Finite Group Databasebeta

← #22600700

Subgroups — #22600700

Order 9088.

388 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 123 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#17986709C211yesmaximal, characteristic, complemented
2#17986652C211yesmaximal, characteristic
3#17986395C211yesmaximal, characteristic, complemented
4#17986328C212yesmaximal, complemented
5#17986319C212yesmaximal, complemented
6#17986290C211yesmaximal, characteristic, complemented
7#17986231C212yesmaximal, complemented
8C4×#11765852C211yesmaximal, characteristic, complemented
9#17986064C211yesmaximal, characteristic, complemented
10#106×D71C212yesmaximal, complemented
11#475×C71C211yesmaximal, characteristic, complemented
12C2×#11765866—21no—
13C2×#11765852C2211yescharacteristic
14C2×C8×D71C2212yes—
15#14556128C2211yescharacteristic
16#14556120—21no—
17#14556113—22no—
18#14556106—22no—
19#14556104—21no—
20#14556104C2212yes—
21#14556103C2212yes—
22C4×D284—21no—
23C4×D284C2211yescharacteristic
24C4×D284C2211yescharacteristic
25C4⋊C4×D71C2212yes—
26C4⋊C4×D71C2212yes—
27#14556060C2212yes—
28#14556059C2212yes—
29#14556059C2211yescharacteristic
30#14556059C2212yes—
31#14556052C2211yescharacteristic
32#14556050C2212yes—
33#14556048C2211yescharacteristic
34(C2×D71)⋊C8C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C4⋊Q8×C71C2211yescharacteristic
37C4×Q8×C71C2211yescharacteristic
38#106×C71C2212yes—
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41#11765866—44no—
42C2×Dic142—21no—
43C2×D284—22no—
44C2×D284D411yescharacteristic
45C2×D284C2311yescharacteristic
46C2×C4×D71—22no—
47C2×C4×D71—21no—
48C2×C4×D71—22no—
49C2×C4×D71C2312yes—
50#11765852—22no—
51C2×C71⋊C8C2311yescharacteristic
52C2×C71⋊C8C2311yescharacteristic
53C8×D71—24no—
54C4⋊Dic71—21no—
55C4⋊Dic71D411yescharacteristic
56C4⋊Dic71C2312yes—
57C4⋊Dic71C2311yescharacteristic
58Dic71⋊C4—22no—
59Dic71⋊C4—22no—
60Dic71⋊C4—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—21no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4D412yes—
65C2×Q8×C71—21no—
66C2×Q8×C71C2311yescharacteristic
67C2×C8×C71C2312yes—
68C4⋊C4×C71—21no—
69C4⋊C4×C71—21no—
70C4⋊C4×C71C2312yes—
71C4⋊C4×C71C2311yescharacteristic
72C42×C71—21no—
73C42×C71C2311yescharacteristic
74C22×D71—21no—
75C22×D71C2×D412yes—
76Q8×C71—21no—
77Q8×C71—21no—
78Q8×C71—42no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2411yescharacteristic
85C2×Dic71—22no—
86C2×Dic71—21no—
87C2×Dic71C2×D412yes—
88D284—21no—
89D284—21no—
90D284—44no—
91D284SD1614yescomplemented
92C4×D71—22no—
93C4×D71—22no—
94C4×D71—44no—
95C4×D71—42no—
96C4×D71—82no—
97Dic142—81no—
98C8×C71—22no—
99C71⋊C8—21no—
100C71⋊C8#4112yescomplemented
101C22×C71C22×D411yescharacteristic
102C2×D71—22no—
103C2×D71—22no—
104C2×D71—41no—
105C2×D71—41no—
106C2×D71—41no—
107C2×D71C2×SD1614yes—
108C4×C71—21no—
109C4×C71—41no—
110C4×C71—42no—
111C4×C71—41no—
112C4×C71C2×SD1612yes—
113C4×C71C2×#4111yescharacteristic
114C4×C712−1+411yescharacteristic
115Dic71—22no—
116Dic71—41no—
117Dic71—42no—
118C2×C71#54711yescharacteristic
119C2×C71C22×SD1611yescharacteristic
120C2×C71#57711yescharacteristic
121D71—24no—
122D71—81no—
123#3335—711nomaximal, Sylow
124C71#333511yescharacteristic, complemented, Sylow
125#554—711no—
126#547—711no—
127#498—711no—
128#483—712no—
129#481—712no—
130#476—711no—
131#475D7111yescharacteristic, complemented
132#465—712no—
133C4×SD16—711no—
134C8×D4—711no—
135C2×#106—712no—
136C2×#41—1421no—
137C2×SD16—711no—
138C22×C8—712no—
139C4⋊Q8—711no—
140C4⋊Q8C2×D7111yescharacteristic
141#125—1421no—
142#123—1422no—
143C22⋊Q8—712no—
144C22⋊Q8—1422no—
145C22⋊Q8—1421no—
146#121—712no—
147C4×Q8C2×D7111yescharacteristic
148C4×D4—711no—
149C4×D4—711no—
150C4×D4—1421no—
151C2×C4⋊C4—712no—
152C2×C4⋊C4—712no—
153#106—712no—
154#106—711no—
155#106—712no—
156#106—712no—
157#106C2×D7112yes—
158C4⋊C8C2×D7111yescharacteristic
159#103—711no—
160#103C2×D7112yes—
161#102—712no—
162#102—711no—
163C22⋊C8—712no—
164C4×C8—711no—
165#41—2844no—
166C2×Q8—21no—
167C2×Q8—1421no—
168C2×Q8C22×D7111yescharacteristic
169C2×D4—711no—
170C2×D4—711no—
171C2×D4—1422no—
172C22×C4—712no—
173C22×C4—1422no—
174C22×C4—1422no—
175C22×C4—1421no—
176SD16—1422no—
177C2×C8—711no—
178C2×C8—711no—
179C2×C8—1424no—
180C2×C8C22×D7112yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—711no—
184C4⋊C4—712no—
185C4⋊C4—711no—
186C4⋊C4—1422no—
187C4⋊C4—1422no—
188C4⋊C4—1422no—
189C4⋊C4—1421no—
190C4⋊C4C22×D7111yescharacteristic
191C4⋊C4C22×D7112yes—
192C22⋊C4—712no—
193C22⋊C4—1422no—
194C22⋊C4—1422no—
195C22⋊C4—1421no—
196C42—21no—
197C42C22×D7111yescharacteristic
198C23—712no—
199C23—1421no—
200Q8—21no—
201Q8—21no—
202Q8—42no—
203Q8—5681no—
204D4—2844no—
205D4—714no—
206D4—1421no—
207D4—1421no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—2844no—
212C2×C4—2842no—
213C2×C4—5682no—
214C2×C4—712no—
215C2×C4—1422no—
216C2×C4—1421no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4D4×D7111yescharacteristic
220C2×C4D4×D7111yescharacteristic
221C2×C4C23×D7111yescharacteristic
222C8—22no—
223C8—712no—
224C8—1421no—
225C22—2841no—
226C22—2841no—
227C22—2841no—
228C22—714no—
229C22—1422no—
230C22—1422no—
231C22C2×D4×D7111yescharacteristic
232C4—21no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—2841no—
237C4—2842no—
238C4—1422no—
239C4SD16×D7112yes—
240C4#41×D7111yescharacteristic
241C4#1455617611yescharacteristic
242C2—5681no—
243C2—1424no—
244C2#1798665111yescharacteristic
245C2C2×SD16×D7111yescharacteristic
246C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#475×C71C2
socleC22×C71C22×D4
hypercenter#475D71
hypocenterC71#3335
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #17986709C2semidirect
3. #17986395C2semidirect
4. #17986328C2semidirect
5. #17986319C2semidirect
6. #17986290C2semidirect
7. #17986231C2semidirect
8. C4×#11765852C2semidirect
9. #17986064C2semidirect
10. #106×D71C2semidirect
11. #475×C71C2semidirect
91. D284SD16semidirect
100. C71⋊C8#41semidirect
124. C71#3335semidirect
131. #475D71semidirect