Finite Group Databasebeta

← #29597010

Subgroups — #29597010

Order 15616.

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

normal subgroups only

This group has 528 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#21455400C211yesmaximal, characteristic
2C2×#16839800C211yesmaximal, characteristic, complemented
3C2×#16839573C212yesmaximal, complemented
4#21443925C212yesmaximal, complemented
5C61×#2112C211yesmaximal, characteristic, complemented
6C2×#14519799C2211yescharacteristic
7#16840273—21no—
8C2×#14519736—21no—
9#16840003C412yescomplemented
10#16839998C412yescomplemented
11#16839991—22no—
12#16839800—24no—
13#16839800C2211yescharacteristic
14C2×#14519711—21no—
15C2×#14519709C2212yes—
16#16839759—21no—
17#16839573—24no—
18C23×C8×C61C2211yescharacteristic
19C2×M32×C61C2212yes—
20C61×C22⋊C16—22no—
21C2×#11758222—21no—
22#14519799—22no—
23#14519799—22no—
24C2×Dic244C2×C411yescharacteristic
25C2×#11758190—21no—
26C2×D488C2×C411yescharacteristic
27#14519760—41no—
28#14519756—22no—
29#14519751—22no—
30#14519746—41no—
31C4×C22⋊D61—42no—
32#14519736—24no—
33#14519736—21no—
34#14519736—42no—
35C2×C4⋊Dic61—22no—
36C2×C22⋊Dic61—41no—
37C2×C4×Dic61—41no—
38#14519711—24no—
39#14519711C2×C412yes—
40#14519709—21no—
41#14519709—22no—
42#14519709D414yescomplemented
43#14519702—22no—
44#14519699—22no—
45C23×C4×C61D411yescharacteristic
46C22×C8×C61—22no—
47C22×C8×C61—22no—
48C22×C8×C61C2×C412yes—
49C22×C8×C61D411yescharacteristic
50C22×C8×C61D412yes—
51C22×C8×C61C2311yescharacteristic
52M32×C61—22no—
53M32×C61—22no—
54C2×C16×C61—22no—
55C24×C61D811yescharacteristic
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61D811yescharacteristic
62C22×C4×C61SD1612yes—
63C22×C4×C61C2×D411yescharacteristic
64C2×C22⋊D61—42no—
65C2×C22⋊D61—41no—
66C22×Dic61—42no—
67#11758222—41no—
68#11758222—41no—
69#11758222—41no—
70#11758222—41no—
71C2×D244—21no—
72C2×C4×D61—41no—
73C2×Dic122—21no—
74C2×C8×C61—21no—
75C2×C8×C61—21no—
76C2×C8×C61—24no—
77C2×C8×C61—24no—
78C2×C8×C61C22⋊C412yes—
79C2×C8×C61C22×C411yescharacteristic
80C2×C8×C61C2×D412yes—
81C2×C8×C61C2×D411yescharacteristic
82C2×C8×C61#4112yes—
83C22⋊Dic61—44no—
84C22⋊Dic61—42no—
85C22⋊Dic61—41no—
86#11758198—42no—
87C4⋊Dic61—22no—
88C4⋊Dic61—24no—
89C4⋊Dic61—42no—
90#11758196—42no—
91C4×Dic61—44no—
92C4×Dic61—41no—
93Dic244—22no—
94D488—22no—
95#11758190—42no—
96C16×C61—42no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—42no—
100C23×C61#10212yes—
101C23×C61C2×D811yescharacteristic
102C22×D61—41no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61#10212yes—
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C2×C4×C61#12011yescharacteristic
112C2×C4×C61#12311yescharacteristic
113C2×C4×C61C2×SD1611yescharacteristic
114C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C61#13711yescharacteristic
116C22⋊D61—84no—
117C22⋊D61—81no—
118C22⋊D61—81no—
119C2×Dic61—42no—
120C2×Dic61—41no—
121C2×Dic61—44no—
122C2×Dic61—84no—
123D244—41no—
124D244—41no—
125C4×D61—81no—
126C4×D61—81no—
127Dic122—41no—
128Dic122—41no—
129C8×C61—22no—
130C8×C61C4×D412yes—
131C8×C61#12112yes—
132#13002—611nomaximal, Sylow
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—42no—
138C22×C61—42no—
139C22×C61C2×#10211yescharacteristic
140C22×C61#44711yescharacteristic
141C22×C61#48011yescharacteristic
142C2×D61—81no—
143C2×D61—82no—
144C4×C61—22no—
145C4×C61#38611yescharacteristic
146C4×C61#41711yescharacteristic
147C4×C61#44811yescharacteristic
148C4×C61#48111yescharacteristic
149Dic61—84no—
150Dic61—81no—
151Dic61—82no—
152#3050—611no—
153C2×#427—611no—
154C2×#360—612no—
155#2112D6111yescharacteristic, complemented
156#1386—612no—
157C2×C61—21no—
158C2×C61—21no—
159C2×C61—42no—
160C2×C61#187411yescharacteristic
161C2×C61#214512yes—
162D61—161no—
163C2×#135—611no—
164C23×C8C2×D6111yescharacteristic
165#525—1221no—
166C2×#117—1221no—
167C2×M32C2×D6112yes—
168#467—612no—
169#466—612no—
170#465—1222no—
171#427—611no—
172#427—1224no—
173C2×#107—1221no—
174C2×#106—612no—
175#416—1221no—
176#360—1224no—
177C22⋊C16—22no—
178C61#1300211yescharacteristic, complemented, Sylow
179C2×#41—1221no—
180C23×C4D24411yescharacteristic
181#135—1222no—
182#135—1222no—
183C2×Q16—611no—
184C2×SD16—1221no—
185C2×D8—611no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C4×D6112yes—
189C22×C8D24411yescharacteristic
190C22×C8C22⋊D6112yes—
191C22×C8C22×D6111yescharacteristic
192#123—2441no—
193C22⋊Q8—1222no—
194#121—1222no—
195#120—2441no—
196C4×D4—2442no—
197#117—1224no—
198#117—1221no—
199#117—2442no—
200C2×C4⋊C4—1222no—
201C2×C22⋊C4—2441no—
202C2×C42—2441no—
203M32—22no—
204M32—22no—
205C2×C16—22no—
206#107—612no—
207#107—1224no—
208#106—614no—
209#106—1222no—
210#106—1221no—
211#103—1222no—
212#102—1222no—
213C24D48811yescharacteristic
214#41—2441no—
215#41—2441no—
216#41—2441no—
217#41—2441no—
218C2×Q8—1221no—
219C2×D4—1221no—
220C2×D4—2442no—
221C2×D4—2441no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—2442no—
227C22×C4—2441no—
228C22×C4#1175819012yes—
229C22×C4D48811yescharacteristic
230C22×C4#1175819812yes—
231C22×C4C2×D24411yescharacteristic
232Q16—1222no—
233SD16—2442no—
234D8—1222no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8#1175819812yes—
240C2×C8C2×C4×D6111yescharacteristic
241C2×C8D4×D6112yes—
242C2×C8#1175822412yes—
243C2×C8C2×C22⋊D6111yescharacteristic
244C4⋊C4—1222no—
245C4⋊C4—1224no—
246C4⋊C4—2442no—
247C4⋊C4—2442no—
248C22⋊C4—2444no—
249C22⋊C4—2442no—
250C22⋊C4—2441no—
251C22⋊C4—2442no—
252C42—2444no—
253C42—2441no—
254C16—42no—
255C23—21no—
256C23—21no—
257C23—42no—
258C23—2441no—
259C23#1451969912yes—
260C23C2×D48811yescharacteristic
261Q8—2441no—
262Q8—2441no—
263D4—4884no—
264D4—4881no—
265D4—4881no—
266D4—2441no—
267D4—2441no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—4884no—
275C2×C4—4881no—
276C2×C4—4881no—
277C2×C4—2442no—
278C2×C4—2441no—
279C2×C4—2444no—
280C2×C4#1451969912yes—
281C2×C4C2×#1175819811yescharacteristic
282C2×C4#1451974711yescharacteristic
283C2×C4#1451976111yescharacteristic
284C2×C4C2×#1175819011yescharacteristic
285C2×C4#1451980511yescharacteristic
286C2×C4#1451981011yescharacteristic
287C8—22no—
288C8#1451973712yes—
289C8#1451974912yes—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—4882no—
297C22—4881no—
298C22C2×#1451969911yescharacteristic
299C22#1683989811yescharacteristic
300C22#1684007611yescharacteristic
301C4—22no—
302C4—4884no—
303C4—4882no—
304C4—4881no—
305C4#1683963911yescharacteristic
306C4#1683976411yescharacteristic
307C4#1683990311yescharacteristic
308C4#1684008111yescharacteristic
309C2—21no—
310C2—21no—
311C2—42no—
312C2—9761no—
313C2#2144613411yescharacteristic
314C2#2144755612yes—
subgroupG/H
centerC22C2×#14519699
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#2112C2
socleC22×C61C2×#102
prime radical#21455400C2
cosocleC22×C8×C61C23
hypercenter#2112D61
hypocenterC61#13002
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C2×#16839800C2semidirect
3. C2×#16839573C2semidirect
4. #21443925C2semidirect
5. C61×#2112C2semidirect
9. #16840003C4semidirect
10. #16839998C4semidirect
42. #14519709D4semidirect
155. #2112D61semidirect
178. C61#13002semidirect

Maximal subgroups

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

subgroupindex
#214554002
C2×#168398002
C2×#168395732
#214439252
C61×#21122
#1300261