Finite Group Databasebeta

← #21452197

Subgroups — #21452197

Order 7808.

688 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 141 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
2#16839690C212yesmaximal, complemented
3#16839676C212yesmaximal, complemented
4#16839674C212yesmaximal, complemented
5#16839650C212yesmaximal, complemented
6#16839642C212yesmaximal, complemented
7#16839642C211yesmaximal, characteristic, complemented
8#16839618C212yesmaximal, complemented
9C2×#120×C61C211yesmaximal, characteristic
10C22×Dic122C2211yescharacteristic, complemented
11C23×Dic61C2211yescharacteristic, complemented
12#14519764—24no—
13C2×#11758196C2212yescomplemented
14C2×C22⋊Dic61C2212yes—
15C2×C22⋊Dic61C2212yes—
16C2×C22⋊Dic61C2212yes—
17C2×C22⋊Dic61C2212yes—
18C2×C22⋊Dic61C2212yes—
19C2×C22⋊Dic61C2212yes—
20C2×C22⋊Dic61C2212yescomplemented
21C2×C22⋊Dic61C2211yescharacteristic, complemented
22C2×C4×Dic61C2211yescharacteristic, complemented
23#14519684C2212yescomplemented
24#14519684C2212yescomplemented
25#14519684C2212yescomplemented
26#14519684C2212yescomplemented
27C25×C61C2211yescharacteristic
28C22×D4×C61C2212yes—
29C22×D4×C61C2211yescharacteristic
30#120×C61—24no—
31C2×C22⋊C4×C61C2211yescharacteristic
32C2×C22⋊C4×C61C2212yes—
33C24×C61—22no—
34C24×C61—24no—
35C24×C61—42no—
36C24×C61—41no—
37C24×C61—41no—
38C24×C61C2311yescharacteristic
39C24×C61C2311yescharacteristic
40C24×C61C2312yes—
41C2×D4×C61—24no—
42C2×D4×C61—24no—
43C2×D4×C61—41no—
44C2×D4×C61—41no—
45C2×D4×C61—42no—
46C2×D4×C61D414yes—
47C22×C4×C61C2312yes—
48C22×C4×C61C2311yescharacteristic
49C22×Dic61—41no—
50C22×Dic61—42no—
51C22×Dic61C2312yes—
52C22×Dic61C2312yes—
53C22×Dic61C2312yes—
54C22×Dic61C2312yes—
55C2×Dic122—22no—
56C2×Dic122—41no—
57C2×Dic122—41no—
58C22⋊C4×C61—22no—
59C22⋊C4×C61—24no—
60C22⋊Dic61—24no—
61C22⋊Dic61—24no—
62C22⋊Dic61—24no—
63C22⋊Dic61—42no—
64C22⋊Dic61—42no—
65C22⋊Dic61—42no—
66C22⋊Dic61—42no—
67C22⋊Dic61—41no—
68#11758196—42no—
69C4×Dic61—22no—
70C23×C61—22no—
71C23×C61—22no—
72C23×C61—22no—
73C23×C61—22no—
74C23×C61—21no—
75C23×C61—22no—
76C23×C61—22no—
77C23×C61—22no—
78C23×C61—22no—
79C23×C61—21no—
80C23×C61—22no—
81C23×C61—21no—
82C23×C61—21no—
83C23×C61—22no—
84C23×C61—48no—
85C23×C61—44no—
86C23×C61—44no—
87C23×C61—48no—
88C23×C61—44no—
89C23×C61C2×D412yes—
90C23×C61C2×D412yes—
91C23×C61#4114yes—
92C23×C61C2411yescharacteristic
93D4×C61—44no—
94D4×C61—48no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61C2×D412yes—
101C2×Dic61—22no—
102C2×Dic61—22no—
103C2×Dic61—22no—
104C2×Dic61—22no—
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—44no—
114Dic122—44no—
115C22×C61—22no—
116C22×C61—22no—
117C22×C61—24no—
118C22×C61—24no—
119C22×C61—24no—
120C22×C61—24no—
121C22×C61—24no—
122C22×C61—416no—
123C22×C61—42no—
124C22×C61—42no—
125C22×C61—42no—
126C22×C61—42no—
127C22×C61—42no—
128C22×C61—41no—
129C22×C61—41no—
130C22×C61—41no—
131C22×C61—41no—
132C22×C61—42no—
133C22×C61C22×D411yescharacteristic
134C22×C61C2×#4112yes—
135C22×C61H(5,𝔽2)11yescharacteristic
136C22×C61H(5,𝔽2)11yescharacteristic
137C22×C61H(5,𝔽2)12yes—
138C4×C61—22no—
139C4×C61—42no—
140Dic61—42no—
141Dic61—42no—
142Dic61—42no—
143Dic61—42no—
144#2684—611nomaximal, Sylow
145C2×C61—42no—
146C2×C61—42no—
147C2×C61—44no—
148C2×C61#53411yescharacteristic
149C2×C61#53812yes—
150C2×C61#54612yes—
151C2×C61#55111yescharacteristic
152C2×C61H(3,𝔽4)11yescharacteristic
153C2×#124—611no—
154C2×#120D6111yescharacteristic
155#397—612no—
156#394—612no—
157#393—612no—
158#388—612no—
159#386—611no—
160#386—612no—
161#379—612no—
162C61#268411yescharacteristic, complemented, Sylow
163C25C2×D6111yescharacteristic
164C22×Q8—611no—
165C22×D4C2×D6112yes—
166C22×D4C2×D6111yescharacteristic
167C23×C4—611no—
168#124—1224no—
169#120—24no—
170C2×C4⋊C4—612no—
171C2×C22⋊C4—612no—
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—611no—
179C2×C22⋊C4C2×D6112yes—
180C2×C22⋊C4C2×D6111yescharacteristic
181C2×C42—611no—
182#95—612no—
183#95—612no—
184#95—612no—
185#95—612no—
186C24—24no—
187C24—22no—
188C24—42no—
189C24—41no—
190C24—41no—
191C24C22×D6112yes—
192C24C22×D6111yescharacteristic
193C24C22×D6111yescharacteristic
194C2×Q8—1222no—
195C2×Q8—2441no—
196C2×Q8—2441no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4C22⋊D6114yes—
203C22×C4—612no—
204C22×C4—612no—
205C22×C4—612no—
206C22×C4—612no—
207C22×C4—2441no—
208C22×C4—2442no—
209C22×C4C22×D6112yes—
210C22×C4C22×D6111yescharacteristic
211C4⋊C4—2442no—
212C22⋊C4—22no—
213C22⋊C4—24no—
214C22⋊C4—1224no—
215C22⋊C4—1224no—
216C22⋊C4—1224no—
217C22⋊C4—2441no—
218C22⋊C4—2442no—
219C22⋊C4—2442no—
220C22⋊C4—2442no—
221C22⋊C4—2442no—
222C42—1222no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—48no—
238C23—44no—
239C23—44no—
240C23—48no—
241C23—44no—
242C23#1175822414yes—
243C23C2×C22⋊D6112yes—
244C23C2×C22⋊D6112yes—
245C23C23×D6111yescharacteristic
246Q8—2444no—
247D4—44no—
248D4—48no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—1222no—
255C2×C4—1222no—
256C2×C4—1222no—
257C2×C4—1222no—
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—2444no—
267C2×C4C2×C22⋊D6112yes—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—416no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22C22×C22⋊D6111yescharacteristic
287C22C2×#1175822412yes—
288C22#1451982211yescharacteristic
289C22#1451982212yes—
290C22#1451982211yescharacteristic
291C4—22no—
292C4—42no—
293C4—2442no—
294C4—2442no—
295C4—2442no—
296C4—2442no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2#1684032111yescharacteristic
301C2#1684034712yes—
302C2#1684040612yes—
303C2#1684044311yescharacteristic
304C2#1684050511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×#120×C61C2
hypercenterC2×#120D61
hypocenterC61#2684
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#14519764C2semidirect
2. #16839690C2semidirect
3. #16839676C2semidirect
4. #16839674C2semidirect
5. #16839650C2semidirect
6. #16839642C2semidirect
7. #16839642C2semidirect
8. #16839618C2semidirect
10. C22×Dic122C22semidirect
11. C23×Dic61C22semidirect
13. C2×#11758196C22semidirect
20. C2×C22⋊Dic61C22semidirect
21. C2×C22⋊Dic61C22semidirect
22. C2×C4×Dic61C22semidirect
23. #14519684C22semidirect
24. #14519684C22semidirect
25. #14519684C22semidirect
26. #14519684C22semidirect
162. C61#2684semidirect