Finite Group Databasebeta

← #27294955

Subgroups — #27294955

Order 13568.

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

normal subgroups only

This group has 446 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, complemented
2#20273029C211yesmaximal, characteristic, complemented
3#20272739C212yesmaximal, complemented
4#20272712C212yesmaximal
5C53×#2598C211yesmaximal, characteristic
6#16810676—21no—
7#16810486—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×C4×D4×C53C411yescharacteristic, complemented
13C2×#117×C53C2211yescharacteristic
14C53×#400C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#394C411yescharacteristic, complemented
17C53×#392C2211yescharacteristic
18C53×#388—21no—
19C53×#386—21no—
20C53×#386—21no—
21C53×#385—21no—
22C53×#383C411yescharacteristic, complemented
23C2×C22⋊Dic53—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26C4×#88322—41no—
27#13133124—41no—
28#13133124—41no—
29C22×D4×C53—21no—
30C22×D4×C53C2×C411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53D411yescharacteristic
33C4×D4×C53—41no—
34C4×D4×C53—41no—
35#117×C53—41no—
36#117×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53C2311yescharacteristic
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×C53—21no—
57C24×C53C2×D411yescharacteristic
58C2×D4×C53—41no—
59C2×D4×C53—42no—
60C2×D4×C53—42no—
61C2×D4×C53—42no—
62C2×D4×C53—81no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53—42no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—81no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—81no—
91C22⋊C4×C53—81no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—21no—
96C22⋊Dic53—42no—
97C22⋊Dic53—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—22no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53#13512yes—
112D4×C53—42no—
113D4×C53—42no—
114D4×C53—84no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
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—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—84no—
140C2×C4×C53#9911yescharacteristic
141C2×C4×C53#9911yescharacteristic
142C2×Dic53—82no—
143C2×Dic53—82no—
144C2×Dic53—82no—
145#88322—82no—
146#11840—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—22no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—42no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53C2×#9911yescharacteristic
162C22×C53#41611yescharacteristic
163C22×C53#42111yescharacteristic
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—81no—
172C4×C53—81no—
173C4×C53—81no—
174Dic53—162no—
175#2598D5311yescharacteristic
176#1820—531no—
177#1458—531no—
178#1352—532no—
179#1343—532no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—42no—
183C2×C53—82no—
184C2×C53#151611yescharacteristic
185C2×C53#212011yescharacteristic
186C2×C53#212811yescharacteristic
187C2×C4×D4Dic5311yescharacteristic, complemented
188C2×#117C2×D5311yescharacteristic
189#431—1061no—
190#400Dic5311yescharacteristic, complemented
191#399C2×D5311yescharacteristic
192#394Dic5311yescharacteristic, complemented
193#392C2×D5311yescharacteristic
194#388—21no—
195#386—21no—
196#386—21no—
197#385—21no—
198#383Dic5311yescharacteristic, complemented
199#381—1061no—
200#342—532no—
201#342—1061no—
202#324—1061no—
203#323—532no—
204C53#1184011yescharacteristic, complemented, Sylow
205C22×D4—21no—
206C22×D4C2×Dic5311yescharacteristic
207C23×C4—21no—
208C23×C4C22⋊D5311yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211#117—41no—
212#117—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic5311yescharacteristic
215C2×C4⋊C4C2×Dic5311yescharacteristic
216C2×C4⋊C4C22⋊D5311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—1062no—
223C2×C22⋊C4C22⋊D5311yescharacteristic
224C2×C22⋊C4C22×D5311yescharacteristic
225C2×C42C2×Dic5311yescharacteristic
226C2×C42C22⋊D5311yescharacteristic
227C4⋊C8—2121no—
228C22⋊C8—1062no—
229C4×C8—2121no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—2121no—
235#95—2121no—
236#95C2×Dic5311yescharacteristic
237#95C2×Dic5311yescharacteristic
238C24—21no—
239C24C2×C22⋊D5311yescharacteristic
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—81no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—2122no—
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22⋊Dic5311yescharacteristic
260C22×C4C22×Dic5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262C2×C8—2122no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—2122no—
276C22⋊C4—2122no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C23—21no—
282C23—21no—
283C23—22no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊Dic5311yescharacteristic
293C23#1313323712yes—
294D4—42no—
295D4—42no—
296D4—84no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—84no—
322C2×C4—4242no—
323C2×C4—4242no—
324C2×C4—4242no—
325C2×C4#1313313211yescharacteristic
326C2×C4#1313313211yescharacteristic
327C8—4242no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22C2×#1313313211yescharacteristic
343C22#1681061911yescharacteristic
344C22#1681064011yescharacteristic
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—8482no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—82no—
360C2#2027327111yescharacteristic
361C2#2027635611yescharacteristic
362C2#2027638711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2598C2
socleC22×C53C2×#99
prime radicalC53×#392C22
cosocleC2×C22⋊C4×C53C23
hypercenter#2598D53
hypocenterC53#11840
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2598

Complements

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

NH ≅ G/Ntype
1. #20274792C2semidirect
2. #20273029C2semidirect
3. #20272739C2semidirect
12. C2×C4×D4×C53C4semidirect
14. C53×#400C4semidirect
16. C53×#394C4semidirect
22. C53×#383C4semidirect
187. C2×C4×D4Dic53semidirect
190. #400Dic53semidirect
192. #394Dic53semidirect
198. #383Dic53semidirect
204. C53#11840semidirect

Maximal subgroups

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

subgroupindex
#202747922
#202730292
#202727392
#202727122
C53×#25982
#1184053