Finite Group Databasebeta

← #28299788

Subgroups — #28299788

Order 13568.

568 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 126 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#20290570C211yesmaximal, characteristic
2#20290376C211yesmaximal, characteristic
3#20289536C214yesmaximal, complemented
4#20283068C211yesmaximal, characteristic
5C22×D53⋊C8—21no—
6C2×C4⋊#88324—21no—
7C2×#13133274—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133217C411yescharacteristic
12C2×#13133197C412yes—
13C2×C42×D53C2211yescharacteristic
14#16810528C2212yes—
15#16810522—21no—
16#16810522—21no—
17#16810480—21no—
18#16810480—21no—
19C53×#384C411yescharacteristic
20C2×D53⋊C8—24no—
21C2×D53⋊C8—42no—
22C2×D53⋊C8—42no—
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26#13133274—42no—
27C22×Dic106C2×C412yes—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31#13133217—42no—
32#13133197—44no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—22no—
36C2×#11719921—22no—
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53C2311yescharacteristic
40C42×D53—42no—
41#13133124—22no—
42#13133124—22no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C42×C53C2×C411yescharacteristic
47C23×D53C22⋊C411yescharacteristic
48C22×C4×C53—22no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—22no—
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C2×D411yescharacteristic
55C22×Dic53C2×D411yescharacteristic
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—22no—
59C2×C4×D53—22no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×C4×D53SD1612yes—
64C2×C4×D53SD1612yes—
65C2×C4×D53Q1612yes—
66C2×C4×D53Q1612yes—
67C2×Dic106—24no—
68C2×Dic106—42no—
69C2×Dic106—42no—
70C2×#88324—24no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76D53⋊C8—48no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C42×C53—22no—
81#11719921—44no—
82C4⋊Dic53—44no—
83C4⋊Dic53—42no—
84C4⋊Dic53—42no—
85C4×Dic53—22no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—21no—
93C22×D53#10012yes—
94C22×D53#10312yes—
95C22×D53#10312yes—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53#9912yes—
104C2×C4×C53#10312yes—
105C2×C4×C53#10312yes—
106C2×C4×C53#10412yes—
107C2×C4×C53#13112yes—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53C2×SD1611yescharacteristic
119C2×Dic53C2×SD1611yescharacteristic
120C2×Dic53C2×Q1611yescharacteristic
121C2×Dic53C2×Q1611yescharacteristic
122C4×D53—24no—
123C4×D53—24no—
124C4×D53—42no—
125C4×D53—42no—
126C4×D53—44no—
127Dic106—84no—
128#88324—44no—
129#88324—82no—
130#7667—531nomaximal, Sylow
131C22×C53#40811yescharacteristic
132C22×C53C2×#9911yescharacteristic
133C22×C53C2×#10011yescharacteristic
134C22×C53C2×#10311yescharacteristic
135C22×C53C2×#10311yescharacteristic
136C22×C53C2×#10411yescharacteristic
137C22×C53#42111yescharacteristic
138C2×D53—22no—
139C2×D53—22no—
140C2×D53—22no—
141C2×D53—22no—
142C2×D53—22no—
143C2×D53—22no—
144C2×D53#33314yes—
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—82no—
148C4×C53#32814yes—
149Dic53—21no—
150Dic53—21no—
151Dic53—21no—
152Dic53—21no—
153Dic53—41no—
154Dic53—82no—
155#2857—531no—
156#1797—531no—
157C2×#336—531no—
158#1272—534no—
159C2×C53#147111yescharacteristic
160C2×C53#148111yescharacteristic
161C2×C53#149311yescharacteristic
162C2×C53C2×#32811yescharacteristic
163C2×C53#152011yescharacteristic
164C2×C53#153011yescharacteristic
165C2×C53C2×#33311yescharacteristic
166D53—24no—
167C23×C8—1061no—
168C2×C4⋊Q8—531no—
169C2×C22⋊Q8—532no—
170C22×C42—531no—
171C2×C4⋊C8—1061no—
172C2×C22⋊C8—1061no—
173#391—532no—
174#389—1061no—
175#389—1061no—
176#384#615711yescharacteristic
177#380—1061no—
178#380—1061no—
179#336—532no—
180#336—532no—
181#336—1061no—
182C53#766711yescharacteristic, complemented, Sylow
183C22×Q8—532no—
184C23×C4—531no—
185C23×C4—531no—
186C23×C4—531no—
187C22×C8—1062no—
188C22×C8—1064no—
189C22×C8—1062no—
190C22×C8—2122no—
191C22×C8—2122no—
192C4⋊Q8—2122no—
193C22⋊Q8—2124no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4C2×#615712yes—
200C2×C22⋊C4—1062no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42—2122no—
205C2×C42C2×#615711yescharacteristic
206C4⋊C8—2122no—
207C22⋊C8—2122no—
208#95—1062no—
209#95—1062no—
210C24—531no—
211C2×Q8—1064no—
212C2×Q8—2122no—
213C2×Q8—2122no—
214C22×C4—22no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—532no—
218C22×C4—532no—
219C22×C4—532no—
220C22×C4—532no—
221C22×C4—531no—
222C22×C4—1062no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—1062no—
226C22×C4—1062no—
227C22×C4—2121no—
228C22×C4—2121no—
229C22×C4—2121no—
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—
239C2×C8—2128no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—2122no—
244C4⋊C4—2124no—
245C4⋊C4—2122no—
246C22⋊C4—2124no—
247C42—22no—
248C42—1062no—
249C42—2121no—
250C42—2121no—
251C23—532no—
252C23—532no—
253C23—532no—
254C23—1061no—
255C23—1061no—
256C23—1061no—
257C23—1061no—
258C23C2×C22⋊#615711yescharacteristic
259Q8—4244no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—531no—
268C2×C4—531no—
269C2×C4—531no—
270C2×C4—531no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1064no—
276C2×C4—1064no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2124no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4#1313327612yes—
287C2×C4#1313328012yes—
288C2×C4#1313328012yes—
289C2×C4#1313328112yes—
290C2×C4#1313330512yes—
291C8—4242no—
292C8—2124no—
293C22—534no—
294C22—1062no—
295C22—1062no—
296C22—1062no—
297C22—1062no—
298C22—1062no—
299C22—1062no—
300C22#1681159811yescharacteristic
301C22C2×#1313327611yescharacteristic
302C22D53⋊#10011yescharacteristic
303C22C2×#1313328011yescharacteristic
304C22C2×#1313328011yescharacteristic
305C22C2×#1313328111yescharacteristic
306C22#1681162111yescharacteristic
307C4—22no—
308C4—22no—
309C4—82no—
310C4—1061no—
311C4—1061no—
312C4—1061no—
313C4—1061no—
314C4—4242no—
315C4—2121no—
316C4#1681147614yes—
317C2—1064no—
318C2#2028999211yescharacteristic
319C2#2029002511yescharacteristic
320C2#2029006511yescharacteristic
321C2C2×#1681147611yescharacteristic
322C2#2029015411yescharacteristic
323C2#2029018511yescharacteristic
324C2D53⋊#33311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7667
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #20289536C2semidirect
182. C53#7667semidirect

Maximal subgroups

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

subgroupindex
#202905702
#202903762
#202895362
#202830682
#766753