Finite Group Databasebeta

← #22598294

Subgroups — #22598294

Order 9088.

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

normal subgroups only

This group has 418 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#17986566C211yesmaximal, characteristic
2#17986480C211yesmaximal, characteristic, complemented
3#17986322C212yesmaximal, complemented
4#17986321C212yesmaximal, complemented
5#17986316C212yesmaximal, complemented
6#17986311C212yesmaximal, complemented
7#17986039C211yesmaximal, characteristic, complemented
8#17986037C211yesmaximal, characteristic, complemented
9#17986004C211yesmaximal, characteristic, complemented
10#17985965C211yesmaximal, characteristic, complemented
11C2×#102×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C2×#11765849C2211yescharacteristic
14#14556124—21no—
15#14556115—21no—
16#14556099C2212yes—
17#14556099C2212yes—
18#14556097—22no—
19#14556093—21no—
20#14556090—21no—
21C4×C22⋊D71—22no—
22#14556087C2211yescharacteristic
23#14556087C2211yescharacteristic
24#14556086C2212yescomplemented
25C4⋊C4×D71C2212yescomplemented
26#14556062C2211yescharacteristic, complemented
27#14556062C2211yescharacteristic, complemented
28#14556061C2212yescomplemented
29#14556060C2211yescharacteristic, complemented
30#14556060C2211yescharacteristic, complemented
31#14556059C2212yescomplemented
32#14556049C2212yes—
33#14556049C2212yes—
34(C2×D71)⋊C8C2212yescomplemented
35(C2×D71)⋊C8C2212yescomplemented
36C22×D4×C71C2211yescharacteristic
37C22×C8×C71C2211yescharacteristic
38C2×C4⋊C4×C71C2211yescharacteristic
39#102×C71C2212yes—
40#102×C71C2212yes—
41#11765865—22no—
42#11765865—22no—
43C2×Dic142D411yescharacteristic
44C2×D284D411yescharacteristic
45C2×C22⋊D71—22no—
46C2×C22⋊D71—22no—
47C2×C22⋊D71D412yes—
48C2×C4×D71—22no—
49C2×C4×D71C2312yes—
50#11765849—22no—
51C2×C71⋊C8C2312yes—
52C4⋊Dic71C2311yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2311yescharacteristic
55C4⋊Dic71C2311yescharacteristic
56Dic71⋊C4—22no—
57C22⋊Dic71—21no—
58C22⋊Dic71—21no—
59C22⋊Dic71—22no—
60C22⋊Dic71—22no—
61(C2×D71)⋊C4—22no—
62C4×Dic71—21no—
63C4×Dic71—21no—
64C4×Dic71—22no—
65C24×C71—21no—
66C2×D4×C71—21no—
67C2×D4×C71—22no—
68C2×D4×C71—22no—
69C2×D4×C71C2312yes—
70C22×C4×C71—21no—
71C22×C4×C71C2311yescharacteristic
72C2×C8×C71—22no—
73C2×C8×C71C2311yescharacteristic
74C2×C8×C71C2311yescharacteristic
75C4⋊C4×C71—21no—
76C4⋊C4×C71C2312yes—
77C23×C71—22no—
78C23×C71—42no—
79C23×C71—81no—
80C23×C71—81no—
81C23×C71C2×D411yescharacteristic
82C22×D71C2×D412yes—
83D4×C71—22no—
84D4×C71—22no—
85D4×C71—42no—
86C2×C4×C71—22no—
87C2×C4×C71—81no—
88C2×C4×C71C2×D411yescharacteristic
89C2×C4×C71#4112yes—
90C2×C4×C71#4112yes—
91C2×C4×C71C2411yescharacteristic
92C22⋊D71—24no—
93C22⋊D71—82no—
94C22⋊D71—82no—
95C2×Dic71—21no—
96C2×Dic71—21no—
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71C2×D412yes—
100D284—22no—
101C4×D71—22no—
102C4×D71—22no—
103C4×D71—82no—
104Dic142—22no—
105C8×C71—21no—
106C8×C71—21no—
107C71⋊C8—22no—
108C22×C71—21no—
109C22×C71—21no—
110C22×C71—42no—
111C22×C71—42no—
112C22×C71—42no—
113C22×C71—84no—
114C22×C71#12312yes—
115C22×C71C22×D411yescharacteristic
116C2×D71—22no—
117C2×D71—22no—
118C2×D71—22no—
119C4×C71—42no—
120C4×C71#12312yes—
121C4×C71C2×#4111yescharacteristic
122C4×C71C2×#4111yescharacteristic
123Dic71—22no—
124Dic71—41no—
125Dic71—41no—
126Dic71—41no—
127Dic71—41no—
128C2×C71—22no—
129C2×C71—82no—
130C2×C71C2×#12311yescharacteristic
131C2×C71#57611yescharacteristic
132C2×C71#57711yescharacteristic
133D71—42no—
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×#102D7111yescharacteristic, complemented
146#408—711no—
147C2×#41—711no—
148C22×D4C2×D7111yescharacteristic
149C2×M16—711no—
150C22×C8C2×D7111yescharacteristic
151#127—1421no—
152#124—1421no—
153#121—712no—
154#121—712no—
155#120—1422no—
156C4×Q8—1421no—
157C4×D4—1421no—
158C4×D4—1422no—
159#117—711no—
160#117—711no—
161#117—712no—
162C2×C4⋊C4—712no—
163C2×C4⋊C4C2×D7111yescharacteristic
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#102C2×D7112yes—
173#102C2×D7112yes—
174C22⋊C8—712no—
175C22⋊C8—712no—
176C24—21no—
177#41—1422no—
178#41—1422no—
179C2×Q8—711no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—712no—
184C2×D4—711no—
185C2×D4—1422no—
186C2×D4—1422no—
187C2×D4C22×D7112yes—
188C22×C4—21no—
189C22×C4—712no—
190C22×C4—1422no—
191C22×C4C22×D7111yescharacteristic
192M16—1422no—
193C2×C8—22no—
194C2×C8—712no—
195C2×C8C22×D7111yescharacteristic
196C2×C8C22×D7111yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—711no—
199C4⋊C4—711no—
200C4⋊C4—711no—
201C4⋊C4—711no—
202C4⋊C4—1422no—
203C4⋊C4C22×D7112yes—
204C22⋊C4—1422no—
205C22⋊C4—1422no—
206C22⋊C4—1422no—
207C22⋊C4—1421no—
208C22⋊C4—1421no—
209C42—1422no—
210C42—1421no—
211C42—1421no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—712no—
217C23D4×D7111yescharacteristic
218Q8—1422no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—5682no—
223D4—5682no—
224D4—1424no—
225D4—1422no—
226C2×C4—22no—
227C2×C4—81no—
228C2×C4—5682no—
229C2×C4—712no—
230C2×C4—1421no—
231C2×C4—1421no—
232C2×C4—1421no—
233C2×C4—1421no—
234C2×C4—1422no—
235C2×C4—1422no—
236C2×C4D4×D7111yescharacteristic
237C2×C4#1176586412yes—
238C2×C4#1176586512yes—
239C2×C4C23×D7111yescharacteristic
240C8—21no—
241C8—21no—
242C8—1422no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—1422no—
250C22—1422no—
251C22—1422no—
252C22#1455611012yes—
253C22C2×D4×D7111yescharacteristic
254C4—42no—
255C4—2841no—
256C4—2841no—
257C4—2841no—
258C4—2841no—
259C4—1422no—
260C4#1455611012yes—
261C4C2×#1176586411yescharacteristic
262C4C2×#1176586511yescharacteristic
263C2—22no—
264C2—82no—
265C2—2842no—
266C2C2×#1455611011yescharacteristic
267C2#1798680011yescharacteristic
268C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#102×C71C2
socleC22×C71C22×D4
hypercenterC2×#102D71
hypocenterC71#3096
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. #17986480C2semidirect
3. #17986322C2semidirect
4. #17986321C2semidirect
5. #17986316C2semidirect
6. #17986311C2semidirect
7. #17986039C2semidirect
8. #17986037C2semidirect
9. #17986004C2semidirect
10. #17985965C2semidirect
11. C2×#102×C71C2semidirect
24. #14556086C22semidirect
25. C4⋊C4×D71C22semidirect
26. #14556062C22semidirect
27. #14556062C22semidirect
28. #14556061C22semidirect
29. #14556060C22semidirect
30. #14556060C22semidirect
31. #14556059C22semidirect
34. (C2×D71)⋊C8C22semidirect
35. (C2×D71)⋊C8C22semidirect
135. C71#3096semidirect
145. C2×#102D71semidirect