Finite Group Databasebeta

← #21455220

Subgroups — #21455220

Order 7808.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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#16840598C211yesmaximal, characteristic, complemented
2#16840329C212yesmaximal, complemented
3#16839948C218yesmaximal, complemented
4#16839736C211yesmaximal, characteristic, complemented
5#16839735C212yesmaximal, complemented
6C61×#535C211yesmaximal, characteristic, complemented
7#14519825—21no—
8#14519824—22no—
9#14519823—22no—
10#14519822—21no—
11#14519822C2214yes—
12C2×#11758222—22no—
13C2×#11758222C2212yes—
14#41×D61—24no—
15#14519773—22no—
16#14519764—22no—
17#14519748—24no—
18#14519746C2218yescomplemented
19#14519736C2212yescomplemented
20#14519692C2214yescomplemented
21#14519691C2218yescomplemented
22C2×#41×C61C2211yescharacteristic
23C22×D4×C61C2211yescharacteristic, complemented
24#127×C61—21no—
25#124×C61—21no—
26#121×C61—22no—
27#120×C61C2214yescomplemented
28#117×C61C2211yescharacteristic, complemented
29C24×C61C2312yes—
30#41×C61—22no—
31#41×C61—22no—
32C2×Q8×C61D411yescharacteristic, complemented
33C2×D4×C61—24no—
34C2×D4×C61—21no—
35C2×D4×C61—21no—
36C2×D4×C61—22no—
37C2×D4×C61—22no—
38C2×D4×C61D411yescharacteristic, complemented
39C2×D4×C61C2312yes—
40C22×C4×C61D412yescomplemented
41C22×C4×C61C2311yescharacteristic
42C2×C22⋊D61—24no—
43C2×C22⋊D61—28no—
44C2×C22⋊D61C2314yes—
45#11758226—24no—
46Q8×D61—24no—
47#11758224—24no—
48#11758224—28no—
49D4×D61—24no—
50D4×D61—28no—
51#11758222—24no—
52#11758222—24no—
53#11758222—22no—
54#11758222—22no—
55#11758222—24no—
56#11758222—22no—
57#11758222—22no—
58C2×D244—22no—
59C2×D244D412yescomplemented
60C2×C4×D61—24no—
61C2×C4×D61D414yescomplemented
62C2×Dic122—22no—
63C2×Dic122D412yescomplemented
64C4⋊C4×C61—21no—
65C22⋊C4×C61—24no—
66C22⋊C4×C61C2312yes—
67C42×C61—21no—
68C22⋊Dic61—28no—
69C22⋊Dic61C2314yes—
70C4⋊Dic61—22no—
71C4×Dic61—22no—
72C23×C61—24no—
73C23×C61—22no—
74C23×C61—82no—
75C23×C61C2×D412yes—
76C23×C61C2411yescharacteristic
77C22×D61—24no—
78C22×D61C2×D414yes—
79Q8×C61—22no—
80D4×C61—24no—
81D4×C61—22no—
82D4×C61—41no—
83D4×C61—42no—
84D4×C61—41no—
85D4×C61—84no—
86C2×C4×C61—22no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61C2×D412yes—
90C2×C4×C61C2×D412yes—
91C2×C4×C61C2×D412yes—
92C2×C4×C61C2×D411yescharacteristic
93C2×C4×C61C2×D411yescharacteristic
94C22⋊D61—28no—
95C22⋊D61—24no—
96C22⋊D61—24no—
97C22⋊D61—88no—
98C2×Dic61—24no—
99C2×Dic61—24no—
100C2×Dic61C2×D414yes—
101D244—24no—
102D244—22no—
103D244—22no—
104C4×D61—28no—
105C4×D61—24no—
106C4×D61—24no—
107Dic122—24no—
108Dic122—22no—
109Dic122—22no—
110C22×C61—22no—
111C22×C61—41no—
112C22×C61—42no—
113C22×C61—48no—
114C22×C61—44no—
115C22×C61—42no—
116C22×C61C22×D412yes—
117C22×C61C22×D411yescharacteristic
118C2×D61—24no—
119C2×D61—44no—
120C2×D61—84no—
121C4×C61—22no—
122C4×C61—42no—
123C4×C61#12011yescharacteristic
124C4×C61#12012yes—
125C4×C61#12011yescharacteristic
126Dic61—24no—
127Dic61—44no—
128#3027—611nomaximal, Sylow
129C2×C61—22no—
130C2×C61—21no—
131C2×C61—42no—
132C2×C61—82no—
133C2×C61C2×#12011yescharacteristic
134D61—44no—
135#585—611no—
136#535—612no—
137#535D6111yescharacteristic, complemented
138B(4,𝔽2)—618no—
139#410—611no—
140#410—612no—
141C61#302711yescharacteristic, complemented, Sylow
1422−1+4—1222no—
1432−1+4—1221no—
144H(5,𝔽2)—614no—
145H(5,𝔽2)—1222no—
146H(5,𝔽2)—1221no—
147C2×#41—612no—
148C2×#41—1224no—
149C2×#41—1222no—
150C2×#41C2×D6111yescharacteristic
151C22×D4C2×D6111yescharacteristic, complemented
152#127—21no—
153#127—1222no—
154#124—21no—
155#124—1222no—
156#121—22no—
157#121—1224no—
158#120—618no—
159#120C2×D6114yescomplemented
160#117—612no—
161#117C2×D6111yescharacteristic, complemented
162#99—614no—
163#99—618no—
164C24C22×D6112yes—
165#41—22no—
166#41—22no—
167#41—1224no—
168#41—1224no—
169#41—1228no—
170#41—1224no—
171#41—1224no—
172#41—1222no—
173#41—1222no—
174#41—1222no—
175#41—1222no—
176#41—1224no—
177C2×Q8—612no—
178C2×Q8—1222no—
179C2×Q8—1224no—
180C2×Q8C22⋊D6111yescharacteristic, complemented
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—614no—
187C2×D4—612no—
188C2×D4—1222no—
189C2×D4—1228no—
190C2×D4—1224no—
191C2×D4—1224no—
192C2×D4—1228no—
193C2×D4C22⋊D6111yescharacteristic, complemented
194C2×D4C22×D6112yes—
195C22×C4—614no—
196C22×C4—1224no—
197C22×C4C22⋊D6112yescomplemented
198C22×C4C22×D6111yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—1222no—
201C22⋊C4—24no—
202C22⋊C4—614no—
203C22⋊C4—1228no—
204C22⋊C4C22×D6112yes—
205C42—21no—
206C42—1222no—
207C23—24no—
208C23—22no—
209C23—82no—
210C23—614no—
211C23—1224no—
212C23C2×C22⋊D6112yes—
213C23C23×D6111yescharacteristic
214Q8—22no—
215Q8—1224no—
216Q8—1222no—
217Q8—1222no—
218D4—24no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—84no—
224D4—1224no—
225D4—1228no—
226D4—1224no—
227D4—1224no—
228D4—1222no—
229D4—1222no—
230D4—4888no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—614no—
235C2×C4—1228no—
236C2×C4—1224no—
237C2×C4—1224no—
238C2×C4—1224no—
239C2×C4—1224no—
240C2×C4D4×D6112yes—
241C2×C4D4×D6112yes—
242C2×C4C2×C22⋊D6112yes—
243C2×C4C2×C22⋊D6111yescharacteristic
244C2×C4C2×C22⋊D6111yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—1224no—
252C22—4884no—
253C22—2444no—
254C22C2×D4×D6112yes—
255C22C22×C22⋊D6111yescharacteristic
256C4—22no—
257C4—42no—
258C4—1224no—
259C4—2444no—
260C4#1451974511yescharacteristic
261C4#1451974511yescharacteristic
262C4#1451974512yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—2444no—
268C2C2×#1451974511yescharacteristic
subgroupG/H
centerC2C2×#14519745
commutatorC23×C61C24
FrattiniC23C23×D61
FittingC61×#535C2
socleC2×C61C2×#120
hypercenter#535D61
hypocenterC61#3027
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #16840598C2semidirect
2. #16840329C2semidirect
3. #16839948C2semidirect
4. #16839736C2semidirect
5. #16839735C2semidirect
6. C61×#535C2semidirect
18. #14519746C22semidirect
19. #14519736C22semidirect
20. #14519692C22semidirect
21. #14519691C22semidirect
23. C22×D4×C61C22semidirect
27. #120×C61C22semidirect
28. #117×C61C22semidirect
32. C2×Q8×C61D4semidirect
38. C2×D4×C61D4semidirect
40. C22×C4×C61D4semidirect
59. C2×D244D4semidirect
61. C2×C4×D61D4semidirect
63. C2×Dic122D4semidirect
137. #535D61semidirect
141. C61#3027semidirect
151. C22×D4C2×D61semidirect
159. #120C2×D61semidirect
161. #117C2×D61semidirect
180. C2×Q8C22⋊D61semidirect
193. C2×D4C22⋊D61semidirect
197. C22×C4C22⋊D61semidirect