Finite Group Databasebeta

← #27294372

Subgroups — #27294372

Order 13568.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274448C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272739C212yesmaximal
4#20272736C212yesmaximal
5C53×#2396C211yesmaximal, characteristic
6C2×#13133176—21no—
7#16810676—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—
23#13133176—24no—
24#13133176—42no—
25C2×C4⋊Dic53—41no—
26C2×C22⋊Dic53—22no—
27C2×C4×Dic53—41no—
28C4⋊#88322—41no—
29C22⋊#88322—22no—
30C4×#88322—41no—
31C22×Q8×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35#117×C53—41no—
36#117×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53C2311yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53—21no—
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53D411yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×Q8×C53—42no—
60C2×Q8×C53—41no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53SD1612yes—
73C22×C4×C53Q1612yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—41no—
93C42×C53—41no—
94C42×C53—42no—
95C22⋊Dic53—24no—
96C22⋊Dic53—42no—
97C4⋊Dic53—44no—
98C4×Dic53—44no—
99C2×#88322—42no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53C2×SD1611yescharacteristic
106C23×C53C2×Q1611yescharacteristic
107Q8×C53—42no—
108Q8×C53—42no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
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—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53#9911yescharacteristic
136C2×C4×C53#9911yescharacteristic
137C2×C4×C53#10312yes—
138C2×C4×C53#10312yes—
139C2×C4×C53#10412yes—
140C2×Dic53—44no—
141C2×Dic53—82no—
142C2×Dic53—82no—
143#88322—82no—
144#11715—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—81no—
154C22×C53C2×#9911yescharacteristic
155C22×C53C2×#10311yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C4×C53—22no—
158C4×C53—42no—
159C4×C53—41no—
160C4×C53—42no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53#32812yes—
167C4×C53#32812yes—
168Dic53—84no—
169#2396D5311yescharacteristic
170#1740—531no—
171#1458—531no—
172#1352—532no—
173#1351—532no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—41no—
177C2×C53—41no—
178C2×C53C2×#32811yescharacteristic
179C2×C53#212011yescharacteristic
180C2×C53#212611yescharacteristic
181C2×C4×Q8Dic5311yescharacteristic, complemented
182C2×#117—1061no—
183C2×#117C2×D5311yescharacteristic
184#431—1061no—
185#400Dic5311yescharacteristic, complemented
186#399C2×D5311yescharacteristic
187#398—21no—
188#396—21no—
189#395Dic5311yescharacteristic, complemented
190#393C2×D5311yescharacteristic
191C4×C4⋊C4—21no—
192C4×C4⋊C4Dic5311yescharacteristic, complemented
193C4×C22⋊C4—21no—
194#342—532no—
195#342—1061no—
196#324—1061no—
197#323—532no—
198C53#1171511yescharacteristic, complemented, Sylow
199C22×Q8C2×Dic5311yescharacteristic
200C23×C4C22⋊D5311yescharacteristic
201C4×Q8—41no—
202C4×Q8—41no—
203#117—41no—
204#117—41no—
205#117—1064no—
206#117—2122no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—2121no—
212C2×C4⋊C4C2×Dic5311yescharacteristic
213C2×C4⋊C4C2×Dic5311yescharacteristic
214C2×C4⋊C4C22⋊D5311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1062no—
218C2×C22⋊C4C22⋊D5311yescharacteristic
219C2×C22⋊C4C22×D5311yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—2121no—
223C2×C42C2×Dic5311yescharacteristic
224C2×C42C22⋊D5311yescharacteristic
225C4⋊C8—2121no—
226C22⋊C8—1062no—
227C4×C8—2121no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×Dic5311yescharacteristic
233#95C2×Dic5311yescharacteristic
234C24C2×C22⋊D5311yescharacteristic
235C2×Q8—42no—
236C2×Q8—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—2122no—
245C22×C4#1171992412yes—
246C22×C4#1171992512yes—
247C22×C4C22⋊Dic5311yescharacteristic
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22⋊Dic5311yescharacteristic
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22×Dic5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253C2×C8—2122no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—2124no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—1064no—
267C22⋊C4—2122no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—41no—
273C42—41no—
274C42—42no—
275C42—2124no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic5311yescharacteristic
281C23C2×#1171992411yescharacteristic
282C23C2×#1171992511yescharacteristic
283Q8—42no—
284Q8—42no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—4242no—
312C2×C4—4242no—
313C2×C4—2124no—
314C2×C4#1313313211yescharacteristic
315C2×C4#1313313211yescharacteristic
316C2×C4#1313314112yes—
317C2×C4#1313314112yes—
318C2×C4#1313314412yes—
319C8—4242no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#1313313211yescharacteristic
330C22C2×#1313314111yescharacteristic
331C22C2×#1313314411yescharacteristic
332C4—22no—
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—4244no—
342C4#1681036012yes—
343C4#1681036012yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1681036011yescharacteristic
349C2#2027635611yescharacteristic
350C2#2027638011yescharacteristic
subgroupG/H
centerC2×C4#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#11715
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ 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
181. C2×C4×Q8Dic53semidirect
185. #400Dic53semidirect
189. #395Dic53semidirect
192. C4×C4⋊C4Dic53semidirect
198. C53#11715semidirect

Maximal subgroups

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

subgroupindex
#202744482
#202730292
#202727392
#202727362
C53×#23962
#1171553