Finite Group Databasebeta

← #27294713

Subgroups — #27294713

Order 13568.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13133132C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272739C214yesmaximal, complemented
4C53×#2542C211yesmaximal, characteristic
5C2×#13133132—21no—
6#16810587—21no—
7C4×C22⋊Dic53—21no—
8#16810390C2212yes—
9#16810349—21no—
10#16810347C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic, complemented
12C2×C22⋊Q8×C53—21no—
13C2×C22⋊Q8×C53—21no—
14C2×C4×D4×C53C2211yescharacteristic
15C53×#399C412yescomplemented
16C53×#399C2212yes—
17C53×#390—21no—
18C53×#384—21no—
19C4×C4⋊C4×C53C411yescharacteristic, complemented
20C2×#11719917—41no—
21C2×C22⋊Dic53—22no—
22C2×C4×Dic53—41no—
23#13133132—24no—
24C22⋊#88322—22no—
25C22⋊#88322—41no—
26#13133124—41no—
27C22×Q8×C53—21no—
28C22×D4×C53D411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C4⋊Q8×C53—41no—
32C4⋊Q8×C53—41no—
33C22⋊Q8×C53—44no—
34C22⋊Q8×C53—44no—
35C4×D4×C53—41no—
36C4×D4×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53C2×C412yes—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53C2311yescharacteristic
47C2×C42×C53—21no—
48C2×C42×C53C2×C411yescharacteristic
49#95×C53C2×C412yes—
50C24×C53C22⋊C411yescharacteristic
51C24×C53C2×D411yescharacteristic
52C2×Q8×C53—42no—
53C2×Q8×C53—42no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×D4×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—41no—
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—42no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—44no—
76C4⋊C4×C53—44no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—44no—
82C22⋊C4×C53—44no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—42no—
86C22⋊Dic53—24no—
87C22⋊Dic53—42no—
88C4×Dic53—82no—
89#11719917—82no—
90C2×#88322—42no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53#13712yes—
102Q8×C53—82no—
103Q8×C53—82no—
104D4×C53—44no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—24no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53#9911yescharacteristic
126C2×C4×C53#10412yes—
127C2×Dic53—44no—
128C2×Dic53—82no—
129C2×Dic53—82no—
130#88322—82no—
131#11788—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53—22no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53C2×#9911yescharacteristic
145C22×C53#41911yescharacteristic
146C22×C53C2×#10411yescharacteristic
147C4×C53—21no—
148C4×C53—22no—
149C4×C53—44no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—82no—
154C4×C53#35312yes—
155Dic53—84no—
156#2542D5311yescharacteristic
157C4×#99—531no—
158#1465—531no—
159#1352—534no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53—81no—
164C2×C53#150811yescharacteristic
165C2×C53#212311yescharacteristic
166C2×C53C2×#35311yescharacteristic
167C2×C4⋊Q8Dic5311yescharacteristic, complemented
168C2×C22⋊Q8—21no—
169C2×C22⋊Q8—21no—
170C2×C4×D4C2×D5311yescharacteristic
171C2×#99—1061no—
172#407—1061no—
173#399Dic5312yescomplemented
174#399C2×D5312yes—
175#390—21no—
176#384—21no—
177C4×C4⋊C4Dic5311yescharacteristic, complemented
178C4×C22⋊C4—1061no—
179#342—532no—
180#324—1061no—
181#323—532no—
182C53#1178811yescharacteristic, complemented, Sylow
183C22×Q8—21no—
184C22×D4C22⋊D5311yescharacteristic
185C23×C4C22⋊D5311yescharacteristic
186C23×C4C22⋊D5311yescharacteristic
187C2×M16—2121no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic5312yes—
199C2×C4⋊C4C22⋊D5311yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4C2×Dic5311yescharacteristic
204C2×C22⋊C4C22×D5311yescharacteristic
205C2×C42—21no—
206C2×C42—2121no—
207C2×C42C2×Dic5311yescharacteristic
208#99—1064no—
209C22⋊C8—1062no—
210C22⋊C8—2121no—
211#95—2121no—
212#95C2×Dic5312yes—
213C24C22⋊Dic5311yescharacteristic
214C24C2×C22⋊D5311yescharacteristic
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—2122no—
231C22×C4C22⋊Dic5311yescharacteristic
232C22×C4C22⋊Dic5311yescharacteristic
233C22×C4C22⋊Dic5311yescharacteristic
234C22×C4C22×Dic5311yescharacteristic
235C22×C4C2×C22⋊D5311yescharacteristic
236M16—4242no—
237C2×C8—2122no—
238C4⋊C4—24no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—1064no—
249C22⋊C4—2122no—
250C42—21no—
251C42—21no—
252C42—42no—
253C42—4242no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊Dic5311yescharacteristic
264C23#1313324812yes—
265Q8—82no—
266Q8—82no—
267D4—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—4242no—
288C2×C4—4242no—
289C2×C4—2124no—
290C2×C4#1313313211yescharacteristic
291C2×C4#1313313211yescharacteristic
292C2×C4#1313314412yes—
293C8—4242no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22C2×#1313313211yescharacteristic
307C22#1681063211yescharacteristic
308C22C2×#1313314411yescharacteristic
309C4—22no—
310C4—21no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—82no—
316C4—4244no—
317C4#1681041612yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#2027323511yescharacteristic
323C2#2027636911yescharacteristic
324C2C2×#1681041611yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2542C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2542D53
hypocenterC53#11788
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #20272739C2semidirect
11. C2×C4⋊Q8×C53C4semidirect
15. C53×#399C4semidirect
19. C4×C4⋊C4×C53C4semidirect
167. C2×C4⋊Q8Dic53semidirect
173. #399Dic53semidirect
177. C4×C4⋊C4Dic53semidirect
182. C53#11788semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730532
#202727392
C53×#25422
#1178853