Finite Group Databasebeta

← #21456821

Subgroups — #21456821

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×#14519766C211yesmaximal, characteristic
3#16840155C212yesmaximal, complemented
4#16840123C212yesmaximal, complemented
5#16839987C212yesmaximal, complemented
6#16839986C212yesmaximal, complemented
7#16839918C211yesmaximal, characteristic, complemented
8#16839915C211yesmaximal, characteristic
9#16839912C211yesmaximal, characteristic
10#16839816C211yesmaximal, characteristic, complemented
11C61×#451C211yesmaximal, characteristic, complemented
12C22×D244C2211yescharacteristic
13C2×#11758202C2212yescomplemented
14C2×#11758190C2212yes—
15C2×#11758201C2212yes—
16#14519776C2211yescharacteristic, complemented
17#14519775C2211yescharacteristic, complemented
18#14519767—21no—
19#14519766—21no—
20#14519766—21no—
21#14519766C2212yes—
22#14519766C2212yes—
23#14519756C2211yescharacteristic
24#14519754—21no—
25Q8×Dic61C2211yescharacteristic, complemented
26#14519742—21no—
27#14519742C2211yescharacteristic, complemented
28#14519733—21no—
29C2×#11758198—21no—
30C2×#11758198—21no—
31C2×C4×Dic61C2211yescharacteristic
32#14519706C2212yescomplemented
33#14519701C2212yescomplemented
34#14519699C2212yes—
35#14519697C2212yes—
36C22⋊#92418C2211yescharacteristic
37#14519687C2212yescomplemented
38C22×Q8×C61C2211yescharacteristic
39C2×Q16×C61C2212yescomplemented
40C22⋊Q8×C61C2211yescharacteristic
41#103×C61C2212yescomplemented
42C22⋊C8×C61C2211yescharacteristic
43C23×D61—21no—
44C2×Q8×C61—21no—
45C2×Q8×C61—42no—
46C2×Q8×C61C2312yes—
47C2×Q8×C61C2311yescharacteristic
48C22×C4×C61—21no—
49C22×C4×C61C2311yescharacteristic
50C22×Dic61D411yescharacteristic
51C22×Dic61D411yescharacteristic
52C2×D244—21no—
53C2×D244—42no—
54C2×D244C2312yes—
55C2×Dic122—21no—
56C2×Dic122C2311yescharacteristic
57Q16×C61—42no—
58C2×C8×C61C2312yes—
59C4⋊C4×C61—21no—
60C4⋊C4×C61C2311yescharacteristic
61C22⋊C4×C61—21no—
62C22⋊Dic61—21no—
63#11758202—42no—
64#11758201—42no—
65#11758198—22no—
66#11758198—22no—
67#11758198—22no—
68#11758198—22no—
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—
83#11758190—42no—
84C23×C61C2×D411yescharacteristic
85C22×D61—22no—
86C22×D61—42no—
87C22×D61—42no—
88C22×D61—42no—
89Q8×C61—22no—
90Q8×C61—22no—
91Q8×C61—22no—
92Q8×C61—42no—
93C2×C4×C61—22no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—22no—
97C2×C4×C61—42no—
98C2×C4×C61C2×D411yescharacteristic
99C2×C4×C61C2411yescharacteristic
100C2×Dic61—22no—
101C2×Dic61—21no—
102C2×Dic61—21no—
103C2×Dic61—41no—
104C2×Dic61C2×D411yescharacteristic
105C2×Dic61C2×D411yescharacteristic
106C2×Dic61C2×D411yescharacteristic
107C2×Dic61C2×D411yescharacteristic
108D244—22no—
109D244—22no—
110D244—42no—
111Dic122—22no—
112Dic122—42no—
113C8×C61—22no—
114#92418—22no—
115C22×C61—21no—
116C22×C61—21no—
117C22×C61#13712yes—
118C22×C61C22×D411yescharacteristic
119C2×D61—42no—
120C2×D61—42no—
121C2×D61—42no—
122C2×D61—84no—
123C4×C61—21no—
124C4×C61—42no—
125C4×C61—41no—
126C4×C61—41no—
127C4×C61H(5,𝔽2)12yes—
128Dic61—21no—
129Dic61—21no—
130Dic61—21no—
131Dic61—21no—
132Dic61—41no—
133Dic61—41no—
134#3215—611nomaximal, Sylow
135C2×C61—22no—
136C2×C61#53511yescharacteristic
137C2×C61C2×#13711yescharacteristic
138C2×C61#57511yescharacteristic
139D61—82no—
140#533—611no—
141C2×#124—611no—
142#497—612no—
143#489—612no—
144#464—612no—
145#464—612no—
146#451—611no—
147#451D6111yescharacteristic, complemented
148#450—611no—
149#450—611no—
150#432—611no—
151C61#321511yescharacteristic, complemented, Sylow
152C22×Q8C2×D6111yescharacteristic
153C22×D4—611no—
154C2×Q16—612no—
155C2×Q16C2×D6112yescomplemented
156C2×SD16—612no—
157C2×SD16—612no—
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#103C2×D6112yescomplemented
178#102—612no—
179#102—612no—
180C22⋊C8—611no—
181C22⋊C8C2×D6111yescharacteristic
182#97—612no—
183C24—1221no—
184C2×Q8—21no—
185C2×Q8—42no—
186C2×Q8—611no—
187C2×Q8—1221no—
188C2×Q8C22×D6112yes—
189C2×Q8C22×D6111yescharacteristic
190C2×D4—612no—
191C2×D4—1221no—
192C2×D4—2442no—
193C22×C4—21no—
194C22×C4—611no—
195C22×C4—611no—
196C22×C4C22×D6111yescharacteristic
197Q16—42no—
198Q16—2442no—
199SD16—2442no—
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—1222no—
225C23—2442no—
226C23—2442no—
227C23—2442no—
228C23D4×D6111yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—22no—
232Q8—42no—
233Q8—1222no—
234Q8—2442no—
235D4—1222no—
236D4—1222no—
237D4—2442no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—611no—
244C2×C4—611no—
245C2×C4—611no—
246C2×C4—611no—
247C2×C4—1222no—
248C2×C4—1221no—
249C2×C4—1221no—
250C2×C4—2441no—
251C2×C4D4×D6111yescharacteristic
252C2×C4C23×D6111yescharacteristic
253C8—22no—
254C8—1222no—
255C22—21no—
256C22—21no—
257C22—4884no—
258C22—2442no—
259C22—2442no—
260C22—2442no—
261C22#1451980912yes—
262C22C2×D4×D6111yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—1221no—
268C4—1221no—
269C4—1221no—
270C4—1221no—
271C4—2441no—
272C4—2441no—
273C4#1451982212yes—
274C2—22no—
275C2—4882no—
276C2#1684032711yescharacteristic
277C2C2×#1451980911yescharacteristic
278C2#1684056111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#451C2
socleC22×C61C22×D4
hypercenter#451D61
hypocenterC61#3215
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #16840314C2semidirect
3. #16840155C2semidirect
4. #16840123C2semidirect
5. #16839987C2semidirect
6. #16839986C2semidirect
7. #16839918C2semidirect
10. #16839816C2semidirect
11. C61×#451C2semidirect
13. C2×#11758202C22semidirect
16. #14519776C22semidirect
17. #14519775C22semidirect
25. Q8×Dic61C22semidirect
27. #14519742C22semidirect
32. #14519706C22semidirect
33. #14519701C22semidirect
37. #14519687C22semidirect
39. C2×Q16×C61C22semidirect
41. #103×C61C22semidirect
147. #451D61semidirect
151. C61#3215semidirect
155. C2×Q16C2×D61semidirect
177. #103C2×D61semidirect