Finite Group Databasebeta

← #28299979

Subgroups — #28299979

Order 13568.

388 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 96 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#20290643C211yesmaximal, characteristic
2#20290630C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5#20283074C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133207—21no—
13#16810522C2211yescharacteristic
14#16810521C411yescharacteristic
15#16810520—21no—
16#16810519C2211yescharacteristic
17#16810518—21no—
18#16810492C411yescharacteristic
19#16810492C411yescharacteristic
20#16810487—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—42no—
26#13133207—44no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53C2311yescharacteristic
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49#95×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C2×#88324—42no—
62C2×#88324—42no—
63C2×#88324—42no—
64C2×#88324—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C4⋊C4×C53—22no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—41no—
70C42×C53—22no—
71C4⋊Dic53—22no—
72C4⋊Dic53—42no—
73C4⋊Dic53—41no—
74#11719919—42no—
75#11719919—42no—
76#11719919—42no—
77#11719919—42no—
78#11719919—42no—
79#11719919—42no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—41no—
83C4×Dic53—41no—
84C4×Dic53—41no—
85C4×Dic53SD1612yes—
86C4×Dic53Q1612yes—
87C23×C53C2×C22⋊C411yescharacteristic
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53#10312yes—
102C2×C4×C53#13112yes—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53#10112yes—
118C2×Dic53#10312yes—
119C2×Dic53C2×SD1611yescharacteristic
120C2×Dic53C2×Q1611yescharacteristic
121C2×Dic53AGL(1,ℤ/8)12yes—
122#88324—82no—
123#88324—82no—
124#7720—531nomaximal, Sylow
125C22×C53#40811yescharacteristic
126C22×C53#41011yescharacteristic
127C22×C53C2×#10111yescharacteristic
128C22×C53C2×#10311yescharacteristic
129C22×C53#41811yescharacteristic
130C22×C53#42111yescharacteristic
131C22×C53#42111yescharacteristic
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—41no—
135C4×C53—81no—
136C4×C53—81no—
137Dic53—22no—
138Dic53—22no—
139Dic53—21no—
140Dic53—21no—
141Dic53—22no—
142Dic53—81no—
143Dic53—81no—
144#2859—531no—
145#1839—531no—
146#1831—531no—
147#1271—532no—
148#1271—532no—
149C2×C53#147011yescharacteristic
150C2×C53#148111yescharacteristic
151C2×C53#149611yescharacteristic
152C2×C53#151411yescharacteristic
153C2×C53#151811yescharacteristic
154C2×C53#153411yescharacteristic
155C2×C53#154911yescharacteristic
156C2×#125—1061no—
157C2×C4⋊C8—1061no—
158#389—531no—
159#389—531no—
160#389—531no—
161#389—1061no—
162#389—1061no—
163#389#615711yescharacteristic
164#384—531no—
165#384—531no—
166#382—1061no—
167C43—531no—
168#336—532no—
169#336—532no—
170#336—1061no—
171#336—1061no—
172#336—1061no—
173C53#772011yescharacteristic, complemented, Sylow
174C22×C8—1062no—
175C22×C8—1062no—
176#125—2124no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4C2×#615711yescharacteristic
188C2×C4⋊C4C2×#615711yescharacteristic
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—531no—
193C2×C42—531no—
194C2×C42—531no—
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—2122no—
197#95—21no—
198#95—1061no—
199#95—1061no—
200#95—1061no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4C22⋊#615711yescharacteristic
210C22×C4C22⋊#615711yescharacteristic
211C22×C4C22×#615711yescharacteristic
212C2×C8—2122no—
213C2×C8—2122no—
214C2×C8—2122no—
215C2×C8—2122no—
216C2×C8—2122no—
217C2×C8—2122no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—1062no—
222C4⋊C4—2121no—
223C4⋊C4—2122no—
224C4⋊C4—2122no—
225C4⋊C4—2122no—
226C4⋊C4—2122no—
227C4⋊C4—2122no—
228C4⋊C4—2122no—
229C4⋊C4—2122no—
230C42—22no—
231C42—532no—
232C42—532no—
233C42—1062no—
234C42—1062no—
235C42—2121no—
236C42—2121no—
237C42—2121no—
238C23C2×C22⋊#615711yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—532no—
253C2×C4—532no—
254C2×C4—531no—
255C2×C4—531no—
256C2×C4—532no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4#1313328012yes—
272C2×C4#1313330412yes—
273C8—4242no—
274C8—4242no—
275C22#1681159611yescharacteristic
276C22#1681160211yescharacteristic
277C22D53⋊#10111yescharacteristic
278C22C2×#1313328011yescharacteristic
279C22#1681161511yescharacteristic
280C22#1681162011yescharacteristic
281C22#1681162011yescharacteristic
282C4—22no—
283C4—22no—
284C4—41no—
285C4—81no—
286C4—81no—
287C4—1062no—
288C4—1061no—
289C4—1061no—
290C4—1062no—
291C4—1062no—
292C4—4241no—
293C4—4241no—
294C2#2028998711yescharacteristic
295C2#2029002311yescharacteristic
296C2#2029007411yescharacteristic
297C2#2029013411yescharacteristic
298C2#2029014811yescharacteristic
299C2#2029019811yescharacteristic
300C2#2029023111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7720
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
173. C53#7720semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202906302
#202895342
#202895332
#202830742
#772053