Finite Group Databasebeta

← #27294975

Subgroups — #27294975

Order 13568.

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

normal subgroups only

This group has 432 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#20272736C212yesmaximal, complemented
4#20272712C212yesmaximal
5C53×#2610C211yesmaximal, characteristic
6#16810676—21no—
7#16810486—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×#124×C53C411yescharacteristic, complemented
13C2×#117×C53C2211yescharacteristic
14C53×#395C411yescharacteristic, complemented
15C53×#394C411yescharacteristic, complemented
16C53×#393C2211yescharacteristic
17C53×#392C2211yescharacteristic
18C53×#390—21no—
19C53×#388—21no—
20C53×#388—21no—
21C53×#386—21no—
22C53×#376C411yescharacteristic, complemented
23C2×C22⋊Dic53—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26C4×#88322—41no—
27#13133124—41no—
28#13133124—41no—
29C22×Q8×C53C2×C411yescharacteristic
30C22×D4×C53—21no—
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33#124×C53—41no—
34#124×C53—41no—
35#117×C53—41no—
36#117×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53—21no—
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×C53—21no—
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
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×Q8×C53—41no—
59C2×Q8×C53—81no—
60C2×D4×C53—42no—
61C2×D4×C53—41no—
62C2×D4×C53—42no—
63C2×D4×C53—81no—
64C2×D4×C53—81no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—81no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—81no—
90C22⋊C4×C53—81no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—42no—
96C22⋊Dic53—42no—
97C22⋊Dic53—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
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—
112Q8×C53—82no—
113D4×C53—84no—
114D4×C53—82no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
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—21no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
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#9911yescharacteristic
140C2×C4×C53#9911yescharacteristic
141C2×Dic53—82no—
142C2×Dic53—82no—
143C2×Dic53—82no—
144#88322—82no—
145#11844—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—22no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53C2×#9911yescharacteristic
161C22×C53#41611yescharacteristic
162C22×C53#42111yescharacteristic
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—42no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—81no—
172C4×C53—81no—
173Dic53—162no—
174#2610D5311yescharacteristic
175#1820—531no—
176#1458—531no—
177#1351—532no—
178#1343—532no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—42no—
182C2×C53—161no—
183C2×C53#151611yescharacteristic
184C2×C53#212211yescharacteristic
185C2×C53#212611yescharacteristic
186C2×#124Dic5311yescharacteristic, complemented
187C2×#117C2×D5311yescharacteristic
188#431—1061no—
189#395Dic5311yescharacteristic, complemented
190#394Dic5311yescharacteristic, complemented
191#393C2×D5311yescharacteristic
192#392C2×D5311yescharacteristic
193#390—21no—
194#388—21no—
195#388—21no—
196#386—21no—
197#381—1061no—
198#376Dic5311yescharacteristic, complemented
199#342—532no—
200#342—1061no—
201#324—1061no—
202#323—532no—
203C53#1184411yescharacteristic, complemented, Sylow
204C22×Q8C2×Dic5311yescharacteristic
205C22×D4—21no—
206C22×D4C2×Dic5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208#124—41no—
209#124—41no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×Dic5311yescharacteristic
214C2×C4⋊C4C22⋊D5311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—1062no—
222C2×C22⋊C4C22⋊D5311yescharacteristic
223C2×C22⋊C4C22×D5311yescharacteristic
224C2×C42—21no—
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—2121no—
234#95—2121no—
235#95C2×Dic5311yescharacteristic
236#95C2×Dic5311yescharacteristic
237C24—21no—
238C24C2×C22⋊D5311yescharacteristic
239C2×Q8—41no—
240C2×Q8—81no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—81no—
245C2×D4—81no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—2122no—
254C22×C4C22⋊Dic5311yescharacteristic
255C22×C4C22⋊Dic5311yescharacteristic
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22×Dic5311yescharacteristic
259C22×C4C2×C22⋊D5311yescharacteristic
260C2×C8—2122no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C22⋊C4—2122no—
274C22⋊C4—2122no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—42no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊Dic5311yescharacteristic
292C23#1313323712yes—
293Q8—82no—
294D4—84no—
295D4—82no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
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—4242no—
321C2×C4—4242no—
322C2×C4—4242no—
323C2×C4#1313313211yescharacteristic
324C2×C4#1313313211yescharacteristic
325C8—4242no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22C2×#1313313211yescharacteristic
341C22#1681061911yescharacteristic
342C22#1681064011yescharacteristic
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—8482no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—161no—
358C2#2027327111yescharacteristic
359C2#2027636511yescharacteristic
360C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2610C2
socleC22×C53C2×#99
prime radicalC53×#392C22
cosocleC2×C22⋊C4×C53C23
hypercenter#2610D53
hypocenterC53#11844
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #20274792C2semidirect
2. #20273029C2semidirect
3. #20272736C2semidirect
12. C2×#124×C53C4semidirect
14. C53×#395C4semidirect
15. C53×#394C4semidirect
22. C53×#376C4semidirect
186. C2×#124Dic53semidirect
189. #395Dic53semidirect
190. #394Dic53semidirect
198. #376Dic53semidirect
203. C53#11844semidirect

Maximal subgroups

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

subgroupindex
#202747922
#202730292
#202727362
#202727122
C53×#26102
#1184453