Finite Group Databasebeta

← #22598299

Subgroups — #22598299

Order 9088.

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

normal subgroups only

This group has 418 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#17986563C211yesmaximal, characteristic
2#17986482C211yesmaximal, characteristic, complemented
3#17986324C212yesmaximal, complemented
4#17986324C212yesmaximal, complemented
5#17986318C212yesmaximal, complemented
6#17986313C212yesmaximal, complemented
7#17986041C211yesmaximal, characteristic, complemented
8#17986037C211yesmaximal, characteristic, complemented
9#17986005C211yesmaximal, characteristic, complemented
10C2×#14556050C211yesmaximal, characteristic, complemented
11#408×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13#14556124—21no—
14#14556117—21no—
15#14556102C2212yes—
16#14556102C2212yes—
17#14556096—22no—
18Q8×Dic71—21no—
19D4×Dic71—22no—
20D4×Dic71—21no—
21#14556087C2211yescharacteristic
22#14556087C2211yescharacteristic
23#14556087C2212yescomplemented
24C2×C4⋊Dic71C2211yescharacteristic
25C2×C4⋊Dic71C2212yescomplemented
26#14556062C2211yescharacteristic, complemented
27#14556062C2212yescomplemented
28#14556062C2211yescharacteristic, complemented
29#14556060C2211yescharacteristic, complemented
30#14556060C2212yescomplemented
31#14556060C2211yescharacteristic, complemented
32#14556050C2212yes—
33#14556050C2212yes—
34#14556050C2212yes—
35#14556050C2212yes—
36C2×#41×C71C2211yescharacteristic
37C2×M16×C71C2211yescharacteristic
38C22×C8×C71C2211yescharacteristic
39C22⋊C8×C71C2212yescomplemented
40C22⋊C8×C71C2212yescomplemented
41C23×D71—21no—
42C2×D284—22no—
43C2×D284—22no—
44C2×D284—21no—
45C2×D284C2312yes—
46C2×C22⋊D71—22no—
47C2×C22⋊D71—22no—
48C22×Dic71—21no—
49C22×Dic71—22no—
50C4⋊Dic71—21no—
51C4⋊Dic71—22no—
52C4⋊Dic71C2312yes—
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2311yescharacteristic
55C4⋊Dic71C2311yescharacteristic
56C4⋊Dic71C2311yescharacteristic
57C22⋊Dic71—21no—
58C22⋊Dic71—21no—
59C22⋊Dic71—22no—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—22no—
62C4×Dic71—21no—
63C4×Dic71—21no—
64C4×Dic71—22no—
65#41×C71—22no—
66#41×C71—22no—
67C2×Q8×C71D411yescharacteristic
68C2×D4×C71D412yes—
69C2×D4×C71D411yescharacteristic
70C22×C4×C71C2312yes—
71C22×C4×C71C2311yescharacteristic
72M16×C71—22no—
73C2×C8×C71—22no—
74C2×C8×C71C2311yescharacteristic
75C2×C8×C71C2311yescharacteristic
76C2×C8×C71C2312yes—
77C23×C71C2×D412yes—
78C23×C71C2×D411yescharacteristic
79C22×D71—22no—
80C22×D71—42no—
81C22×D71—81no—
82C22×D71—81no—
83Q8×C71—22no—
84D4×C71—24no—
85D4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71C2×D412yes—
89C2×C4×C71C2×D411yescharacteristic
90C2×C4×C71#4112yes—
91C2×C4×C71#4112yes—
92C2×C4×C71C2411yescharacteristic
93C22⋊D71—82no—
94C22⋊D71—82no—
95C2×Dic71—21no—
96C2×Dic71—21no—
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71—22no—
100C2×Dic71—81no—
101C2×Dic71—82no—
102D284—22no—
103D284—22no—
104D284—42no—
105C8×C71—21no—
106C8×C71—21no—
107C8×C71—22no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71—22no—
111C22×C71—22no—
112C22×C71—22no—
113C22×C71#12312yes—
114C22×C71C22×D411yescharacteristic
115C2×D71—42no—
116C2×D71—42no—
117C2×D71—42no—
118C2×D71—84no—
119C4×C71—22no—
120C4×C71#12312yes—
121C4×C71C2×#4111yescharacteristic
122C4×C71C2×#4111yescharacteristic
123Dic71—41no—
124Dic71—42no—
125Dic71—41no—
126Dic71—41no—
127Dic71—41no—
128C2×C71—22no—
129C2×C71—42no—
130C2×C71C2×#12311yescharacteristic
131C2×C71#57611yescharacteristic
132C2×C71#57711yescharacteristic
133D71—82no—
134#3096—711nomaximal, Sylow
135C71#309611yescharacteristic, complemented, Sylow
136#535—711no—
137#520—711no—
138#482—712no—
139#482—712no—
140#481—712no—
141#480—712no—
142#428—711no—
143#427—711no—
144#418—711no—
145C2×#102—711no—
146#408D7111yescharacteristic, complemented
147C2×#41C2×D7111yescharacteristic
148C22×D4—711no—
149C2×M16C2×D7111yescharacteristic
150C22×C8C2×D7111yescharacteristic
151#127—1421no—
152#124—1421no—
153#121—712no—
154#121—712no—
155#120—1422no—
156C4×Q8—1421no—
157C4×D4—1422no—
158C4×D4—1421no—
159#117—711no—
160#117—712no—
161#117—711no—
162C2×C4⋊C4—711no—
163C2×C4⋊C4—712no—
164#107—711no—
165#107—711no—
166#107—712no—
167#106—711no—
168#106—711no—
169#106—712no—
170#102—712no—
171#102—712no—
172#102—712no—
173#102—712no—
174C22⋊C8C2×D7112yescomplemented
175C22⋊C8C2×D7112yescomplemented
176C24—1421no—
177#41—22no—
178#41—22no—
179C2×Q8D28411yescharacteristic
180C2×D4—712no—
181C2×D4—1422no—
182C2×D4—1422no—
183C2×D4—1421no—
184C2×D4—1422no—
185C2×D4—1422no—
186C2×D4D28412yes—
187C2×D4D28411yescharacteristic
188C22×C4—1422no—
189C22×C4—1421no—
190C22×C4C22×D7112yes—
191C22×C4C22×D7111yescharacteristic
192M16—22no—
193C2×C8—22no—
194C2×C8C22×D7111yescharacteristic
195C2×C8C22×D7112yes—
196C2×C8C22×D7111yescharacteristic
197C4⋊C4—712no—
198C4⋊C4—711no—
199C4⋊C4—711no—
200C4⋊C4—711no—
201C4⋊C4—711no—
202C4⋊C4—1422no—
203C4⋊C4—1421no—
204C22⋊C4—1422no—
205C22⋊C4—1421no—
206C22⋊C4—1421no—
207C22⋊C4—1422no—
208C22⋊C4—1422no—
209C42—1422no—
210C42—1421no—
211C42—1421no—
212C23—2842no—
213C23—5681no—
214C23—5681no—
215C23—1422no—
216C23C2×D28412yes—
217C23C2×D28411yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—2842no—
222D4—5682no—
223D4—5682no—
224D4—1422no—
225D4—1422no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—5681no—
229C2×C4—5682no—
230C2×C4—1421no—
231C2×C4—1421no—
232C2×C4—1421no—
233C2×C4—1422no—
234C2×C4—1421no—
235C2×C4C2×D28412yes—
236C2×C4C2×D28411yescharacteristic
237C2×C4#1176586412yes—
238C2×C4#1176586412yes—
239C2×C4C23×D7111yescharacteristic
240C8—21no—
241C8—21no—
242C8—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—2842no—
249C22—2842no—
250C22—2842no—
251C22—5684no—
252C22#1455611212yes—
253C22C22×D28411yescharacteristic
254C4—22no—
255C4—2842no—
256C4—2841no—
257C4—2841no—
258C4—2841no—
259C4—2841no—
260C4#1455611212yes—
261C4C2×#1176586411yescharacteristic
262C4C2×#1176586411yescharacteristic
263C2—22no—
264C2—42no—
265C2—5682no—
266C2C2×#1455611211yescharacteristic
267C2#1798680411yescharacteristic
268C2#1798681111yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#408×C71C2
socleC22×C71C22×D4
hypercenter#408D71
hypocenterC71#3096
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #17986482C2semidirect
3. #17986324C2semidirect
4. #17986324C2semidirect
5. #17986318C2semidirect
6. #17986313C2semidirect
7. #17986041C2semidirect
8. #17986037C2semidirect
9. #17986005C2semidirect
10. C2×#14556050C2semidirect
11. #408×C71C2semidirect
23. #14556087C22semidirect
25. C2×C4⋊Dic71C22semidirect
26. #14556062C22semidirect
27. #14556062C22semidirect
28. #14556062C22semidirect
29. #14556060C22semidirect
30. #14556060C22semidirect
31. #14556060C22semidirect
39. C22⋊C8×C71C22semidirect
40. C22⋊C8×C71C22semidirect
135. C71#3096semidirect
146. #408D71semidirect
174. C22⋊C8C2×D71semidirect
175. C22⋊C8C2×D71semidirect