Finite Group Databasebeta

← #27295246

Subgroups — #27295246

Order 13568.

372 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 103 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#20274711C211yesmaximal, characteristic
2#20273039C211yesmaximal, characteristic
3#20272745C212yesmaximal
4#20272742C212yesmaximal
5C53×#2581C211yesmaximal, characteristic
6#16810676—21no—
7#16810647—21no—
8C2×#13133136—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—
25C4⋊#88322—41no—
26C4⋊#88322—41no—
27#13133136—42no—
28C22⋊#88322—22no—
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—41no—
58C2×Q8×C53—41no—
59C2×Q8×C53—42no—
60C2×Q8×C53—41no—
61C2×Q8×C53SD1612yes—
62C2×Q8×C53Q1612yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—42no—
90#11719917—42no—
91#11719917—42no—
92C2×#88322—42no—
93C2×#88322—42no—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—42no—
96Q8×C53—42no—
97Q8×C53—42no—
98Q8×C53—42no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
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—41no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53#9911yescharacteristic
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53#10312yes—
126C2×C4×C53#10312yes—
127C2×C4×C53C2×SD1611yescharacteristic
128C2×C4×C53C2×Q1611yescharacteristic
129#88322—82no—
130#88322—82no—
131#11905—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53C2×#9911yescharacteristic
135C22×C53C2×#10311yescharacteristic
136C22×C53#42111yescharacteristic
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—22no—
141C4×C53—22no—
142C4×C53—21no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—42no—
146C4×C53—41no—
147C4×C53—81no—
148C4×C53—81no—
149C4×C53—81no—
150#2581D5311yescharacteristic
151#1801—531no—
152#1461—531no—
153#1354—532no—
154#1353—532no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53#152011yescharacteristic
158C2×C53#212011yescharacteristic
159C2×C53#213311yescharacteristic
160C2×C4×Q8Dic5311yescharacteristic
161C2×C4×Q8C2×D5311yescharacteristic
162#431—1061no—
163#423—1061no—
164C2×#101—1061no—
165#400Dic5311yescharacteristic
166#400C2×D5311yescharacteristic
167#395Dic5311yescharacteristic
168#395C2×D5311yescharacteristic
169#391—21no—
170#389—21no—
171#387—21no—
172#387—21no—
173#382Dic5311yescharacteristic
174#343—532no—
175#324—532no—
176#324—1061no—
177C53#1190511yescharacteristic, complemented, Sylow
178C22×Q8C2×Dic5311yescharacteristic
179C22×Q8C22⋊D5311yescharacteristic
180C2×M16—1062no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C2×Dic5311yescharacteristic
190C2×C4⋊C4C2×Dic5311yescharacteristic
191C2×C4⋊C4C22⋊D5311yescharacteristic
192C2×C4⋊C4C22×D5311yescharacteristic
193C2×C42—21no—
194C2×C42C2×Dic5311yescharacteristic
195C2×C42C22⋊D5311yescharacteristic
196C2×C42C22⋊D5311yescharacteristic
197C4⋊C8—2121no—
198C4⋊C8—2121no—
199C4⋊C8—2121no—
200#101—2122no—
201C22⋊C8—1062no—
202C4×C8—2121no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95C2×Dic5311yescharacteristic
211#95C2×Dic5311yescharacteristic
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8#1171992412yes—
217C2×Q8#1171992512yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4C22⋊Dic5311yescharacteristic
223C22×C4C22⋊Dic5311yescharacteristic
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22⋊Dic5311yescharacteristic
226C22×C4C22×Dic5311yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228C22×C4C2×C22⋊D5311yescharacteristic
229M16—2122no—
230M16—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—42no—
249C23C2×C22⋊Dic5311yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—42no—
253Q8—42no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4#1313313211yescharacteristic
279C2×C4#1313313211yescharacteristic
280C2×C4#1313314112yes—
281C2×C4#1313314112yes—
282C2×C4C2×#1171992411yescharacteristic
283C2×C4C2×#1171992511yescharacteristic
284C8—4242no—
285C8—4242no—
286C22—21no—
287C22—21no—
288C22C2×#1313313211yescharacteristic
289C22C2×#1313314111yescharacteristic
290C22#1681064011yescharacteristic
291C4—21no—
292C4—22no—
293C4—22no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C2—21no—
305C2—21no—
306C2#2027328711yescharacteristic
307C2#2027635611yescharacteristic
308C2#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#11905
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
177. C53#11905semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730392
#202727452
#202727422
C53×#25812
#1190553