Finite Group Databasebeta

← #21456864

Subgroups — #21456864

Order 7808.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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#16840366C211yesmaximal, characteristic, complemented
2#16840330C211yesmaximal, characteristic, complemented
3#16840138C211yesmaximal, characteristic, complemented
4#16840127C211yesmaximal, characteristic, complemented
5#16840116C211yesmaximal, characteristic, complemented
6#16839979C212yesmaximal, complemented
7#16839955C212yesmaximal, complemented
8#16839906C212yesmaximal, complemented
9#16839894C212yesmaximal, complemented
10#16839824C211yesmaximal, characteristic, complemented
11C61×#496C211yesmaximal, characteristic, complemented
12C2×#11758226C2211yescharacteristic
13C2×D4×D61C2211yescharacteristic
14C2×#11758199C2212yes—
15C2×#11758199C2212yescomplemented
16#14519774C2211yescharacteristic
17#14519773C2211yescharacteristic
18#14519769C2211yescharacteristic, complemented
19#14519766C2211yescharacteristic
20#14519758—22no—
21#14519750—22no—
22#14519749—22no—
23#14519748C2212yes—
24#14519747—22no—
25D4×Dic61C2212yescomplemented
26#14519734C2211yescharacteristic
27#14519706C2212yescomplemented
28#14519701C2212yescomplemented
29#14519699C2212yes—
30#14519699C2212yescomplemented
31#14519698C2212yes—
32#14519689C2212yes—
33C4×#92418C2211yescharacteristic, complemented
34C2×SD16×C61C2212yescomplemented
35C2×D8×C61C2212yes—
36#127×C61C2211yescharacteristic, complemented
37#124×C61C2211yescharacteristic
38#97×C61C2211yescharacteristic, complemented
39C23×D61—21no—
40C2×Q8×C61C2311yescharacteristic
41C2×D4×C61—22no—
42C2×D4×C61C2311yescharacteristic
43C2×D4×C61C2312yes—
44C2×C22⋊D61—22no—
45C2×C22⋊D61—21no—
46C2×C22⋊D61—22no—
47C22×Dic61—22no—
48#11758226—42no—
49D4×D61—42no—
50C2×D244—21no—
51C2×D244—22no—
52C2×D244C2311yescharacteristic
53C2×D244C2311yescharacteristic
54C2×C4×D61—21no—
55C2×C4×D61C2311yescharacteristic
56SD16×C61—42no—
57D8×C61—42no—
58C2×C8×C61C2312yes—
59C22⋊C4×C61—22no—
60C42×C61C2311yescharacteristic
61C22⋊Dic61—22no—
62C22⋊Dic61—22no—
63#11758199—42no—
64#11758199—42no—
65#11758198—22no—
66#11758198D412yes—
67C4⋊Dic61C2312yes—
68#11758196—22no—
69#11758196D412yes—
70C4×Dic61C2311yescharacteristic
71C2×#92418C2312yes—
72C23×C61—22no—
73C23×C61—21no—
74C22×D61—21no—
75C22×D61—21no—
76C22×D61—42no—
77C22×D61—82no—
78C22×D61C2×D411yescharacteristic
79Q8×C61—22no—
80D4×C61—22no—
81D4×C61—22no—
82D4×C61—22no—
83D4×C61—82no—
84C2×C4×C61—21no—
85C2×C4×C61C2×D412yes—
86C2×C4×C61C2411yescharacteristic
87C22⋊D61—42no—
88C22⋊D61—82no—
89C22⋊D61—82no—
90C2×Dic61—22no—
91C2×Dic61—82no—
92C2×Dic61C2×D412yes—
93C2×Dic61C2×D411yescharacteristic
94D244—22no—
95D244—22no—
96D244—42no—
97D244—82no—
98C4×D61—22no—
99C4×D61—42no—
100C8×C61—22no—
101#92418—22no—
102C22×C61—41no—
103C22×C61—42no—
104C22×C61—42no—
105C22×C61—42no—
106C22×C61—42no—
107C22×C61C22×D411yescharacteristic
108C2×D61—21no—
109C2×D61—22no—
110C2×D61—41no—
111C2×D61—41no—
112C2×D61—42no—
113C2×D61—42no—
114C2×D61—84no—
115C4×C61—22no—
116C4×C61—41no—
117C4×C61H(5,𝔽2)12yes—
118Dic61—21no—
119Dic61—22no—
120Dic61—42no—
121#3220—611nomaximal, Sylow
122C2×C61—81no—
123C2×C61—82no—
124C2×C61#53511yescharacteristic
125C2×C61#57612yes—
126D61—41no—
127D61—81no—
128D61—81no—
129#540—611no—
130#535—611no—
131#496D6111yescharacteristic, complemented
132#493—611no—
133#490—611no—
134#488—611no—
135#463—612no—
136#459—612no—
137#449—612no—
138#447—612no—
139#433—611no—
140C61#322011yescharacteristic, complemented, Sylow
141C2×#41—611no—
142C22×D4—611no—
143C2×SD16C2×D6112yescomplemented
144C2×D8—612no—
145C2×D8—612no—
146C2×D8C2×D6112yes—
147#127—611no—
148#127—611no—
149#127C2×D6111yescharacteristic, complemented
150#125—611no—
151#124—611no—
152#124C2×D6111yescharacteristic
153#123—1222no—
154#121—612no—
155#121—1222no—
156#121—1222no—
157#120—1222no—
158C4×D4—612no—
159#117—611no—
160C4⋊C8—612no—
161#103—612no—
162#102—612no—
163#102—612no—
164#102—612no—
165C22⋊C8—612no—
166#97C2×D6111yescharacteristic, complemented
167C4×C8—611no—
168C24—1221no—
169#41—2442no—
170C2×Q8C22×D6111yescharacteristic
171C2×D4—22no—
172C2×D4—611no—
173C2×D4—611no—
174C2×D4—1221no—
175C2×D4—1222no—
176C2×D4—1221no—
177C2×D4—1222no—
178C2×D4—1222no—
179C2×D4—2442no—
180C2×D4C22×D6111yescharacteristic
181C2×D4C22×D6112yes—
182C22×C4—611no—
183C22×C4—1221no—
184C22×C4—1222no—
185SD16—42no—
186D8—42no—
187D8—2442no—
188D8—2442no—
189C2×C8—612no—
190C2×C8C22×D6112yes—
191C4⋊C4—612no—
192C4⋊C4—612no—
193C4⋊C4—1222no—
194C22⋊C4—22no—
195C22⋊C4—612no—
196C22⋊C4—1222no—
197C22⋊C4—1222no—
198C22⋊C4—1222no—
199C42—611no—
200C42C22×D6111yescharacteristic
201C23—21no—
202C23—22no—
203C23—611no—
204C23—1221no—
205C23—1221no—
206C23—4882no—
207C23—2442no—
208Q8—22no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—82no—
213D4—1222no—
214D4—1222no—
215D4—4882no—
216D4—4882no—
217D4—4882no—
218D4—2442no—
219D4—2442no—
220C2×C4—21no—
221C2×C4—612no—
222C2×C4—611no—
223C2×C4—1222no—
224C2×C4—1222no—
225C2×C4—4882no—
226C2×C4—2442no—
227C2×C4D4×D6112yes—
228C2×C4C23×D6111yescharacteristic
229C8—22no—
230C8—1222no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—41no—
235C22—42no—
236C22—1222no—
237C22—1221no—
238C22—4884no—
239C22—2442no—
240C22—2442no—
241C22—2441no—
242C22—2441no—
243C22C2×D4×D6111yescharacteristic
244C4—22no—
245C4—41no—
246C4—1222no—
247C4—1221no—
248C4—2442no—
249C4#1451982212yes—
250C2—81no—
251C2—82no—
252C2—4881no—
253C2—4881no—
254C2—2441no—
255C2#1684032511yescharacteristic
256C2#1684056612yes—
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#496C2
socleC22×C61C22×D4
hypercenter#496D61
hypocenterC61#3220
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #16840366C2semidirect
2. #16840330C2semidirect
3. #16840138C2semidirect
4. #16840127C2semidirect
5. #16840116C2semidirect
6. #16839979C2semidirect
7. #16839955C2semidirect
8. #16839906C2semidirect
9. #16839894C2semidirect
10. #16839824C2semidirect
11. C61×#496C2semidirect
15. C2×#11758199C22semidirect
18. #14519769C22semidirect
25. D4×Dic61C22semidirect
27. #14519706C22semidirect
28. #14519701C22semidirect
30. #14519699C22semidirect
33. C4×#92418C22semidirect
34. C2×SD16×C61C22semidirect
36. #127×C61C22semidirect
38. #97×C61C22semidirect
131. #496D61semidirect
140. C61#3220semidirect
143. C2×SD16C2×D61semidirect
149. #127C2×D61semidirect
166. #97C2×D61semidirect