Finite Group Databasebeta

← #27294731

Subgroups — #27294731

Order 13568.

378 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#13133136C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272745C212yesmaximal
4#20272742C212yesmaximal
5C53×#2581C211yesmaximal, characteristic
6#16810680—21no—
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×C4×Q8×C53C411yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#400C411yescharacteristic
15C53×#400C2211yescharacteristic
16C53×#395C411yescharacteristic
17C53×#395C2211yescharacteristic
18C53×#391—21no—
19C53×#389—21no—
20C53×#387—21no—
21C53×#387—21no—
22C53×#382C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25#13133136—24no—
26C22⋊#88322—22no—
27#13133128—41no—
28#13133128—41no—
29C4×#88322—41no—
30C22×Q8×C53C2×C411yescharacteristic
31C22×Q8×C53D411yescharacteristic
32C4×Q8×C53—41no—
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35C4×Q8×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C42×C53—21no—
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48#95×C53—21no—
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53—41no—
60C2×Q8×C53—42no—
61C2×Q8×C53—41no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—42no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—42no—
89#11719917—24no—
90#11719917—42no—
91C2×#88322—42no—
92C2×#88322—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—44no—
95Q8×C53—42no—
96Q8×C53—42no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#9911yescharacteristic
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#10412yes—
125C2×C4×C53#13712yes—
126#88322—44no—
127#88322—82no—
128#11792—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53C2×#9911yescharacteristic
132C22×C53#41911yescharacteristic
133C22×C53C2×#10411yescharacteristic
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—42no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—81no—
144C4×C53—81no—
145C4×C53—81no—
146#2581D5311yescharacteristic
147C4×#101—531no—
148#1467—531no—
149#1354—532no—
150#1353—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53#150811yescharacteristic
154C2×C53#212011yescharacteristic
155C2×C53#213311yescharacteristic
156C2×C4×Q8Dic5311yescharacteristic
157C2×C4×Q8C2×D5311yescharacteristic
158#432—1061no—
159C2×#101—1061no—
160C4×M16—1061no—
161#400Dic5311yescharacteristic
162#400C2×D5311yescharacteristic
163#395Dic5311yescharacteristic
164#395C2×D5311yescharacteristic
165#391—21no—
166#389—21no—
167#387—21no—
168#387—21no—
169#382Dic5311yescharacteristic
170#343—532no—
171#324—532no—
172#324—1061no—
173C53#1179211yescharacteristic, complemented, Sylow
174C22×Q8C2×Dic5311yescharacteristic
175C22×Q8C22⋊D5311yescharacteristic
176C2×M16—1062no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic5311yescharacteristic
186C2×C4⋊C4C2×Dic5311yescharacteristic
187C2×C4⋊C4C22⋊D5311yescharacteristic
188C2×C4⋊C4C22×D5311yescharacteristic
189C2×C42—21no—
190C2×C42C2×Dic5311yescharacteristic
191C2×C42C22⋊D5311yescharacteristic
192C2×C42C22⋊D5311yescharacteristic
193C4⋊C8—2121no—
194#101—1064no—
195C22⋊C8—1062no—
196#97—2121no—
197#97—2121no—
198C4×C8—2121no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95C2×Dic5311yescharacteristic
207#95C2×Dic5311yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—41no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C22⋊Dic5311yescharacteristic
218C22×C4C22⋊Dic5311yescharacteristic
219C22×C4C22⋊Dic5311yescharacteristic
220C22×C4C22⋊Dic5311yescharacteristic
221C22×C4C22×Dic5311yescharacteristic
222C22×C4C2×C22⋊D5311yescharacteristic
223C22×C4C2×C22⋊D5311yescharacteristic
224M16—1064no—
225M16—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C23C2×C22⋊Dic5311yescharacteristic
245Q8—44no—
246Q8—42no—
247Q8—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4#1313313211yescharacteristic
274C2×C4#1313313211yescharacteristic
275C2×C4#1313314412yes—
276C2×C4#1313324812yes—
277C8—4242no—
278C8—2124no—
279C22—21no—
280C22—21no—
281C22C2×#1313313211yescharacteristic
282C22#1681063211yescharacteristic
283C22C2×#1313314411yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4—81no—
296C2—21no—
297C2—21no—
298C2#2027323511yescharacteristic
299C2#2027635611yescharacteristic
300C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2581C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2581D53
hypocenterC53#11792
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2581

Complements

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

NH ≅ G/Ntype
173. C53#11792semidirect

Maximal subgroups

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

subgroupindex
C4×#131331362
#202730592
#202727452
#202727422
C53×#25812
#1179253