Finite Group Databasebeta

← #30604979

Subgroups — #30604979

Order 15616.

436 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 68 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#21461518C211yesmaximal, characteristic
2#21461072C211yesmaximal, characteristic
3#21460816C212yesmaximal
4#21460799C212yesmaximal
5#21451519C211yesmaximal, characteristic
6C2×#14519856—21no—
7#16840782—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840608—21no—
11#16840606C2212yes—
12C2×#14519734C2211yescharacteristic
13#16839700C411yescharacteristic, complemented
14#16839697C2211yescharacteristic
15#16839693C411yescharacteristic, complemented
16#16839688C2211yescharacteristic
17#16839657—21no—
18#16839652—21no—
19#16839643—21no—
20#16839633—21no—
21#16839608C411yescharacteristic, complemented
22C2×#125×C61C411yescharacteristic, complemented
23#14519856—24no—
24#14519856—42no—
25C2×C4⋊#6923—41no—
26C2×C22⋊#6923—22no—
27C2×C4×#6923—41no—
28#14519844—41no—
29C22⋊#92420—22no—
30#14519832—41no—
31C22×C4×D61D411yescharacteristic
32#14519734—41no—
33#14519734—41no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196C2×C411yescharacteristic
37C2×#11758196D411yescharacteristic
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198D411yescharacteristic
41C2×#11758198C2311yescharacteristic
42C2×C4×Dic61—21no—
43C2×C4×Dic61D411yescharacteristic
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684C2×C411yescharacteristic
50#14519684C2×C411yescharacteristic
51#125×C61—41no—
52#125×C61—41no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C42×C61C2×C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22×C411yescharacteristic
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C2×D411yescharacteristic
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—22no—
75C22⋊#6923—24no—
76C22⋊#6923—42no—
77C2×#92420—42no—
78C4⋊#6923—44no—
79C4×#6923—44no—
80C4⋊C4×C61—42no—
81C4⋊C4×C61—41no—
82C4⋊C4×C61—41no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C42×C61—21no—
86C42×C61—21no—
87#11758198—41no—
88#11758198—41no—
89#11758198—42no—
90#11758198—41no—
91#11758198—41no—
92#11758196—41no—
93#11758196—41no—
94#11758196—42no—
95#11758196—42no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C4×Dic61—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61AGL(1,ℤ/8)11yescharacteristic
105C22×D61#13711yescharacteristic
106C2×#6923—44no—
107C2×#6923—82no—
108C2×#6923—82no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61#9911yescharacteristic
122C2×C4×C61#9911yescharacteristic
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—22no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61#10412yes—
136C4×D61—22no—
137C4×D61—22no—
138C4×D61—22no—
139C4×D61—22no—
140#92420—82no—
141#11760—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61C2×#9911yescharacteristic
145C22×C61#41711yescharacteristic
146C22×C61C2×#10411yescharacteristic
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—41no—
150C2×D61—41no—
151C2×D61—41no—
152C2×D61—41no—
153C2×D61—81no—
154#6923—84no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—81no—
160C4×C61—81no—
161Dic61—22no—
162Dic61—42no—
163Dic61—41no—
164Dic61—41no—
165Dic61—81no—
166#2611—611no—
167#1740—611no—
168#1464—611no—
169#1352—612no—
170#1344—612no—
171C2×C61—21no—
172C2×C61—21no—
173C2×C61#150411yescharacteristic
174C2×C61#212311yescharacteristic
175C2×C61#212611yescharacteristic
176D61—41no—
177D61—41no—
178C2×#125#692311yescharacteristic, complemented
179C2×#117—611no—
180C2×#117—1221no—
181#432—1221no—
182#400—611no—
183#399—611no—
184#398—611no—
185#397—611no—
186#389—1221no—
187#388—1221no—
188#387—1221no—
189#385—1221no—
190#376—611no—
191#342—612no—
192#342—1221no—
193#324—1221no—
194#323—612no—
195C61#1176011yescharacteristic, complemented, Sylow
196C23×C4—611no—
197#125—41no—
198#125—41no—
199#117—1224no—
200#117—2441no—
201#117—2441no—
202#117—2442no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—2441no—
210C2×C4⋊C4C2×#692311yescharacteristic
211C2×C4⋊C4C2×#692311yescharacteristic
212C2×C22⋊C4—611no—
213C2×C22⋊C4—611no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—1221no—
216C2×C22⋊C4—1221no—
217C2×C42—611no—
218C2×C42—1221no—
219C2×C42—2441no—
220C2×C42C2×#692311yescharacteristic
221C4⋊C8—2441no—
222C22⋊C8—1222no—
223#97—2441no—
224#95—611no—
225#95—611no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C24—611no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—1222no—
238C22×C4—1221no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—1221no—
244C22×C4—2442no—
245C22×C4C22⋊#692311yescharacteristic
246C22×C4C22⋊#692311yescharacteristic
247C22×C4C22×#692311yescharacteristic
248C2×C8—2442no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C4⋊C4—2442no—
257C4⋊C4—2442no—
258C4⋊C4—2444no—
259C22⋊C4—1224no—
260C22⋊C4—2441no—
261C22⋊C4—2441no—
262C22⋊C4—2442no—
263C22⋊C4—2441no—
264C22⋊C4—2441no—
265C22⋊C4—2442no—
266C42—21no—
267C42—21no—
268C42—1221no—
269C42—1221no—
270C42—2442no—
271C42—2444no—
272C23—611no—
273C23—611no—
274C23—1221no—
275C23—1221no—
276C23—2441no—
277C23—2441no—
278C23C2×C22⋊#692311yescharacteristic
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—612no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—1222no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1222no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—4882no—
302C2×C4—4882no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2444no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4#1451983511yescharacteristic
312C2×C4#1451983511yescharacteristic
313C8—4882no—
314C22—21no—
315C22—21no—
316C22—1221no—
317C22—1221no—
318C22—4881no—
319C22—2441no—
320C22—2441no—
321C22—2441no—
322C22—2441no—
323C22C2×#1451983511yescharacteristic
324C22#1684075311yescharacteristic
325C22D61⋊#10411yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—1222no—
333C4—4881no—
334C4—4884no—
335C4—2442no—
336C4—2441no—
337C4—2441no—
338C2—21no—
339C2—21no—
340C2—2441no—
341C2—2441no—
342C2#2146120511yescharacteristic
343C2#2146208111yescharacteristic
344C2#2146209011yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#125×C61C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenterC2×#125#6923
hypocenterC61#11760
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
13. #16839700C4semidirect
15. #16839693C4semidirect
21. #16839608C4semidirect
22. C2×#125×C61C4semidirect
178. C2×#125#6923semidirect
195. C61#11760semidirect

Maximal subgroups

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

subgroupindex
#214615182
#214610722
#214608162
#214607992
#214515192
#1176061