Finite Group Databasebeta

← #27297203

Subgroups — #27297203

Order 13568.

362 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 59 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#20275882C211yesmaximal, characteristic
2#20273739C211yesmaximal, characteristic
3#20273705C211yesmaximal, characteristic
4#20272739C211yesmaximal, characteristic
5#20272715C211yesmaximal, characteristic
6#20272645C211yesmaximal, characteristic
7C53×#2771C211yesmaximal, characteristic
8#16811249C2211yescharacteristic
9#16810958—21no—
10#16810953—21no—
11#16810486—21no—
12#16810390C2211yescharacteristic
13#16810389—21no—
14#16810359—21no—
15#16810359—21no—
16#16810347C2211yescharacteristic
17#16810347C2211yescharacteristic
18C53×#400—21no—
19C53×#400—21no—
20C53×#399C2211yescharacteristic
21C53×#397C2211yescharacteristic
22C53×#389—21no—
23C53×#388—21no—
24C53×#387—21no—
25C53×#385—21no—
26C53×#381C2211yescharacteristic
27#13133196—21no—
28#13133196—21no—
29#13133194—41no—
30#13133194—41no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33#13133150—41no—
34#13133148—41no—
35#13133140—41no—
36#13133140—41no—
37C22⋊#88322—21no—
38C22⋊#88322—21no—
39#13133124—41no—
40#13133124—41no—
41C23×C4×C53D411yescharacteristic
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53C2311yescharacteristic
52C2×C42×C53—21no—
53#95×C53—21no—
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×C53D411yescharacteristic
61#95×C53D411yescharacteristic
62C24×C53C2×D411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—41no—
78C2×Dic106—21no—
79C2×Dic106—21no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C42×C53—81no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94C22⋊Dic53—41no—
95C22⋊Dic53—41no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98#11719919—41no—
99#11719919—41no—
100C2×#88322—41no—
101C2×#88322—41no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53#12011yescharacteristic
107C23×C53#13511yescharacteristic
108C23×C53#13711yescharacteristic
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
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—41no—
131C2×C4×C53—41no—
132C2×C4×C53—81no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—81no—
136C2×Dic53—81no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139C2×Dic53—81no—
140Dic106—81no—
141Dic106—81no—
142#88322—81no—
143#88322—81no—
144#12265—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×C53#45211yescharacteristic
155C22×C53#45511yescharacteristic
156C22×C53B(4,𝔽2)11yescharacteristic
157C4×C53—42no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—81no—
161C4×C53—81no—
162C4×C53—81no—
163C4×C53—81no—
164C4×C53—81no—
165Dic53—81no—
166Dic53—81no—
167Dic53—161no—
168#2771D5311yescharacteristic
169#2030—531no—
170#1614—531no—
171#1608—531no—
172#1352—531no—
173#1344—531no—
174#1317—531no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53—41no—
178C2×C53—41no—
179C2×C53#160611yescharacteristic
180C2×C53#219811yescharacteristic
181C2×C53#220111yescharacteristic
182#543—531no—
183#484—1061no—
184#483—1061no—
185#400—21no—
186#400—21no—
187#399C2×D5311yescharacteristic
188#397C2×D5311yescharacteristic
189#389—21no—
190#388—21no—
191#387—21no—
192#385—21no—
193#381—1061no—
194#381C2×D5311yescharacteristic
195#342—531no—
196#342—1061no—
197#328—1061no—
198#328—1061no—
199#323—531no—
200#323—531no—
201C53#1226511yescharacteristic, complemented, Sylow
202C23×C4C22⋊D5311yescharacteristic
203C22⋊Q8—1061no—
204C22⋊Q8—1061no—
205C22⋊Q8—2121no—
206C22⋊Q8—2121no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22⋊D5311yescharacteristic
212C2×C4⋊C4C22⋊D5311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4C22⋊D5311yescharacteristic
218C2×C22⋊C4C22×D5311yescharacteristic
219C2×C42—21no—
220#107—2121no—
221#106—2121no—
222#103—2121no—
223#103—2121no—
224C22⋊C8—1061no—
225C22⋊C8—1061no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—2121no—
234#95—2121no—
235#95C22⋊D5311yescharacteristic
236#95C22⋊D5311yescharacteristic
237C24C2×C22⋊D5311yescharacteristic
238C2×Q8—1061no—
239C2×Q8—1061no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—2121no—
253C22×C4C2×C22⋊D5311yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255C2×C8—2121no—
256C2×C8—2121no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—2121no—
273C22⋊C4—2121no—
274C22⋊C4—2121no—
275C22⋊C4—2121no—
276C42—81no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23#1313318611yescharacteristic
282C23#1313323711yescharacteristic
283C23#1313324711yescharacteristic
284Q8—4241no—
285Q8—4241no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
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—42no—
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—81no—
310C2×C4—4241no—
311C2×C4—4241no—
312C2×C4—4241no—
313C2×C4—4241no—
314C2×C4—4241no—
315C2×C4—2121no—
316C2×C4—2121no—
317C8—4241no—
318C8—4241no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22#1681078511yescharacteristic
329C22#1681080111yescharacteristic
330C22#1681081011yescharacteristic
331C4—42no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—8481no—
340C4—4241no—
341C4—4241no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2#2027369511yescharacteristic
347C2#2027671111yescharacteristic
348C2#2027672811yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2771C2
socleC22×C53B(4,𝔽2)
prime radicalC24×C53C2×D4
hypercenter#2771D53
hypocenterC53#12265
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2771

Complements

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

NH ≅ G/Ntype
201. C53#12265semidirect

Maximal subgroups

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

subgroupindex
#202758822
#202737392
#202737052
#202727392
#202727152
#202726452
C53×#27712
#1226553