Finite Group Databasebeta

← #21456831

Subgroups — #21456831

Order 7808.

390 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 111 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#16840317C211yesmaximal, characteristic, complemented
2#16840311C211yesmaximal, characteristic
3#16840157C211yesmaximal, characteristic, complemented
4#16840156C211yesmaximal, characteristic, complemented
5#16840124C211yesmaximal, characteristic, complemented
6#16839971C212yesmaximal, complemented
7#16839961C212yesmaximal, complemented
8#16839927C212yesmaximal, complemented
9#16839925C212yesmaximal, complemented
10#16839817C211yesmaximal, characteristic, complemented
11C61×#489C211yesmaximal, characteristic, complemented
12C2×#11758226C2211yescharacteristic
13C2×#11758224C2211yescharacteristic
14C2×Dic244C2212yes—
15C2×#11758202C2212yes—
16C2×#11758190C2212yes—
17C2×#11758200C2212yes—
18#14519777C2211yescharacteristic
19#14519775C2211yescharacteristic
20#14519773C2211yescharacteristic
21#14519768—21no—
22#14519765C2211yescharacteristic
23#14519757C2212yes—
24#14519753—21no—
25#14519750—21no—
26#14519750—21no—
27Q8×Dic61—21no—
28#14519742C2212yes—
29#14519737—21no—
30#14519737—21no—
31#14519735—21no—
32C42×D61C2211yescharacteristic
33#14519706C2212yes—
34#14519700C2212yes—
35#14519697C2212yes—
36#14519689C2212yes—
37#14519688C2211yescharacteristic
38C4⋊Q8×C61C2211yescharacteristic
39#124×C61C2211yescharacteristic
40#103×C61C2212yes—
41#102×C61C2212yes—
42#97×C61C2211yescharacteristic
43C2×Q8×C61—21no—
44C2×Q8×C61C2311yescharacteristic
45C2×D4×C61C2311yescharacteristic
46C2×C22⋊D61—21no—
47C2×C22⋊D61—21no—
48C2×C22⋊D61—21no—
49C22×Dic61—21no—
50#11758226—42no—
51#11758224—42no—
52C2×D244—21no—
53C2×D244C2311yescharacteristic
54C2×C4×D61—21no—
55C2×C4×D61D411yescharacteristic
56C2×C4×D61D411yescharacteristic
57C2×C4×D61C2311yescharacteristic
58C2×Dic122C2311yescharacteristic
59C2×Dic122C2312yes—
60C2×C8×C61C2312yes—
61C4⋊C4×C61—21no—
62C4⋊C4×C61C2312yes—
63C22⋊C4×C61—21no—
64C22⋊C4×C61—21no—
65C42×C61C2311yescharacteristic
66#11758202—42no—
67#11758200—42no—
68#11758198—22no—
69#11758198—21no—
70#11758198—21no—
71C4⋊Dic61—21no—
72C4⋊Dic61—21no—
73C4⋊Dic61—22no—
74#11758196—21no—
75#11758196—21no—
76#11758196—22no—
77C4×Dic61—22no—
78C4×Dic61D411yescharacteristic
79C4×Dic61D411yescharacteristic
80C4×Dic61C2311yescharacteristic
81C2×#92418C2312yes—
82Dic244—42no—
83#11758190—42no—
84C23×C61—21no—
85C22×D61—21no—
86C22×D61C2×D411yescharacteristic
87Q8×C61—22no—
88Q8×C61—42no—
89D4×C61—22no—
90C2×C4×C61—22no—
91C2×C4×C61—21no—
92C2×C4×C61C2×D411yescharacteristic
93C2×C4×C61C2×D411yescharacteristic
94C2×C4×C61C2411yescharacteristic
95C22⋊D61—42no—
96C22⋊D61—42no—
97C22⋊D61—42no—
98C2×Dic61—22no—
99C2×Dic61—21no—
100C2×Dic61—42no—
101C2×Dic61C2×D411yescharacteristic
102C2×Dic61C2×D411yescharacteristic
103C2×Dic61C2×D411yescharacteristic
104D244—22no—
105D244—42no—
106C4×D61—22no—
107C4×D61—22no—
108C4×D61—41no—
109C4×D61—42no—
110Dic122—22no—
111Dic122—22no—
112Dic122—22no—
113C8×C61—22no—
114#92418—22no—
115C22×C61—41no—
116C22×C61—41no—
117C22×C61—41no—
118C22×C61C22×D411yescharacteristic
119C2×D61—21no—
120C2×D61—21no—
121C2×D61—21no—
122C2×D61—41no—
123C2×D61—41no—
124C2×D61—41no—
125C4×C61—21no—
126C4×C61—21no—
127C4×C61—41no—
128C4×C61—42no—
129C4×C61H(5,𝔽2)12yes—
130Dic61—21no—
131Dic61—21no—
132Dic61—42no—
133Dic61—41no—
134Dic61#13712yes—
135#3216—611nomaximal, Sylow
136C2×C61—81no—
137C2×C61#53511yescharacteristic
138C2×C61C2×#13711yescharacteristic
139C2×C61#57511yescharacteristic
140D61—41no—
141D61—81no—
142#533—611no—
143#532—611no—
144#497—611no—
145#497—611no—
146#489—611no—
147#489D6111yescharacteristic, complemented
148#462—612no—
149#460—612no—
150#452—612no—
151#452—612no—
152#432—611no—
153C61#321611yescharacteristic, complemented, Sylow
154C2×#41—611no—
155C2×#41—611no—
156C2×Q16—612no—
157C2×Q16—612no—
158C2×SD16—612no—
159C2×SD16—612no—
160C4⋊Q8—611no—
161C4⋊Q8—611no—
162C4⋊Q8C2×D6111yescharacteristic
163#127—611no—
164#125—1221no—
165#124—611no—
166#124C2×D6111yescharacteristic
167C22⋊Q8—612no—
168C22⋊Q8—1221no—
169#121—1221no—
170#121—1221no—
171C4×Q8—612no—
172C4×Q8—1221no—
173C4×D4—1221no—
174C4×D4—1221no—
175#117—1221no—
176C2×C42—611no—
177C4⋊C8—612no—
178#103—612no—
179#103C2×D6112yes—
180#102—612no—
181#102C2×D6112yes—
182C22⋊C8—612no—
183#97—611no—
184#97C2×D6111yescharacteristic
185#41—2442no—
186#41—2442no—
187C2×Q8—21no—
188C2×Q8—611no—
189C2×Q8—612no—
190C2×Q8C22×D6111yescharacteristic
191C2×D4—611no—
192C2×D4—1221no—
193C2×D4—1221no—
194C2×D4—1221no—
195C2×D4—1221no—
196C2×D4C22×D6111yescharacteristic
197C22×C4—611no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—1221no—
201C22×C4—1221no—
202Q16—2442no—
203Q16—2442no—
204SD16—2442no—
205SD16—2442no—
206C2×C8—612no—
207C2×C8C22×D6112yes—
208C4⋊C4—21no—
209C4⋊C4—1221no—
210C4⋊C4—1221no—
211C4⋊C4—1222no—
212C4⋊C4—1222no—
213C4⋊C4—1221no—
214C4⋊C4—1221no—
215C4⋊C4C22×D6112yes—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—1222no—
219C22⋊C4—1221no—
220C22⋊C4—1221no—
221C42—611no—
222C42—611no—
223C42—611no—
224C42—1222no—
225C42C22×D6111yescharacteristic
226C23—21no—
227C23—611no—
228C23—1221no—
229Q8—22no—
230Q8—42no—
231Q8—1222no—
232Q8—1222no—
233Q8—1222no—
234D4—22no—
235D4—1222no—
236D4—2442no—
237D4—2442no—
238D4—2442no—
239D4—2442no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—611no—
243C2×C4—611no—
244C2×C4—611no—
245C2×C4—1222no—
246C2×C4—1222no—
247C2×C4—1222no—
248C2×C4—1221no—
249C2×C4—2441no—
250C2×C4—2442no—
251C2×C4—2442no—
252C2×C4D4×D6111yescharacteristic
253C2×C4D4×D6111yescharacteristic
254C2×C4C23×D6111yescharacteristic
255C8—22no—
256C8—1222no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—1221no—
261C22—1221no—
262C22—1221no—
263C22—2441no—
264C22—2441no—
265C22—2441no—
266C22C2×D4×D6111yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—41no—
271C4—612no—
272C4—1221no—
273C4—1221no—
274C4—2441no—
275C4—2442no—
276C4#1451982312yes—
277C2—81no—
278C2—4881no—
279C2—2441no—
280C2#1684033011yescharacteristic
281C2#137×D6111yescharacteristic
282C2#1684055811yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#489C2
socleC22×C61C22×D4
hypercenter#489D61
hypocenterC61#3216
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #16840317C2semidirect
3. #16840157C2semidirect
4. #16840156C2semidirect
5. #16840124C2semidirect
6. #16839971C2semidirect
7. #16839961C2semidirect
8. #16839927C2semidirect
9. #16839925C2semidirect
10. #16839817C2semidirect
11. C61×#489C2semidirect
147. #489D61semidirect
153. C61#3216semidirect