Finite Group Databasebeta

← #22600289

Subgroups — #22600289

Order 9088.

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

normal subgroups only

This group has 408 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#17986715C211yesmaximal, characteristic, complemented
2#17986669C211yesmaximal, characteristic, complemented
3#17986322C212yesmaximal, complemented
4#17986291C211yesmaximal, characteristic, complemented
5#17986234C212yesmaximal, complemented
6#17986191C211yesmaximal, characteristic, complemented
7#17986145C212yesmaximal, complemented
8C4×#11765854C211yesmaximal, characteristic, complemented
9C8×D284C211yesmaximal, characteristic, complemented
10#17985992C212yesmaximal, complemented
11#486×C71C211yesmaximal, characteristic, complemented
12C2×#11765864C2212yes—
13C2×#11765854C2211yescharacteristic
14C2×#11765850C2212yes—
15C2×C8×D71C2212yes—
16#14556124—21no—
17#14556117—22no—
18#14556103—21no—
19#14556103C2212yes—
20#14556100—22no—
21#14556099C2212yes—
22C4×Dic142C2211yescharacteristic
23C4×D284C2211yescharacteristic
24C4×D284C2211yescharacteristic
25D4×Dic71—21no—
26#14556086—21no—
27#14556086C2212yes—
28#14556061C2212yes—
29#14556060C2211yescharacteristic
30C4⋊(C71⋊C8)C2211yescharacteristic
31#14556053C2211yescharacteristic
32#14556051C2212yes—
33#14556050C2211yescharacteristic
34#14556049C2212yes—
35(C2×D71)⋊C8C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37#127×C71C2211yescharacteristic
38#102×C71C2212yes—
39#102×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41#11765864—24no—
42#11765864—42no—
43C2×Dic142C2311yescharacteristic
44C2×D284—22no—
45C2×D284D411yescharacteristic
46C2×D284C2311yescharacteristic
47C2×C22⋊D71—22no—
48C2×C22⋊D71—22no—
49C22×Dic71—22no—
50C2×C4×D71—21no—
51C2×C4×D71C2312yes—
52#11765854—22no—
53#11765850—42no—
54C2×C71⋊C8C2312yes—
55C8×D71—42no—
56C4⋊Dic71D411yescharacteristic
57C4⋊Dic71C2311yescharacteristic
58C4⋊Dic71C2311yescharacteristic
59Dic71⋊C4—21no—
60C22⋊Dic71—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—21no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4D412yes—
65C4×Dic71—21no—
66C4×Dic71—22no—
67C2×Q8×C71—21no—
68C2×D4×C71—21no—
69C2×D4×C71—21no—
70C2×D4×C71C2312yes—
71C2×C8×C71C2311yescharacteristic
72C2×C8×C71C2311yescharacteristic
73C4⋊C4×C71—21no—
74C4⋊C4×C71C2312yes—
75C42×C71C2311yescharacteristic
76C23×C71—22no—
77C22×D71—21no—
78C22×D71C2×D412yes—
79Q8×C71—81no—
80D4×C71—22no—
81D4×C71—22no—
82D4×C71—42no—
83D4×C71—81no—
84C2×C4×C71—22no—
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2411yescharacteristic
88C22⋊D71—44no—
89C22⋊D71—82no—
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—44no—
94C2×Dic71C2×D412yes—
95D284—22no—
96D284—21no—
97D284—21no—
98D284—82no—
99C4×D71—22no—
100C4×D71—22no—
101C4×D71—42no—
102Dic142—21no—
103Dic142#4112yes—
104C8×C71—21no—
105C8×C71—21no—
106C71⋊C8—22no—
107C22×C71—42no—
108C22×C71—42no—
109C22×C71—42no—
110C22×C71C22×D411yescharacteristic
111C2×D71—22no—
112C2×D71—22no—
113C2×D71—22no—
114C2×D71—41no—
115C2×D71—41no—
116C2×D71—41no—
117C4×C71—21no—
118C4×C71—42no—
119C4×C71#13512yes—
120C4×C71C2×#4111yescharacteristic
121C4×C71H(5,𝔽2)11yescharacteristic
122Dic71—22no—
123Dic71—22no—
124Dic71—41no—
125Dic71—41no—
126C2×C71—82no—
127C2×C71#54611yescharacteristic
128C2×C71C2×#13511yescharacteristic
129C2×C71#57611yescharacteristic
130D71—42no—
131D71—81no—
132#3305—711nomaximal, Sylow
133C71#330511yescharacteristic, complemented, Sylow
134#555—711no—
135#550—711no—
136#486D7111yescharacteristic, complemented
137#482—712no—
138#477—711no—
139#466—712no—
140#459—711no—
141#449—712no—
142C4×SD16—711no—
143C8×D4—711no—
144#416—712no—
145C2×#41—712no—
146C2×SD16—711no—
147C2×D8—712no—
148C22×C8—712no—
149C4⋊Q8C2×D7111yescharacteristic
150#127—1421no—
151#127C2×D7111yescharacteristic
152#124—1422no—
153#121—712no—
154#121—712no—
155#121—1422no—
156#121—1421no—
157C4×Q8—711no—
158C4×D4—711no—
159C4×D4—711no—
160C4×D4—1421no—
161#117—712no—
162#117—1421no—
163#107—712no—
164#106—711no—
165C4⋊C8—711no—
166#103—711no—
167#103—712no—
168#102—712no—
169#102—711no—
170#102C2×D7112yes—
171#102C2×D7112yes—
172C22⋊C8—712no—
173C4×C8C2×D7111yescharacteristic
174#41—2842no—
175#41—1424no—
176C2×Q8—21no—
177C2×Q8—711no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—711no—
181C2×D4—711no—
182C2×D4—1422no—
183C2×D4—1422no—
184C2×D4—1422no—
185C2×D4C22×D7112yes—
186C22×C4—712no—
187C22×C4—1422no—
188C22×C4—1421no—
189SD16—1422no—
190D8—2842no—
191C2×C8—2842no—
192C2×C8—712no—
193C2×C8C22×D7111yescharacteristic
194C2×C8C22×D7111yescharacteristic
195C4⋊C4—21no—
196C4⋊C4—711no—
197C4⋊C4—711no—
198C4⋊C4—711no—
199C4⋊C4—1421no—
200C4⋊C4C22×D7112yes—
201C22⋊C4—712no—
202C22⋊C4—1422no—
203C22⋊C4—1422no—
204C22⋊C4—1422no—
205C22⋊C4—1421no—
206C42—1422no—
207C42—1421no—
208C42C22×D7111yescharacteristic
209C23—22no—
210C23—712no—
211C23—1421no—
212Q8—81no—
213Q8—712no—
214Q8—1421no—
215D4—22no—
216D4—22no—
217D4—42no—
218D4—81no—
219D4—2844no—
220D4—5682no—
221D4—5682no—
222D4—1421no—
223D4—1421no—
224D4—1422no—
225C2×C4—22no—
226C2×C4—2842no—
227C2×C4—2844no—
228C2×C4—712no—
229C2×C4—1421no—
230C2×C4—1421no—
231C2×C4—1421no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4D4×D7111yescharacteristic
235C2×C4D4×D7111yescharacteristic
236C2×C4C23×D7111yescharacteristic
237C8—21no—
238C8—21no—
239C8—1422no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—2841no—
244C22—2841no—
245C22—2841no—
246C22—1422no—
247C22—1422no—
248C22—1422no—
249C22C2×D4×D7111yescharacteristic
250C4—21no—
251C4—42no—
252C4—2841no—
253C4—2841no—
254C4—1422no—
255C4—1422no—
256C4#1455614912yes—
257C4#41×D7111yescharacteristic
258C4#1455617411yescharacteristic
259C2—82no—
260C2—2842no—
261C2—5681no—
262C2#1798664211yescharacteristic
263C2C2×#1455614911yescharacteristic
264C2#1798680011yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#486×C71C2
socleC22×C71C22×D4
hypercenter#486D71
hypocenterC71#3305
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #17986715C2semidirect
2. #17986669C2semidirect
3. #17986322C2semidirect
4. #17986291C2semidirect
5. #17986234C2semidirect
6. #17986191C2semidirect
7. #17986145C2semidirect
8. C4×#11765854C2semidirect
9. C8×D284C2semidirect
10. #17985992C2semidirect
11. #486×C71C2semidirect
133. C71#3305semidirect
136. #486D71semidirect