Finite Group Databasebeta

← #27358992

Subgroups — #27358992

Order 13568.

906 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 207 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#20285856C212yesmaximal
2#20279700C211yesmaximal, characteristic
3#20273723C218yesmaximal
4#20273214C212yesmaximal
5#20273039C211yesmaximal, characteristic
6C53×#3502C211yesmaximal, characteristic
7C2×#13133217C2211yescharacteristic
8C2×#13133215—21no—
9#16811019—22no—
10#16810973—22no—
11#16810951C2218yes—
12#16810908—24no—
13#16810676C2212yes—
14#16810553C2214yes—
15#16810528—21no—
16#16810521—22no—
17C4×C4⋊Dic53C2211yescharacteristic
18#16810360C2218yes—
19#16810349C2214yes—
20Q82×C53—22no—
21C53×#557—22no—
22C53×#557—24no—
23C53×#544—21no—
24C53×#544C2214yes—
25C53×#533—22no—
26C53×#533C2212yes—
27C53×#519—21no—
28C2×C4×Q8×C53C2211yescharacteristic
29C22×Dic106—21no—
30#13133217—22no—
31#13133217—22no—
32#13133215—44no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2312yes—
36C2×#11719919—24no—
37C2×C4×Dic53—21no—
38#13133148—28no—
39C4⋊#88322—22no—
40#13133141—28no—
41C22⋊#88322C2314yes—
42C4×#88322—22no—
43#13133124—24no—
44#13133124C2312yes—
45C22×Q8×C53D411yescharacteristic
46C4⋊Q8×C53—22no—
47C4⋊Q8×C53—22no—
48C4⋊Q8×C53—24no—
49C4⋊Q8×C53—28no—
50C4⋊Q8×C53D412yes—
51C4⋊Q8×C53D412yes—
52#125×C53—24no—
53#125×C53—24no—
54#125×C53—28no—
55#125×C53—22no—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53—24no—
58C22⋊Q8×C53C2314yes—
59C4×Q8×C53—24no—
60C4×Q8×C53—24no—
61C4×Q8×C53—24no—
62C4×Q8×C53—24no—
63C4×Q8×C53—22no—
64C4×Q8×C53—22no—
65C4×Q8×C53—22no—
66C4×Q8×C53D414yes—
67#117×C53—24no—
68#117×C53—22no—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C4⋊C4×C53C2312yes—
71C2×C42×C53D412yes—
72C2×C42×C53C2311yescharacteristic
73C2×Q8×C53—24no—
74C2×Q8×C53—22no—
75C2×Q8×C53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53SD1614yes—
78C2×Q8×C53C2×D414yes—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2411yescharacteristic
84C22×Dic53—22no—
85C22×Dic53—22no—
86C2×Dic106—42no—
87C2×Dic106—42no—
88C2×Dic106—42no—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—28no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—24no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53C2×D414yes—
101C22⋊C4×C53—24no—
102C42×C53—28no—
103C42×C53—22no—
104C42×C53—22no—
105C42×C53—24no—
106C42×C53C2×D412yes—
107C42×C53C2×D412yes—
108C4⋊Dic53—24no—
109C4⋊Dic53—24no—
110C4⋊Dic53—41no—
111C4⋊Dic53—41no—
112#11719919—48no—
113C4×Dic53—42no—
114C2×#88322—24no—
115C23×C53C22×D411yescharacteristic
116Q8×C53—28no—
117Q8×C53—44no—
118Q8×C53—44no—
119Q8×C53—84no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—24no—
125C2×C4×C53—24no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53#12012yes—
130C2×C4×C53#12011yescharacteristic
131C2×C4×C53C2×SD1614yes—
132C2×C4×C53C2×SD1612yes—
133C2×C4×C53#13712yes—
134C2×C4×C53C22×D412yes—
135C2×Dic53—24no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
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—44no—
155C4×C53#44814yes—
156C4×C53#45612yes—
157Dic53—44no—
158Dic53—82no—
159#3502D5311yescharacteristic
160#3229—532no—
161#2546—531no—
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—22no—
171#557—22no—
172#557—24no—
173#544—21no—
174#544C2×D5314yes—
175#533—22no—
176#533C2×D5312yes—
177C2×C4⋊Q8—531no—
178C2×C4⋊Q8—1061no—
179#519—21no—
180C2×C4×Q8C2×D5311yescharacteristic
181#498—1062no—
182#487—1062no—
183#483—538no—
184#475—1064no—
185#431—532no—
186#398—534no—
187#391—1061no—
188#389—1062no—
189C4×C4⋊C4—531no—
190#328—538no—
191#324—534no—
192C53#1841011yescharacteristic, complemented, Sylow
193C22×Q8—1061no—
194C22×Q8C22⋊D5311yescharacteristic
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—24no—
198C4⋊Q8—28no—
199C4⋊Q8—1062no—
200C4⋊Q8—1062no—
201C4⋊Q8—2124no—
202C4⋊Q8C22⋊D5312yes—
203C4⋊Q8C22⋊D5312yes—
204#125—24no—
205#125—24no—
206#125—28no—
207#125—22no—
208C22⋊Q8—22no—
209C22⋊Q8—24no—
210C22⋊Q8C22×D5314yes—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C22⋊D5314yes—
219#117—24no—
220#117—22no—
221C2×C4⋊C4—532no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—1064no—
225C2×C4⋊C4C22⋊D5311yescharacteristic
226C2×C4⋊C4C22×D5312yes—
227C2×C42—1061no—
228C2×C42C22⋊D5312yes—
229C2×C42C22×D5311yescharacteristic
230#106—1068no—
231C4⋊C8—1062no—
232#103—1068no—
233C22⋊C8—534no—
234C4×C8—1062no—
235#95—532no—
236#95—1064no—
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—2122no—
242C2×Q8—2122no—
243C2×Q8—2122no—
244C2×Q8#1171992414yes—
245C2×Q8C2×C22⋊D5314yes—
246C22×C4—1062no—
247C22×C4—1062no—
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—24no—
256C4⋊C4—24no—
257C4⋊C4—28no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—1064no—
266C4⋊C4—1064no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C4⋊C4—2128no—
270C4⋊C4C2×C22⋊D5314yes—
271C22⋊C4—24no—
272C42—28no—
273C42—22no—
274C42—22no—
275C42—24no—
276C42—2122no—
277C42C2×C22⋊D5312yes—
278C42C2×C22⋊D5312yes—
279C23C22×C22⋊D5311yescharacteristic
280Q8—28no—
281Q8—44no—
282Q8—44no—
283Q8—84no—
284Q8—4244no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—1064no—
294C2×C4—2122no—
295C2×C4—2122no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4#1313318611yescharacteristic
300C2×C4#1313318612yes—
301C2×C4#1313318611yescharacteristic
302C2×C4C2×#1171992414yes—
303C2×C4C2×#1171992412yes—
304C2×C4#1313324812yes—
305C2×C4C22×C22⋊D5312yes—
306C8—2124no—
307C22—21no—
308C22—21no—
309C22C2×#1313318611yescharacteristic
310C22C22×#1171992411yescharacteristic
311C22C2×#1313324811yescharacteristic
312C4—21no—
313C4—22no—
314C4—24no—
315C4—44no—
316C4—42no—
317C4—44no—
318C4—4242no—
319C4—2124no—
320C4#1681076214yes—
321C4#1681080612yes—
322C2—21no—
323C2—21no—
324C2C2×#1681076211yescharacteristic
325C2C2×#1681080611yescharacteristic
326C2#2028412111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3502C2
socleC22×C53C2×#120
prime radicalC23×C53C22×D4
hypercenter#3502D53
hypocenterC53#18410
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

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
#202858562
#202797002
#202737232
#202732142
#202730392
C53×#35022
#1841053