Finite Group Databasebeta

← #27277501

Subgroups — #27277501

Order 13568.

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

normal subgroups only

This group has 526 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#20277121C211yesmaximal, characteristic
2#20274478C212yesmaximal
3#20272597C212yesmaximal, complemented
4#20272596C211yesmaximal, characteristic, complemented
5C53×#1295C211yesmaximal, characteristic, complemented
6C2×#13133175C411yescharacteristic
7C2×#13133175C411yescharacteristic
8C2×C42×D53C2211yescharacteristic
9C2×#13133129—22no—
10C2×#13133128C2212yes—
11#16810492C411yescharacteristic
12#16810492C411yescharacteristic
13#16810489C2211yescharacteristic
14#16810482—22no—
15#16810474—22no—
16#16810377—22no—
17C2×#97×C53C2212yes—
18C53×#382C2211yescharacteristic
19C22×C4×D53C2×C411yescharacteristic
20C22×C4×D53D412yes—
21C22×#88322—22no—
22#13133175—42no—
23#13133175—42no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×#11719921—21no—
29C2×#11719921—21no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34C42×D53—22no—
35C42×D53—21no—
36C42×D53—21no—
37#13133129—28no—
38#13133128—22no—
39#13133128—22no—
40#13133124—22no—
41C22×C8×C53—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C42×C53C2311yescharacteristic
45#97×C53—22no—
46#97×C53—22no—
47#95×C53—22no—
48C23×D53C22⋊C411yescharacteristic
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22×C411yescharacteristic
52C22×C4×C53C2×D412yes—
53C22×Dic53—21no—
54C22×Dic53—21no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C412yes—
57C2×C4×D53—22no—
58C2×C4×D53—21no—
59C2×C4×D53—21no—
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×C4×D53—22no—
63C2×C4×D53—24no—
64C2×C4×D53—42no—
65C2×C8×C53—24no—
66C2×C8×C53—42no—
67C2×C8×C53—42no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C42×C53#4112yes—
71C42×C53#4111yescharacteristic
72C42×C53#4111yescharacteristic
73#11719921—42no—
74#11719921—42no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77#11719919—44no—
78#11719919—44no—
79C4×Dic53—21no—
80C4×Dic53—21no—
81C4×Dic53—22no—
82C4×Dic53—22no—
83C4×Dic53—42no—
84C4×Dic53—42no—
85C2×#88322—24no—
86C2×#88322—42no—
87C2×#88322—42no—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53#10112yes—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53#11711yescharacteristic
103C2×C4×C53#11711yescharacteristic
104C2×C4×C53#12312yes—
105C2×C4×C53#12312yes—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53#10012yes—
115C2×Dic53#10414yes—
116C4×D53—24no—
117C4×D53—24no—
118C4×D53—44no—
119C4×D53—44no—
120C8×C53—44no—
121#88322—44no—
122#8721—531nomaximal, Sylow
123C22×C53#38111yescharacteristic
124C22×C53C2×#10011yescharacteristic
125C22×C53C2×#10111yescharacteristic
126C22×C53C2×#10412yes—
127C22×C53#42112yes—
128C2×D53—22no—
129C2×D53—24no—
130C2×D53—41no—
131C2×D53—42no—
132C2×D53—41no—
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—22no—
136C4×C53—22no—
137C4×C53—81no—
138C4×C53—81no—
139Dic53—22no—
140Dic53—24no—
141Dic53—81no—
142Dic53—81no—
143#2285—531no—
144#1749—532no—
145#1295—532no—
146#1295—531no—
147#1295D5311yescharacteristic, complemented
148C2×C53#152511yescharacteristic
149C2×C53#153112yes—
150C2×C53#153611yescharacteristic
151C2×C53#182111yescharacteristic
152C2×C53#182612yes—
153D53—42no—
154C2×#117—531no—
155C2×#117—531no—
156C22×C42—531no—
157C2×C22⋊C8—1062no—
158C2×#97—532no—
159C2×#97C2×D5312yes—
160#384—531no—
161#384—531no—
162#382—531no—
163#382C2×D5311yescharacteristic
164#380—1062no—
165C4×C4⋊C4—1062no—
166#336—1062no—
167C53#872111yescharacteristic, complemented, Sylow
168C23×C4—531no—
169C23×C4—532no—
170C22×C8—22no—
171C22×C8—1062no—
172#117—2122no—
173#117—2122no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—1062no—
177C2×C4⋊C4—1062no—
178C2×C4⋊C4C4×D5311yescharacteristic
179C2×C4⋊C4C4×D5311yescharacteristic
180C2×C22⋊C4—1061no—
181C2×C22⋊C4—1061no—
182C2×C42—531no—
183C2×C42—532no—
184C2×C42—1062no—
185C2×C42—1061no—
186C2×C42—1061no—
187C2×C42—1061no—
188C2×C42—1061no—
189C2×C42C22×D5311yescharacteristic
190C22⋊C8—1068no—
191#97—22no—
192#97—22no—
193#97—1062no—
194#97—1062no—
195#95—22no—
196#95—1062no—
197C24—531no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—531no—
201C22×C4—532no—
202C22×C4—1062no—
203C22×C4—1064no—
204C22×C4—1062no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4—1062no—
208C22×C4—1062no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—2122no—
212C22×C4C2×C4×D5311yescharacteristic
213C22×C4D4×D5312yes—
214C2×C8—24no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—1064no—
218C2×C8—2122no—
219C2×C8—2122no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—2121no—
223C4⋊C4—2121no—
224C4⋊C4—2124no—
225C4⋊C4—2124no—
226C22⋊C4—2122no—
227C22⋊C4—2122no—
228C42—1061no—
229C42—1061no—
230C42—1062no—
231C42—1062no—
232C42—2122no—
233C42—2122no—
234C42#1171994512yes—
235C42#1171994711yescharacteristic
236C42#1171994911yescharacteristic
237C23—532no—
238C23—1061no—
239C23—1062no—
240C23—1062no—
241C23—1061no—
242C23C22⋊C4×D5311yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—532no—
252C2×C4—534no—
253C2×C4—1064no—
254C2×C4—1064no—
255C2×C4—1061no—
256C2×C4—1062no—
257C2×C4—1061no—
258C2×C4—1062no—
259C2×C4—2124no—
260C2×C4—2124no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4#1313317411yescharacteristic
266C2×C4#1313317511yescharacteristic
267C2×C4#1313319912yes—
268C2×C4#1313320212yes—
269C8—44no—
270C8—2124no—
271C22—1062no—
272C22—1064no—
273C22—2122no—
274C22—2121no—
275C22—2121no—
276C22#1681048511yescharacteristic
277C22#100×D5311yescharacteristic
278C22#101×D5311yescharacteristic
279C22#104×D5312yes—
280C22#1681063812yes—
281C4—21no—
282C4—21no—
283C4—22no—
284C4—22no—
285C4—81no—
286C4—81no—
287C4—1062no—
288C4—1064no—
289C4—4241no—
290C4—4241no—
291C2—2122no—
292C2#2027330511yescharacteristic
293C2#2027332812yes—
294C2#2027334811yescharacteristic
295C2#2027479411yescharacteristic
296C2#2027481312yes—
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1295C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1295D53
hypocenterC53#8721
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
3. #20272597C2semidirect
4. #20272596C2semidirect
5. C53×#1295C2semidirect
147. #1295D53semidirect
167. C53#8721semidirect

Maximal subgroups

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

subgroupindex
#202771212
#202744782
#202725972
#202725962
C53×#12952
#872153