Finite Group Databasebeta

← #21450781

Subgroups — #21450781

Order 7808.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14519749C212yesmaximal, complemented
2C2×C4×C22⋊D61C211yesmaximal, characteristic, complemented
3#16839693C211yesmaximal, characteristic, complemented
4#16839688C211yesmaximal, characteristic, complemented
5#16839676C211yesmaximal, characteristic, complemented
6#16839670C211yesmaximal, characteristic, complemented
7#16839650C212yesmaximal, complemented
8#16839642C212yesmaximal, complemented
9#16839638C212yesmaximal, complemented
10C2×#14519684C211yesmaximal, characteristic, complemented
11C2×#121×C61C211yesmaximal, characteristic, complemented
12C22×C22⋊D61C2212yes—
13C22×C22⋊D61C2211yescharacteristic
14C23×Dic61C2212yes—
15C22×C4×D61C2211yescharacteristic
16#14519749—28no—
17C4×C22⋊D61—24no—
18C2×#11758196C2211yescharacteristic
19C2×#11758196C2212yes—
20C2×C22⋊Dic61C2211yescharacteristic
21C2×C22⋊Dic61C2212yes—
22C2×C22⋊Dic61C2212yes—
23C2×C22⋊Dic61C2212yes—
24C2×#11758198C2211yescharacteristic
25C2×#11758198C2212yes—
26C2×C4×Dic61C2211yescharacteristic
27#14519684C2211yescharacteristic
28#14519684C2212yes—
29#14519684C2211yescharacteristic
30#14519684C2212yes—
31#14519684C2212yes—
32C22×D4×C61C2212yes—
33C22×D4×C61C2211yescharacteristic
34C23×C4×C61C2211yescharacteristic
35#121×C61—24no—
36C2×C4⋊C4×C61C2211yescharacteristic
37C2×C22⋊C4×C61C2212yes—
38C24×C61C2312yes—
39C24×C61C2311yescharacteristic
40C23×D61C2311yescharacteristic
41C2×D4×C61—24no—
42C2×D4×C61—24no—
43C2×D4×C61—24no—
44C2×D4×C61—22no—
45C2×D4×C61—42no—
46C22×C4×C61—42no—
47C22×C4×C61D414yescomplemented
48C22×C4×C61C2312yes—
49C22×C4×C61C2311yescharacteristic
50C22×C4×C61C2311yescharacteristic
51C2×C22⋊D61—22no—
52C2×C22⋊D61—22no—
53C2×C22⋊D61—24no—
54C2×C22⋊D61—24no—
55C2×C22⋊D61—42no—
56C2×C22⋊D61D414yescomplemented
57C22×Dic61—24no—
58C22×Dic61—24no—
59C22×Dic61—42no—
60C22×Dic61C2311yescharacteristic
61C22×Dic61C2312yes—
62C22×Dic61C2311yescharacteristic
63C22×Dic61C2311yescharacteristic
64C22×Dic61C2312yes—
65C2×C4×D61—22no—
66C2×C4×D61—42no—
67C4⋊C4×C61—22no—
68C22⋊C4×C61—24no—
69C22⋊Dic61—22no—
70C22⋊Dic61—24no—
71C22⋊Dic61—24no—
72C22⋊Dic61—42no—
73#11758198—24no—
74#11758198—22no—
75#11758196—22no—
76#11758196—24no—
77C4×Dic61—22no—
78C23×C61—22no—
79C23×C61—22no—
80C23×C61—22no—
81C23×C61—22no—
82C23×C61—22no—
83C23×C61—22no—
84C23×C61—22no—
85C23×C61—22no—
86C23×C61—22no—
87C23×C61C2×D412yes—
88C23×C61C2×D412yes—
89C23×C61#4112yes—
90C23×C61C2411yescharacteristic
91C22×D61—21no—
92C22×D61—21no—
93C22×D61—22no—
94C22×D61—22no—
95C22×D61C2×D412yes—
96D4×C61—44no—
97D4×C61—48no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—44no—
104C2×C4×C61C2×D412yes—
105C2×C4×C61C2×D412yes—
106C22⋊D61—28no—
107C22⋊D61—48no—
108C2×Dic61—21no—
109C2×Dic61—22no—
110C2×Dic61—21no—
111C2×Dic61—22no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—22no—
117C2×Dic61—22no—
118C2×Dic61—22no—
119C2×Dic61—48no—
120C2×Dic61C2×D412yes—
121C2×Dic61#4114yes—
122C4×D61—44no—
123C22×C61—22no—
124C22×C61—22no—
125C22×C61—24no—
126C22×C61—24no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61—42no—
131C22×C61—42no—
132C22×C61—42no—
133C22×C61—42no—
134C22×C61#12114yes—
135C22×C61C22×D411yescharacteristic
136C22×C61C22×D411yescharacteristic
137C22×C61C2×#4112yes—
138C22×C61C2×#4111yescharacteristic
139C22×C61H(5,𝔽2)12yes—
140C2×D61—22no—
141C2×D61—24no—
142C2×D61—41no—
143C2×D61—41no—
144C2×D61—42no—
145C4×C61—22no—
146C4×C61—22no—
147C4×C61—42no—
148Dic61—22no—
149Dic61—24no—
150Dic61—41no—
151Dic61—41no—
152Dic61—42no—
153#2543—611nomaximal, Sylow
154C2×C61—24no—
155C2×C61—82no—
156C2×C61C2×#12111yescharacteristic
157C2×C61#53211yescharacteristic
158C2×C61#53411yescharacteristic
159C2×C61#54612yes—
160C2×C61#55312yes—
161D61—42no—
162C2×#121—612no—
163C2×#121D6111yescharacteristic, complemented
164C2×C4×D4—611no—
165#398—611no—
166#397—611no—
167#394—611no—
168#392—611no—
169#388—612no—
170#386—612no—
171#386—612no—
172C2×#95—611no—
173C61#254311yescharacteristic, complemented, Sylow
174C22×D4—611no—
175C22×D4—612no—
176C22×D4C2×D6112yes—
177C22×D4C2×D6111yescharacteristic
178C23×C4—612no—
179C23×C4—611no—
180C23×C4C2×D6111yescharacteristic
181#121—24no—
182#121—1228no—
183C4×D4—1224no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—612no—
186C2×C4⋊C4C2×D6111yescharacteristic
187C2×C22⋊C4—611no—
188C2×C22⋊C4—612no—
189C2×C22⋊C4—612no—
190C2×C22⋊C4—611no—
191C2×C22⋊C4—612no—
192C2×C22⋊C4—612no—
193C2×C22⋊C4C2×D6112yes—
194C2×C42—611no—
195#95—612no—
196#95—611no—
197#95—611no—
198#95—612no—
199#95—612no—
200C24—611no—
201C24C22×D6112yes—
202C24C22×D6111yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—614no—
209C2×D4—1224no—
210C2×D4—1222no—
211C2×D4—1222no—
212C2×D4—1224no—
213C2×D4—2442no—
214C22×C4—42no—
215C22×C4—612no—
216C22×C4—611no—
217C22×C4—612no—
218C22×C4—611no—
219C22×C4—611no—
220C22×C4—1224no—
221C22×C4—1224no—
222C22×C4—1222no—
223C22×C4—2442no—
224C22×C4—2442no—
225C22×C4C22⋊D6114yescomplemented
226C22×C4C22×D6112yes—
227C22×C4C22×D6111yescharacteristic
228C22×C4C22×D6111yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—1224no—
231C4⋊C4—1222no—
232C22⋊C4—24no—
233C22⋊C4—1224no—
234C22⋊C4—1224no—
235C22⋊C4—1224no—
236C22⋊C4—1222no—
237C22⋊C4—1222no—
238C22⋊C4—2442no—
239C42—1222no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—612no—
250C23—1221no—
251C23—1222no—
252C23—1221no—
253C23—1222no—
254C23D4×D6112yes—
255C23#1175822412yes—
256C23C2×C22⋊D6112yes—
257C23C23×D6111yescharacteristic
258D4—44no—
259D4—48no—
260D4—1228no—
261D4—2448no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—612no—
269C2×C4—614no—
270C2×C4—1222no—
271C2×C4—1221no—
272C2×C4—1221no—
273C2×C4—1222no—
274C2×C4—1222no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1222no—
278C2×C4—1222no—
279C2×C4—1222no—
280C2×C4—1222no—
281C2×C4—2444no—
282C2×C4—2448no—
283C2×C4C2×C22⋊D6112yes—
284C2×C4C2×C22⋊D6112yes—
285C22—24no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—1222no—
297C22—1224no—
298C22—2442no—
299C22—2441no—
300C22—2441no—
301C22#1451974814yes—
302C22C2×D4×D6111yescharacteristic
303C22C22×C22⋊D6111yescharacteristic
304C22#41×D6112yes—
305C22C2×#1175822411yescharacteristic
306C22#1451982212yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—1222no—
311C4—1224no—
312C4—2441no—
313C4—2441no—
314C4—2442no—
315C2—24no—
316C2—82no—
317C2—2442no—
318C2C2×#1451974811yescharacteristic
319C2#1684030811yescharacteristic
320C2#1684032111yescharacteristic
321C2#1684040112yes—
322C2#1684045512yes—
subgroupG/H
center, FrattiniC23C23×D61
commutator, socleC23×C61C24
FittingC2×#121×C61C2
hypercenterC2×#121D61
hypocenterC61#2543
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14519749C2semidirect
2. C2×C4×C22⋊D61C2semidirect
3. #16839693C2semidirect
4. #16839688C2semidirect
5. #16839676C2semidirect
6. #16839670C2semidirect
7. #16839650C2semidirect
8. #16839642C2semidirect
9. #16839638C2semidirect
10. C2×#14519684C2semidirect
11. C2×#121×C61C2semidirect
47. C22×C4×C61D4semidirect
56. C2×C22⋊D61D4semidirect
163. C2×#121D61semidirect
173. C61#2543semidirect
225. C22×C4C22⋊D61semidirect