Finite Group Databasebeta

← #27300090

Subgroups — #27300090

Order 13568.

500 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 95 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#20277258C211yesmaximal, characteristic
2C2×#16810419C212yesmaximal
3#20274709C212yesmaximal, complemented
4#20272842C211yesmaximal, characteristic, complemented
5C53×#1394C211yesmaximal, characteristic, complemented
6C2×#13133206C2211yescharacteristic
7C2×Q8×Dic53C2211yescharacteristic
8C2×#13133179C411yescharacteristic
9#16810646—22no—
10C2×#13133136C2212yes—
11#16810525C411yescharacteristic
12#16810515—22no—
13#16810488C411yescharacteristic
14#16810470—22no—
15C42×Dic53C411yescharacteristic
16#16810419—24no—
17#16810391—22no—
18C2×#101×C53C2212yes—
19C53×#391C2211yescharacteristic
20C22×D212C2×C411yescharacteristic
21C22×C4×D53—21no—
22C2×#11719917—22no—
23#13133206—22no—
24#13133206D412yes—
25#13133206Q812yes—
26Q8×Dic53—41no—
27Q8×Dic53—41no—
28#13133179—41no—
29#13133179—41no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53D412yes—
32C2×#11719921—22no—
33C2×#11719921—21no—
34C2×#11719921—22no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D412yes—
40#13133146—42no—
41#13133146—42no—
42#13133136—22no—
43#13133136—22no—
44#13133124—22no—
45C22×Q8×C53C2311yescharacteristic
46C2×M16×C53—22no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49#101×C53—22no—
50#101×C53—22no—
51#95×C53—22no—
52C23×D53—21no—
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×Q8×C53C2×D411yescharacteristic
56C2×Q8×C53C2×Q811yescharacteristic
57C2×Q8×C53#4112yes—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D412yes—
62C22×Dic53—21no—
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C22⋊C412yes—
65C22×Dic53C22⋊C411yescharacteristic
66C2×D212—21no—
67C2×D212—41no—
68C2×D212—41no—
69C2×D212C4⋊C412yes—
70C2×C4×D53—44no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73M16×C53—42no—
74M16×C53—42no—
75C2×C8×C53—42no—
76C4⋊C4×C53—41no—
77C42×C53—41no—
78#11719921—24no—
79#11719921—24no—
80#11719921—42no—
81#11719921—44no—
82C4⋊Dic53—41no—
83C4⋊Dic53—42no—
84C4×Dic53—21no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C4×Dic53—42no—
90C4×Dic53—42no—
91C4×Dic53C4⋊C412yes—
92#11719917—42no—
93#11719917—42no—
94C2×#88322—42no—
95C23×C53C2×C22⋊C411yescharacteristic
96C22×D53—22no—
97C22×D53—42no—
98C22×D53—42no—
99C22×D53—41no—
100C22×D53—41no—
101Q8×C53—41no—
102Q8×C53—41no—
103Q8×C53—42no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53C2×C4⋊C411yescharacteristic
114C2×C4×C53#11711yescharacteristic
115C2×C4×C53#12112yes—
116C2×C4×C53C22⋊Q812yes—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—21no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—42no—
125C2×Dic53#9911yescharacteristic
126C2×Dic53#9911yescharacteristic
127D212—42no—
128D212—42no—
129C4×D53—82no—
130C4×D53—82no—
131C8×C53—82no—
132#88322—82no—
133#12801—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53#38011yescharacteristic
137C22×C53C2×#9911yescharacteristic
138C22×C53#41011yescharacteristic
139C2×D53—44no—
140C2×D53—42no—
141C2×D53—82no—
142C2×D53—81no—
143C2×D53—81no—
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—22no—
148C4×C53—42no—
149C4×C53—81no—
150Dic53—21no—
151Dic53—42no—
152Dic53—42no—
153Dic53—41no—
154Dic53#35512yescomplemented
155C4×#124—531no—
156C2×#355—532no—
157#1801—532no—
158#1394—531no—
159#1394D5311yescharacteristic, complemented
160C2×C53—21no—
161C2×C53—21no—
162C2×C53#179911yescharacteristic
163C2×C53C2×#35511yescharacteristic
164C2×C53#213211yescharacteristic
165D53—82no—
166C2×#124—531no—
167C2×C4×Q8—531no—
168C2×C4×D4—531no—
169#423—1062no—
170C2×#101—532no—
171C2×#101C2×D5312yes—
172#391C2×D5311yescharacteristic
173#390—531no—
174#388—1062no—
175#382—531no—
176C4×C22⋊C4—1062no—
177C43—531no—
178#355—1064no—
179#343—1062no—
180C53#1280111yescharacteristic, complemented, Sylow
181C22×Q8C22×D5311yescharacteristic
182C22×D4—531no—
183C23×C4—1061no—
184C2×M16—22no—
185C2×M16—1062no—
186#124—532no—
187#124—532no—
188#124—1062no—
189C4×Q8—2121no—
190C4×Q8—2121no—
191C4×D4—2121no—
192C4×D4—2121no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—532no—
195C2×C4⋊C4C4×D5311yescharacteristic
196C2×C22⋊C4—1062no—
197C2×C22⋊C4—1061no—
198C2×C22⋊C4—1062no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—532no—
202C2×C42—1061no—
203C2×C42—1061no—
204C2×C42C4×D5311yescharacteristic
205C4⋊C8—2122no—
206C4⋊C8—2122no—
207#101—22no—
208#101—22no—
209#101—1062no—
210#101—1062no—
211#95—22no—
212#95—1062no—
213C24—1061no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8#1171994512yes—
217C2×Q8D4×D5311yescharacteristic
218C2×Q8Q8×D5311yescharacteristic
219C2×D4—532no—
220C2×D4—1061no—
221C2×D4—2121no—
222C2×D4—2121no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—531no—
226C22×C4—532no—
227C22×C4—531no—
228C22×C4—1061no—
229C22×C4—2121no—
230C22×C4—2121no—
231C22×C4—2124no—
232C22×C4C2×C4×D5311yescharacteristic
233C22×C4D4×D5312yes—
234M16—42no—
235M16—42no—
236M16—2122no—
237M16—2122no—
238C2×C8—42no—
239C2×C8—2122no—
240C4⋊C4—41no—
241C4⋊C4—2121no—
242C4⋊C4—2122no—
243C22⋊C4—1064no—
244C22⋊C4—1064no—
245C22⋊C4—2124no—
246C22⋊C4—2122no—
247C42—41no—
248C42—532no—
249C42—1062no—
250C42—1062no—
251C42—1061no—
252C42—1061no—
253C42—1061no—
254C42—2122no—
255C42—2122no—
256C23—1062no—
257C23—2122no—
258C23—2122no—
259C23—2121no—
260C23—2121no—
261C23C22⋊C4×D5311yescharacteristic
262Q8—42no—
263Q8—41no—
264Q8—41no—
265D4—2122no—
266D4—2122no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—531no—
277C2×C4—531no—
278C2×C4—1062no—
279C2×C4—1061no—
280C2×C4—1062no—
281C2×C4—1062no—
282C2×C4—4242no—
283C2×C4—4242no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2122no—
288C2×C4C4⋊C4×D5311yescharacteristic
289C2×C4#1313317411yescharacteristic
290C2×C4#1313319012yes—
291C2×C4#1313319512yes—
292C8—82no—
293C8—4242no—
294C22—21no—
295C22—21no—
296C22—4242no—
297C22—4241no—
298C22—4241no—
299C22—2124no—
300C22—2122no—
301C22#1681048211yescharacteristic
302C22#99×D5311yescharacteristic
303C22#1681059511yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—42no—
309C4—81no—
310C4—532no—
311C4—1061no—
312C4—2122no—
313C4—2121no—
314C4—2122no—
315C2—21no—
316C2—21no—
317C2—4242no—
318C2#2027470211yescharacteristic
319C2#355×D5311yescharacteristic
320C2#2027639911yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#1394C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC22×Q8×C53C23
hypercenter#1394D53
hypocenterC53#12801
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #20274709C2semidirect
4. #20272842C2semidirect
5. C53×#1394C2semidirect
154. Dic53#355semidirect
159. #1394D53semidirect
180. C53#12801semidirect

Maximal subgroups

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

subgroupindex
#202772582
C2×#168104192
#202747092
#202728422
C53×#13942
#1280153