Finite Group Databasebeta

← #21449467

Subgroups — #21449467

Order 7808.

1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×D4×D61C211yesmaximal, characteristic
2C2×#14519766C211yesmaximal, characteristic, complemented
3C2×#14519753C211yesmaximal, characteristic, complemented
4#16839677C214yesmaximal, complemented
5#16839665C211yesmaximal, characteristic, complemented
6#16839664C211yesmaximal, characteristic, complemented
7#16839617C214yesmaximal, complemented
8C4×#11758198C211yesmaximal, characteristic, complemented
9C2×#124×C61C211yesmaximal, characteristic, complemented
10C24×D61C2212yes—
11C22×D244C2211yescharacteristic
12C22×C22⋊D61C2212yes—
13C2×D4×D61—24no—
14C2×D4×D61—24no—
15C2×D4×D61—24no—
16C22×C4×D61C2211yescharacteristic
17#14519766—24no—
18#14519753—24no—
19C2×C4⋊Dic61C2211yescharacteristic, complemented
20C2×#11758196C2212yescomplemented
21C2×C22⋊Dic61C2214yescomplemented
22C2×#11758198C2212yes—
23C2×#11758198C2212yes—
24C2×#11758198C2214yescomplemented
25C2×#11758198C2214yescomplemented
26C2×C4×Dic61C2211yescharacteristic, complemented
27#14519684C2212yescomplemented
28C22×Q8×C61C2211yescharacteristic, complemented
29C22×D4×C61C2211yescharacteristic
30#124×C61—24no—
31C2×C22⋊C4×C61C2214yescomplemented
32C2×C42×C61C2211yescharacteristic, complemented
33C24×C61C2312yes—
34C23×D61—24no—
35C23×D61—24no—
36C23×D61—24no—
37C23×D61—28no—
38C23×D61—44no—
39C23×D61C2312yes—
40C23×D61C2311yescharacteristic
41C2×Q8×C61—22no—
42C2×Q8×C61—22no—
43C2×Q8×C61—22no—
44C2×D4×C61—22no—
45C2×D4×C61—22no—
46C2×D4×C61D414yes—
47C22×C4×C61C2312yes—
48C22×C4×C61C2312yes—
49C22×C4×C61C2311yescharacteristic
50C2×C22⋊D61—24no—
51C2×C22⋊D61—24no—
52C2×C22⋊D61—24no—
53C22×Dic61C2312yes—
54C22×Dic61C2312yes—
55C22×Dic61C2311yescharacteristic
56D4×D61—232no—
57C2×D244—22no—
58C2×D244—22no—
59C2×D244D414yes—
60C2×C4×D61—22no—
61C2×C4×D61—22no—
62C2×C4×D61D414yes—
63C22⋊C4×C61—28no—
64C42×C61—22no—
65C22⋊Dic61—44no—
66#11758198—24no—
67#11758198—28no—
68#11758198—24no—
69#11758198—44no—
70C4⋊Dic61—22no—
71#11758196—24no—
72C4×Dic61—22no—
73C23×C61—22no—
74C23×C61—22no—
75C23×C61—24no—
76C23×C61—24no—
77C23×C61C2×D414yes—
78C23×C61C2411yescharacteristic
79C22×D61—21no—
80C22×D61—21no—
81C22×D61—22no—
82C22×D61—216no—
83C22×D61—22no—
84C22×D61—22no—
85C22×D61—24no—
86C22×D61—24no—
87C22×D61—416no—
88C22×D61—416no—
89C22×D61—48no—
90C22×D61—48no—
91C22×D61C2×D414yes—
92C22×D61C2×D412yes—
93C22×D61#4114yes—
94Q8×C61—44no—
95D4×C61—28no—
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×C61C2×D412yes—
103C2×C4×C61C2×D412yes—
104C2×C4×C61C2×D412yes—
105C22⋊D61—216no—
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—22no—
111C2×Dic61—24no—
112C2×Dic61—22no—
113C2×Dic61C2×D412yes—
114D244—28no—
115C4×D61—28no—
116C22×C61—24no—
117C22×C61—28no—
118C22×C61—42no—
119C22×C61—42no—
120C22×C61—44no—
121C22×C61C22×D411yescharacteristic
122C22×C61C22×D411yescharacteristic
123C22×C61C22×D411yescharacteristic
124C22×C61C2×#4112yes—
125C22×C61H(5,𝔽2)12yes—
126C2×D61—22no—
127C2×D61—22no—
128C2×D61—24no—
129C2×D61—24no—
130C2×D61—24no—
131C2×D61—28no—
132C2×D61—432no—
133C2×D61—42no—
134C2×D61—42no—
135C2×D61—44no—
136C2×D61#12414yescomplemented
137C4×C61—42no—
138C4×C61—42no—
139C4×C61#12014yes—
140Dic61—22no—
141Dic61—42no—
142Dic61—42no—
143#2435—611nomaximal, Sylow
144C2×C61—44no—
145C2×C61C2×#12011yescharacteristic
146C2×C61C2×#12411yescharacteristic
147C2×C61#53511yescharacteristic
148C2×C61#54612yes—
149C2×C61#54612yes—
150D61—24no—
151D61—44no—
152C23×D4—611no—
153C2×#124—611no—
154C2×#124D6111yescharacteristic, complemented
155C2×C22⋊Q8—611no—
156#394—614no—
157#390—611no—
158#390—611no—
159#379—614no—
160C4×C22⋊C4—611no—
161C61#243511yescharacteristic, complemented, Sylow
162C25—612no—
163C22×Q8C2×D6111yescharacteristic, complemented
164C22×D4—612no—
165C22×D4—611no—
166C22×D4—1224no—
167C22×D4—1224no—
168C22×D4—1224no—
169C22×D4C2×D6111yescharacteristic
170C23×C4—611no—
171#124—24no—
172#124—1224no—
173C22⋊Q8—1224no—
174C2×C4⋊C4—611no—
175C2×C4⋊C4—612no—
176C2×C22⋊C4—612no—
177C2×C22⋊C4—614no—
178C2×C22⋊C4—612no—
179C2×C22⋊C4—614no—
180C2×C22⋊C4—614no—
181C2×C22⋊C4C2×D6114yescomplemented
182C2×C42—611no—
183C2×C42C2×D6111yescharacteristic, complemented
184#95—612no—
185C24—611no—
186C24—612no—
187C24—1224no—
188C24—1224no—
189C24—1228no—
190C24—1224no—
191C24—2444no—
192C24C22×D6112yes—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—614no—
199C2×D4—1224no—
200C2×D4—12232no—
201C2×D4—1224no—
202C2×D4—1224no—
203C2×D4—1222no—
204C2×D4—1222no—
205C2×D4C22⋊D6114yes—
206C22×C4—611no—
207C22×C4—614no—
208C22×C4—612no—
209C22×C4—612no—
210C22×C4—1222no—
211C22×C4—1222no—
212C22×C4C22×D6112yes—
213C22×C4C22×D6112yes—
214C22×C4C22×D6111yescharacteristic
215C4⋊C4—1224no—
216C4⋊C4—1222no—
217C22⋊C4—28no—
218C22⋊C4—1224no—
219C22⋊C4—1224no—
220C22⋊C4—1228no—
221C22⋊C4—2444no—
222C22⋊C4—2444no—
223C42—22no—
224C42—1222no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—612no—
230C23—614no—
231C23—614no—
232C23—12216no—
233C23—1221no—
234C23—1221no—
235C23—1222no—
236C23—1222no—
237C23—1222no—
238C23—1224no—
239C23—1224no—
240C23—24416no—
241C23—2448no—
242C23—24416no—
243C23—2448no—
244C23C2×C22⋊D6114yes—
245C23C23×D6111yescharacteristic
246Q8—44no—
247D4—28no—
248D4—12216no—
249D4—1228no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—612no—
257C2×C4—1221no—
258C2×C4—1221no—
259C2×C4—1228no—
260C2×C4—1222no—
261C2×C4—1222no—
262C2×C4—1222no—
263C2×C4—1224no—
264C2×C4—1222no—
265C2×C4D4×D6112yes—
266C2×C4D4×D6112yes—
267C2×C4C2×C22⋊D6112yes—
268C22—24no—
269C22—28no—
270C22—42no—
271C22—42no—
272C22—44no—
273C22—614no—
274C22—1222no—
275C22—1222no—
276C22—1224no—
277C22—1224no—
278C22—1224no—
279C22—1228no—
280C22—24432no—
281C22—2442no—
282C22—2442no—
283C22—2444no—
284C22C2×D4×D6111yescharacteristic
285C22C2×D4×D6111yescharacteristic
286C22C22×C22⋊D6111yescharacteristic
287C22#41×D6112yes—
288C22#1451982312yes—
289C4—42no—
290C4—42no—
291C4—1222no—
292C4—2442no—
293C4—2442no—
294C4#1451974514yes—
295C2—44no—
296C2—1224no—
297C2—2444no—
298C2C2×#1451974511yescharacteristic
299C2#124×D6111yescharacteristic
300C2#1684033011yescharacteristic
301C2#1684040012yes—
302C2#1684040512yes—
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×#124×C61C2
hypercenterC2×#124D61
hypocenterC61#2435
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#14519766C2semidirect
3. C2×#14519753C2semidirect
4. #16839677C2semidirect
5. #16839665C2semidirect
6. #16839664C2semidirect
7. #16839617C2semidirect
8. C4×#11758198C2semidirect
9. C2×#124×C61C2semidirect
19. C2×C4⋊Dic61C22semidirect
20. C2×#11758196C22semidirect
21. C2×C22⋊Dic61C22semidirect
24. C2×#11758198C22semidirect
25. C2×#11758198C22semidirect
26. C2×C4×Dic61C22semidirect
27. #14519684C22semidirect
28. C22×Q8×C61C22semidirect
31. C2×C22⋊C4×C61C22semidirect
32. C2×C42×C61C22semidirect
136. C2×D61#124semidirect
154. C2×#124D61semidirect
161. C61#2435semidirect
163. C22×Q8C2×D61semidirect
181. C2×C22⋊C4C2×D61semidirect
183. C2×C42C2×D61semidirect