Finite Group Databasebeta

← #28308486

Subgroups — #28308486

Order 13568.

450 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 102 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#20290922C211yesmaximal, characteristic, complemented
2#20290265C211yesmaximal, characteristic, complemented
3#20289670C212yesmaximal, complemented
4#20289669C212yesmaximal, complemented
5C4×D8×D53C211yesmaximal, characteristic
6C2×#13133292—21no—
7#16811642—21no—
8#16811511—21no—
9#16811509C2212yescomplemented
10#16811486C2212yescomplemented
11#16811485—21no—
12C2×D8×D53C411yescharacteristic
13C4×D4×D53—21no—
14C4×D424C411yescharacteristic
15D8×Dic53C2211yescharacteristic
16#16810705C2211yescharacteristic
17C4×#11719922—21no—
18#16810703—21no—
19#107×D53C411yescharacteristic
20#102×D53—21no—
21C4×C8×D53C2211yescharacteristic
22C4×D8×C53C411yescharacteristic
23#13133292—24no—
24C2×#11719910—22no—
25C4⋊#88324—22no—
26#13133274—41no—
27C4×#88324—41no—
28C2×D4×D53—21no—
29C22×C4×D53—41no—
30C2×D424C2×C411yescharacteristic
31C2×#11719922—21no—
32D8×D53—22no—
33D8×D53C814yescomplemented
34C2×C8×D53C2×C411yescharacteristic
35C2×C8×D53D411yescharacteristic
36C4×D212—21no—
37D4×Dic53—21no—
38#13133179—21no—
39C4×C22⋊D53—41no—
40#13133177—41no—
41C4⋊C4×D53—21no—
42C22⋊C4×D53—41no—
43C42×D53D411yescharacteristic
44#13133151C2×C411yescharacteristic
45#13133150—21no—
46#13133139—21no—
47#13133138—21no—
48#13133137—21no—
49C4×#88322D411yescharacteristic
50C8×Dic53D411yescharacteristic
51C8×Dic53C2311yescharacteristic
52C2×D8×C53C2×C411yescharacteristic
53C4×D4×C53—21no—
54#107×C53C2×C411yescharacteristic
55#102×C53—21no—
56C4×C8×C53C2×C411yescharacteristic
57C23×D53—41no—
58C2×D4×C53—21no—
59C22×C4×C53—41no—
60C2×C22⋊D53—41no—
61C22×Dic53—41no—
62D4×D53—24no—
63D4×D53—42no—
64C2×D212—21no—
65C2×C4×D53—41no—
66C2×C4×D53—41no—
67C2×C4×D53—41no—
68C2×C4×D53—84no—
69C2×C4×D53C22⋊C411yescharacteristic
70C2×C4×D53D811yescharacteristic
71C2×C4×D53SD1611yescharacteristic
72C2×#88324—42no—
73D8×C53—21no—
74D8×C53C2×C812yes—
75C2×C8×C53C22⋊C411yescharacteristic
76C2×C8×C53C22×C411yescharacteristic
77C4⋊C4×C53—21no—
78C22⋊C4×C53—41no—
79C42×C53C22⋊C411yescharacteristic
80C22⋊Dic53—41no—
81#11719922—24no—
82#11719921—41no—
83#11719921—41no—
84C4⋊Dic53—21no—
85#11719919—41no—
86C4×Dic53D811yescharacteristic
87C4×Dic53SD1611yescharacteristic
88C4×Dic53C2×D411yescharacteristic
89C2×#88322C22⋊C411yescharacteristic
90C2×#88322C2×D411yescharacteristic
91D424—21no—
92D424C2×C812yes—
93C8×D53—22no—
94C8×D53C2×C812yes—
95C8×D53M1612yes—
96#11719910—24no—
97C23×C53—41no—
98C22×D53—41no—
99C22×D53—44no—
100C22×D53—82no—
101C22×D53—82no—
102C22×D53#10211yescharacteristic
103D4×C53—22no—
104D4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—82no—
107C2×C4×C53#10211yescharacteristic
108C2×C4×C53#10211yescharacteristic
109C2×C4×C53C2×C22⋊C411yescharacteristic
110C22⋊D53—44no—
111C2×Dic53—41no—
112C2×Dic53—82no—
113C2×Dic53#10211yescharacteristic
114C2×Dic53C2×D811yescharacteristic
115C2×Dic53C2×SD1611yescharacteristic
116D212—22no—
117D212—41no—
118C4×D53—22no—
119C4×D53—41no—
120C4×D53—82no—
121C4×D53—82no—
122C4×D53—82no—
123C4×D53C22⋊C812yes—
124C4×D53#10412yes—
125#88324—82no—
126C8×C53—21no—
127C8×C53C22×C811yescharacteristic
128C8×C53C2×M1611yescharacteristic
129#88322—21no—
130#88322C22⋊C812yes—
131#11593—531nomaximal, Sylow
132C22×C53—42no—
133C22×C53—81no—
134C22×C53—81no—
135C22×C53C2×#10211yescharacteristic
136C2×D53—21no—
137C2×D53—21no—
138C2×D53—42no—
139C2×D53—88no—
140C2×D53—81no—
141C2×D53—81no—
142C2×D53D4⋊C812yes—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—81no—
146C4×C53C2×C22⋊C811yescharacteristic
147C4×C53C2×#10411yescharacteristic
148Dic53—21no—
149Dic53—81no—
150Dic53D4⋊C812yes—
151Dic53#50912yes—
152C2×C4×D8—531no—
153C4×M32—531no—
154#1564—531no—
155#1331—532no—
156#1331—532no—
157C2×C53—82no—
158C2×C53C2×D4⋊C811yescharacteristic
159C2×C53#214011yescharacteristic
160C2×C53#214311yescharacteristic
161D53—22no—
162D53—82no—
163C22×D8—531no—
164C2×C4×D4—1061no—
165C2×M32—1061no—
166C4×D8—531no—
167C4×D8—531no—
168C4×D8—531no—
169C4×D8—1061no—
170C4×D8—1061no—
171C4×D8#615711yescharacteristic
172#431—1061no—
173C2×#107—531no—
174C2×#102—1061no—
175C2×C4×C8—531no—
176#346—1061no—
177C4×C16—532no—
178#335—532no—
179#334—1061no—
180C53#1159311yescharacteristic, complemented, Sylow
181C22×D4—1061no—
182C23×C4—2121no—
183C2×D8—534no—
184C2×D8—531no—
185C2×D8—1062no—
186C2×D8—1061no—
187C2×D8C2×#615711yescharacteristic
188C22×C8—531no—
189C22×C8—531no—
190C4×D4—21no—
191C4×D4—1061no—
192C4×D4—1061no—
193C4×D4—1061no—
194C4×D4—2121no—
195C4×D4—2121no—
196C2×C4⋊C4—1061no—
197C2×C22⋊C4—2121no—
198C2×C42—531no—
199M32—1064no—
200C2×C16—1062no—
201#107—531no—
202#107—1061no—
203#107C2×#615711yescharacteristic
204C4⋊C8—1062no—
205#102—21no—
206#102—1061no—
207#102—1061no—
208#102—1061no—
209C22⋊C8—2121no—
210C4×C8—531no—
211C4×C8—531no—
212C4×C8—531no—
213C4×C8—2121no—
214C4×C8C2×#615711yescharacteristic
215C24—2121no—
216C2×D4—21no—
217C2×D4—1064no—
218C2×D4—1061no—
219C2×D4—2122no—
220C2×D4—2121no—
221C22×C4—41no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—4244no—
226C22×C4—2121no—
227C22×C4—2121no—
228C22×C4—2121no—
229C22×C4—2121no—
230D8—21no—
231D8—532no—
232D8—1064no—
233D8—1061no—
234D8D53⋊C812yes—
235C2×C8—531no—
236C2×C8—532no—
237C2×C8—532no—
238C2×C8—531no—
239C2×C8—1062no—
240C2×C8—2122no—
241C2×C8C22⋊#615711yescharacteristic
242C2×C8C22×#615711yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—1061no—
245C4⋊C4—2121no—
246C22⋊C4—41no—
247C22⋊C4—2121no—
248C22⋊C4—2121no—
249C22⋊C4—2121no—
250C42—531no—
251C42—531no—
252C42—531no—
253C42C22⋊#615711yescharacteristic
254C16—1064no—
255C23—41no—
256C23—531no—
257C23—4242no—
258C23—4242no—
259C23—2124no—
260C23—2121no—
261D4—22no—
262D4—41no—
263D4—1062no—
264D4—2124no—
265D4—2121no—
266C2×C4—41no—
267C2×C4—82no—
268C2×C4—531no—
269C2×C4—531no—
270C2×C4—532no—
271C2×C4—532no—
272C2×C4—531no—
273C2×C4—1062no—
274C2×C4—4242no—
275C2×C4—4242no—
276C2×C4—4242no—
277C2×C4—4242no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4#1313327911yescharacteristic
281C2×C4#1313327911yescharacteristic
282C2×C4C2×C22⋊#615711yescharacteristic
283C8—21no—
284C8—532no—
285C8—1061no—
286C8—4242no—
287C8C2×D53⋊C811yescharacteristic
288C8D53⋊M1611yescharacteristic
289C22—42no—
290C22—81no—
291C22—81no—
292C22—532no—
293C22—1061no—
294C22—1061no—
295C22—4248no—
296C22—4241no—
297C22—4241no—
298C22—2122no—
299C22C2×#1313327911yescharacteristic
300C4—21no—
301C4—21no—
302C4—81no—
303C4—532no—
304C4—532no—
305C4—1061no—
306C4—4241no—
307C4D53⋊(C22⋊C8)11yescharacteristic
308C4D53⋊#10411yescharacteristic
309C2—82no—
310C2—1062no—
311C2—4242no—
312C2D53⋊(D4⋊C8)11yescharacteristic
313C2#2029102411yescharacteristic
314C2#2029102911yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC8×C53C22×C8
FrattiniC2×C8C22×#6157
FittingC4×D8×C53C4
socleC22×C53C2×#102
prime radicalC2×D8×D53C4
cosocleC8×Dic53C23
hypercenterC4×D8#6157
hypocenterC53#11593
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20290922C2semidirect
2. #20290265C2semidirect
3. #20289670C2semidirect
4. #20289669C2semidirect
9. #16811509C22semidirect
10. #16811486C22semidirect
33. D8×D53C8semidirect
180. C53#11593semidirect

Maximal subgroups

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

subgroupindex
#202909222
#202902652
#202896702
#202896692
C4×D8×D532
#1159353