Finite Group Databasebeta

← #28303344

Subgroups — #28303344

Order 13568.

588 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 110 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#20289891C211yesmaximal, characteristic
3#20289588C214yesmaximal
4#20277120C211yesmaximal, characteristic
5C2×#13133274C2212yes—
6C2×(C2×#6157)⋊C4—21no—
7C2×(C2×#6157)⋊C4—21no—
8C2×(C2×#6157)⋊C4C2212yes—
9#16811489—21no—
10#16811489—21no—
11C2×#13133175—21no—
12C2×C42×D53C2211yescharacteristic
13#16810493C411yescharacteristic, complemented
14#16810488—21no—
15#16810480—21no—
16#16810480C2212yes—
17C4⋊C4×Dic53—21no—
18C4⋊C4×Dic53C412yescomplemented
19C53×#384C411yescharacteristic, complemented
20C23×#6157—22no—
21C22×#88324—22no—
22#13133274—24no—
23(C2×#6157)⋊C4—24no—
24(C2×#6157)⋊C4—24no—
25(C2×#6157)⋊C4—42no—
26C22×C4×D53D411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×C4×D53C2311yescharacteristic
29#13133175—44no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C412yes—
33C2×#11719921—22no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C42×D53—24no—
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×D53C2×D411yescharacteristic
46C22×#6157—24no—
47C22×#6157—42no—
48C22×#6157—42no—
49C22×#6157—42no—
50C22×#6157—42no—
51C22×#6157—42no—
52C22×#6157—42no—
53C22×C4×C53—22no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C2×C4×D53—22no—
62C2×C4×D53—22no—
63C2×C4×D53—22no—
64C2×C4×D53—22no—
65C2×C4×D53—41no—
66C2×C4×D53—41no—
67C2×C4×D53—41no—
68C2×C4×D53SD1612yes—
69C2×C4×D53Q1612yes—
70C2×C4×D53#4114yes—
71C2×#88324—42no—
72C2×#88324—42no—
73C2×#88324—42no—
74C4⋊C4×C53—42no—
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—22no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—44no—
86C23×C53C2×C22⋊C411yescharacteristic
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53#12312yes—
92C22×D53C2×SD1611yescharacteristic
93C22×D53C2×Q1611yescharacteristic
94C22×D53AGL(1,ℤ/8)12yes—
95C2×#6157—44no—
96C2×#6157—44no—
97C2×#6157—44no—
98C2×#6157—82no—
99C2×#6157—82no—
100C2×#6157—82no—
101C2×#6157—82no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53#10312yes—
112C2×C4×C53#10412yes—
113C2×C4×C53#11712yes—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53#10312yes—
125C2×Dic53#12312yes—
126C4×D53—24no—
127C4×D53—24no—
128C4×D53—44no—
129C4×D53—44no—
130#88324—82no—
131#8717—531nomaximal, Sylow
132C22×C53#38111yescharacteristic
133C22×C53#41011yescharacteristic
134C22×C53#41011yescharacteristic
135C22×C53C2×#10311yescharacteristic
136C22×C53#41811yescharacteristic
137C22×C53C2×#10411yescharacteristic
138C22×C53#42111yescharacteristic
139C2×D53—22no—
140C2×D53—22no—
141C2×D53—21no—
142C2×D53—21no—
143C2×D53—22no—
144C2×D53—22no—
145C2×D53—21no—
146C2×D53—21no—
147C2×D53#48112yes—
148C2×D53#48412yes—
149#6157—84no—
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—82no—
154Dic53—22no—
155Dic53—22no—
156Dic53—41no—
157Dic53—82no—
158#2285—531no—
159#1817—531no—
160#1434—531no—
161#1294—534no—
162C2×C53#150311yescharacteristic
163C2×C53#150511yescharacteristic
164C2×C53#151411yescharacteristic
165C2×C53#151811yescharacteristic
166C2×C53#182011yescharacteristic
167C2×C53#182411yescharacteristic
168C2×C53#182611yescharacteristic
169D53—22no—
170D53—22no—
171C2×#117—1061no—
172C22×C42—531no—
173C2×C22⋊C8—532no—
174#384—531no—
175#384#615711yescharacteristic, complemented
176#382—1061no—
177#380—532no—
178#380—1061no—
179C4×C4⋊C4—532no—
180C4×C4⋊C4—1061no—
181C2×#95—532no—
182C2×#95—1061no—
183C2×#95—1061no—
184#336—1061no—
185#336—1061no—
186C53#871711yescharacteristic, complemented, Sylow
187C23×C4—531no—
188C23×C4—531no—
189C23×C4—531no—
190C23×C4—1062no—
191C22×C8—1062no—
192#117—2124no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—532no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4C2×#615712yes—
199C2×C22⋊C4—1062no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—1064no—
204C2×C42—1062no—
205C2×C42C2×#615711yescharacteristic
206C22⋊C8—1064no—
207#95—1062no—
208#95—1062no—
209#95—1064no—
210#95—1064no—
211#95—2122no—
212C24—531no—
213C22×C4—22no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—532no—
217C22×C4—532no—
218C22×C4—534no—
219C22×C4—531no—
220C22×C4—1062no—
221C22×C4—1062no—
222C22×C4—1062no—
223C22×C4—1062no—
224C22×C4—1064no—
225C22×C4—1062no—
226C22×C4—2121no—
227C22×C4—2121no—
228C22×C4—2121no—
229C22×C4—2122no—
230C22×C4—2122no—
231C22×C4—2122no—
232C22×C4—2122no—
233C22×C4—2122no—
234C22×C4—2122no—
235C22×C4C22⋊#615711yescharacteristic
236C22×C4C22⋊#615711yescharacteristic
237C22×C4C22×#615711yescharacteristic
238C2×C8—2122no—
239C2×C8—2122no—
240C2×C8—2122no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—2122no—
245C4⋊C4—2122no—
246C4⋊C4—2122no—
247C22⋊C4—2124no—
248C42—22no—
249C42—1062no—
250C42—1062no—
251C42—1062no—
252C42—2124no—
253C23—532no—
254C23—531no—
255C23—531no—
256C23—532no—
257C23—1061no—
258C23—1061no—
259C23—1061no—
260C23—1061no—
261C23C2×C22⋊#615711yescharacteristic
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—532no—
272C2×C4—532no—
273C2×C4—1064no—
274C2×C4—1064no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—4242no—
283C2×C4—4242no—
284C2×C4—4242no—
285C2×C4—4242no—
286C2×C4—2124no—
287C2×C4—2124no—
288C2×C4—2124no—
289C2×C4—2124no—
290C2×C4—2124no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4#1313328012yes—
295C2×C4#1313328112yes—
296C2×C4#1313329612yes—
297C8—4242no—
298C22—532no—
299C22—532no—
300C22—1061no—
301C22—1061no—
302C22—1062no—
303C22—1062no—
304C22—1062no—
305C22—1062no—
306C22—1061no—
307C22—1061no—
308C22#1681156611yescharacteristic
309C22#1681160311yescharacteristic
310C22#1681160411yescharacteristic
311C22C2×#1313328011yescharacteristic
312C22#1681161511yescharacteristic
313C22C2×#1313328111yescharacteristic
314C22#1681162211yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—82no—
319C4—1062no—
320C4—1062no—
321C4—4244no—
322C4—4242no—
323C4—2121no—
324C2—1062no—
325C2—1062no—
326C2#2029009811yescharacteristic
327C2#2029010411yescharacteristic
328C2#2029013611yescharacteristic
329C2#2029014911yescharacteristic
330C2#2029061311yescharacteristic
331C2#2029061911yescharacteristic
332C2#2029062111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenter#384#6157
hypocenterC53#8717
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
13. #16810493C4semidirect
18. C4⋊C4×Dic53C4semidirect
19. C53×#384C4semidirect
175. #384#6157semidirect
186. C53#8717semidirect

Maximal subgroups

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

subgroupindex
#202906072
#202898912
#202895882
#202771202
#871753