Finite Group Databasebeta

← #28299864

Subgroups — #28299864

Order 13568.

410 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290501C211yesmaximal, characteristic
2C4×#13133284C211yesmaximal, characteristic
3#20289535C212yesmaximal
4#20289533C212yesmaximal
5#20283094C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8C2×#13133272—21no—
9C2×C4×#88324—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12#16810530C411yescharacteristic
13#16810530C2211yescharacteristic
14#16810522C2211yescharacteristic
15#16810521C411yescharacteristic
16#16810519C411yescharacteristic
17#16810510—21no—
18#16810508—21no—
19#16810505—21no—
20#16810490—21no—
21C42×Dic53C2211yescharacteristic
22C53×#391C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—24no—
26C4⋊#88324—24no—
27#13133272—42no—
28C4×#88324—42no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53C2311yescharacteristic
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47C22×Q8×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52C2×Q8×C53—22no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×#88324—24no—
67C2×#88324—24no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C4⋊C4×C53—22no—
73C42×C53—41no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78#11719919—42no—
79#11719919—42no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—22no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C4×Dic53—41no—
86C4×Dic53SD1612yes—
87C4×Dic53Q1612yes—
88C23×C53C2×C22⋊C411yescharacteristic
89Q8×C53—82no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53#10312yes—
103C2×C4×C53#10412yes—
104C2×C4×C53#13112yes—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53#10312yes—
120C2×Dic53#10412yes—
121C2×Dic53C2×SD1611yescharacteristic
122C2×Dic53C2×Q1611yescharacteristic
123C2×Dic53AGL(1,ℤ/8)11yescharacteristic
124C2×Dic53#13711yescharacteristic
125#88324—44no—
126#88324—44no—
127#7689—531nomaximal, Sylow
128C22×C53#40811yescharacteristic
129C22×C53#41011yescharacteristic
130C22×C53#41311yescharacteristic
131C22×C53C2×#10311yescharacteristic
132C22×C53#41711yescharacteristic
133C22×C53C2×#10411yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C4×C53—22no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—81no—
139C4×C53—81no—
140Dic53—22no—
141Dic53—22no—
142Dic53—21no—
143Dic53—21no—
144Dic53—21no—
145Dic53—21no—
146Dic53—81no—
147Dic53—81no—
148#2863—531no—
149#1767—531no—
150C4×C4⋊C8—531no—
151#1272—532no—
152#1271—532no—
153C2×C53#147311yescharacteristic
154C2×C53#148311yescharacteristic
155C2×C53#149311yescharacteristic
156C2×C53#150311yescharacteristic
157C2×C53#150911yescharacteristic
158C2×C53#152411yescharacteristic
159C2×C53#155011yescharacteristic
160C2×C4⋊C8—1061no—
161C2×C4⋊C8—1061no—
162C2×#97—1061no—
163C2×C4×C8—1061no—
164#391—531no—
165#391—531no—
166#391#615711yescharacteristic
167#389—531no—
168#389—531no—
169#389—531no—
170#387—1061no—
171#387—1061no—
172#387—1061no—
173#383—1061no—
174C43—531no—
175#336—532no—
176#336—532no—
177C53#768911yescharacteristic, complemented, Sylow
178C22×Q8C2×#615711yescharacteristic
179C22×C8—1062no—
180C22×C8—1062no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4C2×#615711yescharacteristic
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—531no—
193C2×C42C2×#615711yescharacteristic
194C4⋊C8—1064no—
195C4⋊C8—1064no—
196#97—2122no—
197C4×C8—2122no—
198#95—21no—
199#95—21no—
200#95—1061no—
201#95—1061no—
202#95—1061no—
203#95—1061no—
204#95—1061no—
205#95—1061no—
206#95—1061no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4C22⋊#615711yescharacteristic
219C22×C4C22⋊#615711yescharacteristic
220C22×C4C22×#615711yescharacteristic
221C2×C8—1064no—
222C2×C8—1064no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C4⋊C4—22no—
228C4⋊C4—2121no—
229C4⋊C4—2121no—
230C4⋊C4—2122no—
231C4⋊C4—2121no—
232C4⋊C4—2121no—
233C4⋊C4—2122no—
234C42—41no—
235C42—532no—
236C42—532no—
237C42—1061no—
238C42—1062no—
239C42—1061no—
240C42—1062no—
241C42—1062no—
242C42—2121no—
243C23C2×C22⋊#615711yescharacteristic
244Q8—82no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—532no—
258C2×C4—531no—
259C2×C4—531no—
260C2×C4—531no—
261C2×C4—531no—
262C2×C4—532no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4#1313328012yes—
278C2×C4#1313328212yes—
279C2×C4#1313330412yes—
280C8—2124no—
281C8—2124no—
282C22#1681159711yescharacteristic
283C22#1681160211yescharacteristic
284C22#1681161011yescharacteristic
285C22C2×#1313328011yescharacteristic
286C22#1681161411yescharacteristic
287C22D53⋊#10411yescharacteristic
288C22C2×#1313328211yescharacteristic
289C4—22no—
290C4—22no—
291C4—22no—
292C4—81no—
293C4—81no—
294C4—1062no—
295C4—1061no—
296C4—1061no—
297C4—1061no—
298C4—1061no—
299C4—1062no—
300C4—4241no—
301C4—4241no—
302C2#2028999611yescharacteristic
303C2#2029003111yescharacteristic
304C2#2029006411yescharacteristic
305C2#2029009711yescharacteristic
306C2#2029011911yescharacteristic
307C2#2029016811yescharacteristic
308C2#2029023311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#391#6157
hypocenterC53#7689
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
177. C53#7689semidirect

Maximal subgroups

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

subgroupindex
#202905012
C4×#131332842
#202895352
#202895332
#202830942
#768953