Finite Group Databasebeta

← #28300005

Subgroups — #28300005

Order 13568.

494 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290607C211yesmaximal, characteristic
2#20290572C211yesmaximal, characteristic
3#20289533C214yesmaximal, complemented
4#20283068C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133274—21no—
7C2×#13133274—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133217C411yescharacteristic
12C2×#13133197—21no—
13C2×C42×D53C2211yescharacteristic
14#16810528—21no—
15#16810522—21no—
16#16810522C2212yes—
17#16810480—21no—
18#16810480C412yes—
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—42no—
23#13133274—42no—
24#13133274—42no—
25C22×Dic106—21no—
26C22×C4×D53C2×C411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29#13133217—42no—
30#13133197—42no—
31#13133197—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53C2×C412yes—
35C2×#11719921—22no—
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53C2311yescharacteristic
39C42×D53—42no—
40#13133124—22no—
41#13133124—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53C2×C412yes—
45C2×C42×C53C2×C411yescharacteristic
46C23×D53C22⋊C411yescharacteristic
47C22×C4×C53—22no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—22no—
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C2×D411yescharacteristic
54C22×Dic53C2×D411yescharacteristic
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—22no—
59C2×C4×D53—41no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×C4×D53D812yes—
64C2×C4×D53SD1612yes—
65C2×Dic106—42no—
66C2×Dic106—42no—
67C2×Dic106—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C42×C53—22no—
78#11719921—44no—
79C4⋊Dic53—42no—
80C4⋊Dic53—42no—
81C4⋊Dic53—42no—
82C4⋊Dic53—42no—
83C4×Dic53—22no—
84C4×Dic53—41no—
85C4×Dic53—41no—
86C23×C53C2×C22⋊C411yescharacteristic
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—21no—
93C22×D53#10212yes—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
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#10212yes—
105C2×C4×C53#13112yes—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53#10112yes—
116C2×Dic53C2×D811yescharacteristic
117C2×Dic53C2×SD1611yescharacteristic
118C2×Dic53#13712yes—
119C4×D53—42no—
120C4×D53—42no—
121C4×D53—44no—
122C4×D53—44no—
123Dic106—84no—
124#88324—82no—
125#88324—82no—
126#7728—531nomaximal, Sylow
127C22×C53#40811yescharacteristic
128C22×C53C2×#9911yescharacteristic
129C22×C53C2×#10111yescharacteristic
130C22×C53C2×#10211yescharacteristic
131C22×C53#41911yescharacteristic
132C22×C53#42111yescharacteristic
133C22×C53#42111yescharacteristic
134C2×D53—22no—
135C2×D53—22no—
136C2×D53—22no—
137C2×D53—41no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—41no—
141C4×C53—22no—
142C4×C53—22no—
143C4×C53—22no—
144C4×C53—82no—
145Dic53—22no—
146Dic53—22no—
147Dic53—21no—
148Dic53—21no—
149Dic53—82no—
150#2857—531no—
151#1817—531no—
152#1797—531no—
153#1271—534no—
154C2×C53#147211yescharacteristic
155C2×C53#148011yescharacteristic
156C2×C53#149711yescharacteristic
157C2×C53#151111yescharacteristic
158C2×C53#151911yescharacteristic
159C2×C53#153711yescharacteristic
160C2×C53#154611yescharacteristic
161D53—42no—
162C2×C4⋊Q8—531no—
163C2×C22⋊Q8—1061no—
164C22×C42—531no—
165C2×C4⋊C8—1061no—
166C2×C22⋊C8—1061no—
167C2×C22⋊C8—1061no—
168#391—1061no—
169#389—532no—
170#389—1061no—
171#384#615711yescharacteristic
172#380—532no—
173#380—1061no—
174#336—532no—
175#336—532no—
176#336—1061no—
177C53#772811yescharacteristic, complemented, Sylow
178C22×Q8—1061no—
179C23×C4—531no—
180C23×C4—531no—
181C23×C4—531no—
182C22×C8—1062no—
183C22×C8—1062no—
184C4⋊Q8—2122no—
185C22⋊Q8—2122no—
186C22⋊Q8—2122no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—532no—
190C2×C4⋊C4—1062no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4C2×#615712yes—
193C2×C22⋊C4—1062no—
194C2×C42—531no—
195C2×C42—531no—
196C2×C42—531no—
197C2×C42—2122no—
198C2×C42C2×#615711yescharacteristic
199C4⋊C8—2122no—
200C22⋊C8—2122no—
201C22⋊C8—2122no—
202#95—1062no—
203#95—1062no—
204C24—531no—
205C2×Q8—2122no—
206C2×Q8—2122no—
207C2×Q8—2122no—
208C22×C4—22no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—532no—
213C22×C4—532no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4—1062no—
218C22×C4—1062no—
219C22×C4—2121no—
220C22×C4—2121no—
221C22×C4—2121no—
222C22×C4—2121no—
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22⋊#615711yescharacteristic
225C22×C4C22×#615711yescharacteristic
226C2×C8—2122no—
227C2×C8—2122no—
228C2×C8—2122no—
229C2×C8—2122no—
230C2×C8—2122no—
231C2×C8—2122no—
232C4⋊C4—24no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—2122no—
236C4⋊C4—2122no—
237C4⋊C4—2122no—
238C4⋊C4—2122no—
239C22⋊C4—2124no—
240C42—22no—
241C42—1062no—
242C42—2121no—
243C42—2121no—
244C23—532no—
245C23—1061no—
246C23—1061no—
247C23—1061no—
248C23—1061no—
249C23—1061no—
250C23—1061no—
251C23C2×C22⋊#615711yescharacteristic
252Q8—4244no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—532no—
263C2×C4—531no—
264C2×C4—531no—
265C2×C4—532no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—2122no—
273C2×C4—2122no—
274C2×C4—2124no—
275C2×C4—2124no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4#1313327612yes—
280C2×C4#1313327912yes—
281C2×C4#1313330412yes—
282C8—4242no—
283C8—4242no—
284C22—1062no—
285C22—1062no—
286C22—1062no—
287C22—2121no—
288C22—2121no—
289C22—2121no—
290C22—2121no—
291C22#1681159711yescharacteristic
292C22C2×#1313327611yescharacteristic
293C22D53⋊#10111yescharacteristic
294C22C2×#1313327911yescharacteristic
295C22#1681161611yescharacteristic
296C22#1681162111yescharacteristic
297C22#1681162211yescharacteristic
298C4—22no—
299C4—22no—
300C4—22no—
301C4—82no—
302C4—1062no—
303C4—1061no—
304C4—1061no—
305C4—1062no—
306C4—4242no—
307C2—2122no—
308C2#2028999311yescharacteristic
309C2#2029002111yescharacteristic
310C2#2029007811yescharacteristic
311C2#2029012311yescharacteristic
312C2#2029015211yescharacteristic
313C2#2029020711yescharacteristic
314C2#2029022511yescharacteristic
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#7728
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. #20289533C2semidirect
177. C53#7728semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202905722
#202895332
#202830682
#772853