Finite Group Databasebeta

← #27277558

Subgroups — #27277558

Order 13568.

418 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277318C211yesmaximal, characteristic
2C2×#16810349C211yesmaximal, characteristic
3#20272562C212yesmaximal
4#20272561C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7#16810509—21no—
8#16810509—21no—
9#16810509—21no—
10#16810509—21no—
11#16810486C411yescharacteristic
12#16810486C411yescharacteristic
13#16810483C411yescharacteristic
14#16810483C411yescharacteristic
15C4×C22⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17#16810349—21no—
18#16810349—21no—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53C2212yes—
21C53×#324—21no—
22C53×#324—21no—
23C22×#88322—22no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C22⋊Dic53C2×C411yescharacteristic
27C2×C22⋊Dic53C2×C411yescharacteristic
28C2×C22⋊Dic53C2×C411yescharacteristic
29C2×C22⋊Dic53C2×C411yescharacteristic
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C22⋊#88322—22no—
33C22⋊#88322—22no—
34#13133124—21no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C22×C8×C53—22no—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53—22no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C4⋊C4×C53D411yescharacteristic
52C22⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54C24×C53C22×C411yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—81no—
62C22×C4×C53—81no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4112yes—
70C22×C4×C53#4112yes—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C2×C8×C53—42no—
76C2×C8×C53—82no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C22⋊Dic53—22no—
82C22⋊Dic53—22no—
83C22⋊Dic53—22no—
84C22⋊Dic53—22no—
85C4⋊Dic53—81no—
86C4⋊Dic53—81no—
87C4×Dic53—81no—
88C4×Dic53—81no—
89C2×#88322—42no—
90C2×#88322—82no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53#11711yescharacteristic
95C23×C53#11711yescharacteristic
96C23×C53#13112yes—
97C23×C53#13112yes—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—81no—
115C2×C4×C53—81no—
116C2×C4×C53#12312yes—
117C2×C4×C53#12312yes—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C8×C53—82no—
131#88322—82no—
132#8737—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53#38111yescharacteristic
142C22×C53#40811yescharacteristic
143C22×C53#40811yescharacteristic
144C22×C53#43312yes—
145C22×C53#43312yes—
146C4×C53—22no—
147C4×C53—22no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—41no—
152Dic53—81no—
153Dic53—81no—
154Dic53—81no—
155Dic53—81no—
156#2313—531no—
157C2×#324—531no—
158C2×#324D5311yescharacteristic
159#1282—532no—
160#1282—532no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—22no—
164C2×C53—22no—
165C2×C53#181811yescharacteristic
166C2×C53#182811yescharacteristic
167C2×C53#183011yescharacteristic
168C22×C4⋊C4C2×D5311yescharacteristic
169C2×C22⋊C8—532no—
170C2×C22⋊C8C2×D5312yes—
171#387—1061no—
172#387—1061no—
173#387—1061no—
174#387—1061no—
175#381—531no—
176#381—531no—
177#380—531no—
178#380—531no—
179C4×C22⋊C4—531no—
180C4×C22⋊C4—531no—
181#324—21no—
182#324—21no—
183#324—1061no—
184#324—1061no—
185C53#873711yescharacteristic, complemented, Sylow
186C23×C4C4×D5311yescharacteristic
187C23×C4C4×D5311yescharacteristic
188C23×C4C22×D5311yescharacteristic
189C22×C8—22no—
190C22×C8—1062no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4D21211yescharacteristic
196C2×C4⋊C4D21211yescharacteristic
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C4⋊C4C22⋊D5311yescharacteristic
199C2×C22⋊C4—531no—
200C2×C22⋊C4—531no—
201C2×C22⋊C4—531no—
202C2×C22⋊C4—531no—
203C2×C42—1061no—
204C2×C42—1061no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—1062no—
208C22⋊C8—1062no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217C24C2×C4×D5311yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—1061no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4—1061no—
230C22×C4#1171992111yescharacteristic
231C22×C4#1171992111yescharacteristic
232C22×C4#1171992111yescharacteristic
233C22×C4#1171992111yescharacteristic
234C22×C4C2×D21211yescharacteristic
235C22×C4#1171994712yes—
236C22×C4#1171994712yes—
237C22×C4C2×C22⋊D5311yescharacteristic
238C2×C8—42no—
239C2×C8—82no—
240C2×C8—4242no—
241C2×C8—2122no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—4241no—
247C4⋊C4—4241no—
248C22⋊C4—1062no—
249C22⋊C4—1062no—
250C22⋊C4—1062no—
251C22⋊C4—1062no—
252C42—4241no—
253C42—4241no—
254C23—41no—
255C23—41no—
256C23C2×#1171992111yescharacteristic
257C23#1313317311yescharacteristic
258C23#1313317311yescharacteristic
259C23#1313322312yes—
260C23#1313322512yes—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4#1313319812yes—
292C2×C4#1313320112yes—
293C8—82no—
294C8—4242no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—42no—
303C22#1681048411yescharacteristic
304C22#1681058911yescharacteristic
305C22#1681059111yescharacteristic
306C22#1681068112yes—
307C22#1681068112yes—
308C4—22no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—4241no—
315C4—4241no—
316C4—4241no—
317C4—4241no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2027478411yescharacteristic
323C2#2027482111yescharacteristic
324C2#2027482711yescharacteristic
subgroupG/H
centerC22#16810484
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC22×C53#381
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#8737
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
185. C53#8737semidirect

Maximal subgroups

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

subgroupindex
#202773182
C2×#168103492
#202725622
#202725612
C2×C53×#3242
#873753