Finite Group Databasebeta

← #27293186

Subgroups — #27293186

Order 13568.

526 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 93 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#20277300C211yesmaximal, characteristic
2#20273026C212yesmaximal
3#20272645C211yesmaximal, characteristic
4#20272644C212yesmaximal
5C53×#1317C211yesmaximal, characteristic
6C23×Dic106C411yescharacteristic
7C2×#13133173C411yescharacteristic
8C2×#13133173C2211yescharacteristic
9#16810673—22no—
10#16810527—22no—
11#16810527—22no—
12#16810486C411yescharacteristic
13#16810484C411yescharacteristic
14#16810484C2211yescharacteristic
15#16810348—22no—
16#16810347C2212yes—
17C53×#381C2211yescharacteristic
18C53×#323C2212yes—
19C22×Dic106—21no—
20C22×Dic106—21no—
21C22×Dic106—22no—
22C22×Dic106—42no—
23C22×Dic106—42no—
24C22×Dic106C814yescomplemented
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27#13133173—42no—
28#13133173—42no—
29C2×#11719919—21no—
30C2×#11719919D412yes—
31C2×C22⋊Dic53C2×C411yescharacteristic
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C4×Dic53—21no—
34C2×C4×Dic53D412yes—
35#13133146—42no—
36C22⋊#88322—22no—
37C8×Dic53—42no—
38#13133124—22no—
39#13133124—22no—
40#13133124—22no—
41C23×C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C22⋊C8×C53—22no—
45#95×C53—22no—
46C24×C53C22×C411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—41no—
51C22×C4×C53—41no—
52C22×C4×C53C2×C812yes—
53C22×C4×C53C2×D412yes—
54C22×Dic53—22no—
55C22×Dic53—22no—
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53—21no—
59C22×Dic53—41no—
60C22×Dic53—41no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C412yes—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C2×C812yes—
65C22×Dic53C2×C812yes—
66C22×Dic53M1612yes—
67C2×Dic106—24no—
68C2×Dic106—24no—
69C2×Dic106—42no—
70C2×Dic106—42no—
71C2×Dic106—48no—
72C2×Dic106—42no—
73C2×Dic106—42no—
74C2×Dic106—48no—
75C2×C8×C53—42no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80#11719919—42no—
81#11719919—42no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—42no—
85C2×#88322—42no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53C22×C811yescharacteristic
91C23×C53C2×M1611yescharacteristic
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—42no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—24no—
104C2×Dic53—24no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—42no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—41no—
114C2×Dic53C22⋊C812yes—
115C2×Dic53C22⋊C812yes—
116C2×Dic53#9912yes—
117C2×Dic53#10012yes—
118Dic106—48no—
119Dic106—48no—
120C8×C53—82no—
121#88322—82no—
122#11428—531nomaximal, Sylow
123C22×C53—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—41no—
127C22×C53—41no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53C2×C22⋊C811yescharacteristic
131C22×C53C2×#9911yescharacteristic
132C22×C53C2×#10011yescharacteristic
133C4×C53—42no—
134C4×C53—42no—
135C4×C53—81no—
136Dic53—22no—
137Dic53—42no—
138Dic53—42no—
139Dic53—81no—
140Dic53#32314yes—
141#2310—531no—
142#1458—532no—
143#1317—531no—
144#1317—532no—
145#1317D5311yescharacteristic
146C2×C53—21no—
147C2×C53—21no—
148C2×C53—41no—
149C2×C53—41no—
150C2×C53C2×#32311yescharacteristic
151C2×C53#212312yes—
152C23×Q8—531no—
153C2×#117—531no—
154C2×#117—531no—
155#431—1062no—
156#391—1062no—
157#391—1062no—
158#381—531no—
159#381—531no—
160#381—531no—
161#381C2×D5311yescharacteristic
162#324—1062no—
163#323—532no—
164#323C2×D5312yes—
165C53#1142811yescharacteristic, complemented, Sylow
166C22×Q8—534no—
167C22×Q8—1061no—
168C22×Q8—1061no—
169C22×Q8—1062no—
170C22×Q8—2122no—
171C22×Q8—2122no—
172C23×C4—531no—
173C23×C4—531no—
174C23×C4C4×D5311yescharacteristic
175#117—2122no—
176#117—2122no—
177C2×C4⋊C4—532no—
178C2×C4⋊C4—1061no—
179C2×C22⋊C4—531no—
180C2×C22⋊C4—531no—
181C2×C22⋊C4C4×D5311yescharacteristic
182C2×C22⋊C4C22×D5311yescharacteristic
183C2×C42—532no—
184C2×C42—1061no—
185C4⋊C8—2122no—
186C22⋊C8—22no—
187C22⋊C8—1062no—
188C4×C8—2122no—
189#95—22no—
190#95—1062no—
191#95—1062no—
192#95—1062no—
193C24C2×C4×D5311yescharacteristic
194C2×Q8—1064no—
195C2×Q8—1064no—
196C2×Q8—2122no—
197C2×Q8—2122no—
198C2×Q8—2128no—
199C2×Q8—2128no—
200C2×Q8—2122no—
201C2×Q8—2122no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—531no—
208C22×C4—531no—
209C22×C4—532no—
210C22×C4—532no—
211C22×C4—532no—
212C22×C4—532no—
213C22×C4—1062no—
214C22×C4—1062no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—2121no—
219C22×C4—2121no—
220C22×C4C8×D5312yes—
221C22×C4D4×D5312yes—
222C2×C8—42no—
223C2×C8—2122no—
224C4⋊C4—2122no—
225C4⋊C4—2122no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—2121no—
229C22⋊C4—2121no—
230C42—1062no—
231C42—1062no—
232C42—2122no—
233C23—21no—
234C23—21no—
235C23—42no—
236C23C22⋊C4×D5311yescharacteristic
237C23C2×C8×D5311yescharacteristic
238C23M16×D5311yescharacteristic
239Q8—2128no—
240Q8—2128no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—532no—
251C2×C4—532no—
252C2×C4—532no—
253C2×C4—532no—
254C2×C4—1064no—
255C2×C4—1064no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—2122no—
261C2×C4—2121no—
262C2×C4—2122no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2122no—
266C2×C4—2122no—
267C8—82no—
268C8—4242no—
269C22—22no—
270C22—21no—
271C22—21no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22C22⋊C8×D5311yescharacteristic
277C22#99×D5311yescharacteristic
278C22#100×D5311yescharacteristic
279C4—42no—
280C4—42no—
281C4—81no—
282C4—534no—
283C4—1062no—
284C4—4241no—
285C4—2122no—
286C4—2122no—
287C2—21no—
288C2—21no—
289C2—41no—
290C2—41no—
291C2#323×D5311yescharacteristic
292C2#2027636712yes—
subgroupG/H
centerC22#99×D53
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1317C2
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1317D53
hypocenterC53#11428
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1317

Complements

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

NH ≅ G/Ntype
24. C22×Dic106C8semidirect
165. C53#11428semidirect

Maximal subgroups

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

subgroupindex
#202773002
#202730262
#202726452
#202726442
C53×#13172
#1142853