Finite Group Databasebeta

← #28300004

Subgroups — #28300004

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#20289536C214yesmaximal, complemented
4#20283068C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133274—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
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—42no—
23#13133274—42no—
24#13133274—42no—
25C22×Dic106C2×C412yes—
26C22×C4×D53C2×C411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29#13133217—42no—
30#13133197—44no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—22no—
34C2×#11719921—22no—
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53C2311yescharacteristic
38C42×D53—42no—
39#13133124—22no—
40#13133124—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C42×C53C2×C411yescharacteristic
45C23×D53C22⋊C411yescharacteristic
46C22×C4×C53—22no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C2×D411yescharacteristic
53C22×Dic53C2×D411yescharacteristic
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—41no—
59C2×C4×D53—41no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×C4×D53SD1612yes—
63C2×C4×D53Q1612yes—
64C2×Dic106—24no—
65C2×Dic106—42no—
66C2×Dic106—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—42no—
76C42×C53—22no—
77#11719921—44no—
78C4⋊Dic53—42no—
79C4⋊Dic53—42no—
80C4⋊Dic53—44no—
81C4×Dic53—22no—
82C4×Dic53—41no—
83C4×Dic53—41no—
84C23×C53C2×C22⋊C411yescharacteristic
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53#10312yes—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53#9912yes—
102C2×C4×C53#10312yes—
103C2×C4×C53#13112yes—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53#10012yes—
114C2×Dic53C2×SD1611yescharacteristic
115C2×Dic53C2×Q1611yescharacteristic
116C2×Dic53#13712yes—
117C4×D53—42no—
118C4×D53—42no—
119C4×D53—44no—
120C4×D53—44no—
121Dic106—84no—
122#88324—82no—
123#88324—82no—
124#7727—531nomaximal, Sylow
125C22×C53#40811yescharacteristic
126C22×C53C2×#9911yescharacteristic
127C22×C53C2×#10011yescharacteristic
128C22×C53C2×#10311yescharacteristic
129C22×C53#41911yescharacteristic
130C22×C53#42111yescharacteristic
131C22×C53#42111yescharacteristic
132C2×D53—22no—
133C2×D53—22no—
134C2×D53—22no—
135C2×D53—41no—
136C2×D53—41no—
137C2×D53—41no—
138C2×D53—41no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—22no—
142C4×C53—82no—
143Dic53—22no—
144Dic53—22no—
145Dic53—21no—
146Dic53—21no—
147Dic53—82no—
148#2857—531no—
149#1817—531no—
150#1797—531no—
151#1272—534no—
152C2×C53#147111yescharacteristic
153C2×C53#148111yescharacteristic
154C2×C53#149711yescharacteristic
155C2×C53#151111yescharacteristic
156C2×C53#152011yescharacteristic
157C2×C53#153611yescharacteristic
158C2×C53#154811yescharacteristic
159D53—42no—
160C2×C4⋊Q8—531no—
161C2×C22⋊Q8—532no—
162C22×C42—531no—
163C2×C4⋊C8—1061no—
164C2×C22⋊C8—1061no—
165C2×C22⋊C8—1061no—
166#391—532no—
167#389—1061no—
168#389—1061no—
169#384#615711yescharacteristic
170#380—1061no—
171#380—1061no—
172#336—532no—
173#336—532no—
174#336—1061no—
175C53#772711yescharacteristic, complemented, Sylow
176C22×Q8—532no—
177C23×C4—531no—
178C23×C4—531no—
179C23×C4—531no—
180C22×C8—1062no—
181C22×C8—1062no—
182C4⋊Q8—2122no—
183C22⋊Q8—2124no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—1062no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4C2×#615712yes—
190C2×C22⋊C4—1062no—
191C2×C42—531no—
192C2×C42—531no—
193C2×C42—531no—
194C2×C42—2122no—
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—2122no—
197C22⋊C8—2122no—
198C22⋊C8—2122no—
199#95—1062no—
200#95—1062no—
201C24—531no—
202C2×Q8—1064no—
203C2×Q8—2122no—
204C2×Q8—2122no—
205C22×C4—22no—
206C22×C4—531no—
207C22×C4—531no—
208C22×C4—531no—
209C22×C4—532no—
210C22×C4—532no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4—2121no—
217C22×C4—2121no—
218C22×C4—2121no—
219C22×C4—2121no—
220C22×C4C22⋊#615711yescharacteristic
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22×#615711yescharacteristic
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C2×C8—2122no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—2122no—
233C4⋊C4—2124no—
234C4⋊C4—2122no—
235C22⋊C4—2124no—
236C42—22no—
237C42—1062no—
238C42—2121no—
239C42—2121no—
240C23—532no—
241C23—1061no—
242C23—1061no—
243C23—1061no—
244C23—1061no—
245C23—1061no—
246C23—1061no—
247C23C2×C22⋊#615711yescharacteristic
248Q8—4244no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—532no—
259C2×C4—531no—
260C2×C4—531no—
261C2×C4—532no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4—2124no—
271C2×C4—2124no—
272C2×C4—2122no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4#1313327612yes—
276C2×C4#1313328012yes—
277C2×C4#1313330512yes—
278C8—4242no—
279C8—4242no—
280C22—1062no—
281C22—1062no—
282C22—1062no—
283C22—2121no—
284C22—2121no—
285C22—2121no—
286C22—2121no—
287C22#1681159811yescharacteristic
288C22C2×#1313327611yescharacteristic
289C22D53⋊#10011yescharacteristic
290C22C2×#1313328011yescharacteristic
291C22#1681161611yescharacteristic
292C22#1681162111yescharacteristic
293C22#1681162111yescharacteristic
294C4—22no—
295C4—22no—
296C4—22no—
297C4—82no—
298C4—1062no—
299C4—1061no—
300C4—1061no—
301C4—1062no—
302C4—4242no—
303C2—2122no—
304C2#2028999211yescharacteristic
305C2#2029002511yescharacteristic
306C2#2029007911yescharacteristic
307C2#2029012311yescharacteristic
308C2#2029015411yescharacteristic
309C2#2029020411yescharacteristic
310C2#2029022911yescharacteristic
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#7727
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
175. C53#7727semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202905722
#202895362
#202830682
#772753