Finite Group Databasebeta

← #22598333

Subgroups — #22598333

Order 9088.

398 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986580C211yesmaximal, characteristic
2#17986479C211yesmaximal, characteristic, complemented
3#17986342C212yesmaximal, complemented
4#17986331C212yesmaximal, complemented
5#17986326C212yesmaximal, complemented
6#17986319C212yesmaximal, complemented
7C2×#14556059C211yesmaximal, characteristic, complemented
8#17986001C211yesmaximal, characteristic, complemented
9#17985993C211yesmaximal, characteristic, complemented
10#17985967C211yesmaximal, characteristic, complemented
11#428×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C2×#11765865C2211yescharacteristic
14C22×C71⋊C8C2211yescharacteristic
15#14556108—21no—
16#14556108C2212yes—
17#14556103—21no—
18#14556103C2212yes—
19#14556102—22no—
20C4×Dic142—21no—
21C4×D284—21no—
22(C22⋊D71)⋊C4—22no—
23#14556086C2212yes—
24C2×C4⋊Dic71C2211yescharacteristic
25C4⋊C4×D71C2212yes—
26#14556059C2212yes—
27#14556059C2212yes—
28#14556053C2212yes—
29#14556051C2211yescharacteristic
30#14556051C2211yescharacteristic
31#14556050C2212yes—
32#14556048C2211yescharacteristic
33#14556048C2211yescharacteristic
34(C2×D71)⋊C8C2212yes—
35(C2×D71)⋊C8C2212yes—
36C2×M16×C71C2211yescharacteristic
37#117×C71C2211yescharacteristic
38C2×C4⋊C4×C71C2211yescharacteristic
39#107×C71C2212yes—
40#106×C71C2212yes—
41#11765865—22no—
42#11765865—22no—
43#11765865—22no—
44#11765865—22no—
45C2×Dic142D411yescharacteristic
46C2×Dic142C2311yescharacteristic
47C2×D284—22no—
48C2×D284D411yescharacteristic
49C2×D284C2311yescharacteristic
50C2×C22⋊D71—21no—
51C2×C22⋊D71D412yes—
52C22×Dic71—21no—
53C2×C4×D71—21no—
54C2×C4×D71—22no—
55C2×C4×D71C2312yes—
56C2×C71⋊C8—22no—
57C2×C71⋊C8C2311yescharacteristic
58C2×C71⋊C8C2311yescharacteristic
59C4⋊Dic71—21no—
60C4⋊Dic71—22no—
61C4⋊Dic71C2312yes—
62Dic71⋊C4—22no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4—22no—
65(C2×D71)⋊C4—22no—
66C4×Dic71—22no—
67C22×C4×C71—21no—
68C22×C4×C71C2311yescharacteristic
69M16×C71—22no—
70C2×C8×C71C2312yes—
71C4⋊C4×C71—21no—
72C4⋊C4×C71C2311yescharacteristic
73C4⋊C4×C71C2312yes—
74C4⋊C4×C71C2311yescharacteristic
75C22⋊C4×C71—21no—
76C42×C71—21no—
77C23×C71C2×D411yescharacteristic
78C22×D71—21no—
79C22×D71C2×D412yes—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71—81no—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71#4112yes—
86C2×C4×C71#4112yes—
87C2×C4×C71C2411yescharacteristic
88C22⋊D71—24no—
89C22⋊D71—42no—
90C2×Dic71—22no—
91C2×Dic71—21no—
92C2×Dic71—81no—
93C2×Dic71C2×D412yes—
94D284—22no—
95D284—21no—
96D284—21no—
97D284—82no—
98C4×D71—22no—
99C4×D71—22no—
100C4×D71—42no—
101C4×D71—82no—
102Dic142—22no—
103Dic142—21no—
104Dic142—21no—
105C8×C71—22no—
106C71⋊C8—21no—
107C71⋊C8—21no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71#12312yes—
111C22×C71C22×D411yescharacteristic
112C2×D71—22no—
113C2×D71—22no—
114C2×D71—22no—
115C2×D71—41no—
116C2×D71—41no—
117C2×D71—41no—
118C4×C71—41no—
119C4×C71—42no—
120C4×C71—41no—
121C4×C71#12312yes—
122C4×C71C2×#4111yescharacteristic
123C4×C71C2×#4111yescharacteristic
124Dic71—22no—
125Dic71—41no—
126Dic71—42no—
127C2×C71—22no—
128C2×C71C2×#12311yescharacteristic
129C2×C71#57711yescharacteristic
130C2×C71#57711yescharacteristic
131D71—42no—
132D71—81no—
133#3099—711nomaximal, Sylow
134C71#309911yescharacteristic, complemented, Sylow
135#537—711no—
136#520—711no—
137#485—712no—
138#483—712no—
139#482—712no—
140#481—712no—
141#428D7111yescharacteristic, complemented
142C2×#106—711no—
143#417—711no—
144#416—711no—
145#408—711no—
146C2×#41—711no—
147C2×#41—711no—
148C2×M16C2×D7111yescharacteristic
149C22×C8—711no—
150C22⋊Q8—712no—
151C22⋊Q8—1421no—
152#121—712no—
153#121—1421no—
154#121—1422no—
155C4×Q8—1421no—
156C4×D4—1421no—
157C4×D4—1422no—
158#117—712no—
159#117C2×D7111yescharacteristic
160C2×C4⋊C4—711no—
161C2×C4⋊C4—712no—
162C2×C4⋊C4C2×D7111yescharacteristic
163#107C2×D7112yes—
164#106—712no—
165#106—712no—
166#106C2×D7112yes—
167#103—712no—
168#103—711no—
169#103—711no—
170#102—712no—
171#102—711no—
172#102—711no—
173C22⋊C8—712no—
174C22⋊C8—712no—
175#41—1422no—
176#41—1422no—
177#41—1422no—
178#41—1422no—
179C2×Q8—711no—
180C2×Q8—711no—
181C2×D4—712no—
182C2×D4—711no—
183C2×D4—711no—
184C2×D4—1422no—
185C2×D4—1421no—
186C22×C4—21no—
187C22×C4—712no—
188C22×C4—1422no—
189C22×C4—1421no—
190C22×C4—1421no—
191C22×C4C22×D7111yescharacteristic
192M16—22no—
193C2×C8—711no—
194C2×C8—711no—
195C2×C8—1422no—
196C2×C8C22×D7112yes—
197C4⋊C4—21no—
198C4⋊C4—712no—
199C4⋊C4—1422no—
200C4⋊C4—1422no—
201C4⋊C4—1421no—
202C4⋊C4C22×D7111yescharacteristic
203C4⋊C4C22×D7111yescharacteristic
204C4⋊C4C22×D7112yes—
205C22⋊C4—21no—
206C22⋊C4—1422no—
207C22⋊C4—1422no—
208C22⋊C4—1422no—
209C42—21no—
210C42—1422no—
211C23—712no—
212C23—1421no—
213C23D4×D7111yescharacteristic
214Q8—1421no—
215Q8—1421no—
216Q8—1422no—
217D4—2842no—
218D4—5682no—
219D4—1424no—
220D4—1421no—
221D4—1421no—
222D4—1422no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—81no—
227C2×C4—2842no—
228C2×C4—5682no—
229C2×C4—5681no—
230C2×C4—712no—
231C2×C4—1422no—
232C2×C4—1421no—
233C2×C4—1422no—
234C2×C4—1422no—
235C2×C4D4×D7111yescharacteristic
236C2×C4#1176586512yes—
237C2×C4#1176586612yes—
238C2×C4C23×D7111yescharacteristic
239C8—22no—
240C8—1421no—
241C8—1421no—
242C22—21no—
243C22—21no—
244C22—2841no—
245C22—2841no—
246C22—2841no—
247C22—1422no—
248C22—1422no—
249C22—1422no—
250C22#1455611312yes—
251C22C2×D4×D7111yescharacteristic
252C4—41no—
253C4—42no—
254C4—41no—
255C4—2841no—
256C4—2842no—
257C4—1422no—
258C4#1455611312yes—
259C4C2×#1176586511yescharacteristic
260C4C2×#1176586611yescharacteristic
261C2—22no—
262C2—2842no—
263C2—5681no—
264C2C2×#1455611311yescharacteristic
265C2#1798680811yescharacteristic
266C2#1798680911yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#428×C71C2
socleC22×C71C22×D4
hypercenter#428D71
hypocenterC71#3099
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #17986479C2semidirect
3. #17986342C2semidirect
4. #17986331C2semidirect
5. #17986326C2semidirect
6. #17986319C2semidirect
7. C2×#14556059C2semidirect
8. #17986001C2semidirect
9. #17985993C2semidirect
10. #17985967C2semidirect
11. #428×C71C2semidirect
134. C71#3099semidirect
141. #428D71semidirect