Finite Group Databasebeta

← #27302785

Subgroups — #27302785

Order 13568.

396 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 107 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
2C4×#13133136C211yesmaximal, characteristic
3#20272842C212yesmaximal
4#20272842C212yesmaximal
5C53×#2565C211yesmaximal, characteristic
6#16810647—21no—
7C2×#13133136C2212yes—
8C2×#13133136C2212yes—
9C4×#11719917—21no—
10#16810392—21no—
11#16810392—21no—
12C2×#125×C53C411yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#398—21no—
15C53×#398—21no—
16C53×#391C2211yescharacteristic
17C53×#391C2211yescharacteristic
18C53×#387—21no—
19C53×#387—21no—
20C53×#384C411yescharacteristic
21C53×#382C411yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic
23C2×#11719917—22no—
24C2×#11719917—22no—
25C4⋊#88322—41no—
26C4⋊#88322—41no—
27#13133136—22no—
28#13133136—24no—
29#13133136Q814yes—
30#13133128—41no—
31C4×#88322—41no—
32C22×Q8×C53C2311yescharacteristic
33#125×C53—41no—
34#125×C53—41no—
35C4×Q8×C53—41no—
36C4×Q8×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C2×Q8×C53C2×Q812yes—
62C2×Q8×C53#4112yes—
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—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—41no—
92C42×C53—41no—
93#11719917—24no—
94#11719917—42no—
95#11719917—42no—
96#11719917—42no—
97C2×#88322—42no—
98C2×#88322—42no—
99C23×C53C2×C22⋊C411yescharacteristic
100Q8×C53—42no—
101Q8×C53—42no—
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—22no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53C4×Q812yes—
129C2×C4×C53C22⋊Q812yes—
130C2×C4×C53#12411yescharacteristic
131C2×C4×C53C4⋊Q811yescharacteristic
132#88322—44no—
133#88322—82no—
134#13303—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53#39111yescharacteristic
138C22×C53C2×#9911yescharacteristic
139C22×C53#41011yescharacteristic
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—22no—
143C4×C53—21no—
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—42no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—81no—
151C4×C53—81no—
152C4×C53—81no—
153#2565D5311yescharacteristic
154#1801—531no—
155C4×#101—531no—
156#1394—532no—
157#1394—532no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53#199611yescharacteristic
161C2×C53#213211yescharacteristic
162C2×C53#213311yescharacteristic
163C2×#125Dic5311yescharacteristic
164C2×C4×Q8C2×D5311yescharacteristic
165#423—1061no—
166C2×#101—532no—
167C2×#101—532no—
168C4×M16—1061no—
169#398—21no—
170#398—21no—
171#391C2×D5311yescharacteristic
172#391C2×D5311yescharacteristic
173#387—21no—
174#387—21no—
175#384Dic5311yescharacteristic
176#382Dic5311yescharacteristic
177C4×C4⋊C4Dic5311yescharacteristic
178#343—1061no—
179#343—1061no—
180C53#1330311yescharacteristic, complemented, Sylow
181C22×Q8C22×D5311yescharacteristic
182C2×M16—1062no—
183C2×M16—1062no—
184#125—41no—
185#125—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic5311yescharacteristic
194C2×C4⋊C4C2×Dic5311yescharacteristic
195C2×C4⋊C4C2×Dic5311yescharacteristic
196C2×C4⋊C4C22⋊D5311yescharacteristic
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C42C2×Dic5311yescharacteristic
199C2×C42C2×Dic5311yescharacteristic
200C2×C42C2×Dic5311yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C2×C42C22⋊D5311yescharacteristic
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#101—534no—
206#101—1062no—
207#101—1064no—
208#97—2121no—
209C4×C8—2121no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8Q8×D5312yes—
220C2×Q8#1171994912yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4C22⋊Dic5311yescharacteristic
226C22×C4C22⋊Dic5311yescharacteristic
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22×Dic5311yescharacteristic
230C22×C4C2×C22⋊D5311yescharacteristic
231C22×C4C2×C22⋊D5311yescharacteristic
232M16—1064no—
233M16—2122no—
234M16—2122no—
235M16—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—41no—
256C42—41no—
257C23C2×C22⋊Dic5311yescharacteristic
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4#1313313211yescharacteristic
285C2×C4#1313313211yescharacteristic
286C2×C4Q8×Dic5312yes—
287C2×C4#1313319312yes—
288C2×C4#1313320611yescharacteristic
289C2×C4#1313321511yescharacteristic
290C8—4242no—
291C8—2124no—
292C22—21no—
293C22—21no—
294C22#1681053011yescharacteristic
295C22C2×#1313313211yescharacteristic
296C22#1681059711yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—21no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2#2027570111yescharacteristic
313C2#2027640011yescharacteristic
314C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2565C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#2565D53
hypocenterC53#13303
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2565

Complements

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

NH ≅ G/Ntype
180. C53#13303semidirect

Maximal subgroups

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

subgroupindex
#202747112
C4×#131331362
#202728422
C53×#25652
#1330353