Finite Group Databasebeta

← #27294809

Subgroups — #27294809

Order 13568.

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

normal subgroups only

This group has 402 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
1#20274792C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272739C212yesmaximal
4#20272736C212yesmaximal
5C53×#2396C211yesmaximal, characteristic
6#16810680—21no—
7#16810486—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×C4×Q8×C53C411yescharacteristic, complemented
13C2×#117×C53C2211yescharacteristic
14C53×#400C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#398—21no—
17C53×#396—21no—
18C53×#395C411yescharacteristic, complemented
19C53×#393C2211yescharacteristic
20C4×C4⋊C4×C53—21no—
21C4×C4⋊C4×C53C411yescharacteristic, complemented
22C4×C22⋊C4×C53—21no—
23C2×C22⋊Dic53—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26#13133128—41no—
27#13133124—41no—
28#13133124—41no—
29C22×Q8×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C4×Q8×C53—41no—
32C4×Q8×C53—41no—
33#117×C53—41no—
34#117×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53C2311yescharacteristic
46C2×C42×C53—21no—
47C2×C42×C53—21no—
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C2×Q8×C53—42no—
58C2×Q8×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—41no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—42no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—42no—
90C42×C53—41no—
91C42×C53—41no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53#13712yes—
101Q8×C53—42no—
102Q8×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
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—42no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53#9911yescharacteristic
131C2×C4×C53#9911yescharacteristic
132C2×Dic53—82no—
133C2×Dic53—82no—
134C2×Dic53—82no—
135#88322—82no—
136#11810—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—42no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—81no—
144C22×C53C2×#9911yescharacteristic
145C22×C53#41911yescharacteristic
146C22×C53#42111yescharacteristic
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—42no—
150C4×C53—41no—
151C4×C53—42no—
152C4×C53—41no—
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—81no—
157C4×C53—81no—
158Dic53—162no—
159#2396D5311yescharacteristic
160#1820—531no—
161#1464—531no—
162#1352—532no—
163#1351—532no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—42no—
167C2×C53#151111yescharacteristic
168C2×C53#212011yescharacteristic
169C2×C53#212611yescharacteristic
170C2×C4×Q8Dic5311yescharacteristic, complemented
171C2×#117C2×D5311yescharacteristic
172#432—1061no—
173#400Dic5311yescharacteristic, complemented
174#399C2×D5311yescharacteristic
175#398—21no—
176#396—21no—
177#395Dic5311yescharacteristic, complemented
178#393C2×D5311yescharacteristic
179#381—1061no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4Dic5311yescharacteristic, complemented
182C4×C22⋊C4—21no—
183#342—532no—
184#342—1061no—
185#324—1061no—
186#323—532no—
187C53#1181011yescharacteristic, complemented, Sylow
188C22×Q8C2×Dic5311yescharacteristic
189C23×C4C22⋊D5311yescharacteristic
190C4×Q8—41no—
191C4×Q8—41no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic5311yescharacteristic
199C2×C4⋊C4C2×Dic5311yescharacteristic
200C2×C4⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4C22⋊D5311yescharacteristic
205C2×C22⋊C4C22×D5311yescharacteristic
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C2×Dic5311yescharacteristic
209C2×C42C22⋊D5311yescharacteristic
210C4⋊C8—2121no—
211C22⋊C8—1062no—
212#97—2121no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—2121no—
218#95—2121no—
219#95C2×Dic5311yescharacteristic
220#95C2×Dic5311yescharacteristic
221C24C2×C22⋊D5311yescharacteristic
222C2×Q8—42no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—2122no—
233C22×C4C22⋊Dic5311yescharacteristic
234C22×C4C22⋊Dic5311yescharacteristic
235C22×C4C22⋊Dic5311yescharacteristic
236C22×C4C22⋊Dic5311yescharacteristic
237C22×C4C22×Dic5311yescharacteristic
238C22×C4C2×C22⋊D5311yescharacteristic
239C2×C8—2122no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—2122no—
252C22⋊C4—2122no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—42no—
258C42—41no—
259C42—41no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic5311yescharacteristic
265C23#1313324812yes—
266Q8—42no—
267Q8—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
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—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—4242no—
296C2×C4—4242no—
297C2×C4—4242no—
298C2×C4#1313313211yescharacteristic
299C2×C4#1313313211yescharacteristic
300C8—4242no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#1313313211yescharacteristic
309C22#1681063211yescharacteristic
310C22#1681064011yescharacteristic
311C4—21no—
312C4—21no—
313C4—42no—
314C4—41no—
315C4—42no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—8482no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2027324811yescharacteristic
327C2#2027635611yescharacteristic
328C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2396C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2396D53
hypocenterC53#11810
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C53C4semidirect
14. C53×#400C4semidirect
18. C53×#395C4semidirect
21. C4×C4⋊C4×C53C4semidirect
170. C2×C4×Q8Dic53semidirect
173. #400Dic53semidirect
177. #395Dic53semidirect
181. C4×C4⋊C4Dic53semidirect
187. C53#11810semidirect

Maximal subgroups

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

subgroupindex
#202747922
#202730492
#202727392
#202727362
C53×#23962
#1181053