Finite Group Databasebeta

← #22598300

Subgroups — #22598300

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#17986568C211yesmaximal, characteristic
2#17986480C211yesmaximal, characteristic, complemented
3#17986326C212yesmaximal, complemented
4#17986325C212yesmaximal, complemented
5#17986319C212yesmaximal, complemented
6#17986314C212yesmaximal, complemented
7#17986035C211yesmaximal, characteristic, complemented
8#17986005C211yesmaximal, characteristic, complemented
9C2×#14556048C211yesmaximal, characteristic, complemented
10#17985967C211yesmaximal, characteristic, complemented
11#428×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C22×D284C2211yescharacteristic
14C22×C71⋊C8C2211yescharacteristic
15#14556125—21no—
16#14556116—21no—
17#14556103C2212yes—
18#14556103C2212yes—
19#14556098—22no—
20#14556093—21no—
21#14556090—21no—
22C4×C22⋊D71—22no—
23#14556087C2211yescharacteristic
24#14556086C2212yescomplemented
25C4⋊C4×D71C2212yescomplemented
26#14556061C2211yescharacteristic, complemented
27#14556061C2211yescharacteristic, complemented
28#14556059C2211yescharacteristic, complemented
29#14556059C2211yescharacteristic, complemented
30#14556050C2212yes—
31#14556050C2212yes—
32#14556048C2212yes—
33#14556048C2212yes—
34(C2×D71)⋊C8C2212yescomplemented
35(C2×D71)⋊C8C2212yescomplemented
36C2×M16×C71C2211yescharacteristic
37#117×C71C2211yescharacteristic
38C2×C4⋊C4×C71C2211yescharacteristic
39#107×C71C2212yescomplemented
40#106×C71C2212yescomplemented
41C23×D71—21no—
42#11765865—22no—
43#11765865—22no—
44C2×Dic142D411yescharacteristic
45C2×D284—22no—
46C2×D284—22no—
47C2×D284—21no—
48C2×D284—22no—
49C2×D284—22no—
50C2×D284D411yescharacteristic
51C2×D284C2312yes—
52C2×C22⋊D71D412yes—
53C2×C4×D71—22no—
54C2×C4×D71C2312yes—
55C2×C71⋊C8—22no—
56C2×C71⋊C8C2311yescharacteristic
57C2×C71⋊C8C2311yescharacteristic
58C4⋊Dic71C2311yescharacteristic
59C4⋊Dic71C2311yescharacteristic
60Dic71⋊C4—22no—
61C22⋊Dic71—21no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4—22no—
65C4×Dic71—21no—
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—22no—
79C22×D71—42no—
80C22×D71—81no—
81C22×D71—81no—
82C22×D71C2×D412yes—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—81no—
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71#4112yes—
89C2×C4×C71#4112yes—
90C2×C4×C71C2411yescharacteristic
91C22⋊D71—24no—
92C2×Dic71—21no—
93C2×Dic71—21no—
94C2×Dic71C2×D412yes—
95D284—22no—
96D284—22no—
97D284—22no—
98D284—42no—
99D284—82no—
100D284—82no—
101C4×D71—22no—
102C4×D71—22no—
103C4×D71—82no—
104Dic142—22no—
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—42no—
116C2×D71—42no—
117C2×D71—42no—
118C2×D71—84no—
119C4×C71—42no—
120C4×C71—41no—
121C4×C71—41no—
122C4×C71#12312yes—
123C4×C71C2×#4111yescharacteristic
124C4×C71C2×#4111yescharacteristic
125Dic71—22no—
126Dic71—41no—
127Dic71—41no—
128C2×C71—22no—
129C2×C71C2×#12311yescharacteristic
130C2×C71#57611yescharacteristic
131C2×C71#57711yescharacteristic
132D71—42no—
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#428D7111yescharacteristic, complemented
143#427—711no—
144#418—711no—
145C2×#102—711no—
146#408—711no—
147C2×#41—711no—
148C22×D4—711no—
149C2×M16C2×D7111yescharacteristic
150C22×C8—711no—
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—712no—
161#117C2×D7111yescharacteristic
162C2×C4⋊C4—712no—
163C2×C4⋊C4C2×D7111yescharacteristic
164#107—711no—
165#107—711no—
166#107C2×D7112yescomplemented
167#106—711no—
168#106—711no—
169#106C2×D7112yescomplemented
170#102—712no—
171#102—712no—
172#102—712no—
173#102—712no—
174C22⋊C8—712no—
175C22⋊C8—712no—
176C24—1421no—
177#41—1422no—
178#41—1422no—
179C2×Q8—711no—
180C2×D4—712no—
181C2×D4—711no—
182C2×D4—712no—
183C2×D4—1422no—
184C2×D4—1422no—
185C2×D4—1421no—
186C2×D4—1422no—
187C2×D4—1422no—
188C22×C4—21no—
189C22×C4—712no—
190C22×C4—1422no—
191C22×C4C22×D7111yescharacteristic
192M16—22no—
193C2×C8—711no—
194C2×C8—711no—
195C2×C8—1422no—
196C2×C8C22×D7112yes—
197C4⋊C4—21no—
198C4⋊C4—711no—
199C4⋊C4—711no—
200C4⋊C4—1422no—
201C4⋊C4C22×D7112yes—
202C4⋊C4C22×D7111yescharacteristic
203C4⋊C4C22×D7111yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—1422no—
206C22⋊C4—1422no—
207C22⋊C4—1422no—
208C22⋊C4—1421no—
209C42—21no—
210C42—1422no—
211C42—1421no—
212C23—2842no—
213C23—5681no—
214C23—5681no—
215C23—712no—
216C23—1422no—
217C23D4×D7111yescharacteristic
218Q8—1422no—
219D4—2842no—
220D4—5682no—
221D4—5682no—
222D4—1424no—
223D4—1422no—
224D4—1422no—
225D4—1422no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—81no—
230C2×C4—5682no—
231C2×C4—712no—
232C2×C4—1421no—
233C2×C4—1421no—
234C2×C4—1422no—
235C2×C4—1422no—
236C2×C4D4×D7111yescharacteristic
237C2×C4#1176586512yes—
238C2×C4#1176586612yes—
239C2×C4C23×D7111yescharacteristic
240C8—22no—
241C8—1421no—
242C8—1421no—
243C22—21no—
244C22—21no—
245C22—2842no—
246C22—2842no—
247C22—2842no—
248C22—5684no—
249C22—1422no—
250C22—1422no—
251C22—1422no—
252C22#1455611312yes—
253C22C2×D4×D7111yescharacteristic
254C4—42no—
255C4—41no—
256C4—41no—
257C4—2841no—
258C4—2841no—
259C4—1422no—
260C4#1455611312yes—
261C4C2×#1176586511yescharacteristic
262C4C2×#1176586611yescharacteristic
263C2—22no—
264C2—2842no—
265C2—5682no—
266C2C2×#1455611311yescharacteristic
267C2#1798680311yescharacteristic
268C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#428×C71C2
socleC22×C71C22×D4
hypercenter#428D71
hypocenterC71#3096
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. #17986480C2semidirect
3. #17986326C2semidirect
4. #17986325C2semidirect
5. #17986319C2semidirect
6. #17986314C2semidirect
7. #17986035C2semidirect
8. #17986005C2semidirect
9. C2×#14556048C2semidirect
10. #17985967C2semidirect
11. #428×C71C2semidirect
24. #14556086C22semidirect
25. C4⋊C4×D71C22semidirect
26. #14556061C22semidirect
27. #14556061C22semidirect
28. #14556059C22semidirect
29. #14556059C22semidirect
34. (C2×D71)⋊C8C22semidirect
35. (C2×D71)⋊C8C22semidirect
39. #107×C71C22semidirect
40. #106×C71C22semidirect
135. C71#3096semidirect
142. #428D71semidirect
166. #107C2×D71semidirect
169. #106C2×D71semidirect