Finite Group Databasebeta

← #21456819

Subgroups — #21456819

Order 7808.

400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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#16840314C211yesmaximal, characteristic, complemented
2C2×#14519764C211yesmaximal, characteristic
3#16840153C212yesmaximal, complemented
4#16840120C212yesmaximal, complemented
5#16839983C212yesmaximal, complemented
6#16839982C212yesmaximal, complemented
7#16839920C211yesmaximal, characteristic, complemented
8#16839917C211yesmaximal, characteristic, complemented
9#16839913C211yesmaximal, characteristic
10#16839816C211yesmaximal, characteristic, complemented
11C61×#450C211yesmaximal, characteristic
12C22×Dic122C2211yescharacteristic
13C2×Dic244C2212yescomplemented
14C2×#11758202C2212yescomplemented
15C2×#11758200C2212yes—
16#14519776C2211yescharacteristic, complemented
17#14519775C2211yescharacteristic, complemented
18#14519767—21no—
19#14519764—21no—
20#14519764—21no—
21#14519764C2212yes—
22#14519764C2212yes—
23#14519756C2211yescharacteristic
24#14519754—21no—
25Q8×Dic61C2211yescharacteristic, complemented
26#14519742—21no—
27#14519742C2211yescharacteristic, complemented
28#14519733—21no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—21no—
31C2×C4×Dic61C2211yescharacteristic
32#14519706C2212yescomplemented
33#14519702C2212yescomplemented
34#14519700C2212yescomplemented
35#14519698C2212yes—
36C22⋊#92418C2211yescharacteristic
37#14519687C2212yescomplemented
38C22×D4×C61C2211yescharacteristic
39C2×SD16×C61C2212yes—
40C22⋊Q8×C61C2211yescharacteristic
41#102×C61C2212yes—
42C22⋊C8×C61C2211yescharacteristic
43C24×C61—21no—
44C2×Q8×C61C2311yescharacteristic
45C2×D4×C61—21no—
46C2×D4×C61—42no—
47C2×D4×C61C2312yes—
48C22×C4×C61C2311yescharacteristic
49C22×Dic61—21no—
50C22×Dic61D411yescharacteristic
51C22×Dic61D411yescharacteristic
52C2×Dic122—21no—
53C2×Dic122—21no—
54C2×Dic122—42no—
55C2×Dic122C2312yes—
56C2×Dic122C2311yescharacteristic
57SD16×C61—42no—
58C2×C8×C61C2312yes—
59C4⋊C4×C61—21no—
60C4⋊C4×C61C2311yescharacteristic
61C22⋊C4×C61—21no—
62C22⋊Dic61—21no—
63C22⋊Dic61—22no—
64C22⋊Dic61—22no—
65C22⋊Dic61—22no—
66C22⋊Dic61—22no—
67#11758202—42no—
68#11758200—42no—
69C4⋊Dic61—21no—
70C4⋊Dic61C2311yescharacteristic
71#11758196—21no—
72#11758196—21no—
73#11758196—21no—
74#11758196—21no—
75#11758196—21no—
76C4×Dic61—21no—
77C4×Dic61—21no—
78C4×Dic61D411yescharacteristic
79C4×Dic61D411yescharacteristic
80C4×Dic61C2311yescharacteristic
81C4×Dic61C2311yescharacteristic
82C2×#92418C2312yes—
83Dic244—42no—
84C23×C61—22no—
85C23×C61—42no—
86C23×C61—42no—
87C23×C61—42no—
88C23×C61C2×D411yescharacteristic
89Q8×C61—22no—
90D4×C61—22no—
91D4×C61—22no—
92D4×C61—42no—
93C2×C4×C61—22no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61C2×D411yescharacteristic
97C2×C4×C61C2411yescharacteristic
98C2×Dic61—22no—
99C2×Dic61—21no—
100C2×Dic61—21no—
101C2×Dic61—22no—
102C2×Dic61—41no—
103C2×Dic61—42no—
104C2×Dic61C2×D411yescharacteristic
105C2×Dic61C2×D411yescharacteristic
106C2×Dic61C2×D411yescharacteristic
107C2×Dic61C2×D411yescharacteristic
108Dic122—22no—
109Dic122—22no—
110Dic122—22no—
111Dic122—42no—
112Dic122—42no—
113C8×C61—22no—
114#92418—22no—
115C22×C61—21no—
116C22×C61—21no—
117C22×C61—42no—
118C22×C61—42no—
119C22×C61—42no—
120C22×C61—84no—
121C22×C61#13712yes—
122C22×C61C22×D411yescharacteristic
123C4×C61—21no—
124C4×C61—41no—
125C4×C61—41no—
126C4×C61H(5,𝔽2)12yes—
127Dic61—21no—
128Dic61—21no—
129Dic61—21no—
130Dic61—21no—
131Dic61—42no—
132Dic61—41no—
133Dic61—41no—
134#3215—611nomaximal, Sylow
135C2×C61—22no—
136C2×C61—82no—
137C2×C61#53511yescharacteristic
138C2×C61C2×#13711yescharacteristic
139C2×C61#57511yescharacteristic
140#533—611no—
141C2×#124—611no—
142#497—612no—
143#489—612no—
144#464—612no—
145#464—612no—
146#451—611no—
147#451—611no—
148#450—611no—
149#450D6111yescharacteristic
150#432—611no—
151C61#321511yescharacteristic, complemented, Sylow
152C22×Q8—611no—
153C22×D4C2×D6111yescharacteristic
154C2×Q16—612no—
155C2×Q16—612no—
156C2×SD16—612no—
157C2×SD16C2×D6112yes—
158C4⋊Q8—611no—
159C4⋊Q8—611no—
160#125—1221no—
161#124—612no—
162#124—612no—
163#124—1221no—
164#124—1221no—
165C22⋊Q8—611no—
166C22⋊Q8—1221no—
167C22⋊Q8C2×D6111yescharacteristic
168C4×Q8—611no—
169C4×Q8—611no—
170C4×Q8—1221no—
171#117—1221no—
172C2×C22⋊C4—1221no—
173C2×C22⋊C4—1221no—
174C2×C42—611no—
175C4⋊C8—612no—
176#103—612no—
177#103—612no—
178#102—612no—
179#102C2×D6112yes—
180C22⋊C8—611no—
181C22⋊C8C2×D6111yescharacteristic
182#97—612no—
183C24—21no—
184C2×Q8—612no—
185C2×Q8—611no—
186C2×Q8—1221no—
187C2×Q8—1221no—
188C2×Q8—2442no—
189C2×Q8C22×D6111yescharacteristic
190C2×D4—21no—
191C2×D4—42no—
192C2×D4C22×D6112yes—
193C22×C4—611no—
194C22×C4—611no—
195C22×C4—1221no—
196C22×C4C22×D6111yescharacteristic
197Q16—2442no—
198Q16—2442no—
199SD16—42no—
200SD16—2442no—
201C2×C8—612no—
202C2×C8C22×D6112yes—
203C4⋊C4—21no—
204C4⋊C4—611no—
205C4⋊C4—1221no—
206C4⋊C4—1221no—
207C4⋊C4—1221no—
208C4⋊C4—1221no—
209C4⋊C4—1221no—
210C4⋊C4—1221no—
211C4⋊C4C22×D6111yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—1222no—
214C22⋊C4—1222no—
215C22⋊C4—1222no—
216C22⋊C4—1222no—
217C22⋊C4—1221no—
218C42—611no—
219C42—611no—
220C42—611no—
221C42—611no—
222C42—1221no—
223C42—1221no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23D4×D6111yescharacteristic
229Q8—22no—
230Q8—1222no—
231Q8—1222no—
232Q8—1222no—
233Q8—2442no—
234Q8—2442no—
235D4—22no—
236D4—22no—
237D4—42no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—611no—
242C2×C4—611no—
243C2×C4—611no—
244C2×C4—611no—
245C2×C4—1222no—
246C2×C4—1221no—
247C2×C4—1221no—
248C2×C4—1222no—
249C2×C4—2441no—
250C2×C4—2442no—
251C2×C4D4×D6111yescharacteristic
252C2×C4C23×D6111yescharacteristic
253C8—22no—
254C8—1222no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—84no—
261C22#1451980612yes—
262C22C2×D4×D6111yescharacteristic
263C4—21no—
264C4—41no—
265C4—41no—
266C4—1221no—
267C4—1221no—
268C4—1221no—
269C4—1221no—
270C4—2441no—
271C4—2442no—
272C4—2441no—
273C4#1451982212yes—
274C2—22no—
275C2—82no—
276C2#1684032711yescharacteristic
277C2C2×#1451980611yescharacteristic
278C2#1684055811yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#450C2
socleC22×C61C22×D4
hypercenter#450D61
hypocenterC61#3215
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16840314C2semidirect
3. #16840153C2semidirect
4. #16840120C2semidirect
5. #16839983C2semidirect
6. #16839982C2semidirect
7. #16839920C2semidirect
8. #16839917C2semidirect
10. #16839816C2semidirect
13. C2×Dic244C22semidirect
14. C2×#11758202C22semidirect
16. #14519776C22semidirect
17. #14519775C22semidirect
25. Q8×Dic61C22semidirect
27. #14519742C22semidirect
32. #14519706C22semidirect
33. #14519702C22semidirect
34. #14519700C22semidirect
37. #14519687C22semidirect
151. C61#3215semidirect