Finite Group Databasebeta

← #21455096

Subgroups — #21455096

Order 7808.

752 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14519807C211yesmaximal, characteristic, complemented
2C22×#11758200C211yesmaximal, characteristic, complemented
3#16840333C211yesmaximal, characteristic, complemented
4#16839924C214yesmaximal, complemented
5#16839913C214yesmaximal, complemented
6#16839767C211yesmaximal, characteristic, complemented
7#16839758C211yesmaximal, characteristic, complemented
8#16839729C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C61C211yesmaximal, characteristic
10C22×Dic122C2211yescharacteristic, complemented
11#14519807—24no—
12C2×#11758202C2212yescomplemented
13C2×#11758200—24no—
14C2×#11758200C2214yescomplemented
15C2×#11758200C2212yescomplemented
16C2×#11758194C2211yescharacteristic, complemented
17C22×#92418C2211yescharacteristic, complemented
18#14519775—21no—
19#14519764—21no—
20#14519758—22no—
21#14519756C2214yescomplemented
22#14519736C2211yescharacteristic, complemented
23#14519701C2212yescomplemented
24#14519698C2212yescomplemented
25#14519698C2214yescomplemented
26C22⋊#92418C2214yescomplemented
27H(5,𝔽2)×C61—24no—
28H(5,𝔽2)×C61—24no—
29C2×#41×C61—21no—
30C2×#41×C61C2212yes—
31C2×#41×C61C2211yescharacteristic
32C2×#41×C61C2211yescharacteristic
33C22×D4×C61—22no—
34C22×D4×C61—21no—
35C22×D4×C61C2212yes—
36C22×D4×C61C2211yescharacteristic
37C24×C61—22no—
38C24×C61—21no—
39#41×C61—28no—
40#41×C61—24no—
41#41×C61—24no—
42#41×C61—22no—
43#41×C61—22no—
44#41×C61—22no—
45#41×C61D414yes—
46C2×Q8×C61D411yescharacteristic
47C2×Q8×C61C2311yescharacteristic
48C2×D4×C61—216no—
49C2×D4×C61—28no—
50C2×D4×C61—22no—
51C2×D4×C61—21no—
52C2×D4×C61—24no—
53C2×D4×C61—24no—
54C2×D4×C61—22no—
55C2×D4×C61—21no—
56C2×D4×C61—22no—
57C2×D4×C61D412yes—
58C2×D4×C61D411yescharacteristic
59C2×D4×C61D414yes—
60C2×D4×C61C2311yescharacteristic
61C2×D4×C61C2312yes—
62C22×C4×C61—22no—
63C22×C4×C61—21no—
64C22×C4×C61C2312yes—
65C22×C4×C61C2311yescharacteristic
66C22×Dic61—21no—
67C2×Dic122—22no—
68C2×Dic122—22no—
69C2×Dic122—21no—
70C2×Dic122C2312yes—
71C22⋊Dic61—21no—
72C22⋊Dic61—24no—
73#11758202—24no—
74#11758200—28no—
75#11758200—24no—
76C4⋊Dic61C2312yes—
77#11758196—24no—
78C4×Dic61—21no—
79#11758194—22no—
80C2×#92418—22no—
81C2×#92418C2312yes—
82C2×#92418C2312yes—
83C23×C61—28no—
84C23×C61—22no—
85C23×C61—21no—
86C23×C61—24no—
87C23×C61—82no—
88C23×C61—84no—
89C23×C61C2×D412yes—
90C23×C61C2×D411yescharacteristic
91Q8×C61—22no—
92Q8×C61—21no—
93Q8×C61C2×D412yes—
94D4×C61—28no—
95D4×C61—28no—
96D4×C61—24no—
97D4×C61—24no—
98D4×C61—22no—
99D4×C61—24no—
100D4×C61—22no—
101D4×C61—21no—
102D4×C61C2×D412yes—
103D4×C61C2×D414yes—
104C2×C4×C61—28no—
105C2×C4×C61—21no—
106C2×C4×C61—24no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61C2×D412yes—
110C2×C4×C61C2×D411yescharacteristic
111C2×C4×C61C2×D412yes—
112C2×C4×C61C2×D412yes—
113C2×C4×C61C2411yescharacteristic
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—42no—
117Dic122—22no—
118Dic122—22no—
119Dic122—42no—
120#92418—22no—
121#92418—22no—
122C22×C61—22no—
123C22×C61—22no—
124C22×C61—24no—
125C22×C61—22no—
126C22×C61—24no—
127C22×C61—44no—
128C22×C61—42no—
129C22×C61—84no—
130C22×C61—88no—
131C22×C61#12012yes—
132C22×C61C22×D411yescharacteristic
133C4×C61—22no—
134C4×C61—22no—
135C4×C61#12012yes—
136C4×C61C22×D411yescharacteristic
137C4×C61C22×D411yescharacteristic
138Dic61—42no—
139Dic61—42no—
140#3009—611nomaximal, Sylow
141C2×C61—22no—
142C2×C61—42no—
143C2×C61—44no—
144C2×C61—42no—
145C2×C61C2×#12011yescharacteristic
146C2×C61#57712yes—
147C2×H(5,𝔽2)D6111yescharacteristic
148C2×#137—611no—
149C22×SD16—611no—
150#536—611no—
151#452—614no—
152#450—614no—
153#418—611no—
154#416—611no—
155#408—611no—
156C61#300911yescharacteristic, complemented, Sylow
157H(5,𝔽2)—24no—
158H(5,𝔽2)—24no—
159C2×#41—21no—
160C2×#41C2×D6112yes—
161C2×#41C2×D6111yescharacteristic
162C2×#41C2×D6111yescharacteristic
163C22×Q8—611no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4C2×D6112yes—
167C22×D4C2×D6111yescharacteristic
168#137—1224no—
169C2×Q16—612no—
170C2×SD16—614no—
171C2×SD16—612no—
172C2×SD16—1224no—
173C2×M16—611no—
174C22×C8—611no—
175C4⋊Q8—1221no—
176#124—1221no—
177#123—1222no—
178C22⋊Q8—614no—
179#117—611no—
180#103—612no—
181#102—612no—
182#102—614no—
183C22⋊C8—614no—
184C24—22no—
185C24—21no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41C22⋊D6114yes—
193C2×Q8—612no—
194C2×Q8—1221no—
195C2×Q8—1222no—
196C2×Q8—1222no—
197C2×Q8C22⋊D6111yescharacteristic
198C2×Q8C22×D6111yescharacteristic
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4C22⋊D6112yes—
209C2×D4C22⋊D6111yescharacteristic
210C2×D4C22⋊D6114yes—
211C2×D4C22×D6111yescharacteristic
212C2×D4C22×D6112yes—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—1221no—
216C22×C4C22×D6112yes—
217C22×C4C22×D6111yescharacteristic
218Q16—1224no—
219SD16—1228no—
220SD16—1224no—
221M16—1222no—
222C2×C8—612no—
223C2×C8—612no—
224C2×C8—1222no—
225C4⋊C4—612no—
226C4⋊C4—1224no—
227C22⋊C4—1221no—
228C22⋊C4—1224no—
229C42—1221no—
230C23—28no—
231C23—22no—
232C23—21no—
233C23—24no—
234C23—82no—
235C23—84no—
236C23C2×C22⋊D6112yes—
237C23C2×C22⋊D6111yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—1222no—
241Q8—1222no—
242Q8—2442no—
243Q8C2×C22⋊D6112yes—
244D4—28no—
245D4—28no—
246D4—24no—
247D4—24no—
248D4—22no—
249D4—21no—
250D4—22no—
251D4—24no—
252D4C2×C22⋊D6112yes—
253D4C2×C22⋊D6114yes—
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—1222no—
260C2×C4—1222no—
261C2×C4—2442no—
262C2×C4C2×C22⋊D6112yes—
263C2×C4C2×C22⋊D6112yes—
264C2×C4C2×C22⋊D6112yes—
265C2×C4C2×C22⋊D6111yescharacteristic
266C2×C4C23×D6111yescharacteristic
267C8—1222no—
268C8—1222no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—22no—
273C22—24no—
274C22—44no—
275C22—42no—
276C22—84no—
277C22—88no—
278C22#1451974612yes—
279C22C22×C22⋊D6111yescharacteristic
280C4—22no—
281C4—22no—
282C4—2442no—
283C4—2442no—
284C4#1451974612yes—
285C4C22×C22⋊D6111yescharacteristic
286C4C22×C22⋊D6111yescharacteristic
287C2—22no—
288C2—42no—
289C2—44no—
290C2—42no—
291C2C2×#1451974611yescharacteristic
292C2#1684057312yes—
subgroupG/H
centerC22C22×C22⋊D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC2×H(5,𝔽2)×C61C2
socleC22×C61C22×D4
hypercenterC2×H(5,𝔽2)D61
hypocenterC61#3009
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#14519807C2semidirect
2. C22×#11758200C2semidirect
3. #16840333C2semidirect
4. #16839924C2semidirect
5. #16839913C2semidirect
6. #16839767C2semidirect
7. #16839758C2semidirect
8. #16839729C2semidirect
10. C22×Dic122C22semidirect
12. C2×#11758202C22semidirect
14. C2×#11758200C22semidirect
15. C2×#11758200C22semidirect
16. C2×#11758194C22semidirect
17. C22×#92418C22semidirect
21. #14519756C22semidirect
22. #14519736C22semidirect
23. #14519701C22semidirect
24. #14519698C22semidirect
25. #14519698C22semidirect
26. C22⋊#92418C22semidirect
156. C61#3009semidirect