Finite Group Databasebeta

← #28305703

Subgroups — #28305703

Order 13568.

698 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 134 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#20290632C211yesmaximal, characteristic, complemented
2#20290429C211yesmaximal, characteristic, complemented
3C2×#16811471C212yesmaximal, complemented
4C2×D4⋊#88324C212yesmaximal, complemented
5#20277279C211yesmaximal, characteristic
6C2×C4⋊#88324C2212yes—
7C2×C4×#88324C2212yes—
8#16811489—22no—
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—22no—
26C4×#88324D414yescomplemented
27C22×D212C2×C411yescharacteristic
28C22×C22⋊D53—22no—
29C23×Dic53—21no—
30C22×C4×D53—21no—
31#13133213—22no—
32#13133213C814yescomplemented
33D4×Dic53—22no—
34D4×Dic53—22no—
35#13133179—22no—
36#13133179—22no—
37#13133177—48no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—22no—
40C2×C22⋊Dic53—21no—
41C2×#11719921—22no—
42C2×#11719921—21no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D412yes—
45C2×C4×Dic53D412yes—
46C2×C4×Dic53C2311yescharacteristic
47C22×D4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53—22no—
50C2×C42×C53C2×C411yescharacteristic
51C24×C53—21no—
52C23×D53—21no—
53C2×D4×C53—22no—
54C2×D4×C53—21no—
55C2×D4×C53—22no—
56C2×D4×C53C2×C812yes—
57C22×C4×C53—21no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C2×C22⋊D53—28no—
61C2×C22⋊D53—44no—
62C2×C22⋊D53—44no—
63C22×Dic53—21no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53—44no—
67C22×Dic53C22⋊C412yes—
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×D212—22no—
71C2×D212—22no—
72C2×D212—21no—
73C2×D212C2×C812yes—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—44no—
77C2×#88324—24no—
78C2×#88324—44no—
79C2×#88324—42no—
80C2×#88324—44no—
81C4⋊C4×C53—22no—
82C22⋊C4×C53—44no—
83C42×C53—22no—
84C22⋊Dic53—42no—
85#11719921—44no—
86#11719921—42no—
87C4⋊Dic53—22no—
88#11719919—44no—
89C4×Dic53—24no—
90C4×Dic53—42no—
91C4×Dic53C2×C812yes—
92C4×Dic53M1612yes—
93C4×Dic53C2×D412yes—
94C4×Dic53#4112yes—
95C23×C53—22no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53C2×C22⋊C411yescharacteristic
101C22×D53—22no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—42no—
105C22×D53—42no—
106D4×C53—24no—
107D4×C53—42no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53C4×D412yes—
114C2×C4×C53C22×C811yescharacteristic
115C2×C4×C53C2×M1611yescharacteristic
116C2×C4×C53#13112yes—
117C22⋊D53—416no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—41no—
122C2×Dic53—42no—
123C2×Dic53—82no—
124C2×Dic53—82no—
125C2×Dic53C22⋊C814yes—
126C2×Dic53#12112yes—
127C2×Dic53#12411yescharacteristic
128C2×Dic53#12711yescharacteristic
129C2×Dic53AGL(1,ℤ/8)14yes—
130D212—24no—
131D212—42no—
132C4×D53—82no—
133C4×D53—82no—
134#88324—44no—
135#88324—82no—
136#9642—531nomaximal, Sylow
137C22×C53—42no—
138C22×C53—44no—
139C22×C53—81no—
140C22×C53—81no—
141C22×C53—82no—
142C22×C53#39011yescharacteristic
143C22×C53C2×C22⋊C811yescharacteristic
144C22×C53#40811yescharacteristic
145C22×C53#41812yes—
146C22×C53#42112yes—
147C2×D53—42no—
148C2×D53—44no—
149C2×D53—81no—
150C2×D53—81no—
151C2×D53—82no—
152C4×C53—42no—
153C4×C53—81no—
154C4×C53C8×D412yes—
155C4×C53#43612yes—
156Dic53—22no—
157Dic53—24no—
158Dic53—81no—
159Dic53#49012yes—
160Dic53#49612yes—
161C4×#127—531no—
162#1831—531no—
163#1744—531no—
164C2×D4⋊C8—532no—
165C2×D4⋊C8—532no—
166C2×C53—82no—
167C2×C53#147812yes—
168C2×C53#149612yes—
169C2×C53#195411yescharacteristic
170C2×C53#196111yescharacteristic
171C2×C53#196311yescharacteristic
172D53—82no—
173C2×#127—531no—
174C2×C4×D4—531no—
175C2×C4×D4—531no—
176C2×C4×D4—1062no—
177C2×C4⋊C8—532no—
178C2×C4×C8—532no—
179#390—531no—
180#390—1062no—
181#390#615711yescharacteristic
182#384—531no—
183C43—531no—
184#336—1062no—
185#336—1062no—
186D4⋊C8—1064no—
187D4⋊C8—1064no—
188C53#964211yescharacteristic, complemented, Sylow
189C22×D4—531no—
190C22×D4—1062no—
191C22×D4C2×#615711yescharacteristic
192C23×C4—1061no—
193C23×C4—1061no—
194C22×C8—1062no—
195C22×C8—1062no—
196#127—534no—
197#127—1062no—
198C4×D4—1062no—
199C4×D4—1062no—
200C4×D4—1062no—
201C4×D4—1062no—
202C4×D4—2128no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1062no—
205C2×C4⋊C4C2×#615711yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1061no—
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1062no—
210C2×C42—531no—
211C2×C42—531no—
212C2×C42—532no—
213C2×C42—532no—
214C2×C42C2×#615711yescharacteristic
215C4⋊C8—1062no—
216C4⋊C8—1062no—
217C4×C8—534no—
218C4×C8—1062no—
219C24—21no—
220C24—1061no—
221C2×D4—22no—
222C2×D4—21no—
223C2×D4—22no—
224C2×D4—532no—
225C2×D4—1068no—
226C2×D4—1062no—
227C2×D4—1061no—
228C2×D4—1062no—
229C2×D4—2124no—
230C2×D4—2124no—
231C2×D4D53⋊C812yes—
232C22×C4—21no—
233C22×C4—531no—
234C22×C4—532no—
235C22×C4—531no—
236C22×C4—1061no—
237C22×C4—2121no—
238C22×C4—2121no—
239C22×C4—2124no—
240C22×C4—2121no—
241C22×C4—2121no—
242C22×C4—2124no—
243C22×C4C22⋊#615712yes—
244C22×C4C22×#615711yescharacteristic
245C2×C8—1064no—
246C2×C8—2122no—
247C2×C8—2124no—
248C2×C8—2124no—
249C4⋊C4—22no—
250C4⋊C4—1062no—
251C4⋊C4—2124no—
252C22⋊C4—44no—
253C22⋊C4—2124no—
254C22⋊C4—2122no—
255C22⋊C4—2122no—
256C42—22no—
257C42—532no—
258C42—532no—
259C42—532no—
260C42—532no—
261C42—1064no—
262C42—2122no—
263C23—22no—
264C23—41no—
265C23—41no—
266C23—42no—
267C23—42no—
268C23—1062no—
269C23—2121no—
270C23—2121no—
271C23—2122no—
272C23—2122no—
273C23C2×C22⋊#615711yescharacteristic
274D4—24no—
275D4—42no—
276D4—1064no—
277D4—21216no—
278D4—2122no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—531no—
285C2×C4—531no—
286C2×C4—534no—
287C2×C4—532no—
288C2×C4—534no—
289C2×C4—1062no—
290C2×C4—1062no—
291C2×C4—1062no—
292C2×C4—4242no—
293C2×C4—4242no—
294C2×C4—4242no—
295C2×C4—4242no—
296C2×C4—2121no—
297C2×C4—2122no—
298C2×C4D4×#615712yes—
299C2×C4C2×D53⋊C811yescharacteristic
300C2×C4D53⋊M1611yescharacteristic
301C2×C4#1313330412yes—
302C8—4242no—
303C8—2124no—
304C22—42no—
305C22—44no—
306C22—82no—
307C22—81no—
308C22—81no—
309C22—4242no—
310C22—4241no—
311C22—4241no—
312C22—2122no—
313C22—2124no—
314C22#1681157511yescharacteristic
315C22D53⋊(C22⋊C8)11yescharacteristic
316C22#1681159611yescharacteristic
317C22#1681161512yes—
318C22#1681162012yes—
319C4—42no—
320C4—81no—
321C4—532no—
322C4—532no—
323C4—1062no—
324C4—1064no—
325C4—4241no—
326C4#1681165012yes—
327C4#1681165612yes—
328C2—82no—
329C2—4242no—
330C2#2029001512yes—
331C2#2029007412yes—
332C2#2029083811yescharacteristic
333C2#2029084711yescharacteristic
334C2#2029085211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133213C4
cosocleC2×C4×Dic53C23
hypercenter#390#6157
hypocenterC53#9642
LCSG ▷ C2×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. #20290632C2semidirect
2. #20290429C2semidirect
3. C2×#16811471C2semidirect
4. C2×D4⋊#88324C2semidirect
26. C4×#88324D4semidirect
32. #13133213C8semidirect
188. C53#9642semidirect

Maximal subgroups

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

subgroupindex
#202906322
#202904292
C2×#168114712
C2×D4⋊#883242
#202772792
#964253