Finite Group Databasebeta

← #27281737

Subgroups — #27281737

Order 13568.

474 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283353C211yesmaximal, characteristic
2#20274792C211yesmaximal, characteristic
3#20273248C212yesmaximal
4#20272568C212yesmaximal
5C53×#2499C211yesmaximal, characteristic
6#16810684—24no—
7#16810647—21no—
8#16810587C2212yes—
9C4×#11719917—21no—
10#16810486C2212yes—
11#16810390—21no—
12#16810390—21no—
13#16810360—24no—
14C2×C22⋊Q8×C53C2211yescharacteristic
15C2×#117×C53C2211yescharacteristic
16C53×#397—21no—
17C53×#397—21no—
18C53×#391C411yescharacteristic, complemented
19C53×#391C411yescharacteristic, complemented
20C53×#387—21no—
21C53×#387—21no—
22C53×#384C411yescharacteristic, complemented
23C53×#382C411yescharacteristic, complemented
24C4×C22⋊C4×C53C2211yescharacteristic
25C2×#11719917—22no—
26C2×C22⋊Dic53—22no—
27C4⋊#88322—24no—
28C22⋊#88322—22no—
29C22⋊#88322D414yes—
30#13133128—22no—
31C4×#88322—22no—
32#13133124—22no—
33#13133124—22no—
34C22×Q8×C53C2×C411yescharacteristic
35C23×C4×C53C2311yescharacteristic
36C22⋊Q8×C53—22no—
37C22⋊Q8×C53—22no—
38#117×C53—22no—
39#117×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C24×C53C2×D411yescharacteristic
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D412yes—
78C22×C4×C53#4112yes—
79C22×Dic53—42no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—22no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—41no—
96C42×C53—41no—
97C22⋊Dic53—42no—
98C22⋊Dic53—42no—
99#11719917—42no—
100#11719917—42no—
101C2×#88322—24no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107C23×C53#12112yes—
108Q8×C53—82no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
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—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C4×D412yes—
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×Dic53—82no—
139C2×Dic53—82no—
140C2×Dic53—82no—
141#88322—44no—
142#9444—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53#38611yescharacteristic
151C22×C53#41011yescharacteristic
152C22×C53#41011yescharacteristic
153C4×C53—21no—
154C4×C53—22no—
155C4×C53—21no—
156C4×C53—42no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—81no—
162C4×C53—81no—
163C4×C53—81no—
164Dic53—162no—
165#2921—531no—
166#2499D5311yescharacteristic
167#1820—531no—
168#1511—532no—
169#1284—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53#188411yescharacteristic
174C2×C53#189911yescharacteristic
175C2×C53#190911yescharacteristic
176C2×C22⋊Q8C2×D5311yescharacteristic
177C2×#117C2×D5311yescharacteristic
178#433—1064no—
179#423—1061no—
180#407—532no—
181C4×M16—1061no—
182#397—21no—
183#397—21no—
184#391Dic5311yescharacteristic, complemented
185#391Dic5311yescharacteristic, complemented
186#387—21no—
187#387—21no—
188#384Dic5311yescharacteristic, complemented
189#382Dic5311yescharacteristic, complemented
190#381—532no—
191C4×C22⋊C4C2×D5311yescharacteristic
192#342—1061no—
193#342—1061no—
194#328—1064no—
195C53#944411yescharacteristic, complemented, Sylow
196C22×Q8C2×Dic5311yescharacteristic
197C23×C4C22×D5311yescharacteristic
198C2×M16—1062no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201#117—22no—
202#117—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×Dic5311yescharacteristic
207C2×C4⋊C4C2×Dic5311yescharacteristic
208C2×C4⋊C4C2×Dic5311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—1062no—
212C2×C22⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4C22⋊D5311yescharacteristic
214C2×C42C2×Dic5311yescharacteristic
215C2×C42C2×Dic5311yescharacteristic
216C2×C42C22⋊D5311yescharacteristic
217C2×C42C22⋊D5311yescharacteristic
218C4⋊C8—1064no—
219C22⋊C8—534no—
220C22⋊C8—1062no—
221#97—1062no—
222C4×C8—1062no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—1062no—
231#95—1062no—
232C24C2×C22⋊D5311yescharacteristic
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—2122no—
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C22⋊Dic5311yescharacteristic
245C22×C4C22⋊Dic5311yescharacteristic
246C22×C4C22⋊Dic5311yescharacteristic
247C22×C4D4×D5312yes—
248C22×C4#1171994712yes—
249C22×C4C22×Dic5311yescharacteristic
250C22×C4C2×C22⋊D5311yescharacteristic
251M16—2122no—
252M16—2122no—
253C2×C8—1064no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—41no—
272C42—41no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊Dic5311yescharacteristic
278C23#1313318912yes—
279Q8—82no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
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—4242no—
307C2×C4—4242no—
308C2×C4—4242no—
309C2×C4D4×Dic5312yes—
310C2×C4#1313318511yescharacteristic
311C2×C4#1313319811yescharacteristic
312C8—2124no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22#1681050311yescharacteristic
321C22#1681059711yescharacteristic
322C22#1681059711yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—8482no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2027509911yescharacteristic
339C2#2027517411yescharacteristic
340C2#2027523811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2499C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2499D53
hypocenterC53#9444
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2499

Complements

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

NH ≅ G/Ntype
18. C53×#391C4semidirect
19. C53×#391C4semidirect
22. C53×#384C4semidirect
23. C53×#382C4semidirect
184. #391Dic53semidirect
185. #391Dic53semidirect
188. #384Dic53semidirect
189. #382Dic53semidirect
195. C53#9444semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202747922
#202732482
#202725682
C53×#24992
#944453