Finite Group Databasebeta

← #28309563

Subgroups — #28309563

Order 13568.

440 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290579C211yesmaximal, characteristic
2#20289950C211yesmaximal, characteristic
3#20289714C214yesmaximal, complemented
4#20279026C211yesmaximal, characteristic
5#16811626—21no—
6C2×#13133277—21no—
7D53⋊(C22⋊C8)—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811466—21no—
11C2×C4×C22⋊D53C2211yescharacteristic
12#16810561C2212yes—
13#16810558C412yes—
14#16810519—21no—
15#16810502—21no—
16#16810496—21no—
17#16810483C411yescharacteristic
18#16810480—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C2×#11719940—22no—
21C2×D53⋊C8—41no—
22#13133284—41no—
23C4⋊#88324—41no—
24#13133277—42no—
25#13133274—41no—
26C22⋊#88324—22no—
27C22×C22⋊D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C4×C22⋊D53—41no—
30C4×C22⋊D53—41no—
31C2×C4⋊Dic53—21no—
32C2×#11719919—22no—
33C2×#11719919C2311yescharacteristic
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53D411yescharacteristic
36C2×#11719921—22no—
37C2×#11719921C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39#13133124—22no—
40#13133124—22no—
41#13133124C2×C412yes—
42C23×C4×C53—21no—
43C23×C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—41no—
46C2×C4⋊C4×C53—41no—
47C2×C4⋊C4×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C23×D53C22⋊C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53—82no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C2×C22⋊D53—41no—
59C2×C22⋊D53—41no—
60C2×C22⋊D53SD1612yes—
61C2×C22⋊D53Q1612yes—
62C22×Dic53—22no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×C4×D53—21no—
67C2×C4×D53—21no—
68C2×C4×D53—42no—
69#11719940—42no—
70#11719940—42no—
71C2×#88324—42no—
72C2×#88324—42no—
73D53⋊C8—82no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C22⋊Dic53—41no—
78C22⋊Dic53—42no—
79#11719921—41no—
80#11719921—82no—
81C4⋊Dic53—42no—
82#11719919—41no—
83#11719919—82no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—41no—
89C23×C53—41no—
90C23×C53#10312yes—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—41no—
95C22×D53—41no—
96C22×D53#10312yes—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—84no—
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#9911yescharacteristic
108C22⋊D53—42no—
109C22⋊D53—42no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53C2×SD1611yescharacteristic
120C2×Dic53C2×Q1611yescharacteristic
121C4×D53—42no—
122C4×D53—81no—
123#88324—82no—
124#88324—82no—
125#11903—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—22no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—41no—
132C22×C53—41no—
133C22×C53C2×#9911yescharacteristic
134C22×C53C2×#10311yescharacteristic
135C22×C53#42111yescharacteristic
136C2×D53—22no—
137C2×D53—41no—
138C2×D53—41no—
139C2×D53—42no—
140C2×D53—81no—
141C4×C53—21no—
142C4×C53—41no—
143C4×C53—82no—
144C4×C53#35212yes—
145Dic53—21no—
146Dic53—21no—
147Dic53—41no—
148Dic53—41no—
149Dic53—82no—
150#2487—531no—
151#1800—531no—
152#1459—531no—
153#1353—534no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53—42no—
157C2×C53#152011yescharacteristic
158C2×C53C2×#35211yescharacteristic
159C2×C53#213411yescharacteristic
160D53—42no—
161C2×C4×D4—531no—
162C22×C4⋊C4#615711yescharacteristic
163#423—1061no—
164C2×#100—1061no—
165C2×C22⋊C8—1061no—
166#400—532no—
167#399—532no—
168#389—1061no—
169#386—1061no—
170#385—1061no—
171#380—531no—
172#380—1061no—
173#343—532no—
174#324—532no—
175#323—1061no—
176C53#1190311yescharacteristic, complemented, Sylow
177C22×D4—531no—
178C23×C4—21no—
179C23×C4—531no—
180C23×C4C2×#615711yescharacteristic
181C2×M16—1062no—
182C22×C8—2121no—
183C4×D4—2121no—
184C4×D4—2121no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—1062no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4C2×#615712yes—
192C2×C22⋊C4—531no—
193C2×C22⋊C4—531no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1061no—
196C2×C42—531no—
197C4⋊C8—2121no—
198C4⋊C8—2121no—
199#100—2122no—
200C22⋊C8—1062no—
201C22⋊C8—2121no—
202#95—532no—
203#95—1062no—
204#95—1062no—
205C24—531no—
206C24C22⋊#615711yescharacteristic
207C2×D4—532no—
208C2×D4—532no—
209C2×D4—2121no—
210C2×D4—2121no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—82no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4—1062no—
223C22×C4—2122no—
224C22×C4C22⋊#615711yescharacteristic
225C22×C4C22×#615711yescharacteristic
226M16—2122no—
227M16—2122no—
228C2×C8—4242no—
229C2×C8—2122no—
230C2×C8—2122no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—4242no—
235C4⋊C4—2121no—
236C4⋊C4—2122no—
237C22⋊C4—4242no—
238C22⋊C4—2121no—
239C22⋊C4—2121no—
240C22⋊C4—2122no—
241C42—1061no—
242C42—1061no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—532no—
248C23—1061no—
249C23—1061no—
250C23—2121no—
251C23—2121no—
252C23#1313328012yes—
253C23C2×C22⋊#615711yescharacteristic
254D4—2122no—
255D4—2122no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—84no—
265C2×C4—531no—
266C2×C4—531no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—4241no—
272C2×C4—2122no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4#1313327511yescharacteristic
279C2×C4#1313327511yescharacteristic
280C8—4242no—
281C8—4242no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—1062no—
290C22—4241no—
291C22—2122no—
292C22—2121no—
293C22—2121no—
294C22C2×#1313327511yescharacteristic
295C22C2×#1313328011yescharacteristic
296C22#1681162211yescharacteristic
297C4—21no—
298C4—41no—
299C4—82no—
300C4—1061no—
301C4—1061no—
302C4—4242no—
303C4—2121no—
304C4—2121no—
305C4#1681152512yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—2122no—
310C2#2029015511yescharacteristic
311C2C2×#1681152511yescharacteristic
312C2#2029101411yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC22×C22⋊D53D4
cosocleC2×#11719919C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11903
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289714C2semidirect
176. C53#11903semidirect

Maximal subgroups

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

subgroupindex
#202905792
#202899502
#202897142
#202790262
#1190353