Finite Group Databasebeta

← #22597697

Subgroups — #22597697

Order 9088.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#17986797C212yesmaximal, complemented
2#17986566C211yesmaximal, characteristic, complemented
3#17986177C214yesmaximal, complemented
4#17986167C214yesmaximal, complemented
5#17986016C212yesmaximal, complemented
6C2×#14556045C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C71C211yesmaximal, characteristic, complemented
8C2×#11765865C2211yescharacteristic, complemented
9#14556159—22no—
10#14556158C2214yescomplemented
11#14556154—22no—
12#14556153C2214yescomplemented
13#14556148—24no—
14C2×#11765849C2212yescomplemented
15#14556124C2212yescomplemented
16#14556115C2212yescomplemented
17#14556099—22no—
18#14556097—22no—
19#14556087C2211yescharacteristic, complemented
20#14556055C2218yescomplemented
21#14556045C2214yescomplemented
22H(5,𝔽2)×C71—24no—
23H(5,𝔽2)×C71—22no—
24H(5,𝔽2)×C71C2214yes—
25C2×#41×C71—22no—
26C2×#41×C71C2212yes—
27C22×D4×C71—22no—
28C22×D4×C71—22no—
29C22×D4×C71C2211yescharacteristic
30#11765865—22no—
31#11765865—22no—
32C2×Dic142C2311yescharacteristic, complemented
33C2×D284C2311yescharacteristic, complemented
34C2×C22⋊D71—21no—
35C2×C22⋊D71—24no—
36C2×C4×D71—21no—
37#11765855—24no—
38#11765853—24no—
39#11765852—24no—
40#11765850—24no—
41#11765849—22no—
42#11765849C2314yescomplemented
43C2×C71⋊C8—22no—
44C4⋊Dic71—21no—
45C22⋊Dic71—21no—
46C22⋊Dic71—24no—
47C4×Dic71C2312yescomplemented
48C24×C71—22no—
49C24×C71D412yes—
50#41×C71—28no—
51#41×C71—28no—
52#41×C71—22no—
53#41×C71—22no—
54#41×C71D414yescomplemented
55#41×C71C2314yes—
56C2×Q8×C71D412yescomplemented
57C2×D4×C71—24no—
58C2×D4×C71—28no—
59C2×D4×C71—28no—
60C2×D4×C71—22no—
61C2×D4×C71—28no—
62C2×D4×C71—28no—
63C2×D4×C71—22no—
64C2×D4×C71—22no—
65C2×D4×C71D412yes—
66C2×D4×C71D412yescomplemented
67C2×D4×C71C2312yes—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71C2311yescharacteristic
71C23×C71—28no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—22no—
75C23×C71—82no—
76C23×C71—84no—
77C23×C71C2×D414yes—
78C23×C71C2×D411yescharacteristic
79C22×D71—21no—
80Q8×C71—22no—
81Q8×C71C2×D414yes—
82D4×C71—216no—
83D4×C71—24no—
84D4×C71—24no—
85D4×C71—24no—
86D4×C71—22no—
87D4×C71—24no—
88D4×C71C2×D414yes—
89C2×C4×C71—28no—
90C2×C4×C71—22no—
91C2×C4×C71—24no—
92C2×C4×C71—24no—
93C2×C4×C71C2×D411yescharacteristic
94C2×C4×C71C2×D412yes—
95C2×C4×C71C2×D412yes—
96C2×C4×C71C2411yescharacteristic
97C22⋊D71—41no—
98C22⋊D71—41no—
99C22⋊D71—84no—
100C2×Dic71—22no—
101C2×Dic71—21no—
102D284—22no—
103C4×D71—42no—
104Dic142—22no—
105C71⋊C8—24no—
106C22×C71—24no—
107C22×C71—24no—
108C22×C71—41no—
109C22×C71—44no—
110C22×C71—44no—
111C22×C71—44no—
112C22×C71—82no—
113C22×C71—88no—
114C22×C71#12011yescharacteristic
115C22×C71#12011yescharacteristic
116C22×C71C22×D411yescharacteristic
117C2×D71—42no—
118C2×D71—41no—
119C4×C71—24no—
120C4×C71#12012yes—
121C4×C71C22×D412yes—
122Dic71—42no—
123Dic71—41no—
124C2×C71—21no—
125C2×C71—21no—
126C2×C71—21no—
127C2×C71—44no—
128C2×C71—44no—
129C2×C71C2×#12011yescharacteristic
130D71—81no—
131#3019—711nomaximal, Sylow
132C71#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)D7111yescharacteristic, complemented
134#575—712no—
135#535—711no—
136#455—714no—
137#453—714no—
138#421—712no—
139C2×#100—711no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D7114yes—
143C2×#41—22no—
144C2×#41—711no—
145C2×#41C2×D7112yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D7111yescharacteristic
149#137—714no—
150#137—1422no—
151AGL(1,ℤ/8)—714no—
152AGL(1,ℤ/8)—1422no—
153#135—1424no—
154C2×M16—712no—
155#127—712no—
156#124—712no—
157#121—1422no—
158#120—1422no—
159#117—711no—
160#104—718no—
161#100—714no—
162C24—22no—
163C24C22⋊D7112yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—1422no—
169#41—1422no—
170#41C22⋊D7114yescomplemented
171#41C22×D7114yes—
172C2×Q8—711no—
173C2×Q8C22⋊D7112yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—711no—
183C2×D4—1421no—
184C2×D4—1424no—
185C2×D4C22⋊D7112yes—
186C2×D4C22⋊D7112yescomplemented
187C2×D4C22×D7112yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—1421no—
191C22×C4C22×D7111yescharacteristic
192Q16—1424no—
193SD16—1424no—
194SD16—1424no—
195D8—1424no—
196M16—714no—
197M16—1422no—
198C2×C8—1422no—
199C4⋊C4—1421no—
200C22⋊C4—1421no—
201C22⋊C4—1424no—
202C42—712no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—1421no—
210C23C2×C22⋊D7114yes—
211C23C2×C22⋊D7111yescharacteristic
212Q8—22no—
213Q8—1422no—
214Q8C2×C22⋊D7114yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—2841no—
222D4—2841no—
223D4—5684no—
224D4—1422no—
225D4C2×C22⋊D7114yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—2842no—
231C2×C4—1422no—
232C2×C4—1421no—
233C2×C4C2×C22⋊D7112yes—
234C2×C4C2×C22⋊D7112yes—
235C2×C4C2×C22⋊D7111yescharacteristic
236C2×C4C23×D7111yescharacteristic
237C8—1424no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—2842no—
247C22—2841no—
248C22#1455609711yescharacteristic
249C22#1455609711yescharacteristic
250C22C22×C22⋊D7111yescharacteristic
251C4—24no—
252C4—2842no—
253C4—2841no—
254C4#1455609712yes—
255C4C22×C22⋊D7112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—5681no—
262C2C2×#1455609711yescharacteristic
subgroupG/H
centerC2C2×#14556097
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×H(5,𝔽2)×C71C2
socleC2×C71C2×#120
hypercenterC2×H(5,𝔽2)D71
hypocenterC71#3019
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #17986797C2semidirect
2. #17986566C2semidirect
3. #17986177C2semidirect
4. #17986167C2semidirect
5. #17986016C2semidirect
6. C2×#14556045C2semidirect
7. C2×H(5,𝔽2)×C71C2semidirect
8. C2×#11765865C22semidirect
10. #14556158C22semidirect
12. #14556153C22semidirect
14. C2×#11765849C22semidirect
15. #14556124C22semidirect
16. #14556115C22semidirect
19. #14556087C22semidirect
20. #14556055C22semidirect
21. #14556045C22semidirect
32. C2×Dic142C23semidirect
33. C2×D284C23semidirect
42. #11765849C23semidirect
47. C4×Dic71C23semidirect
54. #41×C71D4semidirect
56. C2×Q8×C71D4semidirect
66. C2×D4×C71D4semidirect
132. C71#3019semidirect
133. C2×H(5,𝔽2)D71semidirect
170. #41C22⋊D71semidirect
173. C2×Q8C22⋊D71semidirect
186. C2×D4C22⋊D71semidirect