Finite Group Databasebeta

← #21452198

Subgroups — #21452198

Order 7808.

688 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#14519764C211yesmaximal, characteristic, complemented
2C2×#14519745C211yesmaximal, characteristic
3#16839687C212yesmaximal, complemented
4#16839678C212yesmaximal, complemented
5#16839675C212yesmaximal, complemented
6#16839654C212yesmaximal, complemented
7#16839641C212yesmaximal, complemented
8#16839617C212yesmaximal, complemented
9C61×#386C211yesmaximal, characteristic, complemented
10C24×D61C2211yescharacteristic
11C22×Dic122C2211yescharacteristic, complemented
12C22×C22⋊D61C2212yes—
13#14519764—24no—
14#14519745—24no—
15C2×C4⋊Dic61C2212yescomplemented
16C2×C22⋊Dic61C2211yescharacteristic
17C2×C22⋊Dic61C2212yes—
18C2×C22⋊Dic61C2212yescomplemented
19C2×#11758198C2212yes—
20C2×#11758198C2212yes—
21C2×#11758198C2212yes—
22C2×#11758198C2212yes—
23C2×#11758198C2212yes—
24C2×C4×Dic61C2211yescharacteristic, complemented
25#14519684C2212yescomplemented
26#14519684C2212yescomplemented
27#14519684C2212yescomplemented
28C22×D4×C61C2211yescharacteristic
29C23×C4×C61C2211yescharacteristic, complemented
30C2×C22⋊C4×C61C2212yes—
31C2×C22⋊C4×C61C2211yescharacteristic, complemented
32#95×C61C2212yescomplemented
33C24×C61C2311yescharacteristic
34C24×C61C2311yescharacteristic
35C23×D61—22no—
36C23×D61—24no—
37C23×D61—41no—
38C23×D61—41no—
39C23×D61—42no—
40C23×D61C2312yes—
41C2×D4×C61—41no—
42C2×D4×C61—41no—
43C2×D4×C61D414yes—
44C22×C4×C61—41no—
45C22×C4×C61—42no—
46C22×C4×C61C2312yes—
47C22×C4×C61C2312yes—
48C22×C4×C61C2311yescharacteristic
49C2×C22⋊D61—24no—
50C2×C22⋊D61—24no—
51C2×C22⋊D61—42no—
52C22×Dic61C2312yes—
53C22×Dic61C2312yes—
54C22×Dic61C2312yes—
55C2×Dic122—22no—
56C2×Dic122—41no—
57C2×Dic122—41no—
58C22⋊C4×C61—42no—
59C22⋊C4×C61—41no—
60C22⋊Dic61—22no—
61C22⋊Dic61—24no—
62C22⋊Dic61—24no—
63#11758198—24no—
64#11758198—24no—
65#11758198—42no—
66#11758198—42no—
67#11758198—42no—
68C4⋊Dic61—42no—
69C4×Dic61—22no—
70C23×C61—22no—
71C23×C61—22no—
72C23×C61—21no—
73C23×C61—22no—
74C23×C61—22no—
75C23×C61—21no—
76C23×C61—21no—
77C23×C61—21no—
78C23×C61C2×D412yes—
79C23×C61C2×D412yes—
80C23×C61C2411yescharacteristic
81C22×D61—22no—
82C22×D61—22no—
83C22×D61—22no—
84C22×D61—22no—
85C22×D61—22no—
86C22×D61—22no—
87C22×D61—44no—
88C22×D61—48no—
89C22×D61—48no—
90C22×D61—44no—
91C22×D61—44no—
92C22×D61#4114yes—
93D4×C61—44no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—44no—
103C2×C4×C61C2×D412yes—
104C22⋊D61—48no—
105C2×Dic61—22no—
106C2×Dic61—22no—
107C2×Dic61—22no—
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—22no—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—22no—
114Dic122—44no—
115C22×C61—22no—
116C22×C61—22no—
117C22×C61—24no—
118C22×C61—24no—
119C22×C61—42no—
120C22×C61—42no—
121C22×C61—41no—
122C22×C61—41no—
123C22×C61—41no—
124C22×C61—41no—
125C22×C61C22×D411yescharacteristic
126C22×C61C2×#4112yes—
127C22×C61H(5,𝔽2)11yescharacteristic
128C22×C61H(5,𝔽2)11yescharacteristic
129C22×C61H(5,𝔽2)12yes—
130C2×D61—24no—
131C2×D61—24no—
132C2×D61—24no—
133C2×D61—416no—
134C2×D61—42no—
135C2×D61—42no—
136C2×D61—42no—
137C2×D61—42no—
138C4×C61—22no—
139C4×C61—42no—
140C4×C61—42no—
141Dic61—42no—
142Dic61—42no—
143Dic61—42no—
144#2684—611nomaximal, Sylow
145C2×C61—42no—
146C2×C61—42no—
147C2×C61#53411yescharacteristic
148C2×C61#53812yes—
149C2×C61#54612yes—
150C2×C61#55111yescharacteristic
151C2×C61H(3,𝔽4)11yescharacteristic
152D61—44no—
153C2×#124—611no—
154C2×#120—611no—
155#397—612no—
156#394—612no—
157#393—612no—
158#388—612no—
159#386—612no—
160#386D6111yescharacteristic, complemented
161#379—612no—
162C61#268411yescharacteristic, complemented, Sylow
163C25—611no—
164C22×Q8—611no—
165C22×D4—612no—
166C22×D4C2×D6111yescharacteristic
167C23×C4C2×D6111yescharacteristic, complemented
168#124—1224no—
169#120—1224no—
170C2×C4⋊C4—612no—
171C2×C22⋊C4—611no—
172C2×C22⋊C4—612no—
173C2×C22⋊C4—612no—
174C2×C22⋊C4—612no—
175C2×C22⋊C4—612no—
176C2×C22⋊C4—612no—
177C2×C22⋊C4—612no—
178C2×C22⋊C4—612no—
179C2×C22⋊C4C2×D6112yes—
180C2×C22⋊C4C2×D6111yescharacteristic, complemented
181C2×C42—611no—
182#95—612no—
183#95—612no—
184#95—612no—
185#95C2×D6112yescomplemented
186C24—612no—
187C24—1224no—
188C24—1222no—
189C24—2442no—
190C24—2441no—
191C24—2441no—
192C24C22×D6111yescharacteristic
193C24C22×D6111yescharacteristic
194C2×Q8—1222no—
195C2×Q8—2441no—
196C2×Q8—2441no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—1224no—
200C2×D4—1224no—
201C2×D4—2442no—
202C2×D4D24414yes—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—612no—
206C22×C4—612no—
207C22×C4—612no—
208C22×C4C22×D6112yes—
209C22×C4C22×D6112yes—
210C22×C4C22×D6111yescharacteristic
211C4⋊C4—2442no—
212C22⋊C4—42no—
213C22⋊C4—41no—
214C22⋊C4—1224no—
215C22⋊C4—1222no—
216C22⋊C4—1224no—
217C22⋊C4—1224no—
218C22⋊C4—1224no—
219C22⋊C4—2442no—
220C22⋊C4—2442no—
221C22⋊C4—2442no—
222C42—1222no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23—614no—
232C23—1222no—
233C23—1222no—
234C23—1222no—
235C23—1222no—
236C23—1222no—
237C23—1222no—
238C23—2448no—
239C23—2444no—
240C23—2444no—
241C23—2448no—
242C23—2444no—
243C23C2×D24412yes—
244C23C2×D24412yes—
245C23C23×D6111yescharacteristic
246Q8—2444no—
247D4—44no—
248D4—2448no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—1222no—
259C2×C4—1222no—
260C2×C4—1222no—
261C2×C4—1222no—
262C2×C4—1222no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—1222no—
266C2×C4—1222no—
267C2×C4C2×D24412yes—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—1224no—
279C22—1224no—
280C22—1224no—
281C22—24416no—
282C22—2442no—
283C22—2442no—
284C22—2442no—
285C22—2442no—
286C22C22×D24411yescharacteristic
287C22#41×D6112yes—
288C22#1451982211yescharacteristic
289C22#1451982212yes—
290C22#1451982211yescharacteristic
291C4—22no—
292C4—42no—
293C4—42no—
294C4—2442no—
295C4—2442no—
296C4—2442no—
297C2—42no—
298C2—42no—
299C2—2444no—
300C2#1684032411yescharacteristic
301C2#1684034412yes—
302C2#1684040712yes—
303C2#1684044011yescharacteristic
304C2#1684050511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC61×#386C2
hypercenter#386D61
hypocenterC61#2684
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#14519764C2semidirect
3. #16839687C2semidirect
4. #16839678C2semidirect
5. #16839675C2semidirect
6. #16839654C2semidirect
7. #16839641C2semidirect
8. #16839617C2semidirect
9. C61×#386C2semidirect
11. C22×Dic122C22semidirect
15. C2×C4⋊Dic61C22semidirect
18. C2×C22⋊Dic61C22semidirect
24. C2×C4×Dic61C22semidirect
25. #14519684C22semidirect
26. #14519684C22semidirect
27. #14519684C22semidirect
29. C23×C4×C61C22semidirect
31. C2×C22⋊C4×C61C22semidirect
32. #95×C61C22semidirect
160. #386D61semidirect
162. C61#2684semidirect
167. C23×C4C2×D61semidirect
180. C2×C22⋊C4C2×D61semidirect
185. #95C2×D61semidirect