Finite Group Databasebeta

← #27358991

Subgroups — #27358991

Order 13568.

906 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 191 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#20288525C211yesmaximal, characteristic
2#20285859C212yesmaximal
3#20273722C218yesmaximal
4#20273213C212yesmaximal
5#20273039C211yesmaximal, characteristic
6C53×#2546C211yesmaximal, characteristic
7Q8×Dic106—22no—
8#16811352—22no—
9#16811349—24no—
10#16811254—21no—
11#16811254C2214yes—
12#16811180—22no—
13#16811180C2212yes—
14#16811101—21no—
15#16811019—22no—
16#16810975—22no—
17#16810953C2218yes—
18#16810905—24no—
19#16810676C2212yes—
20#16810359C2218yes—
21#16810349C2214yes—
22C2×C4⋊Q8×C53—21no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4×Q8×C53C2211yescharacteristic
25C53×#398C2214yes—
26C53×#391—21no—
27C53×#389—22no—
28C4×C4⋊C4×C53C2211yescharacteristic
29#13133217—22no—
30#13133217—22no—
31#13133217D412yes—
32#13133217D412yes—
33#13133216—24no—
34#13133216—28no—
35#13133209—24no—
36#13133209—22no—
37#13133208—24no—
38#13133208—28no—
39#13133196—22no—
40#13133196—24no—
41#13133196C2314yes—
42Q8×Dic53—24no—
43Q8×Dic53—24no—
44C4×Dic106—24no—
45C4×Dic106—24no—
46C4×Dic106—22no—
47C4×Dic106D414yes—
48#13133176—24no—
49#13133176—22no—
50#13133148—28no—
51C4⋊#88322—22no—
52#13133140—28no—
53C22⋊#88322C2314yes—
54C4×#88322—22no—
55C22×Q8×C53—21no—
56C22×Q8×C53D411yescharacteristic
57C4⋊Q8×C53—22no—
58C4⋊Q8×C53—22no—
59C4⋊Q8×C53—44no—
60C4×Q8×C53—22no—
61C4×Q8×C53—22no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—24no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53C2312yes—
67C2×C4⋊C4×C53C2312yes—
68C2×C42×C53—21no—
69C2×C42×C53D412yes—
70C2×C42×C53C2311yescharacteristic
71#95×C53—24no—
72#95×C53C2312yes—
73C2×Q8×C53—22no—
74C2×Q8×C53—22no—
75C2×Q8×C53—42no—
76C2×Q8×C53—42no—
77C2×Q8×C53—42no—
78C2×Q8×C53SD1614yes—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D412yes—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2411yescharacteristic
86C2×Dic106—22no—
87C2×Dic106—24no—
88C2×Dic106C2×D414yes—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—24no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—48no—
96C42×C53—22no—
97C42×C53—22no—
98C42×C53—42no—
99C42×C53C2×D412yes—
100C42×C53C2×D412yes—
101C22⋊Dic53—24no—
102C4⋊Dic53—24no—
103C4⋊Dic53—24no—
104C4⋊Dic53—24no—
105C4⋊Dic53—22no—
106C4⋊Dic53—22no—
107C4⋊Dic53—22no—
108C4⋊Dic53C2×D414yes—
109#11719919—28no—
110#11719919—24no—
111#11719919—24no—
112C4×Dic53—28no—
113C4×Dic53—24no—
114C2×#88322—24no—
115C23×C53C22×D411yescharacteristic
116Q8×C53—28no—
117Q8×C53—84no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—24no—
122C2×C4×C53—22no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53#12012yes—
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53C2×SD1614yes—
134C2×C4×C53C2×SD1612yes—
135C2×C4×C53#13712yes—
136C2×C4×C53C22×D412yes—
137C2×Dic53—24no—
138C2×Dic53—24no—
139Dic106—44no—
140Dic106—44no—
141Dic106—84no—
142#88322—44no—
143#18410—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53C2×#12011yescharacteristic
147C22×C53C22×SD1611yescharacteristic
148C22×C53C2×#13711yescharacteristic
149C4×C53—24no—
150C4×C53—21no—
151C4×C53—22no—
152C4×C53—44no—
153C4×C53—42no—
154C4×C53—82no—
155C4×C53#44814yes—
156C4×C53#45612yes—
157Dic53—44no—
158Dic53—44no—
159#3502—531no—
160#3229—532no—
161#2546D5311yescharacteristic
162#1611—538no—
163#1503—532no—
164#1461—531no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53C2×#44811yescharacteristic
168C2×C53C2×#45611yescharacteristic
169C2×C53#302811yescharacteristic
170Q82—1062no—
171#557—1064no—
172#557—1062no—
173#544—534no—
174#544—1061no—
175#533—532no—
176#533—1062no—
177C2×C4⋊Q8—21no—
178C2×C4⋊Q8C2×D5311yescharacteristic
179#519—1061no—
180C2×C4×Q8C2×D5311yescharacteristic
181#498—1062no—
182#487—1062no—
183#483—538no—
184#475—1064no—
185#431—532no—
186#398C2×D5314yes—
187#391—21no—
188#389—22no—
189C4×C4⋊C4C2×D5311yescharacteristic
190#328—538no—
191#324—534no—
192C53#1841011yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8C22⋊D5311yescharacteristic
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—44no—
198C4⋊Q8—532no—
199C4⋊Q8—532no—
200C4⋊Q8—1062no—
201C4⋊Q8—1062no—
202C4⋊Q8—1068no—
203C4⋊Q8—1064no—
204#125—1064no—
205#125—1068no—
206#125—1064no—
207#125—1062no—
208C22⋊Q8—534no—
209C22⋊Q8—1062no—
210C22⋊Q8—1064no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—534no—
214C4×Q8—1064no—
215C4×Q8—1064no—
216C4×Q8—1064no—
217C4×Q8—1064no—
218C4×Q8—1062no—
219#117—1062no—
220#117—1064no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22⋊D5311yescharacteristic
225C2×C4⋊C4C22×D5312yes—
226C2×C4⋊C4C22×D5312yes—
227C2×C42—21no—
228C2×C42C22⋊D5312yes—
229C2×C42C22×D5311yescharacteristic
230#106—1068no—
231C4⋊C8—1062no—
232#103—1068no—
233C22⋊C8—534no—
234C4×C8—1062no—
235#95—24no—
236#95C22×D5312yes—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—534no—
243C2×Q8—1064no—
244C2×Q8—1062no—
245C2×Q8#1171992314yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4C2×C22⋊D5311yescharacteristic
249C22×C4C2×C22⋊D5312yes—
250C22×C4C2×C22⋊D5312yes—
251C22×C4C2×C22⋊D5311yescharacteristic
252C22×C4C23×D5311yescharacteristic
253C2×C8—1064no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—48no—
261C4⋊C4—534no—
262C4⋊C4—1064no—
263C4⋊C4—1064no—
264C4⋊C4—1064no—
265C4⋊C4—1068no—
266C4⋊C4—1062no—
267C4⋊C4—1062no—
268C4⋊C4—1064no—
269C4⋊C4—1062no—
270C4⋊C4—1064no—
271C22⋊C4—1064no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—1068no—
276C42—1064no—
277C42D4×D5312yes—
278C42D4×D5312yes—
279C23C22×C22⋊D5311yescharacteristic
280Q8—28no—
281Q8—84no—
282Q8—4244no—
283Q8—2124no—
284Q8—2124no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—1064no—
298C2×C4—1064no—
299C2×C4#1313318511yescharacteristic
300C2×C4#1313318512yes—
301C2×C4#1313318511yescharacteristic
302C2×C4C2×#1171992314yes—
303C2×C4C2×#1171992312yes—
304C2×C4#1313324712yes—
305C2×C4C2×D4×D5312yes—
306C8—2124no—
307C22—21no—
308C22—21no—
309C22C2×#1313318511yescharacteristic
310C22C22×#1171992311yescharacteristic
311C22C2×#1313324711yescharacteristic
312C4—21no—
313C4—22no—
314C4—24no—
315C4—44no—
316C4—42no—
317C4—82no—
318C4—2124no—
319C4—2124no—
320C4#1681076114yes—
321C4#1681080512yes—
322C2—21no—
323C2—21no—
324C2C2×#1681076111yescharacteristic
325C2C2×#1681080511yescharacteristic
326C2#2028412011yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2546C2
socleC22×C53C2×#120
prime radicalC23×C53C22×D4
hypercenter#2546D53
hypocenterC53#18410
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
192. C53#18410semidirect

Maximal subgroups

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

subgroupindex
#202885252
#202858592
#202737222
#202732132
#202730392
C53×#25462
#1841053