Finite Group Databasebeta

← #28309107

Subgroups — #28309107

Order 13568.

482 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#13133275C211yesmaximal, characteristic
2#20289974C211yesmaximal, characteristic
3#20289695C214yesmaximal, complemented
4#20279017C211yesmaximal, characteristic
5#16811644—21no—
6C2×#13133275—21no—
7C4×C22⋊#6157—21no—
8#16811501C2212yes—
9#16811467C2212yes—
10#16811466—21no—
11C2×#13133200—21no—
12C2×C4×C22⋊D53C2211yescharacteristic
13#16810554C412yescomplemented
14#16810549C2212yes—
15#16810519—21no—
16#16810486—21no—
17#16810483C411yescharacteristic, complemented
18#16810480—21no—
19C22×C4⋊C4×C53C411yescharacteristic, complemented
20C2×C22⋊#6157—22no—
21C2×C4×#6157—41no—
22C4⋊#88324—41no—
23#13133275—24no—
24C22⋊#88324—22no—
25#13133272—41no—
26(C2×#6157)⋊C4—41no—
27C22×C22⋊D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29#13133200—44no—
30C4×C22⋊D53—41no—
31C4×C22⋊D53—41no—
32C2×C4⋊Dic53—21no—
33C2×#11719919—22no—
34C2×#11719919C2×C411yescharacteristic
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53D411yescharacteristic
37C2×#11719921—22no—
38C2×#11719921C2311yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40#13133124—22no—
41#13133124—22no—
42#13133124C2×C412yes—
43C23×C4×C53—21no—
44C23×C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—41no—
47C2×C4⋊C4×C53—41no—
48C2×C4⋊C4×C53C2×C412yes—
49C24×C53C22⋊C411yescharacteristic
50C23×D53C2×D411yescharacteristic
51C22×#6157—42no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C2×C22⋊D53—22no—
62C2×C22⋊D53—22no—
63C2×C22⋊D53—41no—
64C22×Dic53—22no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—42no—
71C22⋊#6157—24no—
72C22⋊#6157—42no—
73C2×#88324—42no—
74C4×#6157—82no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C22⋊Dic53—41no—
79C22⋊Dic53—42no—
80#11719921—41no—
81#11719921—44no—
82C4⋊Dic53—42no—
83#11719919—41no—
84#11719919—44no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53#10412yes—
92C23×C53C2×C22⋊C411yescharacteristic
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—41no—
96C22×D53—41no—
97C22×D53AGL(1,ℤ/8)12yes—
98C2×#6157—44no—
99C2×#6157—82no—
100C2×#6157—82no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—84no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C22⋊D53—44no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C4×D53—42no—
124C4×D53—81no—
125#88324—82no—
126#11773—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—22no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53C2×#9911yescharacteristic
135C22×C53#41811yescharacteristic
136C22×C53C2×#10411yescharacteristic
137C2×D53—22no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—42no—
141C2×D53—81no—
142#6157—84no—
143C4×C53—21no—
144C4×C53—41no—
145C4×C53—82no—
146C4×C53#35212yes—
147Dic53—41no—
148Dic53—41no—
149Dic53—41no—
150Dic53—82no—
151#2486—531no—
152C4×#99—531no—
153#1466—531no—
154#1344—534no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—42no—
158C2×C53#150711yescharacteristic
159C2×C53C2×#35211yescharacteristic
160C2×C53#212911yescharacteristic
161D53—42no—
162C2×#123—1061no—
163C2×C4×D4—531no—
164C22×C4⋊C4#615711yescharacteristic, complemented
165#432—1061no—
166C2×#99—1061no—
167#398—532no—
168#397—532no—
169#389—1061no—
170#381—1061no—
171#380—531no—
172#380—1061no—
173C4×C22⋊C4—1061no—
174#342—532no—
175#323—532no—
176#323—1061no—
177C53#1177311yescharacteristic, complemented, Sylow
178C22×D4—531no—
179C23×C4—21no—
180C23×C4—531no—
181C23×C4C2×#615711yescharacteristic
182#123—2124no—
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—1062no—
196C2×C22⋊C4—1061no—
197C2×C42—531no—
198C2×C42—2121no—
199C4⋊C8—2121no—
200#99—1064no—
201C22⋊C8—1062no—
202#97—2121no—
203#95—532no—
204#95—1062no—
205#95—1062no—
206#95—2121no—
207C24—531no—
208C24C22⋊#615711yescharacteristic
209C2×D4—1062no—
210C2×D4—1062no—
211C2×D4—2121no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—1061no—
223C22×C4—1061no—
224C22×C4—1062no—
225C22×C4—2122no—
226C22×C4—2122no—
227C22×C4C22⋊#615711yescharacteristic
228C22×C4C22×#615711yescharacteristic
229C2×C8—2122no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—2121no—
234C4⋊C4—2122no—
235C4⋊C4—2124no—
236C22⋊C4—1064no—
237C22⋊C4—2121no—
238C22⋊C4—2124no—
239C22⋊C4—2121no—
240C22⋊C4—2122no—
241C22⋊C4—2122no—
242C42—1061no—
243C42—1061no—
244C42—4242no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—532no—
250C23—1061no—
251C23—1061no—
252C23—2121no—
253C23—2121no—
254C23#1313328112yes—
255C23C2×C22⋊#615711yescharacteristic
256D4—2124no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—84no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—4241no—
272C2×C4—4242no—
273C2×C4—4242no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2124no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4#1313327511yescharacteristic
282C2×C4#1313327511yescharacteristic
283C8—4242no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—1062no—
292C22—4241no—
293C22—2122no—
294C22—2121no—
295C22—2121no—
296C22C2×#1313327511yescharacteristic
297C22#1681161511yescharacteristic
298C22C2×#1313328111yescharacteristic
299C4—21no—
300C4—41no—
301C4—82no—
302C4—4242no—
303C4—4244no—
304C4—2121no—
305C4—2121no—
306C4—2121no—
307C4#1681152512yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—2122no—
312C2#2029011111yescharacteristic
313C2C2×#1681152511yescharacteristic
314C2#2029099711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC22×C22⋊D53D4
cosocleC2×#11719921C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11773
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. #20289695C2semidirect
13. #16810554C4semidirect
17. #16810483C4semidirect
19. C22×C4⋊C4×C53C4semidirect
164. C22×C4⋊C4#6157semidirect
177. C53#11773semidirect

Maximal subgroups

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

subgroupindex
C4×#131332752
#202899742
#202896952
#202790172
#1177353