Finite Group Databasebeta

← #22600136

Subgroups — #22600136

Order 9088.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986803C212yesmaximal, complemented
2#17986802C212yesmaximal, complemented
3#17986375C211yesmaximal, characteristic, complemented
4#17986268C212yesmaximal, complemented
5#17986267C212yesmaximal, complemented
6#17986168C212yesmaximal, complemented
7#17986166C211yesmaximal, characteristic, complemented
8#17986113C212yesmaximal, complemented
9#453×C71C211yesmaximal, characteristic, complemented
10#14556174C2212yescomplemented
11#14556173C2211yescharacteristic, complemented
12#14556156C2212yescomplemented
13#14556155—22no—
14#14556154—22no—
15#14556154C2212yescomplemented
16#14556153—22no—
17#14556153C2212yescomplemented
18#14556152—22no—
19#14556149—22no—
20#14556148—22no—
21#14556148C2212yes—
22C2×#11765850—22no—
23C2×#11765850C2212yescomplemented
24C2×#11765850C2212yescomplemented
25D8×D71—22no—
26#14556136C2212yescomplemented
27#14556135C2212yescomplemented
28#14556125C2211yescharacteristic, complemented
29#14556063C2212yescomplemented
30#14556056C2212yescomplemented
31#14556045C2211yescharacteristic, complemented
32#14556044C2212yescomplemented
33C4×C71⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C71C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C71C2212yescomplemented
36#127×C71C2211yescharacteristic, complemented
37#104×C71C2212yescomplemented
38#100×C71C2211yescharacteristic, complemented
39#11765866—22no—
40#11765865—22no—
41#11765865C2312yes—
42#11765864—22no—
43C2×D284—22no—
44C2×D284—41no—
45C2×D284C2312yes—
46C2×C22⋊D71—41no—
47D4×D71—22no—
48D4×D71—22no—
49D4×D71—42no—
50#11765855D412yescomplemented
51#11765854—22no—
52#11765853—22no—
53#11765853D412yescomplemented
54#11765852—22no—
55#11765852D412yescomplemented
56D568—22no—
57#11765850—22no—
58#11765850—22no—
59#11765850—22no—
60#11765850—22no—
61#11765850—22no—
62#11765850—22no—
63#11765850D412yescomplemented
64#11765849—22no—
65#11765849C2312yes—
66#11765848—22no—
67C2×C71⋊C8—22no—
68C2×C71⋊C8C2312yes—
69C8×D71—22no—
70#41×C71—22no—
71#41×C71C2312yes—
72C2×D4×C71—22no—
73C2×D4×C71—41no—
74C2×D4×C71—41no—
75C2×D4×C71C2311yescharacteristic
76C2×D4×C71C2311yescharacteristic
77SD16×C71—22no—
78D8×C71—22no—
79M16×C71C2312yes—
80C42×C71C2311yescharacteristic
81C23×C71—21no—
82C23×C71—21no—
83C23×C71—41no—
84C22×D71—22no—
85C22×D71—41no—
86C22×D71—42no—
87Q8×C71C2×D412yes—
88D4×C71—22no—
89D4×C71—22no—
90D4×C71—22no—
91D4×C71—42no—
92D4×C71—82no—
93D4×C71C2×D412yes—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—41no—
97C2×C4×C71C2411yescharacteristic
98C22⋊D71—22no—
99C22⋊D71—42no—
100C22⋊D71—42no—
101C2×Dic71—41no—
102D284—22no—
103D284—22no—
104D284—42no—
105D284—82no—
106D284C2×D412yes—
107C4×D71—22no—
108C4×D71—42no—
109Dic142C2×D412yes—
110C8×C71—22no—
111C71⋊C8—22no—
112C71⋊C8C2×D412yes—
113C71⋊C8C2×D412yes—
114C22×C71—22no—
115C22×C71—41no—
116C22×C71—41no—
117C22×C71—81no—
118C22×C71—81no—
119C22×C71—161no—
120C22×C71H(5,𝔽2)11yescharacteristic
121C2×D71—22no—
122C2×D71—42no—
123C2×D71—82no—
124C2×D71—161no—
125C2×D71—162no—
126C4×C71—22no—
127C4×C71—22no—
128C4×C71C22×D412yes—
129Dic71—22no—
130C2×C71—21no—
131C2×C71—42no—
132C2×C71—81no—
133C2×C71—81no—
134C2×C71D4211yescharacteristic
135D71—42no—
136D71—82no—
137#3292—711nomaximal, Sylow
138C71#329211yescharacteristic, complemented, Sylow
139#576—712no—
140#576—712no—
141#493—711no—
142#472—712no—
143#472—712no—
144#453—711no—
145#453—712no—
146#453D7111yescharacteristic, complemented
147#443—712no—
148H(5,𝔽2)—712no—
149H(5,𝔽2)—711no—
150H(5,𝔽2)C2×D7111yescharacteristic, complemented
151AGL(1,ℤ/8)—712no—
152AGL(1,ℤ/8)—712no—
153AGL(1,ℤ/8)—712no—
154AGL(1,ℤ/8)—1422no—
155AGL(1,ℤ/8)—1422no—
156AGL(1,ℤ/8)—1422no—
157AGL(1,ℤ/8)—1422no—
158AGL(1,ℤ/8)C2×D7112yescomplemented
159#135—712no—
160#135—1422no—
161#135—1422no—
162C2×D8—712no—
163C2×D8—712no—
164C2×D8—1422no—
165C2×D8—1422no—
166#131—712no—
167#131—712no—
168#127—711no—
169#127C2×D7111yescharacteristic, complemented
170#108—712no—
171#104—712no—
172#104C2×D7112yescomplemented
173#100—711no—
174#100—712no—
175#100C2×D7111yescharacteristic, complemented
176C4×C8—711no—
177#41—22no—
178#41—712no—
179#41—1422no—
180#41—1422no—
181#41—1422no—
182#41C22×D7112yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2841no—
187C2×D4—2842no—
188C2×D4—2841no—
189C2×D4—712no—
190C2×D4—1422no—
191C2×D4—1422no—
192C2×D4—1422no—
193C2×D4C22×D7111yescharacteristic
194C2×D4C22×D7111yescharacteristic
195Q16—712no—
196SD16—22no—
197SD16—712no—
198SD16—712no—
199SD16—1422no—
200SD16—1422no—
201SD16—1422no—
202D8—22no—
203D8—712no—
204D8—1422no—
205D8—1422no—
206D8—1422no—
207D8—1422no—
208D8—1422no—
209D8—1422no—
210D8—1422no—
211M16—712no—
212M16—1422no—
213M16—1422no—
214M16C22×D7112yes—
215C2×C8—712no—
216C2×C8—1422no—
217C2×C8—1422no—
218C42C22×D7111yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—2842no—
223C23—2841no—
224C23—1422no—
225Q8—712no—
226Q8D4×D7112yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—2842no—
233D4—2842no—
234D4—2842no—
235D4—5682no—
236D4—712no—
237D4—1422no—
238D4—1422no—
239D4—1422no—
240D4D4×D7112yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—2842no—
245C2×C4—2841no—
246C2×C4—1422no—
247C2×C4C23×D7111yescharacteristic
248C8—22no—
249C8—712no—
250C8—712no—
251C8—1422no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—2842no—
259C22—5682no—
260C22—11362no—
261C22—11361no—
262C22—1422no—
263C22#1455617311yescharacteristic
264C4—22no—
265C4—22no—
266C4—1422no—
267C4C2×D4×D7112yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—2842no—
273C2—5682no—
274C2#1798663111yescharacteristic
subgroupG/H
centerC2#17986631
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#453×C71C2
socleC2×C71D42
hypercenter#453D71
hypocenterC71#3292
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #17986803C2semidirect
2. #17986802C2semidirect
3. #17986375C2semidirect
4. #17986268C2semidirect
5. #17986267C2semidirect
6. #17986168C2semidirect
7. #17986166C2semidirect
8. #17986113C2semidirect
9. #453×C71C2semidirect
10. #14556174C22semidirect
11. #14556173C22semidirect
12. #14556156C22semidirect
15. #14556154C22semidirect
17. #14556153C22semidirect
23. C2×#11765850C22semidirect
24. C2×#11765850C22semidirect
26. #14556136C22semidirect
27. #14556135C22semidirect
28. #14556125C22semidirect
29. #14556063C22semidirect
30. #14556056C22semidirect
31. #14556045C22semidirect
32. #14556044C22semidirect
33. C4×C71⋊C8C22semidirect
34. H(5,𝔽2)×C71C22semidirect
35. AGL(1,ℤ/8)×C71C22semidirect
36. #127×C71C22semidirect
37. #104×C71C22semidirect
38. #100×C71C22semidirect
50. #11765855D4semidirect
53. #11765853D4semidirect
55. #11765852D4semidirect
63. #11765850D4semidirect
138. C71#3292semidirect
146. #453D71semidirect
150. H(5,𝔽2)C2×D71semidirect
158. AGL(1,ℤ/8)C2×D71semidirect
169. #127C2×D71semidirect
172. #104C2×D71semidirect
175. #100C2×D71semidirect