Finite Group Databasebeta

← #30605007

Subgroups — #30605007

Order 15616.

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

normal subgroups only

This group has 420 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#21461524C211yesmaximal, characteristic
2#21461072C211yesmaximal, characteristic
3#21460818C212yesmaximal, complemented
4#21460803C212yesmaximal, complemented
5#21451519C211yesmaximal, characteristic
6C2×D61⋊M16—21no—
7#16840782—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608C2212yes—
11#16840606—21no—
12C2×#14519734C2211yescharacteristic
13#16839700C2211yescharacteristic
14#16839697C411yescharacteristic
15#16839693C2211yescharacteristic
16#16839688C411yescharacteristic
17#16839657—21no—
18#16839652—21no—
19#16839643—21no—
20#16839633—21no—
21#16839608C411yescharacteristic
22C2×#125×C61C411yescharacteristic
23C2×#11758213—41no—
24D61⋊M16—24no—
25D61⋊M16—42no—
26C2×#11758217—22no—
27C2×D61⋊C8—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×#11758196D411yescharacteristic
37C2×#11758196C2311yescharacteristic
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198C2×C411yescharacteristic
41C2×#11758198D411yescharacteristic
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×D61C22⋊C411yescharacteristic
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—21no—
65C22×Dic61—21no—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C22×Dic61C2×D411yescharacteristic
69C2×C4×D61—22no—
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74#11758217—24no—
75#11758217—42no—
76C2×#92420—42no—
77C2×#92420—42no—
78#11758213—44no—
79D61⋊C8—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—42no—
91#11758196—41no—
92#11758196—41no—
93#11758196—42no—
94#11758196—41no—
95#11758196—41no—
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×D61#10412yes—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61#9911yescharacteristic
118C2×C4×C61#9911yescharacteristic
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—22no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61AGL(1,ℤ/8)11yescharacteristic
132C2×Dic61#13711yescharacteristic
133C4×D61—22no—
134C4×D61—22no—
135C4×D61—22no—
136C4×D61—22no—
137#92420—44no—
138#92420—82no—
139#11767—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61C2×#9911yescharacteristic
143C22×C61#41711yescharacteristic
144C22×C61C2×#10411yescharacteristic
145C2×D61—22no—
146C2×D61—41no—
147C2×D61—41no—
148C2×D61—42no—
149C2×D61—81no—
150C4×C61—21no—
151C4×C61—21no—
152C4×C61—41no—
153C4×C61—41no—
154C4×C61—81no—
155C4×C61—81no—
156Dic61—21no—
157Dic61—21no—
158Dic61—42no—
159Dic61—41no—
160Dic61—41no—
161Dic61—81no—
162#2611—611no—
163#1742—611no—
164#1464—611no—
165#1353—612no—
166#1346—612no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61#150411yescharacteristic
170C2×C61#212311yescharacteristic
171C2×C61#213211yescharacteristic
172D61—42no—
173C22×M16—1221no—
174C2×#125#692311yescharacteristic
175C2×#117—611no—
176#432—1221no—
177#400—611no—
178#399—611no—
179#398—611no—
180#397—611no—
181#389—1221no—
182#388—1221no—
183#387—1221no—
184#385—1221no—
185#376—611no—
186#343—612no—
187#343—1221no—
188#324—612no—
189#323—1221no—
190C61#1176711yescharacteristic, complemented, Sylow
191C23×C4—611no—
192C2×M16—1222no—
193C2×M16—1224no—
194C2×M16—2442no—
195C2×M16—2441no—
196C22×C8—2441no—
197#125—41no—
198#125—41no—
199#117—2441no—
200#117—2441no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—611no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4C2×#692311yescharacteristic
208C2×C4⋊C4C2×#692311yescharacteristic
209C2×C22⋊C4—611no—
210C2×C22⋊C4—611no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C42—611no—
214C2×C42—1221no—
215C2×C42C2×#692311yescharacteristic
216C4⋊C8—2441no—
217C22⋊C8—1222no—
218#97—2441no—
219#95—611no—
220#95—611no—
221#95—1221no—
222#95—1221no—
223#95—1221no—
224#95—1221no—
225#95—1221no—
226C24—611no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—611no—
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1222no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4—1221no—
239C22×C4C22⋊#692311yescharacteristic
240C22×C4C22⋊#692311yescharacteristic
241C22×C4C22×#692311yescharacteristic
242M16—1224no—
243M16—2442no—
244M16—2444no—
245C2×C8—2442no—
246C2×C8—2444no—
247C2×C8—2442no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—2441no—
254C4⋊C4—2441no—
255C4⋊C4—2441no—
256C4⋊C4—2441no—
257C4⋊C4—2442no—
258C22⋊C4—2441no—
259C22⋊C4—2441no—
260C22⋊C4—2442no—
261C22⋊C4—2442no—
262C42—21no—
263C42—21no—
264C42—1221no—
265C42—1221no—
266C42—2442no—
267C23—612no—
268C23—1221no—
269C23—1221no—
270C23—2441no—
271C23—2441no—
272C23C2×C22⋊#692311yescharacteristic
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—611no—
286C2×C4—611no—
287C2×C4—1222no—
288C2×C4—1222no—
289C2×C4—1222no—
290C2×C4—1222no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1222no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4#1451983511yescharacteristic
304C2×C4#1451983511yescharacteristic
305C8—4882no—
306C8—2444no—
307C22—21no—
308C22—21no—
309C22—1222no—
310C22—4881no—
311C22—2442no—
312C22—2441no—
313C22—2441no—
314C22C2×#1451983511yescharacteristic
315C22#1684075311yescharacteristic
316C22D61⋊#10411yescharacteristic
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—81no—
322C4—81no—
323C4—1221no—
324C4—1221no—
325C4—4881no—
326C4—2442no—
327C4—2441no—
328C4—2441no—
329C2—21no—
330C2—21no—
331C2—2442no—
332C2#2146120511yescharacteristic
333C2#2146208111yescharacteristic
334C2#2146211011yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#125×C61C4
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC2×#11758196C23
hypercenterC2×#125#6923
hypocenterC61#11767
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
3. #21460818C2semidirect
4. #21460803C2semidirect
190. C61#11767semidirect

Maximal subgroups

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

subgroupindex
#214615242
#214610722
#214608182
#214608032
#214515192
#1176761