Finite Group Databasebeta

← #22598325

Subgroups — #22598325

Order 9088.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#17986482C211yesmaximal, characteristic, complemented
2#17986337C212yesmaximal, complemented
3#17986327C212yesmaximal, complemented
4#17986320C212yesmaximal, complemented
5#17986315C212yesmaximal, complemented
6#17986039C211yesmaximal, characteristic, complemented
7C2×#14556059C211yesmaximal, characteristic, complemented
8#17986000C211yesmaximal, characteristic, complemented
9#17985995C211yesmaximal, characteristic, complemented
10#17985966C211yesmaximal, characteristic, complemented
11#537×C71C211yesmaximal, characteristic
12C2×#11765849C2211yescharacteristic
13C22×C71⋊C8C2211yescharacteristic
14Q8×Dic71—21no—
15D4×Dic71—22no—
16D4×Dic71—21no—
17#14556087C2212yes—
18#14556087C2211yescharacteristic
19C2×C4⋊Dic71C2212yes—
20C2×C4⋊Dic71C2211yescharacteristic
21#14556061C2212yes—
22#14556059C2212yes—
23#14556059C2212yes—
24#14556059C2212yes—
25#14556052C2211yescharacteristic
26#14556052C2212yes—
27#14556052C2211yescharacteristic
28#14556049C2211yescharacteristic
29#14556049C2212yes—
30#14556049C2211yescharacteristic
31C22⋊(C71⋊C8)C2212yes—
32C22⋊(C71⋊C8)C2212yes—
33C2×#41×C71C2211yescharacteristic
34C2×#41×C71C2211yescharacteristic
35C22⋊Q8×C71—21no—
36C22⋊Q8×C71C2212yes—
37#121×C71—21no—
38#121×C71—22no—
39#121×C71C2212yes—
40C2×C4⋊C4×C71C2211yescharacteristic
41C22×Dic71—22no—
42C22×Dic71—21no—
43#11765849—22no—
44C2×C71⋊C8—22no—
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2311yescharacteristic
47C2×C71⋊C8C2312yes—
48C4⋊Dic71—22no—
49C4⋊Dic71—21no—
50C4⋊Dic71C2311yescharacteristic
51C4⋊Dic71C2312yes—
52C4⋊Dic71C2311yescharacteristic
53C22⋊Dic71—22no—
54C22⋊Dic71—21no—
55C4×Dic71—22no—
56C4×Dic71—21no—
57#41×C71—22no—
58#41×C71—22no—
59#41×C71—22no—
60#41×C71—22no—
61C2×Q8×C71D411yescharacteristic
62C2×Q8×C71C2311yescharacteristic
63C2×D4×C71—21no—
64C2×D4×C71—22no—
65C2×D4×C71D412yes—
66C2×D4×C71D411yescharacteristic
67C2×D4×C71C2311yescharacteristic
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71C2312yes—
71C22×C4×C71C2311yescharacteristic
72C4⋊C4×C71—21no—
73C4⋊C4×C71—22no—
74C4⋊C4×C71C2312yes—
75C22⋊C4×C71—22no—
76C22⋊C4×C71—22no—
77C23×C71—21no—
78C23×C71C2×D412yes—
79C23×C71C2×D411yescharacteristic
80Q8×C71—22no—
81Q8×C71—21no—
82Q8×C71—21no—
83D4×C71—24no—
84D4×C71—22no—
85D4×C71—21no—
86D4×C71—21no—
87D4×C71—42no—
88D4×C71—82no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—21no—
93C2×C4×C71—42no—
94C2×C4×C71—81no—
95C2×C4×C71C2×D412yes—
96C2×C4×C71C2×D411yescharacteristic
97C2×C4×C71#4112yes—
98C2×C4×C71#4112yes—
99C2×C4×C71C2411yescharacteristic
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—81no—
104C2×Dic71—82no—
105C71⋊C8—21no—
106C71⋊C8—21no—
107C71⋊C8—22no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71—22no—
111C22×C71—22no—
112C22×C71—22no—
113C22×C71—41no—
114C22×C71—41no—
115C22×C71—41no—
116C22×C71#12312yes—
117C22×C71C22×D411yescharacteristic
118C4×C71—22no—
119C4×C71—41no—
120C4×C71—42no—
121C4×C71#12312yes—
122C4×C71C2×#4111yescharacteristic
123C4×C71C2×#4111yescharacteristic
124Dic71—41no—
125Dic71—42no—
126Dic71—41no—
127C2×C71—22no—
128C2×C71—42no—
129C2×C71—81no—
130C2×C71C2×#12311yescharacteristic
131C2×C71#57711yescharacteristic
132C2×C71#57711yescharacteristic
133#3099—711nomaximal, Sylow
134C71#309911yescharacteristic, complemented, Sylow
135#537D7111yescharacteristic
136#520—711no—
137#485—712no—
138#483—712no—
139#482—712no—
140#481—712no—
141#428—711no—
142C2×#106—711no—
143#417—711no—
144#416—711no—
145#408—711no—
146C2×#41C2×D7111yescharacteristic
147C2×#41C2×D7111yescharacteristic
148C2×M16—711no—
149C22×C8—711no—
150C22⋊Q8—21no—
151C22⋊Q8C2×D7112yes—
152#121—21no—
153#121—22no—
154#121C2×D7112yes—
155C4×Q8—1421no—
156C4×D4—1422no—
157C4×D4—1421no—
158#117—711no—
159#117—712no—
160C2×C4⋊C4—711no—
161C2×C4⋊C4—712no—
162C2×C4⋊C4C2×D7111yescharacteristic
163#107—712no—
164#106—712no—
165#106—712no—
166#106—712no—
167#103—711no—
168#103—711no—
169#103—712no—
170#102—711no—
171#102—711no—
172#102—712no—
173C22⋊C8—712no—
174C22⋊C8—712no—
175#41—22no—
176#41—22no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D7111yescharacteristic
180C2×Q8C22×D7111yescharacteristic
181C2×D4—21no—
182C2×D4—22no—
183C2×D4C22⋊D7112yes—
184C2×D4C22⋊D7111yescharacteristic
185C2×D4C22×D7111yescharacteristic
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—1422no—
189C22×C4—1421no—
190C22×C4C22×D7112yes—
191C22×C4C22×D7111yescharacteristic
192M16—1422no—
193C2×C8—711no—
194C2×C8—711no—
195C2×C8—712no—
196C2×C8—1422no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—711no—
200C4⋊C4—712no—
201C4⋊C4—711no—
202C4⋊C4—1422no—
203C4⋊C4—1421no—
204C4⋊C4C22×D7112yes—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—1422no—
208C22⋊C4—1421no—
209C42—1422no—
210C42—1421no—
211C23—21no—
212C23C2×C22⋊D7112yes—
213C23C2×C22⋊D7111yescharacteristic
214Q8—22no—
215Q8—21no—
216Q8—21no—
217D4—24no—
218D4—21no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—82no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—81no—
229C2×C4—5681no—
230C2×C4—5682no—
231C2×C4—1421no—
232C2×C4—1422no—
233C2×C4—1421no—
234C2×C4C2×C22⋊D7112yes—
235C2×C4C2×C22⋊D7111yescharacteristic
236C2×C4#1176586412yes—
237C2×C4#1176586412yes—
238C2×C4C23×D7111yescharacteristic
239C8—1421no—
240C8—1421no—
241C8—1422no—
242C22—21no—
243C22—21no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22#1455610912yes—
251C22C22×C22⋊D7111yescharacteristic
252C4—22no—
253C4—41no—
254C4—42no—
255C4—2842no—
256C4—2841no—
257C4—2841no—
258C4#1455610912yes—
259C4C2×#1176586411yescharacteristic
260C4C2×#1176586411yescharacteristic
261C2—22no—
262C2—42no—
263C2—81no—
264C2C2×#1455610911yescharacteristic
265C2#1798680711yescharacteristic
266C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#537×C71C2
socleC22×C71C22×D4
hypercenter#537D71
hypocenterC71#3099
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #17986482C2semidirect
2. #17986337C2semidirect
3. #17986327C2semidirect
4. #17986320C2semidirect
5. #17986315C2semidirect
6. #17986039C2semidirect
7. C2×#14556059C2semidirect
8. #17986000C2semidirect
9. #17985995C2semidirect
10. #17985966C2semidirect
134. C71#3099semidirect