Finite Group Databasebeta

← #355×Dic53 #27299272

Subgroups — #355×Dic53 #27299272

Order 13568.

532 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 127 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#124×Dic53C211yesmaximal, characteristic
2C2×#16810420C211yesmaximal, characteristic
3#101×Dic53C212yesmaximal, complemented
4#20272841C212yesmaximal, complemented
5C2×C53×#355C211yesmaximal, characteristic
6C2×Q8×Dic53C2211yescharacteristic
7C2×D4×Dic53C411yescharacteristic
8C2×#13133136C2212yes—
9M16×Dic53—22no—
10#16810530C2211yescharacteristic
11#16810526C411yescharacteristic
12#16810516—22no—
13#16810489C411yescharacteristic
14C22⋊C4×Dic53—22no—
15C42×Dic53C411yescharacteristic
16#16810420C414yescomplemented
17#16810391—22no—
18C2×#124×C53C2211yescharacteristic
19C2×#101×C53C2212yes—
20C53×#355C414yescomplemented
21C23×Dic53—21no—
22C2×#11719917—22no—
23Q8×Dic53—41no—
24Q8×Dic53—41no—
25D4×Dic53—41no—
26D4×Dic53—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53D412yes—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—22no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37#13133136—22no—
38#13133136C2×C414yescomplemented
39#13133127—42no—
40C8×Dic53—42no—
41#13133124—22no—
42#13133124—22no—
43C22×Q8×C53C2311yescharacteristic
44C22×D4×C53C2×C411yescharacteristic
45C2×M16×C53—22no—
46#124×C53—22no—
47#124×C53C2×C412yes—
48#124×C53C2×C412yes—
49C2×C22⋊C4×C53—22no—
50C2×C42×C53C2×C411yescharacteristic
51#101×C53—22no—
52#101×C53C2×C414yescomplemented
53C24×C53—21no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C2×Q8×C53C22×C411yescharacteristic
57C2×Q8×C53C22×C411yescharacteristic
58C2×Q8×C53#4112yes—
59C2×D4×C53—21no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53C4212yes—
63C22×C4×C53—21no—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—44no—
69C22×Dic53—41no—
70C22×Dic53—41no—
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C412yes—
73C22×Dic53C22⋊C411yescharacteristic
74M16×C53—24no—
75M16×C53—42no—
76C2×C8×C53—42no—
77C22⋊C4×C53—24no—
78C22⋊C4×C53—24no—
79C42×C53—21no—
80C42×C53C4212yes—
81C22⋊Dic53—44no—
82C22⋊Dic53—42no—
83C4⋊Dic53—41no—
84C4⋊Dic53—42no—
85C4⋊Dic53—41no—
86C4×Dic53—22no—
87C4×Dic53—22no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—42no—
91C4×Dic53—42no—
92C4×Dic53—41no—
93#11719917—24no—
94#11719917—42no—
95C2×#88322—42no—
96C23×C53—22no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53C2×C22⋊C411yescharacteristic
102Q8×C53—41no—
103Q8×C53—41no—
104Q8×C53—42no—
105D4×C53—42no—
106D4×C53—42no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—21no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53C2×C4211yescharacteristic
115C2×C4×C53#11711yescharacteristic
116C2×C4×C53C4×D412yes—
117C2×C4×C53C4×D412yes—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—21no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—41no—
128C2×Dic53—82no—
129C2×Dic53—82no—
130C2×Dic53#9911yescharacteristic
131C2×Dic53#9911yescharacteristic
132C8×C53—44no—
133#88322—44no—
134C4×#355—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—44no—
138C22×C53—42no—
139C22×C53—82no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53C4×C22⋊C411yescharacteristic
143C22×C53C2×#9911yescharacteristic
144C22×C53#41011yescharacteristic
145C4×C53—22no—
146C4×C53—22no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—42no—
150Dic53—21no—
151Dic53—42no—
152Dic53—42no—
153Dic53—41no—
154Dic53—81no—
155Dic53#35512yesdirect product factor
156C4×#124—531no—
157C2×#355—531no—
158C2×#355D5311yescharacteristic
159C4×#101—532no—
160#1394—532no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—82no—
164C2×C53C4×#9911yescharacteristic
165C2×C53C2×#35511yescharacteristic
166C2×C53#213211yescharacteristic
167C2×#124C2×D5311yescharacteristic
168C2×C4×Q8—531no—
169C2×C4×D4—531no—
170C2×#101—532no—
171C2×#101C2×D5312yes—
172C4×M16—1062no—
173#391—531no—
174#390—531no—
175#388—1062no—
176#382—531no—
177C4×C22⋊C4—1062no—
178C43—531no—
179#355—534no—
180#355Dic5314yesdirect product factor
181#343—1062no—
182C53C4×#35511yescharacteristic, complemented, Sylow
183C22×Q8C22×D5311yescharacteristic
184C22×D4C4×D5311yescharacteristic
185C23×C4—1061no—
186C2×M16—22no—
187C2×M16—1062no—
188#124—22no—
189#124C4×D5312yes—
190#124C2×Dic5312yes—
191C4×Q8—2121no—
192C4×Q8—2121no—
193C4×D4—2121no—
194C4×D4—2121no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—532no—
197C2×C4⋊C4—531no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—1061no—
200C2×C22⋊C4—1062no—
201C2×C42—531no—
202C2×C42—532no—
203C2×C42—531no—
204C2×C42—1061no—
205C2×C42—1061no—
206C2×C42C4×D5311yescharacteristic
207#101—22no—
208#101—534no—
209#101—1062no—
210#101C2×Dic5314yescomplemented
211#97—2122no—
212C4×C8—2122no—
213#95—1062no—
214#95—1062no—
215C24—21no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8C2×C4×D5311yescharacteristic
219C2×Q8#1171994712yes—
220C2×Q8C22×Dic5311yescharacteristic
221C2×D4—21no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4C4×Dic5312yes—
225C22×C4—21no—
226C22×C4—531no—
227C22×C4—532no—
228C22×C4—531no—
229C22×C4—1061no—
230C22×C4—1061no—
231C22×C4—2121no—
232C22×C4—2121no—
233C22×C4—2124no—
234C22×C4C2×C4×D5311yescharacteristic
235C22×C4D4×D5312yes—
236M16—24no—
237M16—42no—
238M16—1064no—
239M16—2122no—
240C2×C8—42no—
241C2×C8—2122no—
242C4⋊C4—2121no—
243C4⋊C4—2122no—
244C4⋊C4—2121no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—2122no—
248C22⋊C4—2124no—
249C42—21no—
250C42—1062no—
251C42—1062no—
252C42—1061no—
253C42—1061no—
254C42—2121no—
255C42—2122no—
256C42—2122no—
257C42C4×Dic5312yes—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23C22⋊C4×D5311yescharacteristic
264Q8—41no—
265Q8—42no—
266Q8—41no—
267D4—42no—
268D4—42no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—531no—
277C2×C4—531no—
278C2×C4—1062no—
279C2×C4—1062no—
280C2×C4—1061no—
281C2×C4—1062no—
282C2×C4—4242no—
283C2×C4—4242no—
284C2×C4—2122no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2122no—
290C2×C4C2×C4×Dic5311yescharacteristic
291C2×C4#1313317311yescharacteristic
292C2×C4#1313317712yes—
293C2×C4D4×Dic5312yes—
294C8—44no—
295C8—2124no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—42no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22C22⋊C4×Dic5311yescharacteristic
304C22#99×D5311yescharacteristic
305C22#1681059511yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—21no—
310C4—42no—
311C4—532no—
312C4—1061no—
313C4—4241no—
314C4—2122no—
315C4—2121no—
316C4—2122no—
317C2—21no—
318C2—21no—
319C2—82no—
320C2#99×Dic5311yescharacteristic
321C2#355×D5311yescharacteristic
322C2#2027639711yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C2×C42
FrattiniC22×Q8C22×D53
FittingC2×C53×#355C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC22×Q8×C53C23
hypercenterC2×#355D53
hypocenterC53C4×#355
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #101×Dic53C2semidirect
4. #20272841C2semidirect
16. #16810420C4semidirect
20. C53×#355C4semidirect
38. #13133136C2×C4semidirect
52. #101×C53C2×C4semidirect
155. Dic53#355direct (N × H)
180. #355Dic53direct (N × H)
182. C53C4×#355semidirect
210. #101C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#124×Dic532
C2×#168104202
#101×Dic532
#202728412
C2×C53×#3552
C4×#35553