Finite Group Databasebeta

← #27297141

Subgroups — #27297141

Order 13568.

378 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 63 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#20276068C211yesmaximal, characteristic
2#20273739C211yesmaximal, characteristic
3#20273739C211yesmaximal, characteristic
4#20272739C211yesmaximal, characteristic
5#20272739C211yesmaximal, characteristic
6#20272663C211yesmaximal, characteristic
7C53×#2807C211yesmaximal, characteristic
8#16811249C2211yescharacteristic
9#16810958—21no—
10#16810958—21no—
11#16810496—21no—
12#16810390C2211yescharacteristic
13#16810389—21no—
14#16810359—21no—
15#16810359—21no—
16#16810347C2211yescharacteristic
17#16810347C2211yescharacteristic
18C53×#400—21no—
19C53×#399C2211yescharacteristic
20C53×#399C2211yescharacteristic
21C53×#398—21no—
22C53×#389—21no—
23C53×#389—21no—
24C53×#388—21no—
25C53×#385—21no—
26C53×#385C2211yescharacteristic
27#13133196—21no—
28#13133196—21no—
29#13133194—41no—
30#13133194—41no—
31C2×#11719919—41no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34#13133150—41no—
35#13133150—41no—
36#13133140—41no—
37#13133140—41no—
38C22⋊#88322—21no—
39C22⋊#88322—21no—
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×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×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—41no—
70C22×C4×C53—81no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—41no—
76C2×Dic106—21no—
77C2×Dic106—21no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—81no—
83C4⋊C4×C53—81no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—81no—
90C22⋊C4×C53Q1612yes—
91C42×C53—81no—
92C22⋊Dic53—22no—
93C22⋊Dic53—41no—
94C22⋊Dic53—41no—
95C22⋊Dic53—41no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98#11719919—41no—
99#11719919—41no—
100#11719919—82no—
101C2×#88322—41no—
102C2×#88322—41no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#12011yescharacteristic
108C23×C53C2×Q1611yescharacteristic
109C23×C53#13711yescharacteristic
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
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—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—81no—
134C2×Dic53—42no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141Dic106—81no—
142Dic106—81no—
143#88322—81no—
144#88322—81no—
145#12255—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—81no—
155C22×C53#45111yescharacteristic
156C22×C53#45611yescharacteristic
157C22×C53B(4,𝔽2)11yescharacteristic
158C4×C53—42no—
159C4×C53—42no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—81no—
163C4×C53—81no—
164C4×C53—81no—
165C4×C53—81no—
166Dic53—82no—
167Dic53—81no—
168Dic53—81no—
169#2807D5311yescharacteristic
170#2062—531no—
171#1614—531no—
172#1614—531no—
173#1352—531no—
174#1352—531no—
175#1326—531no—
176C2×C53—21no—
177C2×C53—21no—
178C2×C53—41no—
179C2×C53—41no—
180C2×C53#160511yescharacteristic
181C2×C53#219811yescharacteristic
182C2×C53#220011yescharacteristic
183#543—531no—
184#484—1061no—
185#484—1061no—
186#400—21no—
187#399C2×D5311yescharacteristic
188#399C2×D5311yescharacteristic
189#398—21no—
190#389—21no—
191#389—21no—
192#388—21no—
193#385—21no—
194#385—1061no—
195#385C2×D5311yescharacteristic
196#342—531no—
197#342—1061no—
198#328—1061no—
199#328—1061no—
200#323—531no—
201#323—531no—
202C53#1225511yescharacteristic, complemented, Sylow
203C23×C4C22⋊D5311yescharacteristic
204C22⋊Q8—1061no—
205C22⋊Q8—1061no—
206C22⋊Q8—2121no—
207C22⋊Q8—2121no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—2121no—
215C2×C4⋊C4C22⋊D5311yescharacteristic
216C2×C4⋊C4C22⋊D5311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4C22⋊D5311yescharacteristic
222C2×C22⋊C4C22×D5311yescharacteristic
223C2×C42—21no—
224#107—2121no—
225#107—2121no—
226#103—2121no—
227#103—2121no—
228C22⋊C8—1061no—
229C22⋊C8—1061no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—2121no—
236#95C22⋊D5311yescharacteristic
237#95C22⋊D5311yescharacteristic
238C24C2×C22⋊D5311yescharacteristic
239C2×Q8—1061no—
240C2×Q8—1061no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—81no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—2121no—
252C22×C4C2×C22⋊D5311yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254C2×C8—2121no—
255C2×C8—2121no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—4242no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C22⋊C4—21no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—81no—
273C22⋊C4—1062no—
274C22⋊C4—2121no—
275C22⋊C4—2121no—
276C22⋊C4—2121no—
277C22⋊C4#1171992512yes—
278C42—81no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23#1313318611yescharacteristic
284C23C2×#1171992511yescharacteristic
285C23#1313324811yescharacteristic
286Q8—4241no—
287Q8—4241no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
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—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—4241no—
313C2×C4—4241no—
314C2×C4—4241no—
315C2×C4—4241no—
316C2×C4—2122no—
317C2×C4—2121no—
318C2×C4—2121no—
319C8—4241no—
320C8—4241no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22#1681077911yescharacteristic
331C22#1681080611yescharacteristic
332C22#1681081011yescharacteristic
333C4—42no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—4242no—
342C4—4241no—
343C4—4241no—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2#2027368911yescharacteristic
349C2#2027671111yescharacteristic
350C2#2027672211yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2807C2
socleC22×C53B(4,𝔽2)
prime radicalC24×C53C2×D4
hypercenter#2807D53
hypocenterC53#12255
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2807

Complements

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

NH ≅ G/Ntype
202. C53#12255semidirect

Maximal subgroups

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

subgroupindex
#202760682
#202737392
#202727392
#202726632
C53×#28072
#1225553