Finite Group Databasebeta

← #29595171

Subgroups — #29595171

Order 15616.

456 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 109 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#21454525C211yesmaximal, characteristic
2#21447461C212yesmaximal
3#21445622C212yesmaximal, complemented
4#21443940C211yesmaximal, characteristic, complemented
5C61×#1392C211yesmaximal, characteristic, complemented
6C2×#14519784C2211yescharacteristic
7C22×#11758189C411yescharacteristic, complemented
8#16840269C2211yescharacteristic
9#16839837—22no—
10#16839830—22no—
11#16839788C411yescharacteristic, complemented
12#16839736C2212yes—
13#16839725C411yescharacteristic, complemented
14#16839720—22no—
15#16839720C411yescharacteristic, complemented
16#16839553C414yescomplemented
17#16839550C414yescomplemented
18#16839534—22no—
19C61×#413C2212yes—
20C61×#408C2211yescharacteristic
21C22×C4×D61C2×C411yescharacteristic
22#14519784—42no—
23C2×#11758194C2×C411yescharacteristic
24C2×#11758194D412yes—
25C2×#11758189—21no—
26C2×#11758189—21no—
27C2×#11758189—42no—
28C22×#92418C2×C411yescharacteristic
29C22×#92418D412yes—
30#14519758C2×C412yes—
31#14519753—21no—
32#14519748—21no—
33#14519745C2×C412yes—
34C4×C22⋊D61—22no—
35#14519736—22no—
36#14519736C2×C411yescharacteristic
37#14519706—22no—
38#14519692—22no—
39#14519692C2×C414yes—
40#14519689—22no—
41#14519687—21no—
42#14519687—42no—
43C8×Dic61—21no—
44C8×Dic61—42no—
45C2×#41×C61C2311yescharacteristic
46C2×M16×C61—22no—
47C2×M16×C61C2×C411yescharacteristic
48C22×C8×C61C2×C411yescharacteristic
49#101×C61—22no—
50#100×C61C2×C414yes—
51C22⋊C8×C61—22no—
52C23×D61C4211yescharacteristic
53#41×C61—41no—
54#41×C61—41no—
55C2×Q8×C61#4111yescharacteristic
56C2×D4×C61C22×C412yes—
57C2×D4×C61#4111yescharacteristic
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D412yes—
60C2×C22⋊D61—22no—
61C22×Dic61C4211yescharacteristic
62C2×C4×D61—21no—
63C2×C4×D61—21no—
64C2×C4×D61—41no—
65C2×C4×D61—41no—
66M16×C61—24no—
67M16×C61—42no—
68M16×C61—41no—
69C2×C8×C61—21no—
70C2×C8×C61—21no—
71C2×C8×C61—42no—
72C2×C8×C61—41no—
73C22⋊Dic61—22no—
74C22⋊Dic61—24no—
75C22⋊Dic61C4212yes—
76#11758198—22no—
77C4⋊Dic61—21no—
78C4⋊Dic61—42no—
79#11758196—22no—
80C4×Dic61—21no—
81C4×Dic61—42no—
82#11758194—41no—
83#11758194—42no—
84C2×#92418—22no—
85C2×#92418—22no—
86C2×#92418—22no—
87C2×#92418C22⋊C411yescharacteristic
88C2×#92418C22⋊C412yes—
89C2×#92418C22⋊C411yescharacteristic
90#11758189—42no—
91#11758189—42no—
92C23×C61C2×C4211yescharacteristic
93C23×C61C4×D412yes—
94C22×D61—21no—
95C22×D61—41no—
96C22×D61—81no—
97Q8×C61—41no—
98D4×C61—42no—
99D4×C61—41no—
100C2×C4×C61—22no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—21no—
104C2×C4×C61C2×C22⋊C411yescharacteristic
105C2×C4×C61#11711yescharacteristic
106C2×C4×C61C4×D412yes—
107C22⋊D61—82no—
108C2×Dic61—22no—
109C2×Dic61—21no—
110C2×Dic61—41no—
111C2×Dic61—44no—
112C4×D61—42no—
113C4×D61—41no—
114C4×D61—41no—
115C8×C61—44no—
116C8×C61—41no—
117C8×C61—41no—
118#92418—21no—
119#92418—22no—
120#92418#9912yes—
121#12624—611nomaximal, Sylow
122C22×C61—22no—
123C22×C61—21no—
124C22×C61—42no—
125C22×C61—41no—
126C22×C61C4×C22⋊C411yescharacteristic
127C2×D61—41no—
128C2×D61—41no—
129C2×D61—41no—
130C2×D61—82no—
131C4×C61—22no—
132C4×C61—21no—
133C4×C61C2×#9911yescharacteristic
134C4×C61#41011yescharacteristic
135Dic61—42no—
136Dic61—41no—
137Dic61—84no—
138#2930—611no—
139#2120—612no—
140#1757—612no—
141#1392D6111yescharacteristic, complemented
142#1391—611no—
143C2×C61—21no—
144C2×C61—42no—
145C2×C61—41no—
146C2×C61C4×#9911yescharacteristic
147D61—81no—
148C2×#131—611no—
149C22×M16—611no—
150#525—611no—
151#436—1222no—
152#435—1222no—
153#424—611no—
154#413C2×D6112yes—
155#410—612no—
156#408C2×D6111yescharacteristic
157#407—611no—
158#405—611no—
159#405—1222no—
160#352—614no—
161AΣL(1,𝔽16)—614no—
162#344—1222no—
163C61#1262411yescharacteristic, complemented, Sylow
164C2×#41C22×D6111yescharacteristic
165C23×C4—611no—
166#131—2442no—
167C2×M16—22no—
168C2×M16—611no—
169C2×M16—612no—
170C2×M16—1221no—
171C2×M16—1221no—
172C2×M16—2442no—
173C2×M16C4×D6111yescharacteristic
174C22×C8—611no—
175C22×C8—612no—
176C22×C8C4×D6111yescharacteristic
177#123—612no—
178C22⋊Q8—1221no—
179#121—1221no—
180#120—612no—
181C4×D4—1222no—
182#117—611no—
183#117—1222no—
184C4⋊C8—1222no—
185#101—22no—
186#100C4×D6114yes—
187#99—614no—
188#99—1222no—
189C22⋊C8—22no—
190C22⋊C8—1222no—
191#97—1221no—
192#97—2442no—
193C4×C8—1221no—
194C4×C8—2442no—
195C24—611no—
196#41—41no—
197#41—41no—
198C2×Q8#1175822411yescharacteristic
199C2×D4—1222no—
200C2×D4C2×C4×D6112yes—
201C2×D4#1175822611yescharacteristic
202C22×C4—611no—
203C22×C4—1221no—
204C22×C4—1221no—
205C22×C4—2441no—
206C22×C4—2441no—
207C22×C4C2×C4×D6111yescharacteristic
208C22×C4D4×D6112yes—
209M16—24no—
210M16—42no—
211M16—41no—
212M16—2442no—
213M16—2442no—
214M16—2441no—
215M16—2442no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—611no—
221C2×C8—611no—
222C2×C8—612no—
223C2×C8—1222no—
224C2×C8—1222no—
225C2×C8—1222no—
226C4⋊C4—1221no—
227C4⋊C4—1222no—
228C4⋊C4—2442no—
229C22⋊C4—612no—
230C22⋊C4—1222no—
231C22⋊C4—1222no—
232C22⋊C4—1224no—
233C42—1221no—
234C42—2442no—
235C23—1221no—
236C23—4881no—
237C23—2441no—
238C23C42×D6111yescharacteristic
239C23#1451973912yes—
240Q8—41no—
241D4—42no—
242D4—41no—
243D4—4882no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—1222no—
249C2×C4—1221no—
250C2×C4—2442no—
251C2×C4—2444no—
252C2×C4—2441no—
253C2×C4—2441no—
254C2×C4—2441no—
255C2×C4C22⋊C4×D6111yescharacteristic
256C2×C4#1451973511yescharacteristic
257C2×C4#1451973712yes—
258C8—44no—
259C8—41no—
260C8—41no—
261C8—612no—
262C8—1222no—
263C8—1221no—
264C22—22no—
265C22—21no—
266C22—42no—
267C22—41no—
268C22—4882no—
269C22—2441no—
270C22—2441no—
271C22—2441no—
272C22#1683960911yescharacteristic
273C4—22no—
274C4—21no—
275C4—4884no—
276C4—2442no—
277C4—2441no—
278C4#99×D6111yescharacteristic
279C4#1683973411yescharacteristic
280C2—21no—
281C2—42no—
282C2—41no—
283C2—4881no—
284C2#2144560611yescharacteristic
subgroupG/H
centerC4#99×D61
commutatorC23×C61C2×C42
FrattiniC2×#41C22×D61
FittingC61×#1392C2
socleC2×C61C4×#99
prime radical#16840269C22
cosocleC2×#41×C61C23
hypercenter#1392D61
hypocenterC61#12624
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1392

Complements

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

NH ≅ G/Ntype
3. #21445622C2semidirect
4. #21443940C2semidirect
5. C61×#1392C2semidirect
7. C22×#11758189C4semidirect
11. #16839788C4semidirect
13. #16839725C4semidirect
15. #16839720C4semidirect
16. #16839553C4semidirect
17. #16839550C4semidirect
141. #1392D61semidirect
163. C61#12624semidirect

Maximal subgroups

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

subgroupindex
#214545252
#214474612
#214456222
#214439402
C61×#13922
#1262461