Finite Group Databasebeta

← #22598723

Subgroups — #22598723

Order 9088.

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

normal subgroups only

This group has 400 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#17986555C211yesmaximal, characteristic, complemented
2#124×D71C211yesmaximal, characteristic
3#17986360C212yesmaximal, complemented
4#17986357C212yesmaximal, complemented
5#17986356C211yesmaximal, characteristic, complemented
6#17986354C211yesmaximal, characteristic, complemented
7#17986353C211yesmaximal, characteristic, complemented
8#17986006C212yesmaximal, complemented
9#17986002C212yesmaximal
10C4×#11765848C211yesmaximal, characteristic, complemented
11#488×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2211yescharacteristic
13C2×D4×D71C2211yescharacteristic
14C2×#11765848C2212yes—
15#14556127C2212yescomplemented
16#14556120C2211yescharacteristic, complemented
17#14556117C2211yescharacteristic
18#14556116C2211yescharacteristic
19#14556115C2211yescharacteristic
20#14556114—22no—
21#14556106—22no—
22#14556093—22no—
23#14556086C2212yes—
24C22⋊C4×D71—22no—
25C42×D71C2211yescharacteristic
26#14556053C2212yescomplemented
27#14556052C2212yescomplemented
28#14556051C2212yescomplemented
29#14556050C2212yes—
30#14556049C2212yes—
31#14556048C2212yes—
32#14556038C2212yescomplemented
33C4×C71⋊C8C2211yescharacteristic, complemented
34#125×C71C2211yescharacteristic, complemented
35#124×C71C2211yescharacteristic
36#103×C71C2212yescomplemented
37#102×C71C2212yes—
38C4×C8×C71C2211yescharacteristic, complemented
39C23×D71—21no—
40C2×Dic142—21no—
41C2×Dic142—22no—
42C2×Dic142C2311yescharacteristic
43Q8×D71—24no—
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—21no—
46D4×D71—24no—
47C2×C4×D71—21no—
48C2×C4×D71D412yes—
49C2×C4×D71C2311yescharacteristic
50#11765848—24no—
51C2×C71⋊C8C2312yes—
52C4⋊Dic71C2312yes—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55C22⋊Dic71—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—22no—
59C4×Dic71—22no—
60C4×Dic71D412yes—
61C4×Dic71C2311yescharacteristic
62C2×Q8×C71C2311yescharacteristic
63C2×D4×C71C2311yescharacteristic
64C2×C8×C71C2312yes—
65C4⋊C4×C71—22no—
66C4⋊C4×C71C2312yes—
67C22⋊C4×C71—22no—
68C42×C71C2311yescharacteristic
69C23×C71—21no—
70C22×D71—21no—
71C22×D71—44no—
72C22×D71—42no—
73C22×D71C2×D411yescharacteristic
74Q8×C71—22no—
75D4×C71—22no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71C2×D412yes—
79C2×C4×C71C2411yescharacteristic
80C22⋊D71—42no—
81C2×Dic71—22no—
82C2×Dic71—21no—
83C2×Dic71C2×D412yes—
84C2×Dic71C2×D411yescharacteristic
85D284—22no—
86C4×D71—42no—
87C4×D71—42no—
88C4×D71#4114yes—
89Dic142—22no—
90Dic142—44no—
91Dic142—42no—
92C8×C71—22no—
93C71⋊C8—22no—
94C22×C71—41no—
95C22×C71—42no—
96C22×C71C22×D411yescharacteristic
97C2×D71—22no—
98C2×D71—41no—
99C2×D71—42no—
100C2×D71—84no—
101C2×D71#12412yes—
102C4×C71—22no—
103C4×C71—41no—
104C4×C71—42no—
105C4×C71C2×#4112yes—
106Dic71—22no—
107Dic71—41no—
108Dic71—42no—
109Dic71#12412yes—
110C2×C71—81no—
111C2×C71C2×#12411yescharacteristic
112C2×C71#57512yes—
113D71—22no—
114D71—81no—
115#3140—711nomaximal, Sylow
116C71#314011yescharacteristic, complemented, Sylow
117#533—711no—
118C2×#124—711no—
119#489—712no—
120#489—712no—
121#488—711no—
122#488—711no—
123#488—711no—
124#488D7111yescharacteristic, complemented
125#419—712no—
126#418—712no—
127C4×M16—711no—
128C22×Q8—711no—
129C22×D4—711no—
130C2×M16—712no—
131C4⋊Q8—712no—
132#125—711no—
133#125C2×D7111yescharacteristic, complemented
134#124—711no—
135#124—711no—
136#124—711no—
137#124—1422no—
138#124C2×D7111yescharacteristic
139C22⋊Q8—1422no—
140C4×Q8—1422no—
141#117—712no—
142C2×C22⋊C4—1422no—
143C2×C42—711no—
144#103—712no—
145#103—712no—
146#103—712no—
147#103C2×D7112yescomplemented
148#102—712no—
149#102—712no—
150#102—712no—
151#102C2×D7112yes—
152#97—712no—
153C4×C8—711no—
154C4×C8C2×D7111yescharacteristic, complemented
155C24—1421no—
156C2×Q8—711no—
157C2×Q8—1422no—
158C2×Q8—1421no—
159C2×Q8—1424no—
160C2×Q8C22×D7111yescharacteristic
161C2×D4—711no—
162C2×D4—1424no—
163C2×D4—1421no—
164C2×D4C22×D7111yescharacteristic
165C22×C4—712no—
166C22×C4—711no—
167C22×C4—1421no—
168M16—1424no—
169C2×C8—712no—
170C2×C8C22×D7112yes—
171C4⋊C4—22no—
172C4⋊C4—712no—
173C4⋊C4—1422no—
174C4⋊C4—1422no—
175C4⋊C4C22×D7112yes—
176C22⋊C4—22no—
177C22⋊C4—1422no—
178C22⋊C4—1422no—
179C22⋊C4—1422no—
180C22⋊C4—1422no—
181C42—712no—
182C42—711no—
183C42—1422no—
184C42C22×D7111yescharacteristic
185C23—21no—
186C23—2842no—
187C23—2844no—
188C23—711no—
189C23—1421no—
190Q8—22no—
191Q8—2842no—
192Q8—2844no—
193Q8—1422no—
194D4—22no—
195D4—2842no—
196D4—1422no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2842no—
200C2×C4—2842no—
201C2×C4—711no—
202C2×C4—714no—
203C2×C4—712no—
204C2×C4—1422no—
205C2×C4—1421no—
206C2×C4D4×D7112yes—
207C2×C4C23×D7111yescharacteristic
208C8—22no—
209C8—1422no—
210C22—41no—
211C22—42no—
212C22—2841no—
213C22—2842no—
214C22—5684no—
215C22—712no—
216C22—1422no—
217C22C2×D4×D7111yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—2841no—
222C4—2842no—
223C4—712no—
224C4—1422no—
225C4#41×D7112yes—
226C2—81no—
227C2—5681no—
228C2—1422no—
229C2#124×D7111yescharacteristic
230C2#1798679612yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#488×C71C2
socleC22×C71C22×D4
hypercenter#488D71
hypocenterC71#3140
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #17986555C2semidirect
3. #17986360C2semidirect
4. #17986357C2semidirect
5. #17986356C2semidirect
6. #17986354C2semidirect
7. #17986353C2semidirect
8. #17986006C2semidirect
10. C4×#11765848C2semidirect
11. #488×C71C2semidirect
15. #14556127C22semidirect
16. #14556120C22semidirect
26. #14556053C22semidirect
27. #14556052C22semidirect
28. #14556051C22semidirect
32. #14556038C22semidirect
33. C4×C71⋊C8C22semidirect
34. #125×C71C22semidirect
36. #103×C71C22semidirect
38. C4×C8×C71C22semidirect
116. C71#3140semidirect
124. #488D71semidirect
133. #125C2×D71semidirect
147. #103C2×D71semidirect
154. C4×C8C2×D71semidirect