Finite Group Databasebeta

← #22599183

Subgroups — #22599183

Order 9088.

418 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986512C212yesmaximal, complemented
2#17986338C212yesmaximal, complemented
3#17986325C212yesmaximal, complemented
4#17986252C211yesmaximal, characteristic, complemented
5#17986247C212yesmaximal, complemented
6#17986160C212yesmaximal, complemented
7#17986147C212yesmaximal, complemented
8#17985962C211yesmaximal, characteristic, complemented
9#470×C71C211yesmaximal, characteristic, complemented
10C2×#11765866C2212yes—
11C22×C4×D71C2211yescharacteristic
12C2×#11765852C2212yescomplemented
13C2×D568C2211yescharacteristic, complemented
14C2×#11765849C2211yescharacteristic
15#14556113—22no—
16#14556104C2212yes—
17#14556103C2212yes—
18#14556102C2212yes—
19#14556098—22no—
20#14556090—22no—
21(C22⋊D71)⋊C4—22no—
22#14556086C2212yes—
23#14556061C2212yescomplemented
24#14556060C2211yescharacteristic, complemented
25#14556052C2212yescomplemented
26#14556050C2212yescomplemented
27#14556048C2212yescomplemented
28(C2×D71)⋊C8C2212yes—
29(C2×D71)⋊C8C2212yes—
30C2×Q16×C71C2211yescharacteristic, complemented
31C2×M16×C71C2211yescharacteristic
32C22⋊Q8×C71C2212yes—
33#106×C71C2211yescharacteristic, complemented
34#103×C71C2212yescomplemented
35C23×D71D411yescharacteristic
36#11765866—42no—
37#11765866—42no—
38C2×D284—22no—
39C2×D284—22no—
40C2×D284C2312yes—
41C2×C22⋊D71—22no—
42C22×Dic71D411yescharacteristic
43C2×C4×D71—22no—
44C2×C4×D71—41no—
45C2×C4×D71—41no—
46C2×C4×D71D412yes—
47C2×C4×D71C2312yes—
48#11765852—42no—
49D568—41no—
50#11765849—41no—
51C2×C71⋊C8C2312yes—
52C4⋊Dic71C2312yes—
53Dic71⋊C4—22no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58C4×Dic71—22no—
59C2×Q8×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61Q16×C71—41no—
62M16×C71—41no—
63C2×C8×C71C2311yescharacteristic
64C2×C8×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C23×C71C2×D411yescharacteristic
69C22×D71—22no—
70C22×D71—41no—
71C22×D71—42no—
72C22×D71C2×D412yes—
73Q8×C71—22no—
74Q8×C71—22no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2411yescharacteristic
81C22⋊D71—44no—
82C2×Dic71—22no—
83C2×Dic71—41no—
84C2×Dic71C2×D412yes—
85D284—22no—
86D284—22no—
87D284—44no—
88D284—44no—
89C4×D71—22no—
90C4×D71—22no—
91C4×D71—42no—
92C4×D71—44no—
93C8×C71—21no—
94C8×C71—21no—
95C71⋊C8—22no—
96C22×C71—21no—
97C22×C71—22no—
98C22×C71C22×D411yescharacteristic
99C2×D71—22no—
100C2×D71—22no—
101C2×D71—22no—
102C2×D71—44no—
103C2×D71—42no—
104C2×D71—42no—
105C2×D71—42no—
106C4×C71—21no—
107C4×C71—42no—
108C4×C71—42no—
109C4×C71H(5,𝔽2)11yescharacteristic
110C4×C71H(5,𝔽2)11yescharacteristic
111Dic71—22no—
112Dic71—42no—
113C2×C71—41no—
114C2×C71#53411yescharacteristic
115C2×C71#57512yes—
116D71—42no—
117D71—82no—
118#3197—711nomaximal, Sylow
119C71#319711yescharacteristic, complemented, Sylow
120#525—712no—
121#485—712no—
122#482—712no—
123#470D7111yescharacteristic, complemented
124#469—711no—
125#468—712no—
126#452—712no—
127#449—712no—
128#407—711no—
129C2×#41—712no—
130C23×C4—711no—
131C2×Q16C2×D7111yescharacteristic, complemented
132C2×SD16—712no—
133C2×D8—711no—
134C2×M16—711no—
135C2×M16C2×D7111yescharacteristic
136#123—1422no—
137C22⋊Q8—712no—
138C22⋊Q8C2×D7112yes—
139#121—712no—
140#121—712no—
141#120—1422no—
142C4×D4—1422no—
143C4×D4—1422no—
144#117—712no—
145#107—712no—
146#106—711no—
147#106C2×D7111yescharacteristic, complemented
148#103—712no—
149#103C2×D7112yescomplemented
150#102—712no—
151#102—712no—
152C22⋊C8—712no—
153C22⋊C8—712no—
154C24—711no—
155#41—2842no—
156#41—2842no—
157C2×Q8C22×D7112yes—
158C2×D4—712no—
159C2×D4—1422no—
160C2×D4—1422no—
161C2×D4—1422no—
162C22×C4—2841no—
163C22×C4—2841no—
164C22×C4—711no—
165C22×C4—712no—
166C22×C4—712no—
167C22×C4—1422no—
168C22×C4C22×D7111yescharacteristic
169Q16—41no—
170SD16—2842no—
171D8—2841no—
172M16—41no—
173M16—2841no—
174C2×C8—712no—
175C2×C8C22×D7111yescharacteristic
176C2×C8C22×D7111yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—712no—
179C4⋊C4—1422no—
180C4⋊C4C22×D7112yes—
181C22⋊C4—22no—
182C22⋊C4—1422no—
183C22⋊C4—1422no—
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C42—1422no—
187C23—2841no—
188C23—2842no—
189C23—712no—
190C23—1422no—
191C23D4×D7111yescharacteristic
192Q8—22no—
193Q8—22no—
194D4—2844no—
195D4—2844no—
196D4—2844no—
197D4—1422no—
198D4—1422no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—2842no—
204C2×C4—2844no—
205C2×C4—2841no—
206C2×C4—712no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4—1422no—
210C2×C4D4×D7111yescharacteristic
211C2×C4C23×D7111yescharacteristic
212C8—21no—
213C8—21no—
214C8—1422no—
215C22—22no—
216C22—21no—
217C22—2844no—
218C22—2842no—
219C22—2842no—
220C22—2842no—
221C22—1422no—
222C22—1422no—
223C22—1422no—
224C22C2×D4×D7111yescharacteristic
225C4—21no—
226C4—42no—
227C4—42no—
228C4—2842no—
229C4—1422no—
230C4#1455617311yescharacteristic
231C4#1455617411yescharacteristic
232C2—41no—
233C2—2842no—
234C2—5682no—
235C2#1798655711yescharacteristic
236C2#1798679812yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#470×C71C2
socleC22×C71C22×D4
hypercenter#470D71
hypocenterC71#3197
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #17986512C2semidirect
2. #17986338C2semidirect
3. #17986325C2semidirect
4. #17986252C2semidirect
5. #17986247C2semidirect
6. #17986160C2semidirect
7. #17986147C2semidirect
8. #17985962C2semidirect
9. #470×C71C2semidirect
12. C2×#11765852C22semidirect
13. C2×D568C22semidirect
23. #14556061C22semidirect
24. #14556060C22semidirect
25. #14556052C22semidirect
26. #14556050C22semidirect
27. #14556048C22semidirect
30. C2×Q16×C71C22semidirect
33. #106×C71C22semidirect
34. #103×C71C22semidirect
119. C71#3197semidirect
123. #470D71semidirect
131. C2×Q16C2×D71semidirect
147. #106C2×D71semidirect
149. #103C2×D71semidirect