—22no—
33#13133213—22no—
34D4×Dic53—22no—
35D4×Dic53—22no—
36#13133179—22no—
37#13133179—22no—
38#13133177—44no—
39#13133177—44no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719919—22no—
42C2×C22⋊Dic53—21no—
43C2×#11719921—22no—
44C2×#11719921—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D412yes—
47C2×C4×Dic53D412yes—
48C2×C4×Dic53C2311yescharacteristic
49C22×D4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—22no—
52C2×C42×C53C2×C411yescharacteristic
53C24×C53—21no—
54C23×D53—21no—
55C2×D4×C53—22no—
56C2×D4×C53—21no—
57C2×D4×C53—22no—
58C2×D4×C53M1612yes—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22×C411yescharacteristic
62C2×C22⋊D53—28no—
63C2×C22⋊D53—44no—
64C2×C22⋊D53—44no—
65C22×Dic53—21no—
66C22×Dic53—41no—
67C22×Dic53—41no—
68C22×Dic53—42no—
69C22×Dic53—42no—
70C22×Dic53C22⋊C412yes—
71C22×Dic53C2×D411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—21no—
76C2×D212M1612yes—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53—42no—
80C2×C4×D53—42no—
81C2×#88324—24no—
82C2×#88324—44no—
83C2×#88324—42no—
84C2×#88324—44no—
85C4⋊C4×C53—22no—
86C22⋊C4×C53—44no—
87C42×C53—22no—
88C22⋊Dic53—42no—
89#11719921—44no—
90#11719921—42no—
91C4⋊Dic53—22no—
92#11719919—44no—
93C4×Dic53—21no—
94C4×Dic53—21no—
95C4×Dic53—42no—
96C4×Dic53D814yes—
97C4×Dic53SD1614yes—
98C4×Dic53C2×D412yes—
99C4×Dic53#4112yes—
100C23×C53—22no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53C2×C22⋊C411yescharacteristic
106C22×D53—22no—
107C22×D53—41no—
108C22×D53—41no—
109C22×D53—42no—
110C22×D53—42no—
111D4×C53—24no—
112D4×C53—42no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53#10214yes—
118C2×C4×C53C4×D412yes—
119C2×C4×C53C2×M1611yescharacteristic
120C2×C4×C53C2×M1611yescharacteristic
121C2×C4×C53#13112yes—
122C22⋊D53—416no—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—42no—
128C2×Dic53—82no—
129C2×Dic53—81no—
130C2×Dic53—81no—
131C2×Dic53#10214yes—
132C2×Dic53#12112yes—
133C2×Dic53#12411yescharacteristic
134C2×Dic53#12711yescharacteristic
135C2×Dic53C2×D812yes—
136C2×Dic53C2×SD1612yes—
137D212—24no—
138D212—42no—
139C4×D53—82no—
140C4×D53—81no—
141C4×D53—81no—
142#88324—44no—
143#88324—82no—
144#9644—531nomaximal, Sylow
145C22×C53—42no—
146C22×C53—44no—
147C22×C53—81no—
148C22×C53—81no—
149C22×C53—82no—
150C22×C53#39011yescharacteristic
151C22×C53#40711yescharacteristic
152C22×C53#40811yescharacteristic
153C22×C53C2×#10212yes—
154C22×C53#42112yes—
155C2×D53—42no—
156C2×D53—44no—
157C2×D53—81no—
158C2×D53—81no—
159C2×D53—82no—
160C4×C53—24no—
161C4×C53—81no—
162C4×C53#43512yes—
163C4×C53#43512yes—
164Dic53—22no—
165Dic53—24no—
166Dic53—81no—
167Dic53#48612yes—
168Dic53#49211yescharacteristic
169Dic53#49311yescharacteristic
170C4×#127—531no—
171#1842—531no—
172#1744—531no—
173C2×D4⋊C8—532no—
174C2×D4⋊C8—532no—
175C2×C53—82no—
176C2×C53#148012yes—
177C2×C53#148612yes—
178C2×C53#195511yescharacteristic
179C2×C53#195711yescharacteristic
180C2×C53#196311yescharacteristic
181D53—82no—
182C2×#127—531no—
183C2×C4×D4—531no—
184C2×C4×D4—531no—
185C2×C4×D4—1062no—
186C2×C4⋊C8—532no—
187C2×C4⋊C8—1062no—
188C2×C4×C8—532no—
189#390—531no—
190#390—1062no—
191#390#615711yescharacteristic
192#384—531no—
193C43—531no—
194#336—1062no—
195D4⋊C8—1064no—
196D4⋊C8—1064no—
197C53#964411yescharacteristic, complemented, Sylow
198C22×D4—531no—
199C22×D4—1062no—
200C22×D4C2×#615711yescharacteristic
201C23×C4—1061no—
202C23×C4—1061no—
203C22×C8—1062no—
204C22×C8—1062no—
205#127—1062no—
206#127—1062no—
207C4×D4—1062no—
208C4×D4—1062no—
209C4×D4—1062no—
210C4×D4—1062no—
211C4×D4—2124no—
212C4×D4—2124no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—1062no—
215C2×C4⋊C4C2×#615711yescharacteristic
216C2×C22⋊C4—22no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1062no—
220C2×C42—532no—
221C2×C42—531no—
222C2×C42—531no—
223C2×C42—532no—
224C2×C42C2×#615711yescharacteristic
225C4⋊C8—1062no—
226C4⋊C8—1062no—
227C4⋊C8—2124no—
228C4×C8—534no—
229C4×C8—1062no—
230C24—21no—
231C24—1061no—
232C2×D4—22no—
233C2×D4—21no—
234C2×D4—22no—
235C2×D4—532no—
236C2×D4—1068no—
237C2×D4—1062no—
238C2×D4—1061no—
239C2×D4—1062no—
240C2×D4—2124no—
241C2×D4—2124no—
242C2×D4#1171993612yes—
243C22×C4—21no—
244C22×C4—531no—
245C22×C4—532no—
246C22×C4—531no—
247C22×C4—1061no—
248C22×C4—2121no—
249C22×C4—2121no—
250C22×C4—2122no—
251C22×C4—2122no—
252C22×C4—2121no—
253C22×C4—2121no—
254C22×C4—2122no—
255C22×C4—2122no—
256C22×C4C22⋊#615712yes—
257C22×C4C22×#615711yescharacteristic
258C2×C8—1064no—
259C2×C8—2122no—
260C2×C8—2124no—
261C2×C8—2124no—
262C4⋊C4—22no—
263C4⋊C4—1062no—
264C4⋊C4—2124no—
265C22⋊C4—44no—
266C22⋊C4—2122no—
267C22⋊C4—2122no—
268C22⋊C4—2124no—
269C42—22no—
270C42—534no—
271C42—534no—
272C42—532no—
273C42—532no—
274C42—1061no—
275C42—1061no—
276C42—2122no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23—1062no—
283C23—2121no—
284C23—2121no—
285C23—2122no—
286C23—2122no—
287C23C2×C22⋊#615711yescharacteristic
288D4—24no—
289D4—42no—
290D4—1064no—
291D4—21216no—
292D4—2122no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—531no—
298C2×C4—531no—
299C2×C4—532no—
300C2×C4—532no—
301C2×C4—532no—
302C2×C4—534no—
303C2×C4—1062no—
304C2×C4—1062no—
305C2×C4—1062no—
306C2×C4—4241no—
307C2×C4—4242no—
308C2×C4—4241no—
309C2×C4—4241no—
310C2×C4—4242no—
311C2×C4—4241no—
312C2×C4—2121no—
313C2×C4—2122no—
314C2×C4#1313327914yes—
315C2×C4D4×#615712yes—
316C2×C4D53⋊M1611yescharacteristic
317C2×C4C2×#1171993611yescharacteristic
318C2×C4#1313330412yes—
319C8—4242no—
320C8—2124no—
321C22—42no—
322C22—44no—
323C22—81no—
324C22—81no—
325C22—82no—
326C22—4241no—
327C22—4241no—
328C22—4242no—
329C22—2122no—
330C22—2124no—
331C22#1681157511yescharacteristic
332C22#1681159411yescharacteristic
333C22#1681159611yescharacteristic
334C22C2×#1313327912yes—
335C22#1681162012yes—
336C4—24no—
337C4—81no—
338C4—531no—
339C4—532no—
340C4—531no—
341C4—1062no—
342C4—1064no—
343C4—4241no—
344C4#1681165212yes—
345C4#1681165512yes—
346C2—82no—
347C2—4242no—
348C2#2029002012yes—
349C2#2029004112yes—
350C2#2029084011yescharacteristic
351C2#2029084311yescharacteristic
352C2#2029085211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#9644
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20290647C2semidirect
2. #20290430C2semidirect
3. C2×#16811471C2semidirect
4. C2×D4⋊#88324C2semidirect
27. C4×#88324D4semidirect
197. C53#9644semidirect

Maximal subgroups

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

subgroupindex
#202906472
#202904302
C2×#168114712
C2×D4⋊#883242
#202772792
#964453