Finite Group Databasebeta

← #22599949

Subgroups — #22599949

Order 9088.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#14556155C212yesmaximal, complemented
2#17986642C212yesmaximal, complemented
3#17986242C212yesmaximal, complemented
4#17986211C212yesmaximal, complemented
5#17986160C212yesmaximal, complemented
6#17986131C212yesmaximal, complemented
7C4×D568C211yesmaximal, characteristic, complemented
8#17986078C211yesmaximal, characteristic, complemented
9#495×C71C211yesmaximal, characteristic, complemented
10C2×#11765866C2212yes—
11C2×D4×D71C2212yescomplemented
12#14556155—28no—
13C2×#11765852C2212yescomplemented
14C2×D568C2211yescharacteristic
15C2×#11765850C2212yes—
16C2×#11765848C2212yescomplemented
17#14556110—22no—
18#14556104C2212yescomplemented
19#14556101—22no—
20#14556099C2212yes—
21#14556098—22no—
22C4×D284C2211yescharacteristic
23C4×D284C2212yescomplemented
24C22⋊C4×D71—22no—
25#14556062C2211yescharacteristic, complemented
26C4⋊(C71⋊C8)C2211yescharacteristic, complemented
27#14556052C2212yescomplemented
28#14556050C2212yescomplemented
29#14556049C2212yescomplemented
30(C2×D71)⋊C8C2212yescomplemented
31C2×Q16×C71C2211yescharacteristic, complemented
32C2×SD16×C71C2212yescomplemented
33C2×D8×C71C2211yescharacteristic
34#124×C71C2212yescomplemented
35C4×C8×C71C2211yescharacteristic, complemented
36C23×D71—22no—
37#11765866—24no—
38#11765866—42no—
39C2×D284—22no—
40C2×D284D411yescharacteristic
41C2×D284C2312yes—
42C2×C22⋊D71—22no—
43C2×C22⋊D71—22no—
44D4×D71—24no—
45D4×D71—42no—
46C2×C4×D71—22no—
47C2×C4×D71—22no—
48C2×C4×D71C2312yes—
49#11765852—24no—
50D568D414yescomplemented
51#11765850—24no—
52#11765848—24no—
53C2×C71⋊C8C2312yes—
54C4⋊Dic71D411yescharacteristic
55C4⋊Dic71C2312yes—
56Dic71⋊C4—22no—
57C22⋊Dic71—22no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4D412yes—
61C2×Q8×C71C2312yes—
62C2×D4×C71C2312yes—
63Q16×C71—41no—
64SD16×C71—24no—
65D8×C71—41no—
66C2×C8×C71C2311yescharacteristic
67C2×C8×C71C2311yescharacteristic
68C22⋊C4×C71—22no—
69C22⋊C4×C71—22no—
70C42×C71C2311yescharacteristic
71C23×C71—22no—
72C22×D71—22no—
73C22×D71—44no—
74C22×D71—44no—
75C22×D71—82no—
76C22×D71C2×D412yes—
77Q8×C71—22no—
78Q8×C71—22no—
79D4×C71—22no—
80D4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2411yescharacteristic
85C22⋊D71—44no—
86C22⋊D71—82no—
87C2×Dic71—22no—
88C2×Dic71C2×D412yes—
89D284—22no—
90D284—41no—
91D284—44no—
92D284C2×D414yes—
93C4×D71—22no—
94C4×D71—22no—
95C4×D71—44no—
96C4×D71—82no—
97C8×C71—21no—
98C8×C71C2×D412yes—
99C71⋊C8—22no—
100C22×C71—42no—
101C22×C71—42no—
102C22×C71—42no—
103C22×C71C22×D411yescharacteristic
104C2×D71—22no—
105C2×D71—24no—
106C2×D71—22no—
107C2×D71—22no—
108C2×D71—42no—
109C2×D71—42no—
110C2×D71—88no—
111C4×C71—21no—
112C4×C71—21no—
113C4×C71—42no—
114C4×C71C22×D411yescharacteristic
115C4×C71H(5,𝔽2)11yescharacteristic
116Dic71—22no—
117Dic71—42no—
118C2×C71—82no—
119C2×C71D4211yescharacteristic
120C2×C71#57511yescharacteristic
121C2×C71#57611yescharacteristic
122D71—42no—
123D71—44no—
124#3273—711nomaximal, Sylow
125C71#327311yescharacteristic, complemented, Sylow
126C2×AGL(1,ℤ/8)—712no—
127#546—712no—
128#495D7111yescharacteristic, complemented
129#468—712no—
130#463—712no—
131#452—712no—
132#447—712no—
133C4×D8—711no—
134#436—711no—
135C2×#41—712no—
136C22×D4—712no—
137AGL(1,ℤ/8)—1428no—
138C2×Q16C2×D7111yescharacteristic, complemented
139C2×SD16—712no—
140C2×SD16C2×D7112yescomplemented
141C2×D8—712no—
142C2×D8—711no—
143C2×D8C2×D7111yescharacteristic
144C2×M16—712no—
145#124C2×D7112yescomplemented
146#123—1422no—
147C22⋊Q8—712no—
148#121—712no—
149#121—1422no—
150#120—1422no—
151C4×D4—711no—
152C4×D4—712no—
153C2×C22⋊C4—1422no—
154#107—711no—
155C4⋊C8—711no—
156#103—712no—
157#102—712no—
158#102—712no—
159C22⋊C8—712no—
160C4×C8C2×D7111yescharacteristic, complemented
161C24—1422no—
162#41—2842no—
163#41—1424no—
164C2×Q8C22×D7112yes—
165C2×D4—2842no—
166C2×D4—711no—
167C2×D4—712no—
168C2×D4—1422no—
169C2×D4—1422no—
170C2×D4—1424no—
171C2×D4—1422no—
172C2×D4C22×D7112yes—
173C22×C4—712no—
174C22×C4—1422no—
175C22×C4—1422no—
176Q16—41no—
177SD16—24no—
178SD16—1424no—
179D8—41no—
180D8—714no—
181D8—1424no—
182M16—1424no—
183C2×C8—712no—
184C2×C8C22×D7111yescharacteristic
185C2×C8C22×D7111yescharacteristic
186C4⋊C4—711no—
187C4⋊C4—712no—
188C4⋊C4—1422no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—712no—
192C22⋊C4—1422no—
193C22⋊C4—1422no—
194C22⋊C4—1422no—
195C42C22×D7111yescharacteristic
196C23—22no—
197C23—2844no—
198C23—2844no—
199C23—5682no—
200C23—712no—
201C23—1422no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—22no—
206D4—2841no—
207D4—2844no—
208D4—2844no—
209D4—5682no—
210D4—714no—
211D4—1422no—
212C2×C4—22no—
213C2×C4—2844no—
214C2×C4—5682no—
215C2×C4—712no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4D4×D7111yescharacteristic
220C2×C4D4×D7111yescharacteristic
221C2×C4C23×D7111yescharacteristic
222C8—21no—
223C8—1422no—
224C8D4×D7112yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—2842no—
229C22—2842no—
230C22—5688no—
231C22—1424no—
232C22—1422no—
233C22—1422no—
234C22—1422no—
235C22C2×D4×D7111yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—2842no—
240C4—1422no—
241C4C2×D4×D7111yescharacteristic
242C4#1455617311yescharacteristic
243C2—82no—
244C2—2844no—
245C2—2842no—
246C2#1798663411yescharacteristic
247C2#1798679611yescharacteristic
248C2#1798680111yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#495×C71C2
socleC22×C71C22×D4
hypercenter#495D71
hypocenterC71#3273
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#14556155C2semidirect
2. #17986642C2semidirect
3. #17986242C2semidirect
4. #17986211C2semidirect
5. #17986160C2semidirect
6. #17986131C2semidirect
7. C4×D568C2semidirect
8. #17986078C2semidirect
9. #495×C71C2semidirect
11. C2×D4×D71C22semidirect
13. C2×#11765852C22semidirect
16. C2×#11765848C22semidirect
18. #14556104C22semidirect
23. C4×D284C22semidirect
25. #14556062C22semidirect
26. C4⋊(C71⋊C8)C22semidirect
27. #14556052C22semidirect
28. #14556050C22semidirect
29. #14556049C22semidirect
30. (C2×D71)⋊C8C22semidirect
31. C2×Q16×C71C22semidirect
32. C2×SD16×C71C22semidirect
34. #124×C71C22semidirect
35. C4×C8×C71C22semidirect
50. D568D4semidirect
125. C71#3273semidirect
128. #495D71semidirect
138. C2×Q16C2×D71semidirect
140. C2×SD16C2×D71semidirect
145. #124C2×D71semidirect
160. C4×C8C2×D71semidirect