Finite Group Databasebeta

← #27299268

Subgroups — #27299268

Order 13568.

628 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 125 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#20277441C211yesmaximal, characteristic
2C2×#16810416C211yesmaximal, characteristic
3#99×Dic53C212yesmaximal, complemented
4#20272839C212yesmaximal, complemented
5C2×C53×#353C211yesmaximal, characteristic
6C2×D4×Dic53C411yescharacteristic, complemented
7C2×D4×Dic53C2211yescharacteristic
8C2×#13133132C2212yes—
9#16810526C411yescharacteristic, complemented
10#16810526C2211yescharacteristic
11#16810503—22no—
12#16810503—22no—
13#16810489C411yescharacteristic, complemented
14C22⋊C4×Dic53—22no—
15#16810469C411yescharacteristic, complemented
16#16810416C414yescomplemented
17#16810389—22no—
18C2×#127×C53C2211yescharacteristic
19C2×#99×C53C2212yes—
20C53×#353C414yescomplemented
21C23×Dic53—21no—
22C23×Dic53D412yes—
23D4×Dic53—42no—
24D4×Dic53—42no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53D412yes—
32C2×C4×Dic53—42no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35#13133132—22no—
36#13133132C2×C414yescomplemented
37#13133124—21no—
38#13133124—21no—
39#13133124—22no—
40#13133124—42no—
41C22×D4×C53—22no—
42C22×D4×C53C2×C411yescharacteristic
43C22×D4×C53C2311yescharacteristic
44#127×C53—22no—
45#127×C53C2×C412yes—
46#127×C53C2×C412yes—
47C2×C22⋊C4×C53—22no—
48C2×C42×C53C2×C411yescharacteristic
49#99×C53—22no—
50#99×C53C2×C414yescomplemented
51C24×C53—21no—
52C24×C53C2×D412yes—
53C2×D4×C53—24no—
54C2×D4×C53—24no—
55C2×D4×C53—21no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53—82no—
61C2×D4×C53—82no—
62C2×D4×C53C4212yescomplemented
63C2×D4×C53C22×C411yescharacteristic
64C2×D4×C53C22×C411yescharacteristic
65C2×D4×C53#4112yes—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53C22×C411yescharacteristic
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—42no—
74C22×Dic53—42no—
75C22×Dic53—82no—
76C22×Dic53—82no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C412yes—
79C22×Dic53C22⋊C411yescharacteristic
80C22⋊C4×C53—24no—
81C22⋊C4×C53—42no—
82C42×C53—21no—
83C42×C53C4212yescomplemented
84C22⋊Dic53—24no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C22⋊Dic53—82no—
89C22⋊Dic53—82no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4×Dic53—22no—
93C4×Dic53—41no—
94C4×Dic53—84no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53C4×D412yes—
106C23×C53C4×D412yes—
107D4×C53—41no—
108D4×C53—41no—
109D4×C53—42no—
110D4×C53—42no—
111D4×C53—42no—
112D4×C53—84no—
113D4×C53—84no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—44no—
120C2×C4×C53—82no—
121C2×C4×C53—82no—
122C2×C4×C53C2×C4211yescharacteristic
123C2×C4×C53#11711yescharacteristic
124C2×Dic53—22no—
125C2×Dic53—21no—
126C2×Dic53—41no—
127C2×Dic53—44no—
128C2×Dic53—41no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—44no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—82no—
137C2×Dic53—84no—
138C2×Dic53—82no—
139C2×Dic53—82no—
140C2×Dic53#9912yes—
141#12628—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—44no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—82no—
153C22×C53—82no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53C4×C22⋊C411yescharacteristic
157C22×C53C2×#9911yescharacteristic
158C22×C53#41011yescharacteristic
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—42no—
162C4×C53—84no—
163Dic53—22no—
164Dic53—42no—
165Dic53—41no—
166Dic53—81no—
167Dic53—84no—
168Dic53—81no—
169#2328—531no—
170C2×#353—531no—
171C2×#353D5311yescharacteristic
172C4×#99—532no—
173#1393—532no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—42no—
177C2×C53—42no—
178C2×C53—82no—
179C2×C53C4×#9911yescharacteristic
180C2×C53#212711yescharacteristic
181C2×C53#212811yescharacteristic
182C2×#127C2×D5311yescharacteristic
183C2×C4×D4—531no—
184C2×C4×D4—531no—
185C2×#99—532no—
186C2×#99C2×D5312yes—
187#390—531no—
188#390—531no—
189#386—1062no—
190#386—1062no—
191#382—531no—
192C4×C22⋊C4—1062no—
193#376—531no—
194#353—534no—
195#353Dic5314yescomplemented
196#342—1062no—
197C53#1262811yescharacteristic, complemented, Sylow
198C22×D4—22no—
199C22×D4C4×D5311yescharacteristic
200C22×D4C22×D5311yescharacteristic
201C23×C4—532no—
202C23×C4—1061no—
203#127—22no—
204#127C4×D5312yes—
205#127C2×Dic5312yes—
206C4×D4—2122no—
207C4×D4—2122no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—1062no—
213C2×C22⋊C4—1062no—
214C2×C22⋊C4—1062no—
215C2×C22⋊C4—1061no—
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—2122no—
219C2×C42C4×D5311yescharacteristic
220#99—22no—
221#99—534no—
222#99—1062no—
223#99C2×Dic5314yescomplemented
224#95—1061no—
225#95—1061no—
226#95—1062no—
227#95—2122no—
228C24—21no—
229C24D4×D5312yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4C4×Dic5312yescomplemented
240C2×D4C2×C4×D5311yescharacteristic
241C2×D4#1171994712yes—
242C2×D4C22×Dic5311yescharacteristic
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—532no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—4242no—
253C22×C4—4242no—
254C22×C4—2122no—
255C22×C4—2122no—
256C22×C4C2×C4×D5311yescharacteristic
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—1064no—
262C22⋊C4—4242no—
263C22⋊C4—4242no—
264C22⋊C4—2122no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C42—21no—
268C42—1062no—
269C42—4244no—
270C42—2121no—
271C42C4×Dic5312yescomplemented
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23C22⋊C4×D5311yescharacteristic
282C23#1313317712yes—
283C23D4×Dic5312yes—
284D4—41no—
285D4—42no—
286D4—41no—
287D4—42no—
288D4—42no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—532no—
300C2×C4—1061no—
301C2×C4—1062no—
302C2×C4—4242no—
303C2×C4—4242no—
304C2×C4—4242no—
305C2×C4—4244no—
306C2×C4—2121no—
307C2×C4—2124no—
308C2×C4—2124no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2122no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2122no—
315C2×C4—2121no—
316C2×C4C2×C4×Dic5311yescharacteristic
317C2×C4#1313317311yescharacteristic
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—44no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—81no—
332C22C22⋊C4×Dic5311yescharacteristic
333C22#99×D5311yescharacteristic
334C22#1681059511yescharacteristic
335C4—21no—
336C4—21no—
337C4—42no—
338C4—84no—
339C4—1062no—
340C4—4241no—
341C4—4244no—
342C4—4241no—
343C4—2122no—
344C4—2121no—
345C2—21no—
346C2—21no—
347C2—42no—
348C2—42no—
349C2—82no—
350C2#99×Dic5311yescharacteristic
351C2#2027638111yescharacteristic
352C2#2027638511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC2×C53×#353C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#353D53
hypocenterC53#12628
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#353

Complements

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

NH ≅ G/Ntype
3. #99×Dic53C2semidirect
4. #20272839C2semidirect
6. C2×D4×Dic53C4semidirect
9. #16810526C4semidirect
13. #16810489C4semidirect
15. #16810469C4semidirect
16. #16810416C4semidirect
20. C53×#353C4semidirect
36. #13133132C2×C4semidirect
50. #99×C53C2×C4semidirect
62. C2×D4×C53C42semidirect
83. C42×C53C42semidirect
195. #353Dic53semidirect
197. C53#12628semidirect
223. #99C2×Dic53semidirect
239. C2×D4C4×Dic53semidirect
271. C42C4×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202774412
C2×#168104162
#99×Dic532
#202728392
C2×C53×#3532
#1262853