Finite Group Databasebeta

← #27293570

Subgroups — #27293570

Order 13568.

412 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 87 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×#13133215C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272660C212yesmaximal
4#20272648C212yesmaximal
5C53×#1467C211yesmaximal, characteristic
6C2×#13133215C411yescharacteristic
7C2×C4×Dic106C411yescharacteristic
8#16810680—21no—
9#16810530C411yescharacteristic
10#16810528C2211yescharacteristic
11#16810518—21no—
12#16810518—21no—
13#16810492C2211yescharacteristic
14C4×#11719919—21no—
15C4×#11719919—21no—
16C42×Dic53C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24#13133215—22no—
25#13133215—22no—
26C4×Dic106—41no—
27C4×Dic106—41no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53C2×C411yescharacteristic
39C4⋊#88322—41no—
40C22⋊#88322—22no—
41#13133128—41no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×Q8×C53C2×C411yescharacteristic
47C4×Q8×C53—41no—
48C4×Q8×C53—41no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53C2311yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C4⋊C8×C53—41no—
56C22⋊C8×C53—22no—
57#97×C53—41no—
58C2×Q8×C53—21no—
59C2×Q8×C53—21no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C2×Dic106—21no—
74C2×Dic106—41no—
75C2×Dic106—41no—
76C2×Dic106M1612yes—
77C2×C8×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C4⋊Dic53—41no—
88#11719919—24no—
89#11719919—24no—
90#11719919—42no—
91#11719919—42no—
92#11719919—42no—
93C4×Dic53—24no—
94C4×Dic53—21no—
95C4×Dic53—42no—
96C4×Dic53—41no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C4×Dic53M1612yes—
100C2×#88322—42no—
101C23×C53C2×C22⋊C411yescharacteristic
102Q8×C53—42no—
103Q8×C53—42no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53#9911yescharacteristic
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53C2×M1611yescharacteristic
122C2×C4×C53C2×M1611yescharacteristic
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135Dic106—42no—
136Dic106—42no—
137C8×C53—82no—
138#88322—82no—
139#11532—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53#40711yescharacteristic
143C22×C53C2×#9911yescharacteristic
144C22×C53#41311yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53#35612yes—
154Dic53—24no—
155Dic53—42no—
156Dic53—42no—
157Dic53—81no—
158C4×C4⋊Q8—531no—
159#1467—531no—
160#1467D5311yescharacteristic
161#1325—532no—
162#1319—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53#146511yescharacteristic
166C2×C53#212611yescharacteristic
167C2×C53C2×#35611yescharacteristic
168C2×C4⋊Q8—531no—
169C2×C4×Q8—531no—
170C2×C4×Q8C2×D5311yescharacteristic
171#432—21no—
172#432—1061no—
173#391—531no—
174#391—531no—
175#389—1061no—
176#389—1061no—
177#384—531no—
178C4×C4⋊C4—1061no—
179C4×C4⋊C4—1061no—
180C43—531no—
181#324—21no—
182#324—532no—
183#324—1061no—
184#324C2×D5312yes—
185C53#1153211yescharacteristic, complemented, Sylow
186C22×Q8—531no—
187C22×Q8C4×D5311yescharacteristic
188C4⋊Q8—1062no—
189C4⋊Q8—1062no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—2121no—
193C4×Q8—2121no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4D21211yescharacteristic
201C2×C4⋊C4C22⋊D5311yescharacteristic
202C2×C4⋊C4C22×D5311yescharacteristic
203C2×C42—531no—
204C2×C42—531no—
205C2×C42—1061no—
206C2×C42—1061no—
207C2×C42—1061no—
208C2×C42C4×D5311yescharacteristic
209C2×C42D21211yescharacteristic
210C2×C42C22⋊D5311yescharacteristic
211C4⋊C8—41no—
212C4⋊C8—2121no—
213C22⋊C8—22no—
214C22⋊C8—1062no—
215#97—41no—
216#97—2121no—
217#95—1061no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—532no—
226C2×Q8—1061no—
227C2×Q8—2121no—
228C2×Q8—2121no—
229C22×C4—1061no—
230C22×C4—1061no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4#1171992111yescharacteristic
234C22×C4#1171992111yescharacteristic
235C22×C4#1171992111yescharacteristic
236C22×C4#1171992111yescharacteristic
237C22×C4C2×C4×D5311yescharacteristic
238C22×C4C2×D21211yescharacteristic
239C22×C4C2×C22⋊D5311yescharacteristic
240C2×C8—42no—
241C2×C8—2122no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—1064no—
246C4⋊C4—1064no—
247C4⋊C4—2121no—
248C4⋊C4—2122no—
249C4⋊C4—2122no—
250C4⋊C4—2122no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—532no—
258C42—1064no—
259C42—1061no—
260C42—2121no—
261C42—2122no—
262C42—2121no—
263C42—2121no—
264C23C2×#1171992111yescharacteristic
265Q8—42no—
266Q8—42no—
267Q8—2122no—
268Q8—2122no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—1062no—
285C2×C4—1062no—
286C2×C4—1062no—
287C2×C4—1062no—
288C2×C4—1062no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4#1313313111yescharacteristic
297C2×C4#1313313111yescharacteristic
298C2×C4M16×D5311yescharacteristic
299C2×C4M16×D5311yescharacteristic
300C8—82no—
301C8—4242no—
302C22—21no—
303C22—21no—
304C22#1681058611yescharacteristic
305C22C2×#1313313111yescharacteristic
306C22#1681060511yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—1064no—
316C4—4241no—
317C4—2122no—
318C4—2122no—
319C4#1681042112yes—
320C2—21no—
321C2—21no—
322C2#2027305111yescharacteristic
323C2#2027637811yescharacteristic
324C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1467C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1467D53
hypocenterC53#11532
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
185. C53#11532semidirect

Maximal subgroups

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

subgroupindex
C4×#131332152
#202730592
#202726602
#202726482
C53×#14672
#1153253