Finite Group Databasebeta

← #29590301

Subgroups — #29590301

Order 15616.

448 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 107 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#21451640C211yesmaximal, characteristic
2#21444133C211yesmaximal, characteristic
3#21443839C214yesmaximal
4C61×#1740C211yesmaximal, characteristic
5C2×#14519777C411yescharacteristic, complemented
6#16839815—21no—
7#16839701—21no—
8#16839698—21no—
9#16839682C412yescomplemented
10#16839674C2212yes—
11#16839660—21no—
12#16839628C411yescharacteristic, complemented
13#16839622—21no—
14#16839488—21no—
15#16839486C2212yes—
16C2×#117×C61—21no—
17C2×#117×C61C2211yescharacteristic
18C61×#342—21no—
19C61×#342C2212yes—
20C22×Dic122C2×C412yes—
21#14519777—41no—
22#14519777—41no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×#11758196—22no—
27C2×C22⋊Dic61—22no—
28C4⋊#92418—41no—
29C22⋊#92418—22no—
30C4×#92418—41no—
31#14519684—22no—
32#14519684—21no—
33#14519684—22no—
34#14519684C2×C412yes—
35C23×C4×C61D411yescharacteristic
36#117×C61—24no—
37#117×C61—41no—
38#117×C61—41no—
39#117×C61—42no—
40C2×C4⋊C4×C61—41no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61D411yescharacteristic
45C2×C22⋊C4×C61C2311yescharacteristic
46C2×C42×C61—41no—
47C2×C42×C61C2×C411yescharacteristic
48C2×C42×C61D411yescharacteristic
49C24×C61C2×D411yescharacteristic
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—22no—
53C22×C4×C61—42no—
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61SD1612yes—
59C22×C4×C61Q1612yes—
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×Dic61—22no—
63C22×Dic61—22no—
64C2×Dic122—42no—
65C2×Dic122—82no—
66C4⋊C4×C61—41no—
67C4⋊C4×C61—41no—
68C4⋊C4×C61—44no—
69C22⋊C4×C61—24no—
70C22⋊C4×C61—41no—
71C22⋊C4×C61—41no—
72C22⋊C4×C61—42no—
73C42×C61—21no—
74C42×C61—21no—
75C42×C61—21no—
76C42×C61—21no—
77C42×C61—44no—
78C22⋊Dic61—82no—
79C4⋊Dic61—42no—
80C4⋊Dic61—41no—
81C4⋊Dic61—41no—
82C4⋊Dic61—42no—
83#11758196—82no—
84C2×#92418—42no—
85C23×C61—21no—
86C23×C61—21no—
87C23×C61—41no—
88C23×C61—41no—
89C23×C61C2×C22⋊C411yescharacteristic
90C23×C61C2×SD1611yescharacteristic
91C23×C61C2×Q1611yescharacteristic
92C2×C4×C61—21no—
93C2×C4×C61—21no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—44no—
107C2×C4×C61—82no—
108C2×C4×C61—82no—
109C2×C4×C61#9911yescharacteristic
110C2×C4×C61#9911yescharacteristic
111C2×C4×C61#10312yes—
112C2×C4×C61#10312yes—
113C2×C4×C61#10412yes—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—42no—
117C2×Dic61—42no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120Dic122—84no—
121#92418—82no—
122#11717—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—41no—
128C22×C61—41no—
129C22×C61—41no—
130C22×C61—41no—
131C22×C61—81no—
132C22×C61C2×#9911yescharacteristic
133C22×C61C2×#10311yescharacteristic
134C22×C61C2×#10411yescharacteristic
135C4×C61—22no—
136C4×C61—41no—
137C4×C61—41no—
138C4×C61—41no—
139C4×C61—41no—
140C4×C61—84no—
141C4×C61#32812yes—
142C4×C61#32812yes—
143Dic61—82no—
144Dic61—82no—
145#2628—611no—
146#1740D6111yescharacteristic
147#1458—611no—
148#1351—614no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61—41no—
152C2×C61—41no—
153C2×C61C2×#32811yescharacteristic
154C2×C61#212211yescharacteristic
155C2×C61#212911yescharacteristic
156C2×C4⋊Q8—611no—
157C2×#117—21no—
158C2×#117C2×D6111yescharacteristic
159#431—1221no—
160#400—1221no—
161#399—1221no—
162#395—612no—
163#393—612no—
164#389—1221no—
165#382—611no—
166#380—1221no—
167#342—21no—
168#342C2×D6112yes—
169#324—1221no—
170#323—612no—
171C61#1171711yescharacteristic, complemented, Sylow
172C22×Q8—612no—
173C23×C4C22⋊D6111yescharacteristic
174C4⋊Q8—2441no—
175C4⋊Q8—2441no—
176#117—24no—
177#117—41no—
178#117—41no—
179#117—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—1222no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4—1221no—
184C2×C4⋊C4—1221no—
185C2×C4⋊C4C4×D6111yescharacteristic
186C2×C4⋊C4D24411yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1222no—
189C2×C22⋊C4C22⋊D6111yescharacteristic
190C2×C22⋊C4C22×D6111yescharacteristic
191C2×C42—41no—
192C2×C42C4×D6111yescharacteristic
193C2×C42D24411yescharacteristic
194C4⋊C8—2441no—
195C22⋊C8—1222no—
196C4×C8—2441no—
197#95—612no—
198#95—1222no—
199#95—1221no—
200#95—1222no—
201C24C2×C22⋊D6111yescharacteristic
202C2×Q8—4882no—
203C2×Q8—2442no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—1222no—
209C22×C4—1222no—
210C22×C4#1175819811yescharacteristic
211C22×C4#1175819811yescharacteristic
212C22×C4#1175819811yescharacteristic
213C22×C4#1175819811yescharacteristic
214C22×C4#1175820012yes—
215C22×C4#1175820212yes—
216C22×C4C2×C4×D6111yescharacteristic
217C22×C4C2×D24411yescharacteristic
218C2×C8—2442no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—44no—
222C4⋊C4—4882no—
223C4⋊C4—2442no—
224C4⋊C4—2441no—
225C4⋊C4—2441no—
226C4⋊C4—2442no—
227C22⋊C4—24no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—4882no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—44no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×#1175819811yescharacteristic
242C23C2×#1175820011yescharacteristic
243C23C2×#1175820211yescharacteristic
244Q8—4884no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—2442no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2442no—
268C2×C4#1451969111yescharacteristic
269C2×C4#1451969111yescharacteristic
270C2×C4#1451970012yes—
271C2×C4#1451970012yes—
272C2×C4#1451970512yes—
273C8—4882no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1451969111yescharacteristic
284C22C2×#1451970011yescharacteristic
285C22C2×#1451970511yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—84no—
292C4—4882no—
293C4—4882no—
294C4#1683949812yes—
295C4#1683949812yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1683949811yescharacteristic
301C2#2144746811yescharacteristic
302C2#2144749311yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1740C2
socleC22×C61C2×#99
prime radicalC24×C61C2×D4
cosocleC2×C22⋊C4×C61C23
hypercenter#1740D61
hypocenterC61#11717
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
5. C2×#14519777C4semidirect
9. #16839682C4semidirect
12. #16839628C4semidirect
171. C61#11717semidirect

Maximal subgroups

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

subgroupindex
#214516402
#214441332
#214438392
C61×#17402
#1171761