Finite Group Databasebeta

← #21456651

Subgroups — #21456651

Order 7808.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 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#16840273C211yesmaximal, characteristic
2C2×#14519756C211yesmaximal, characteristic, complemented
3#16840104C212yesmaximal, complemented
4#16840098C211yesmaximal, characteristic, complemented
5#16840081C211yesmaximal, characteristic, complemented
6#16840020C212yesmaximal, complemented
7#16840008C212yesmaximal, complemented
8#16839926C212yesmaximal, complemented
9#16839920C211yesmaximal, characteristic, complemented
10#16839903C211yesmaximal, characteristic, complemented
11C61×#407C211yesmaximal, characteristic, complemented
12C2×#11758222C2211yescharacteristic
13C22×Dic122C2211yescharacteristic
14C2×Dic244C2212yescomplemented
15C2×#11758190C2212yescomplemented
16#14519760—21no—
17#14519756—21no—
18#14519756—21no—
19#14519756C2212yes—
20#14519756C2212yes—
21#14519756C2211yescharacteristic
22#14519756C2211yescharacteristic
23#14519751C2211yescharacteristic
24#14519751C2211yescharacteristic
25#14519746—21no—
26C4×C22⋊D61—21no—
27C4×C22⋊D61—21no—
28#14519736C2211yescharacteristic
29C2×C4⋊Dic61C2211yescharacteristic
30C2×#11758196—21no—
31C2×C22⋊Dic61—21no—
32#14519711C2212yescomplemented
33#14519709C2212yescomplemented
34#14519702C2212yescomplemented
35#14519702C2212yescomplemented
36#14519702C2212yescomplemented
37#14519699C2212yescomplemented
38C23×C4×C61C2211yescharacteristic
39C2×M16×C61C2212yes—
40C22⋊C8×C61C2211yescharacteristic
41C22⋊C8×C61C2211yescharacteristic
42C22⋊C8×C61C2212yes—
43C24×C61D411yescharacteristic
44C22×C4×C61—42no—
45C22×C4×C61D411yescharacteristic
46C22×C4×C61D411yescharacteristic
47C22×C4×C61D411yescharacteristic
48C22×C4×C61C2311yescharacteristic
49C22×C4×C61C2311yescharacteristic
50C22×C4×C61C2311yescharacteristic
51C2×C22⋊D61—21no—
52C2×C22⋊D61—21no—
53C2×C22⋊D61—21no—
54C22×Dic61—21no—
55C22×Dic61—21no—
56#11758222—42no—
57C2×D244C2311yescharacteristic
58C2×C4×D61—21no—
59C2×Dic122—21no—
60C2×Dic122—42no—
61C2×Dic122C2312yes—
62C2×Dic122C2311yescharacteristic
63M16×C61—42no—
64C2×C8×C61C2312yes—
65C2×C8×C61C2312yes—
66C22⋊Dic61—22no—
67C22⋊Dic61—22no—
68C22⋊Dic61—21no—
69C22⋊Dic61—21no—
70C22⋊Dic61—21no—
71#11758198—21no—
72#11758198—21no—
73C4⋊Dic61—21no—
74C4⋊Dic61C2312yes—
75C4⋊Dic61C2311yescharacteristic
76C4⋊Dic61C2311yescharacteristic
77#11758196—22no—
78#11758196—22no—
79#11758196—21no—
80#11758196—21no—
81C4×Dic61—21no—
82Dic244—42no—
83#11758190—42no—
84C23×C61—22no—
85C23×C61—41no—
86C23×C61—41no—
87C23×C61C2×D411yescharacteristic
88C23×C61C2×D411yescharacteristic
89C23×C61C2×D411yescharacteristic
90C22×D61—21no—
91C2×C4×C61—22no—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—41no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61C2×D411yescharacteristic
98C2×C4×C61C2×D411yescharacteristic
99C2×C4×C61C2×D411yescharacteristic
100C2×C4×C61C2411yescharacteristic
101C22⋊D61—42no—
102C22⋊D61—42no—
103C22⋊D61—42no—
104C2×Dic61—21no—
105C2×Dic61—21no—
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—21no—
109C2×Dic61—42no—
110C2×Dic61—42no—
111D244—22no—
112C4×D61—42no—
113Dic122—22no—
114Dic122—22no—
115Dic122—22no—
116Dic122—42no—
117C8×C61—22no—
118C8×C61—22no—
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—44no—
128C22×C61#13712yes—
129C22×C61C22×D411yescharacteristic
130C2×D61—41no—
131C2×D61—41no—
132C2×D61—41no—
133C4×C61—21no—
134C4×C61—21no—
135C4×C61—21no—
136C4×C61H(5,𝔽2)12yes—
137Dic61—42no—
138Dic61—41no—
139Dic61—41no—
140Dic61—42no—
141Dic61—41no—
142#3196—611nomaximal, Sylow
143C2×C61—22no—
144C2×C61—41no—
145C2×C61—41no—
146C2×C61#53411yescharacteristic
147C2×C61C2×#13711yescharacteristic
148C2×C61#57511yescharacteristic
149D61—81no—
150#525—611no—
151C2×C22⋊Q8—611no—
152#485—612no—
153#484—611no—
154#481—611no—
155#470—612no—
156#468—612no—
157#452—612no—
158#451—611no—
159#448—611no—
160#407D6111yescharacteristic, complemented
161C61#319611yescharacteristic, complemented, Sylow
162C2×#41—611no—
163C22×Q8—611no—
164C23×C4C2×D6111yescharacteristic
165C2×Q16—612no—
166C2×SD16—612no—
167C2×M16C2×D6112yes—
168#123—1221no—
169C22⋊Q8—612no—
170C22⋊Q8—612no—
171C22⋊Q8—611no—
172C22⋊Q8—611no—
173C22⋊Q8—1221no—
174C22⋊Q8—1221no—
175#121—611no—
176#121—611no—
177#120—1221no—
178C4×D4—1221no—
179C4×D4—1221no—
180#117—611no—
181C2×C4⋊C4—611no—
182C2×C4⋊C4—1221no—
183C2×C22⋊C4—1221no—
184#107—612no—
185#106—612no—
186#103—612no—
187#103—612no—
188#103—612no—
189#102—612no—
190C22⋊C8C2×D6111yescharacteristic
191C22⋊C8C2×D6111yescharacteristic
192C22⋊C8C2×D6112yes—
193C24D24411yescharacteristic
194#41—2442no—
195C2×Q8—612no—
196C2×Q8—611no—
197C2×Q8—1221no—
198C2×Q8—2442no—
199C2×D4—611no—
200C2×D4—1221no—
201C2×D4—1221no—
202C2×D4—1221no—
203C22×C4—42no—
204C22×C4—1221no—
205C22×C4—1221no—
206C22×C4—1221no—
207C22×C4D24411yescharacteristic
208C22×C4D24411yescharacteristic
209C22×C4D24411yescharacteristic
210C22×C4C22×D6111yescharacteristic
211C22×C4C22×D6111yescharacteristic
212C22×C4C22×D6111yescharacteristic
213Q16—2442no—
214SD16—2442no—
215M16—42no—
216C2×C8C22×D6112yes—
217C2×C8C22×D6112yes—
218C4⋊C4—612no—
219C4⋊C4—611no—
220C4⋊C4—611no—
221C4⋊C4—1221no—
222C4⋊C4—1222no—
223C4⋊C4—1222no—
224C4⋊C4—1221no—
225C4⋊C4—1221no—
226C22⋊C4—1222no—
227C22⋊C4—1222no—
228C22⋊C4—1221no—
229C22⋊C4—1221no—
230C22⋊C4—1221no—
231C22⋊C4—1221no—
232C22⋊C4—1221no—
233C42—1221no—
234C23—22no—
235C23—41no—
236C23—41no—
237C23—1221no—
238C23C2×D24411yescharacteristic
239C23C2×D24411yescharacteristic
240C23C2×D24411yescharacteristic
241Q8—1222no—
242Q8—1222no—
243Q8—1222no—
244Q8—2442no—
245D4—1222no—
246D4—2442no—
247D4—2442no—
248D4—2442no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—1221no—
256C2×C4—1221no—
257C2×C4—1222no—
258C2×C4—1221no—
259C2×C4—1222no—
260C2×C4—2442no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4C2×D24411yescharacteristic
264C2×C4C2×D24411yescharacteristic
265C2×C4C2×D24411yescharacteristic
266C2×C4C23×D6111yescharacteristic
267C8—22no—
268C8—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—2441no—
279C22—2441no—
280C22—2441no—
281C22#1451981012yes—
282C22C22×D24411yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—2441no—
287C4—2442no—
288C4—2441no—
289C4—2441no—
290C4—2442no—
291C4#1451982212yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—4881no—
296C2#1684032411yescharacteristic
297C2C2×#1451981011yescharacteristic
298C2#1684056211yescharacteristic
subgroupG/H
centerC22C22×D244
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#407C2
socleC22×C61C22×D4
hypercenter#407D61
hypocenterC61#3196
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#14519756C2semidirect
3. #16840104C2semidirect
4. #16840098C2semidirect
5. #16840081C2semidirect
6. #16840020C2semidirect
7. #16840008C2semidirect
8. #16839926C2semidirect
9. #16839920C2semidirect
10. #16839903C2semidirect
11. C61×#407C2semidirect
14. C2×Dic244C22semidirect
15. C2×#11758190C22semidirect
32. #14519711C22semidirect
33. #14519709C22semidirect
34. #14519702C22semidirect
35. #14519702C22semidirect
36. #14519702C22semidirect
37. #14519699C22semidirect
160. #407D61semidirect
161. C61#3196semidirect