Finite Group Databasebeta

← #22598472

Subgroups — #22598472

Order 9088.

380 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986554C211yesmaximal, characteristic
2C4×Q8×D71C211yesmaximal, characteristic
3#17986331C212yesmaximal
4#17986281C211yesmaximal, characteristic, complemented
5#17986210C211yesmaximal, characteristic, complemented
6#17986209C211yesmaximal, characteristic, complemented
7#17986156C212yesmaximal
8#17986105C212yesmaximal, complemented
9#17986103C212yesmaximal, complemented
10#17986055C211yesmaximal, characteristic
11#475×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71C2211yescharacteristic
13C2×Dic284C2212yes—
14C2×#11765855C2212yes—
15#14556128C2211yescharacteristic
16#14556119—21no—
17#14556108C2212yes—
18#14556104—21no—
19Q8×Dic71—21no—
20Q8×Dic71C2211yescharacteristic
21C4×Dic142—21no—
22C4×Dic142C2211yescharacteristic
23#14556093—21no—
24#14556093C2212yes—
25#14556093C2211yescharacteristic
26#14556086—21no—
27C4⋊C4×D71—21no—
28C4⋊C4×D71C2211yescharacteristic
29C42×D71—21no—
30C42×D71C2211yescharacteristic
31#14556059C2212yes—
32C4⋊(C71⋊C8)C2211yescharacteristic
33#14556053C2212yes—
34#14556051C2212yes—
35#14556051C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556038C2212yes—
38C4⋊Q8×C71C2211yescharacteristic
39C4×Q8×C71C2211yescharacteristic
40#106×C71C2212yes—
41C4⋊C8×C71C2211yescharacteristic
42#103×C71C2212yes—
43C2×Dic142—21no—
44C2×Dic142C2312yes—
45C2×Dic142C2311yescharacteristic
46Q8×D71—22no—
47Q8×D71—22no—
48C2×C4×D71—21no—
49C2×C4×D71—21no—
50C2×C4×D71D411yescharacteristic
51C2×C4×D71D411yescharacteristic
52C2×C4×D71C2311yescharacteristic
53Dic284—24no—
54#11765855—24no—
55C2×C71⋊C8C2312yes—
56C4⋊Dic71—21no—
57C4⋊Dic71—21no—
58C4⋊Dic71—21no—
59C4⋊Dic71—22no—
60C4⋊Dic71C2311yescharacteristic
61Dic71⋊C4—21no—
62Dic71⋊C4—21no—
63Dic71⋊C4—21no—
64Dic71⋊C4—22no—
65(C2×D71)⋊C4—22no—
66C4×Dic71—21no—
67C4×Dic71—21no—
68C4×Dic71—21no—
69C4×Dic71—22no—
70C4×Dic71D411yescharacteristic
71C4×Dic71D411yescharacteristic
72C4×Dic71C2311yescharacteristic
73C2×Q8×C71—21no—
74C2×Q8×C71C2311yescharacteristic
75C2×C8×C71C2312yes—
76C4⋊C4×C71—21no—
77C4⋊C4×C71—21no—
78C4⋊C4×C71C2312yes—
79C4⋊C4×C71C2311yescharacteristic
80C42×C71—21no—
81C42×C71C2311yescharacteristic
82C22×D71C2×D411yescharacteristic
83Q8×C71—21no—
84Q8×C71—42no—
85Q8×C71#4112yes—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71C2×D411yescharacteristic
91C2×C4×C71C2411yescharacteristic
92C2×Dic71—21no—
93C2×Dic71—21no—
94C2×Dic71—22no—
95C2×Dic71C2×D411yescharacteristic
96C2×Dic71C2×D411yescharacteristic
97C2×Dic71C2×D411yescharacteristic
98C4×D71—24no—
99C4×D71—22no—
100C4×D71—22no—
101C4×D71—21no—
102C4×D71—21no—
103C4×D71—42no—
104Dic142—24no—
105Dic142—21no—
106Dic142—22no—
107Dic142—22no—
108Dic142#4112yes—
109C8×C71—22no—
110C71⋊C8—22no—
111C22×C71C22×D411yescharacteristic
112C2×D71—21no—
113C2×D71—21no—
114C2×D71#13712yes—
115C4×C71—22no—
116C4×C71—21no—
117C4×C71—41no—
118C4×C71—42no—
119C4×C71#13712yes—
120C4×C71C2×#4111yescharacteristic
121C4×C71C2×#4111yescharacteristic
122Dic71—21no—
123Dic71—22no—
124Dic71—21no—
125Dic71—21no—
126Dic71—42no—
127Dic71—41no—
128C2×C71#52511yescharacteristic
129C2×C71C2×#13711yescharacteristic
130C2×C71C2×#13711yescharacteristic
131D71—22no—
132#3113—711nomaximal, Sylow
133C71#311311yescharacteristic, complemented, Sylow
134#533—711no—
135C2×C4×Q8—711no—
136#483—712no—
137#475—711no—
138#475D7111yescharacteristic, complemented
139#462—711no—
140#462—711no—
141#451—712no—
142#441—712no—
143#441—712no—
144#432—711no—
145C22×Q8—711no—
146C2×Q16—712no—
147C2×Q16—712no—
148C4⋊Q8—711no—
149C4⋊Q8C2×D7111yescharacteristic
150#125—1421no—
151C22⋊Q8—712no—
152C22⋊Q8—1421no—
153C4×Q8—711no—
154C4×Q8—711no—
155C4×Q8—712no—
156C4×Q8—711no—
157C4×Q8—1421no—
158C4×Q8—1421no—
159C4×Q8—1421no—
160C4×Q8C2×D7111yescharacteristic
161#117—1421no—
162C2×C4⋊C4—711no—
163C2×C4⋊C4—1421no—
164C2×C42—711no—
165C2×C42—1421no—
166#106—712no—
167#106C2×D7112yes—
168C4⋊C8—711no—
169C4⋊C8C2×D7111yescharacteristic
170#103—712no—
171#103—712no—
172#103—712no—
173#103C2×D7112yes—
174C22⋊C8—712no—
175#97—712no—
176C2×Q8—21no—
177C2×Q8—711no—
178C2×Q8—712no—
179C2×Q8—1421no—
180C2×Q8—1422no—
181C2×Q8—1422no—
182C2×Q8C22×D7111yescharacteristic
183C22×C4—711no—
184C22×C4—711no—
185C22×C4—711no—
186C22×C4—1421no—
187C22×C4—1421no—
188Q16—1424no—
189Q16—1424no—
190C2×C8—712no—
191C2×C8C22×D7112yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—711no—
195C4⋊C4—1421no—
196C4⋊C4—1421no—
197C4⋊C4—1422no—
198C4⋊C4—1421no—
199C4⋊C4—1422no—
200C4⋊C4—1421no—
201C4⋊C4—1421no—
202C4⋊C4—1421no—
203C4⋊C4C22×D7111yescharacteristic
204C4⋊C4C22×D7112yes—
205C22⋊C4—1422no—
206C42—21no—
207C42—711no—
208C42—711no—
209C42—711no—
210C42—1421no—
211C42—1421no—
212C42—1422no—
213C42—1421no—
214C42C22×D7111yescharacteristic
215C23—711no—
216Q8—21no—
217Q8—42no—
218Q8—712no—
219Q8—1424no—
220Q8—1422no—
221Q8—1422no—
222Q8—1421no—
223Q8#1176586612yes—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—2842no—
228C2×C4—711no—
229C2×C4—711no—
230C2×C4—711no—
231C2×C4—1422no—
232C2×C4—1424no—
233C2×C4—1421no—
234C2×C4—1422no—
235C2×C4—1421no—
236C2×C4—1421no—
237C2×C4—1421no—
238C2×C4—1422no—
239C2×C4D4×D7111yescharacteristic
240C2×C4D4×D7111yescharacteristic
241C2×C4C23×D7111yescharacteristic
242C8—22no—
243C8—1422no—
244C22—712no—
245C22—1421no—
246C22—1421no—
247C22C2×D4×D7111yescharacteristic
248C4—22no—
249C4—21no—
250C4—41no—
251C4—42no—
252C4—2842no—
253C4—2841no—
254C4—1421no—
255C4—1422no—
256C4—1421no—
257C4—1421no—
258C4#1455615712yes—
259C4#41×D7111yescharacteristic
260C4C2×#1176586611yescharacteristic
261C2—1422no—
262C2#1798651211yescharacteristic
263C2#137×D7111yescharacteristic
264C2C2×#1455615711yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#475×C71C2
socleC22×C71C22×D4
hypercenter#475D71
hypocenterC71#3113
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
4. #17986281C2semidirect
5. #17986210C2semidirect
6. #17986209C2semidirect
8. #17986105C2semidirect
9. #17986103C2semidirect
11. #475×C71C2semidirect
133. C71#3113semidirect
138. #475D71semidirect