Finite Group Databasebeta

← #27293692

Subgroups — #27293692

Order 13568.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279023C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272644C214yesmaximal, complemented
4C53×#1466C211yesmaximal, characteristic
5C2×#13133198C412yes—
6C22×C4⋊Dic53C411yescharacteristic
7C2×C22⋊#88322—21no—
8#16810553—21no—
9#16810548—21no—
10#16810521—21no—
11#16810484C2212yes—
12#16810483—21no—
13#16810483C411yescharacteristic
14#16810349—21no—
15#16810347C2212yes—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#432—21no—
18C53×#323—21no—
19C53×#323C2212yes—
20C23×Dic53C2×C412yes—
21C22×#88322—41no—
22#13133198—44no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53—41no—
28C2×#11719919—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53C2×C412yes—
31C22⋊#88322—22no—
32C22⋊#88322—41no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C4×D4×C53—41no—
40C4×D4×C53—41no—
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53D411yescharacteristic
45C4⋊C8×C53—41no—
46C22⋊C8×C53—22no—
47#97×C53—41no—
48C24×C53C22⋊C411yescharacteristic
49C24×C53C22×C411yescharacteristic
50C2×D4×C53—22no—
51C2×D4×C53—22no—
52C2×D4×C53—41no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—24no—
67C22×Dic53—42no—
68C22×Dic53—42no—
69C2×C8×C53—42no—
70C4⋊C4×C53—41no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C42×C53—21no—
74C42×C53—21no—
75C22⋊Dic53—44no—
76C22⋊Dic53—42no—
77C4⋊Dic53—42no—
78C4⋊Dic53—42no—
79C4⋊Dic53—41no—
80C4⋊Dic53—41no—
81#11719919—44no—
82C2×#88322—42no—
83C2×#88322—82no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—41no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53#10012yes—
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53#13112yes—
95D4×C53—44no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—81no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—84no—
119C8×C53—82no—
120#88322—82no—
121#11558—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—41no—
133C22×C53—81no—
134C22×C53#40811yescharacteristic
135C22×C53C2×#9911yescharacteristic
136C22×C53C2×#10011yescharacteristic
137C4×C53—21no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53#35212yes—
143Dic53—82no—
144Dic53—82no—
145#2486—531no—
146#1466D5311yescharacteristic
147#1459—531no—
148#1317—534no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—42no—
152C2×C53—42no—
153C2×C53#147111yescharacteristic
154C2×C53C2×#35211yescharacteristic
155C2×C53#212311yescharacteristic
156C2×#123—532no—
157C2×C4×D4C2×D5311yescharacteristic
158C22×C4⋊C4—531no—
159#432—21no—
160C2×C22⋊C8—1061no—
161#398—1061no—
162#397—1061no—
163#389—1061no—
164#381—532no—
165#380—531no—
166#380—1061no—
167#324—1061no—
168#323—21no—
169#323—532no—
170#323C2×D5312yes—
171C53#1155811yescharacteristic, complemented, Sylow
172C22×D4C4×D5311yescharacteristic
173C23×C4—532no—
174C23×C4C4×D5311yescharacteristic
175C23×C4D21211yescharacteristic
176C22×C8—2121no—
177#123—2124no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4—1062no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4—2121no—
185C2×C4⋊C4—2121no—
186C2×C4⋊C4D21211yescharacteristic
187C2×C22⋊C4—532no—
188C2×C22⋊C4—1062no—
189C2×C22⋊C4C22⋊D5311yescharacteristic
190C2×C22⋊C4C22×D5311yescharacteristic
191C2×C42C22⋊D5311yescharacteristic
192C4⋊C8—41no—
193C22⋊C8—22no—
194C22⋊C8—1062no—
195C22⋊C8—2121no—
196#97—41no—
197#95—1062no—
198#95—1062no—
199#95—1061no—
200C24#1171992111yescharacteristic
201C24C2×C4×D5311yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—1064no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—2122no—
215C22×C4—2122no—
216C22×C4#1171992111yescharacteristic
217C22×C4#1171992111yescharacteristic
218C22×C4#1171992111yescharacteristic
219C22×C4C2×D21211yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221C2×C8—42no—
222C2×C8—4242no—
223C2×C8—2122no—
224C4⋊C4—41no—
225C4⋊C4—2122no—
226C4⋊C4—2124no—
227C4⋊C4—2122no—
228C4⋊C4—2121no—
229C4⋊C4—2121no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—2124no—
233C22⋊C4—2122no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#1313313312yes—
245C23C2×#1171992111yescharacteristic
246C23#1313322512yes—
247D4—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—4244no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4—2122no—
268C2×C4—2122no—
269C2×C4#1313313111yescharacteristic
270C2×C4#1313313111yescharacteristic
271C8—82no—
272C8—4242no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#1681059111yescharacteristic
286C22C2×#1313313111yescharacteristic
287C22C2×#1313313311yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—4242no—
294C4—4242no—
295C4#1681041412yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—42no—
300C2#2027307611yescharacteristic
301C2C2×#1681041411yescharacteristic
302C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC24×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1466C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1466D53
hypocenterC53#11558
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #20272644C2semidirect
171. C53#11558semidirect

Maximal subgroups

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

subgroupindex
#202790232
#202730332
#202726442
C53×#14662
#1155853