Finite Group Databasebeta

← #27300903

Subgroups — #27300903

Order 13568.

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

normal subgroups only

This group has 516 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#20277889C211yesmaximal, characteristic
2#20275104C211yesmaximal, characteristic
3#20274998C212yesmaximal
4#20272832C212yesmaximal
5C2×C53×#354C211yesmaximal, characteristic
6#16811098C2211yescharacteristic
7#16810632—21no—
8C2×#13133134C2211yescharacteristic
9#16810530—21no—
10#16810516—22no—
11#16810503C2212yes—
12#16810498—22no—
13#16810498—22no—
14#16810489—21no—
15#16810389—22no—
16#16810383—21no—
17C2×#124×C53C2211yescharacteristic
18C2×#99×C53C2212yes—
19C53×#354C414yescomplemented
20C23×Dic53D412yes—
21C2×#11719917—21no—
22D4×Dic53—41no—
23D4×Dic53—41no—
24#13133176—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—42no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53D412yes—
32C2×C4×Dic53—21no—
33#13133141—41no—
34#13133141—41no—
35#13133134—22no—
36#13133124—22no—
37#13133124—22no—
38#13133124—22no—
39#13133124—42no—
40C22×Q8×C53D411yescharacteristic
41C22×D4×C53C2311yescharacteristic
42#124×C53—22no—
43#124×C53C2×C412yes—
44#124×C53Q812yes—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C42×C53D411yescharacteristic
48#99×C53—22no—
49#99×C53C2×C414yes—
50C24×C53C2×D412yes—
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53C4⋊C412yescomplemented
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C22×C411yescharacteristic
58C2×D4×C53C2×Q811yescharacteristic
59C2×D4×C53#4112yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—42no—
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—42no—
69C22×Dic53—82no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—24no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—42no—
74C42×C53—21no—
75C42×C53C4⋊C412yescomplemented
76C22⋊Dic53—42no—
77C22⋊Dic53—41no—
78C22⋊Dic53—42no—
79C22⋊Dic53—82no—
80C4⋊Dic53—21no—
81C4⋊Dic53—21no—
82C4⋊Dic53—81no—
83C4⋊Dic53—81no—
84C4⋊Dic53—81no—
85C4⋊Dic53—84no—
86C4×Dic53—41no—
87C4×Dic53—81no—
88#11719917—41no—
89#11719917—41no—
90C2×#88322—41no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53C4×D412yes—
96C23×C53#12011yescharacteristic
97C23×C53C22⋊Q812yes—
98Q8×C53—42no—
99Q8×C53—81no—
100D4×C53—41no—
101D4×C53—41no—
102D4×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—44no—
112C2×C4×C53—82no—
113C2×C4×C53—82no—
114C2×C4×C53C2×C4⋊C411yescharacteristic
115C2×C4×C53#12311yescharacteristic
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—44no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—82no—
126C2×Dic53—82no—
127C2×Dic53—82no—
128#88322—81no—
129#12967—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—82no—
139C22×C53#38511yescharacteristic
140C22×C53B(4,𝔽2)11yescharacteristic
141C22×C53#45811yescharacteristic
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—42no—
145C4×C53—42no—
146C4×C53—84no—
147Dic53—82no—
148Dic53—82no—
149Dic53—81no—
150Dic53—81no—
151#2379—531no—
152C2×#354D5311yescharacteristic
153#1885—531no—
154#1867—532no—
155#1390—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—42no—
159C2×C53—42no—
160C2×C53#186711yescharacteristic
161C2×C53#219911yescharacteristic
162C2×C53#220111yescharacteristic
163C2×#124C2×D5311yescharacteristic
164#518—531no—
165#419—1061no—
166C2×#100—531no—
167C2×#99C2×D5312yes—
168#391—1061no—
169#388—1062no—
170#386—532no—
171#385—1062no—
172#385—1062no—
173#382—1061no—
174#354Dic5314yescomplemented
175#342—1062no—
176#339—1061no—
177C53#1296711yescharacteristic, complemented, Sylow
178C22×Q8D21211yescharacteristic
179C22×D4C22×D5311yescharacteristic
180C23×C4—532no—
181C2×M16—1061no—
182#124—22no—
183#124Dic10612yes—
184#124C2×Dic5312yes—
185C4×D4—2121no—
186C4×D4—2121no—
187#117—1061no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—2122no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—532no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C42—1061no—
198C2×C42D21211yescharacteristic
199#103—2121no—
200#103—2121no—
201#100—1062no—
202#99—22no—
203#99C2×Dic5314yes—
204#95—1062no—
205#95—1062no—
206#95—1062no—
207#95—2122no—
208C24D4×D5312yes—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8C4⋊Dic5312yescomplemented
213C2×D4—41no—
214C2×D4—41no—
215C2×D4C2×Dic10611yescharacteristic
216C2×D4#1171994712yes—
217C2×D4C22×Dic5311yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—1062no—
222C22×C4—1062no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—4242no—
226C22×C4—2122no—
227C22×C4C2×D21211yescharacteristic
228M16—2121no—
229M16—2121no—
230C2×C8—2121no—
231C4⋊C4—1061no—
232C4⋊C4—1061no—
233C4⋊C4—4241no—
234C4⋊C4—4241no—
235C4⋊C4—4241no—
236C4⋊C4—4244no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—42no—
241C22⋊C4—4242no—
242C22⋊C4—2122no—
243C22⋊C4—2121no—
244C22⋊C4—2122no—
245C42—21no—
246C42—4241no—
247C42—2121no—
248C42C4⋊Dic5312yescomplemented
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23D4×Dic5312yes—
254C23#1313318711yescharacteristic
255C23#1313319412yes—
256Q8—42no—
257Q8—81no—
258D4—41no—
259D4—42no—
260D4—41no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—4242no—
273C2×C4—4242no—
274C2×C4—4242no—
275C2×C4—2122no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4—2124no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4C2×C4⋊Dic5311yescharacteristic
285C2×C4#1313320111yescharacteristic
286C8—4241no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—82no—
296C22#1681049811yescharacteristic
297C22#1681081111yescharacteristic
298C22#1681081411yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—84no—
304C4—4242no—
305C4—4241no—
306C4—4242no—
307C4—4241no—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2027499811yescharacteristic
313C2#2027671811yescharacteristic
314C2#2027673011yescharacteristic
subgroupG/H
centerC22#16810811
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC2×C53×#354C2
socleC22×C53B(4,𝔽2)
prime radical, cosocleC22×D4×C53C23
hypercenterC2×#354D53
hypocenterC53#12967
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
19. C53×#354C4semidirect
54. C2×Q8×C53C4⋊C4semidirect
75. C42×C53C4⋊C4semidirect
174. #354Dic53semidirect
177. C53#12967semidirect
212. C2×Q8C4⋊Dic53semidirect
248. C42C4⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
#202778892
#202751042
#202749982
#202728322
C2×C53×#3542
#1296753