Finite Group Databasebeta

← #22600626

Subgroups — #22600626

Order 9088.

374 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986722C211yesmaximal, characteristic
2#17986664C211yesmaximal, characteristic
3#17986349C211yesmaximal, characteristic, complemented
4#17986338C212yesmaximal, complemented
5#17986285C211yesmaximal, characteristic, complemented
6#17986254C212yesmaximal, complemented
7#17986153C212yesmaximal, complemented
8C4×#11765855C211yesmaximal, characteristic, complemented
9#17986076C211yesmaximal, characteristic
10#17986006C212yesmaximal, complemented
11#462×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2212yes—
13C2×#11765855C2211yescharacteristic
14C2×#11765848C2212yes—
15#14556128C2211yescharacteristic
16#14556127—21no—
17#14556123—22no—
18#14556116—21no—
19#14556106—22no—
20#14556104C2212yes—
21#14556104C2212yes—
22C4×Dic142—21no—
23C4×Dic142C2211yescharacteristic
24C4×D284C2211yescharacteristic
25#14556090—21no—
26#14556086C2212yes—
27#14556061C2211yescharacteristic
28#14556060C2212yes—
29#14556053C2212yes—
30#14556052C2212yes—
31#14556052C2211yescharacteristic
32#14556052C2212yes—
33#14556051C2211yescharacteristic
34(C2×D71)⋊C8C2212yes—
35C4×C71⋊C8C2211yescharacteristic
36C2×Q16×C71C2212yes—
37C4⋊Q8×C71C2211yescharacteristic
38C4×Q8×C71C2211yescharacteristic
39C4⋊C8×C71C2211yescharacteristic
40#103×C71C2212yes—
41C2×Dic142—21no—
42C2×Dic142—22no—
43C2×Dic142C2311yescharacteristic
44Q8×D71—24no—
45Q8×D71—42no—
46C2×D284D411yescharacteristic
47C2×C4×D71—22no—
48C2×C4×D71C2312yes—
49#11765855—22no—
50#11765848—42no—
51C2×C71⋊C8C2311yescharacteristic
52C2×C71⋊C8C2311yescharacteristic
53C4⋊Dic71—21no—
54C4⋊Dic71D411yescharacteristic
55C4⋊Dic71C2312yes—
56C4⋊Dic71C2311yescharacteristic
57Dic71⋊C4—21no—
58Dic71⋊C4—22no—
59Dic71⋊C4—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4D412yes—
64C4×Dic71—21no—
65C4×Dic71—22no—
66C2×Q8×C71C2312yes—
67C2×Q8×C71C2311yescharacteristic
68Q16×C71—42no—
69C2×C8×C71C2312yes—
70C4⋊C4×C71—21no—
71C4⋊C4×C71—21no—
72C4⋊C4×C71—21no—
73C4⋊C4×C71C2311yescharacteristic
74C42×C71—21no—
75C42×C71C2311yescharacteristic
76C22×D71C2×D412yes—
77Q8×C71—21no—
78Q8×C71—22no—
79Q8×C71—21no—
80Q8×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2411yescharacteristic
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71—22no—
90C2×Dic71C2×D412yes—
91D284—22no—
92C4×D71—22no—
93C4×D71—22no—
94C4×D71—44no—
95Dic142—21no—
96Dic142—44no—
97Dic142—42no—
98Dic142#4112yes—
99C8×C71—22no—
100C71⋊C8—21no—
101C71⋊C8—21no—
102C22×C71C22×D411yescharacteristic
103C2×D71—22no—
104C2×D71—22no—
105C2×D71—22no—
106C4×C71—21no—
107C4×C71—42no—
108C4×C71—41no—
109C4×C71—41no—
110C4×C71#13712yes—
111C4×C71C2×#4111yescharacteristic
112C4×C71H(5,𝔽2)11yescharacteristic
113Dic71—22no—
114Dic71—22no—
115Dic71—42no—
116Dic71—41no—
117C2×C71#54611yescharacteristic
118C2×C71C2×#13711yescharacteristic
119C2×C71#57811yescharacteristic
120D71—42no—
121#3328—711nomaximal, Sylow
122C71#332811yescharacteristic, complemented, Sylow
123#556—711no—
124D4×Q8—711no—
125#487—711no—
126#485—712no—
127#475—711no—
128#470—712no—
129#462D7111yescharacteristic, complemented
130#451—712no—
131C4×Q16—711no—
132#436—711no—
133#419—712no—
134C22×Q8—712no—
135C2×Q16—711no—
136C2×Q16C2×D7112yes—
137C2×M16—712no—
138C4⋊Q8—711no—
139C4⋊Q8—1421no—
140C4⋊Q8C2×D7111yescharacteristic
141#126—1422no—
142#124—1421no—
143C22⋊Q8—712no—
144C22⋊Q8—712no—
145C22⋊Q8—1422no—
146C4×Q8—711no—
147C4×Q8—1421no—
148C4×Q8C2×D7111yescharacteristic
149C4×D4—711no—
150C4×D4—1421no—
151#117—712no—
152#107—711no—
153#106—712no—
154C4⋊C8C2×D7111yescharacteristic
155#103—712no—
156#103—711no—
157#103—712no—
158#103—712no—
159#103—711no—
160#103C2×D7112yes—
161C22⋊C8—712no—
162C4×C8—711no—
163C2×Q8—2842no—
164C2×Q8—711no—
165C2×Q8—1422no—
166C2×Q8—1424no—
167C2×Q8—1421no—
168C2×Q8C22×D7112yes—
169C2×Q8C22×D7111yescharacteristic
170C2×D4—711no—
171C22×C4—712no—
172C22×C4—1422no—
173Q16—42no—
174Q16—1422no—
175M16—2842no—
176C2×C8—711no—
177C2×C8—711no—
178C2×C8C22×D7112yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—711no—
183C4⋊C4—712no—
184C4⋊C4—711no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4—1421no—
188C4⋊C4—1421no—
189C4⋊C4C22×D7111yescharacteristic
190C22⋊C4—712no—
191C22⋊C4—1422no—
192C22⋊C4—1422no—
193C22⋊C4—1422no—
194C42—21no—
195C42—1422no—
196C42—1421no—
197C42C22×D7111yescharacteristic
198C23—712no—
199Q8—21no—
200Q8—22no—
201Q8—21no—
202Q8—22no—
203Q8—2844no—
204Q8—2842no—
205Q8—712no—
206Q8—1421no—
207D4—1422no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—2844no—
212C2×C4—712no—
213C2×C4—1421no—
214C2×C4—1422no—
215C2×C4—1421no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4D4×D7111yescharacteristic
219C2×C4D4×D7111yescharacteristic
220C2×C4C23×D7111yescharacteristic
221C8—22no—
222C8—1421no—
223C8—1421no—
224C22—1422no—
225C22—1422no—
226C22—1422no—
227C22C2×D4×D7111yescharacteristic
228C4—21no—
229C4—42no—
230C4—41no—
231C4—41no—
232C4—2842no—
233C4—2841no—
234C4—1422no—
235C4—1422no—
236C4#1455616012yes—
237C4#41×D7111yescharacteristic
238C4#1455617311yescharacteristic
239C2—2842no—
240C2#1798664611yescharacteristic
241C2C2×#1455616011yescharacteristic
242C2#1798681511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#462×C71C2
socleC22×C71C22×D4
hypercenter#462D71
hypocenterC71#3328
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #17986349C2semidirect
4. #17986338C2semidirect
5. #17986285C2semidirect
6. #17986254C2semidirect
7. #17986153C2semidirect
8. C4×#11765855C2semidirect
10. #17986006C2semidirect
11. #462×C71C2semidirect
122. C71#3328semidirect
129. #462D71semidirect