Finite Group Databasebeta

← #21456652

Subgroups — #21456652

Order 7808.

414 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 113 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#16840275C211yesmaximal, characteristic
2C2×#14519753C211yesmaximal, characteristic, complemented
3#16840101C212yesmaximal, complemented
4#16840095C211yesmaximal, characteristic, complemented
5#16840081C211yesmaximal, characteristic, complemented
6#16840017C212yesmaximal, complemented
7#16840011C212yesmaximal, complemented
8#16839923C212yesmaximal, complemented
9#16839901C211yesmaximal, characteristic, complemented
10#16839724C211yesmaximal, characteristic, complemented
11C61×#451C211yesmaximal, characteristic, complemented
12C2×#11758226C2211yescharacteristic
13C22×C4×D61C2211yescharacteristic
14C2×#11758201C2212yescomplemented
15C2×#11758189C2212yes—
16#14519762—21no—
17#14519753—21no—
18#14519753—21no—
19#14519753C2212yes—
20#14519753C2212yes—
21#14519753C2211yescharacteristic
22#14519752C2211yescharacteristic
23#14519751C2211yescharacteristic
24#14519747—21no—
25#14519739—21no—
26#14519737—21no—
27#14519735C2211yescharacteristic
28C2×C4⋊Dic61C2211yescharacteristic
29C2×#11758196—21no—
30C2×#11758198—21no—
31#14519710C2212yescomplemented
32#14519709C2212yescomplemented
33#14519701C2212yescomplemented
34#14519701C2212yescomplemented
35#14519699C2212yescomplemented
36C22⋊#92418C2211yescharacteristic
37#14519689C2212yes—
38C22×Q8×C61C2211yescharacteristic
39C2×Q16×C61C2212yescomplemented
40C22⋊Q8×C61C2211yescharacteristic
41#103×C61C2212yescomplemented
42C22⋊C8×C61C2211yescharacteristic
43C23×D61D411yescharacteristic
44C2×Q8×C61—21no—
45C2×Q8×C61—42no—
46C2×Q8×C61C2311yescharacteristic
47C2×Q8×C61C2312yes—
48C22×C4×C61—21no—
49C22×C4×C61C2311yescharacteristic
50C2×C22⋊D61—21no—
51C22×Dic61—21no—
52C22×Dic61D411yescharacteristic
53#11758226—42no—
54C2×D244—21no—
55C2×D244—21no—
56C2×D244C2311yescharacteristic
57C2×C4×D61—21no—
58C2×C4×D61—42no—
59C2×C4×D61D411yescharacteristic
60C2×C4×D61D411yescharacteristic
61C2×C4×D61C2311yescharacteristic
62C2×C4×D61C2311yescharacteristic
63Q16×C61—42no—
64C2×C8×C61C2312yes—
65C4⋊C4×C61—21no—
66C4⋊C4×C61C2311yescharacteristic
67C22⋊C4×C61—21no—
68#11758201—42no—
69#11758198—22no—
70#11758198—22no—
71#11758198—21no—
72#11758198—21no—
73#11758198—21no—
74#11758198—21no—
75C4⋊Dic61—21no—
76C4⋊Dic61C2312yes—
77C4⋊Dic61C2311yescharacteristic
78#11758196—22no—
79#11758196—22no—
80#11758196—21no—
81C4×Dic61—21no—
82C2×#92418C2312yes—
83#11758189—42no—
84C23×C61C2×D411yescharacteristic
85C22×D61—22no—
86C22×D61—21no—
87C22×D61—41no—
88C22×D61—41no—
89C22×D61C2×D411yescharacteristic
90C22×D61C2×D411yescharacteristic
91Q8×C61—22no—
92Q8×C61—22no—
93Q8×C61—22no—
94Q8×C61—42no—
95C2×C4×C61—22no—
96C2×C4×C61—21no—
97C2×C4×C61—22no—
98C2×C4×C61—21no—
99C2×C4×C61—42no—
100C2×C4×C61C2×D411yescharacteristic
101C2×C4×C61C2411yescharacteristic
102C22⋊D61—42no—
103C2×Dic61—21no—
104C2×Dic61—22no—
105C2×Dic61—41no—
106C2×Dic61—42no—
107C2×Dic61C2×D411yescharacteristic
108C2×Dic61C2×D411yescharacteristic
109D244—22no—
110D244—42no—
111D244—42no—
112C4×D61—22no—
113C4×D61—22no—
114C4×D61—41no—
115C4×D61—41no—
116C4×D61—42no—
117C8×C61—22no—
118#92418—22no—
119C22×C61—21no—
120C22×C61—21no—
121C22×C61#13712yes—
122C22×C61C22×D411yescharacteristic
123C2×D61—21no—
124C2×D61—21no—
125C2×D61—21no—
126C2×D61—21no—
127C2×D61—21no—
128C2×D61—21no—
129C2×D61—44no—
130C2×D61—41no—
131C2×D61—41no—
132C2×D61—41no—
133C4×C61—21no—
134C4×C61—41no—
135C4×C61—42no—
136C4×C61—41no—
137C4×C61H(5,𝔽2)12yes—
138Dic61—21no—
139Dic61—21no—
140Dic61—41no—
141Dic61—42no—
142#3196—611nomaximal, Sylow
143C2×C61—22no—
144C2×C61#53411yescharacteristic
145C2×C61C2×#13711yescharacteristic
146C2×C61#57511yescharacteristic
147D61—41no—
148D61—41no—
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#451D6111yescharacteristic, complemented
159#448—611no—
160#407—611no—
161C61#319611yescharacteristic, complemented, Sylow
162C2×#41—611no—
163C22×Q8C2×D6111yescharacteristic
164C23×C4—611no—
165C2×Q16C2×D6112yescomplemented
166C2×SD16—612no—
167C2×M16—612no—
168#123—1221no—
169C22⋊Q8—612no—
170C22⋊Q8—612no—
171C22⋊Q8—611no—
172C22⋊Q8—1221no—
173C22⋊Q8—1221no—
174C22⋊Q8C2×D6111yescharacteristic
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#103C2×D6112yescomplemented
189#102—612no—
190C22⋊C8—611no—
191C22⋊C8—612no—
192C22⋊C8C2×D6111yescharacteristic
193C24—611no—
194#41—2442no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8C22×D6111yescharacteristic
198C2×Q8C22×D6112yes—
199C2×D4—611no—
200C2×D4—1221no—
201C2×D4—1221no—
202C2×D4—1221no—
203C22×C4—21no—
204C22×C4—611no—
205C22×C4—611no—
206C22×C4—611no—
207C22×C4—611no—
208C22×C4—611no—
209C22×C4—1221no—
210C22×C4—1221no—
211C22×C4—2442no—
212C22×C4C22×D6111yescharacteristic
213Q16—42no—
214SD16—2442no—
215M16—2442no—
216C2×C8—612no—
217C2×C8C22×D6112yes—
218C4⋊C4—21no—
219C4⋊C4—612no—
220C4⋊C4—611no—
221C4⋊C4—1222no—
222C4⋊C4—1222no—
223C4⋊C4—1221no—
224C4⋊C4—1221no—
225C4⋊C4C22×D6111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1221no—
228C22⋊C4—1222no—
229C22⋊C4—1222no—
230C22⋊C4—1221no—
231C22⋊C4—1221no—
232C22⋊C4—1221no—
233C42—1221no—
234C23—611no—
235C23—611no—
236C23—1222no—
237C23—1221no—
238C23—2441no—
239C23—2441no—
240C23D4×D6111yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—42no—
245D4—1222no—
246D4—2442no—
247D4—2442no—
248D4—2442no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—611no—
255C2×C4—611no—
256C2×C4—1222no—
257C2×C4—1222no—
258C2×C4—1221no—
259C2×C4—1222no—
260C2×C4—2441no—
261C2×C4—2441no—
262C2×C4—2442no—
263C2×C4—2441no—
264C2×C4—2442no—
265C2×C4D4×D6111yescharacteristic
266C2×C4C23×D6111yescharacteristic
267C8—22no—
268C8—1222no—
269C22—21no—
270C22—21no—
271C22—1221no—
272C22—1221no—
273C22—1221no—
274C22—1221no—
275C22—1221no—
276C22—1221no—
277C22—2444no—
278C22—2441no—
279C22—2441no—
280C22—2441no—
281C22#1451980912yes—
282C22C2×D4×D6111yescharacteristic
283C4—21no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—1221no—
288C4—1221no—
289C4—2441no—
290C4—2442no—
291C4#1451982212yes—
292C2—22no—
293C2—4881no—
294C2—2441no—
295C2—2441no—
296C2#1684032211yescharacteristic
297C2C2×#1451980911yescharacteristic
298C2#1684056111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#451C2
socleC22×C61C22×D4
hypercenter#451D61
hypocenterC61#3196
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#14519753C2semidirect
3. #16840101C2semidirect
4. #16840095C2semidirect
5. #16840081C2semidirect
6. #16840017C2semidirect
7. #16840011C2semidirect
8. #16839923C2semidirect
9. #16839901C2semidirect
10. #16839724C2semidirect
11. C61×#451C2semidirect
14. C2×#11758201C22semidirect
31. #14519710C22semidirect
32. #14519709C22semidirect
33. #14519701C22semidirect
34. #14519701C22semidirect
35. #14519699C22semidirect
39. C2×Q16×C61C22semidirect
41. #103×C61C22semidirect
158. #451D61semidirect
161. C61#3196semidirect
165. C2×Q16C2×D61semidirect
188. #103C2×D61semidirect