Finite Group Databasebeta

← #27297044

Subgroups — #27297044

Order 13568.

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

normal subgroups only

This group has 350 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#20276060C211yesmaximal, characteristic
2#20273722C211yesmaximal, characteristic
3#20273688C211yesmaximal, characteristic
4#20272745C211yesmaximal, characteristic
5#20272721C211yesmaximal, characteristic
6#20272661C211yesmaximal, characteristic
7C53×#2792C211yesmaximal, characteristic
8#16811254C2211yescharacteristic
9#16810953—21no—
10#16810779—21no—
11#16810663—21no—
12#16810392C2211yescharacteristic
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810349C2211yescharacteristic
17#16810349C2211yescharacteristic
18C53×#398—21no—
19C53×#398C2211yescharacteristic
20C53×#395—21no—
21C53×#395C2211yescharacteristic
22C53×#389—21no—
23C53×#389—21no—
24C53×#387—21no—
25C53×#387—21no—
26C53×#384C2211yescharacteristic
27C2×#11719925—41no—
28C2×#11719917—21no—
29#13133216—41no—
30#13133208—41no—
31#13133196—21no—
32#13133196—21no—
33#13133176—21no—
34#13133150—41no—
35#13133148—41no—
36#13133148—41no—
37#13133141—41no—
38#13133140—41no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C22×Q8×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×C53—21no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53C2311yescharacteristic
52C2×C42×C53—21no—
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×C53D411yescharacteristic
61#95×C53D411yescharacteristic
62C2×Q8×C53—41no—
63C2×Q8×C53—81no—
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×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C2×Dic106—21no—
74C2×Dic106—21no—
75C4⋊C4×C53—24no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—81no—
82C4⋊C4×C53—81no—
83C4⋊C4×C53—81no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—81no—
89C22⋊Dic53—41no—
90#11719925—82no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95#11719919—41no—
96#11719919—41no—
97C4×Dic53—41no—
98#11719917—22no—
99#11719917—41no—
100C2×#88322—41no—
101C2×#88322—41no—
102C2×#88322—41no—
103C23×C53#12011yescharacteristic
104Q8×C53—42no—
105Q8×C53—81no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—41no—
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—41no—
129C2×C4×C53—41no—
130C2×C4×C53#13511yescharacteristic
131C2×C4×C53#13711yescharacteristic
132C2×Dic53—41no—
133C2×Dic53—41no—
134Dic106—81no—
135Dic106—81no—
136#88322—42no—
137#88322—81no—
138#88322—81no—
139#12240—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53#45211yescharacteristic
143C22×C53#45611yescharacteristic
144C22×C53B(4,𝔽2)11yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—42no—
148C4×C53—42no—
149C4×C53—42no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—81no—
153C4×C53—81no—
154C4×C53—81no—
155Dic53—81no—
156Dic53—81no—
157#2792D5311yescharacteristic
158#2060—531no—
159#1611—531no—
160#1605—531no—
161#1354—531no—
162#1346—531no—
163#1325—531no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53#160311yescharacteristic
167C2×C53#220011yescharacteristic
168C2×C53#220511yescharacteristic
169#544—531no—
170#483—1061no—
171#451—1061no—
172#428—1061no—
173#398—21no—
174#398C2×D5311yescharacteristic
175#395—21no—
176#395C2×D5311yescharacteristic
177#389—21no—
178#389—21no—
179#387—21no—
180#387—21no—
181#384C2×D5311yescharacteristic
182#343—531no—
183#339—1061no—
184#328—1061no—
185#328—1061no—
186#324—531no—
187#324—531no—
188C53#1224011yescharacteristic, complemented, Sylow
189C22×Q8C22⋊D5311yescharacteristic
190C2×Q16—2121no—
191C2×M16—1061no—
192C4⋊Q8—2121no—
193#125—2121no—
194C22⋊Q8—1061no—
195C22⋊Q8—1061no—
196#117—1061no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C22⋊D5311yescharacteristic
205C2×C4⋊C4C22⋊D5311yescharacteristic
206C2×C4⋊C4C22×D5311yescharacteristic
207C2×C42—21no—
208C2×C42C22⋊D5311yescharacteristic
209#107—2121no—
210#106—2121no—
211#106—2121no—
212#103—2121no—
213#103—2121no—
214C22⋊C8—1061no—
215C22⋊C8—1061no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C22⋊D5311yescharacteristic
223#95C22⋊D5311yescharacteristic
224C2×Q8—41no—
225C2×Q8—81no—
226C2×Q8—1061no—
227C2×Q8—1061no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4C2×C22⋊D5311yescharacteristic
235C22×C4C2×C22⋊D5311yescharacteristic
236C22×C4C2×C22⋊D5311yescharacteristic
237Q16—4242no—
238M16—1062no—
239M16—2121no—
240C2×C8—2121no—
241C2×C8—2121no—
242C2×C8—2121no—
243C4⋊C4—24no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—1061no—
255C4⋊C4—1061no—
256C4⋊C4—1061no—
257C4⋊C4—1061no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C22⋊C4—2121no—
261C42—21no—
262C42—21no—
263C42—81no—
264C42—2121no—
265C23#1313318611yescharacteristic
266Q8—42no—
267Q8—81no—
268Q8—4241no—
269Q8—4241no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4#1313323711yescharacteristic
297C2×C4#1313324711yescharacteristic
298C8—4241no—
299C8—4241no—
300C8—2122no—
301C22—21no—
302C22—21no—
303C22#1681078511yescharacteristic
304C22#1681080611yescharacteristic
305C22#1681081011yescharacteristic
306C4—21no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—4241no—
317C4—4241no—
318C2—21no—
319C2—21no—
320C2#2027367811yescharacteristic
321C2#2027672211yescharacteristic
322C2#2027674511yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2792C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2792D53
hypocenterC53#12240
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2792

Complements

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

NH ≅ G/Ntype
188. C53#12240semidirect

Maximal subgroups

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

subgroupindex
#202760602
#202737222
#202736882
#202727452
#202727212
#202726612
C53×#27922
#1224053