Finite Group Databasebeta

← #27272189

Subgroups — #27272189

Order 13568.

654 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 117 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#20277204C211yesmaximal, characteristic
2#20274694C212yesmaximal, complemented
3#20272541C211yesmaximal, characteristic, complemented
4#20272535C212yesmaximal, complemented
5C53×#1272C211yesmaximal, characteristic, complemented
6C22×#11719949C411yescharacteristic
7C2×#13133175C411yescharacteristic
8C2×C42×D53C2211yescharacteristic
9C2×#13133146—22no—
10C2×#13133129—22no—
11#16810530C411yescharacteristic
12#16810525C2211yescharacteristic
13#16810524C411yescharacteristic
14#16810501—22no—
15#16810470—22no—
16#16810378C2212yes—
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C2×#11719949—22no—
20C2×#11719949—42no—
21C2×#11719949—42no—
22C2×#11719949C814yescomplemented
23C22×D212—21no—
24C22×D212C2×C411yescharacteristic
25C22×C4×D53—21no—
26C22×C4×D53C2×C411yescharacteristic
27C22×C4×D53D412yes—
28C22×#88322—22no—
29#13133175—42no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719921—21no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37C42×D53—42no—
38#13133146—44no—
39#13133129—44no—
40#13133124—22no—
41C22×Q8×C53C2×C411yescharacteristic
42C22×C8×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C42×C53C2311yescharacteristic
45#95×C53—22no—
46C23×D53—21no—
47C23×D53C22⋊C411yescharacteristic
48C2×Q8×C53—21no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53C2×C812yes—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C22×C411yescharacteristic
55C22×C4×C53C2×D412yes—
56C22×Dic53—21no—
57C22×Dic53C22⋊C412yes—
58C22×Dic53C22⋊C411yescharacteristic
59#11719949—48no—
60#11719949—48no—
61C2×D212—24no—
62C2×D212—21no—
63C2×D212—41no—
64C2×D212—41no—
65C2×D212—42no—
66C2×D212—42no—
67C2×D212C2×C812yes—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—24no—
73C2×C4×D53—42no—
74C2×C4×D53—42no—
75C2×C4×D53—42no—
76C2×C4×D53—42no—
77C2×C4×D53C2×C812yes—
78C2×C4×D53M1612yes—
79C2×C8×C53—42no—
80C2×C8×C53—42no—
81C2×C8×C53—42no—
82C4⋊C4×C53—41no—
83C42×C53—41no—
84#11719921—44no—
85#11719921—42no—
86#11719921—42no—
87#11719921—42no—
88C4⋊Dic53—41no—
89C4×Dic53—24no—
90C4×Dic53—41no—
91C4×Dic53—42no—
92C2×#88322—42no—
93C2×#88322—42no—
94C2×#88322—42no—
95C23×C53C2×C22⋊C411yescharacteristic
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53—22no—
101C22×D53—41no—
102C22×D53—41no—
103C22×D53—42no—
104C22×D53—42no—
105C22×D53C22⋊C812yes—
106Q8×C53—42no—
107Q8×C53—42no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53C22×C811yescharacteristic
118C2×C4×C53C2×M1611yescharacteristic
119C2×C4×C53AGL(1,ℤ/8)12yes—
120C2×C4×C53#13712yes—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—22no—
124C2×Dic53—41no—
125C2×Dic53—42no—
126C2×Dic53C22⋊C812yes—
127C2×Dic53#9912yes—
128C2×Dic53#10012yes—
129D212—42no—
130D212—42no—
131D212—48no—
132C4×D53—24no—
133C4×D53—24no—
134C4×D53—44no—
135C4×D53—44no—
136C4×D53—48no—
137C8×C53—82no—
138#88322—82no—
139#7597—531nomaximal, Sylow
140C22×C53C2×C22⋊C811yescharacteristic
141C22×C53C2×#9911yescharacteristic
142C22×C53C2×#10011yescharacteristic
143C22×C53#41712yes—
144C22×C53#42112yes—
145C2×D53—22no—
146C2×D53—24no—
147C2×D53—41no—
148C2×D53—41no—
149C2×D53—42no—
150C2×D53—42no—
151C2×D53—44no—
152C2×D53—81no—
153C2×D53—81no—
154C2×D53—82no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—42no—
160C4×C53—81no—
161Dic53—42no—
162Dic53—81no—
163Dic53#32314yes—
164#2297—531no—
165#1797—532no—
166#1272—531no—
167#1272D5311yescharacteristic, complemented
168#1270—532no—
169C2×C53C2×#32311yescharacteristic
170C2×C53#147712yes—
171C2×C53#151312yes—
172C2×C53#153611yescharacteristic
173C2×C53#156011yescharacteristic
174D53—42no—
175D53—82no—
176C22×#41—531no—
177C2×#117—531no—
178C22×C42—531no—
179C2×C4⋊C8—1062no—
180C2×C22⋊C8—1062no—
181#391—531no—
182#391C2×D5311yescharacteristic
183#390—531no—
184#390—531no—
185#386—1062no—
186C4×C22⋊C4—1062no—
187#336—532no—
188#336C2×D5312yes—
189C53#759711yescharacteristic, complemented, Sylow
190C2×#41—534no—
191C2×#41—1062no—
192C2×#41—2122no—
193C2×#41—2122no—
194C22×Q8C4×D5311yescharacteristic
195C22×D4—531no—
196C22×D4—1061no—
197C23×C4—531no—
198C23×C4—532no—
199C23×C4—1061no—
200C22×C8—22no—
201C22×C8—1062no—
202#117—2122no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4C4×D5311yescharacteristic
205C2×C22⋊C4—1062no—
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1062no—
208C2×C42—532no—
209C2×C42—531no—
210C2×C42—1061no—
211C2×C42—2122no—
212C2×C42C22×D5311yescharacteristic
213C4⋊C8—2124no—
214C22⋊C8—2124no—
215#95—22no—
216#95—1062no—
217C24—531no—
218C24—1061no—
219#41—2128no—
220#41—2128no—
221C2×Q8—21no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8C8×D5312yes—
225C2×D4—532no—
226C2×D4—1064no—
227C2×D4—1061no—
228C2×D4—2121no—
229C2×D4—2121no—
230C2×D4—2122no—
231C2×D4—2122no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—531no—
235C22×C4—532no—
236C22×C4—532no—
237C22×C4—532no—
238C22×C4—1062no—
239C22×C4—1062no—
240C22×C4—1064no—
241C22×C4—1062no—
242C22×C4—1062no—
243C22×C4—1061no—
244C22×C4—2122no—
245C22×C4—2122no—
246C22×C4—2122no—
247C22×C4—2122no—
248C22×C4C2×C4×D5311yescharacteristic
249C22×C4D4×D5312yes—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—2122no—
254C2×C8—2122no—
255C2×C8—2122no—
256C4⋊C4—41no—
257C4⋊C4—2121no—
258C22⋊C4—2124no—
259C22⋊C4—2122no—
260C22⋊C4—2122no—
261C22⋊C4—2122no—
262C42—41no—
263C42—1064no—
264C42—2121no—
265C42—2122no—
266C23—532no—
267C23—1061no—
268C23—1061no—
269C23—1062no—
270C23—1062no—
271C23—1062no—
272C23—2121no—
273C23—2121no—
274C23—2122no—
275C23—2122no—
276C23C22⋊C4×D5311yescharacteristic
277Q8—42no—
278Q8—42no—
279D4—2122no—
280D4—2122no—
281D4—2128no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—532no—
292C2×C4—532no—
293C2×C4—532no—
294C2×C4—1064no—
295C2×C4—1064no—
296C2×C4—1062no—
297C2×C4—1062no—
298C2×C4—1062no—
299C2×C4—2124no—
300C2×C4—2124no—
301C2×C4—2128no—
302C2×C4—2121no—
303C2×C4—2122no—
304C2×C4C2×C8×D5311yescharacteristic
305C2×C4M16×D5311yescharacteristic
306C2×C4#1313324412yes—
307C2×C4#1313324912yes—
308C8—82no—
309C8—4242no—
310C22—1062no—
311C22—1064no—
312C22—4241no—
313C22—4242no—
314C22—4241no—
315C22—2122no—
316C22—2124no—
317C22—2121no—
318C22—2122no—
319C22—2121no—
320C22C22⋊C8×D5311yescharacteristic
321C22#99×D5311yescharacteristic
322C22#100×D5311yescharacteristic
323C22#1681062312yes—
324C22#1681063812yes—
325C4—22no—
326C4—22no—
327C4—21no—
328C4—21no—
329C4—42no—
330C4—81no—
331C4—534no—
332C4—4241no—
333C4—2122no—
334C2—4242no—
335C2—2122no—
336C2#323×D5311yescharacteristic
337C2#2027310112yes—
338C2#2027325712yes—
339C2#2027334811yescharacteristic
340C2#2027344911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53C2×C22⋊C4
prime radicalC22×#11719949C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7597
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #20274694C2semidirect
3. #20272541C2semidirect
4. #20272535C2semidirect
5. C53×#1272C2semidirect
22. C2×#11719949C8semidirect
167. #1272D53semidirect
189. C53#7597semidirect

Maximal subgroups

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

subgroupindex
#202772042
#202746942
#202725412
#202725352
C53×#12722
#759753