Finite Group Databasebeta

← #22599577

Subgroups — #22599577

Order 9088.

328 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986554C211yesmaximal, characteristic, complemented
2#17986528C211yesmaximal, characteristic
3#17986408C211yesmaximal, characteristic, complemented
4#17986405C211yesmaximal, characteristic, complemented
5#17986358C211yesmaximal, characteristic, complemented
6#17986339C212yesmaximal, complemented
7#17986322C212yesmaximal
8#17986304C212yesmaximal, complemented
9#17986297C212yesmaximal, complemented
10#17986054C211yesmaximal, characteristic, complemented
11#489×C71C211yesmaximal, characteristic, complemented
12#14556128C2211yescharacteristic
13#14556121—21no—
14#14556120C2211yescharacteristic
15#14556119C2212yes—
16#14556110—21no—
17#14556108—21no—
18#14556104—21no—
19#14556104C2211yescharacteristic
20#14556099C2211yescharacteristic
21Q8×Dic71—21no—
22Q8×Dic71C2211yescharacteristic
23#14556093—21no—
24D4×Dic71C2211yescharacteristic
25(C22⋊D71)⋊C4—21no—
26#14556086C2212yes—
27#14556084—21no—
28C42×D71C2211yescharacteristic
29#14556062C2212yes—
30#14556061C2212yes—
31#14556060C2212yes—
32#14556059C2212yes—
33Dic71⋊C8C2212yes—
34#14556052C2212yes—
35#14556049C2212yes—
36(C2×D71)⋊C8C2212yes—
37#14556039C2211yescharacteristic
38C4⋊Q8×C71C2211yescharacteristic
39#124×C71C2211yescharacteristic
40#103×C71C2212yes—
41#102×C71C2212yes—
42#97×C71C2211yescharacteristic
43C2×Dic142—21no—
44C2×C22⋊D71—21no—
45C22×Dic71—21no—
46C2×C4×D71D411yescharacteristic
47C2×C4×D71D411yescharacteristic
48C2×C4×D71C2311yescharacteristic
49C2×C71⋊C8C2312yes—
50C4⋊Dic71—21no—
51C4⋊Dic71—22no—
52C4⋊Dic71—21no—
53C4⋊Dic71C2312yes—
54C4⋊Dic71C2311yescharacteristic
55C4⋊Dic71C2311yescharacteristic
56Dic71⋊C4—21no—
57Dic71⋊C4—21no—
58Dic71⋊C4—21no—
59Dic71⋊C4—22no—
60C22⋊Dic71—21no—
61C22⋊Dic71—21no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—21no—
64C4×Dic71—21no—
65C4×Dic71—22no—
66C4×Dic71D411yescharacteristic
67C4×Dic71D411yescharacteristic
68C4×Dic71C2311yescharacteristic
69C2×Q8×C71—21no—
70C2×Q8×C71C2311yescharacteristic
71C2×D4×C71C2311yescharacteristic
72C2×C8×C71C2312yes—
73C4⋊C4×C71—21no—
74C4⋊C4×C71C2312yes—
75C22⋊C4×C71—21no—
76C22⋊C4×C71—21no—
77C42×C71C2311yescharacteristic
78C23×C71—21no—
79C22×D71C2×D411yescharacteristic
80Q8×C71—22no—
81Q8×C71—42no—
82D4×C71—22no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2411yescharacteristic
88C22⋊D71—42no—
89C2×Dic71—22no—
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—42no—
93C2×Dic71C2×D411yescharacteristic
94C2×Dic71C2×D411yescharacteristic
95C2×Dic71C2×D411yescharacteristic
96C4×D71—22no—
97C4×D71—41no—
98C4×D71—41no—
99Dic142—42no—
100C8×C71—22no—
101C71⋊C8—22no—
102C22×C71—41no—
103C22×C71—41no—
104C22×C71—41no—
105C22×C71C22×D411yescharacteristic
106C2×D71—21no—
107C2×D71—21no—
108C2×D71—21no—
109C4×C71—21no—
110C4×C71—21no—
111C4×C71—41no—
112C4×C71—42no—
113C4×C712−1+412yes—
114Dic71—21no—
115Dic71—21no—
116Dic71—21no—
117Dic71—42no—
118Dic71—41no—
119Dic71—41no—
120C2×C71—81no—
121C2×C71#53611yescharacteristic
122C2×C71#57511yescharacteristic
123C2×C71#57511yescharacteristic
124D71—41no—
125#3240—711nomaximal, Sylow
126C71#324011yescharacteristic, complemented, Sylow
127#533—711no—
128#529—711no—
129#501—711no—
130#501—711no—
131#489—711no—
132#489D7111yescharacteristic, complemented
133#485—712no—
134#482—712no—
135#479—712no—
136#478—712no—
137#432—711no—
138C4⋊Q8—711no—
139C4⋊Q8C2×D7111yescharacteristic
140#126—1421no—
141#125—712no—
142#125—711no—
143#124C2×D7111yescharacteristic
144#123—1421no—
145C22⋊Q8—711no—
146C22⋊Q8—1421no—
147C22⋊Q8—1421no—
148#121—711no—
149C4×Q8—711no—
150C4×Q8—1421no—
151C4×Q8—1421no—
152C4×D4—711no—
153C4×D4—1421no—
154#117—712no—
155#117—1421no—
156C2×C42—711no—
157#107—712no—
158#107—712no—
159#106—712no—
160#106—712no—
161C4⋊C8—712no—
162#103—712no—
163#103C2×D7112yes—
164#102—712no—
165#102C2×D7112yes—
166C22⋊C8—712no—
167#97—711no—
168#97C2×D7111yescharacteristic
169C2×Q8—21no—
170C2×Q8—1421no—
171C2×Q8C22×D7111yescharacteristic
172C2×D4—1421no—
173C2×D4C22×D7111yescharacteristic
174C22×C4—711no—
175C22×C4—711no—
176C22×C4—711no—
177C22×C4—1421no—
178C2×C8—712no—
179C2×C8C22×D7112yes—
180C4⋊C4—21no—
181C4⋊C4—712no—
182C4⋊C4—711no—
183C4⋊C4—711no—
184C4⋊C4—1421no—
185C4⋊C4—1422no—
186C4⋊C4—1421no—
187C4⋊C4—1422no—
188C4⋊C4—1421no—
189C4⋊C4—1421no—
190C4⋊C4—1421no—
191C4⋊C4C22×D7112yes—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—1422no—
195C22⋊C4—1421no—
196C22⋊C4—1421no—
197C22⋊C4—1421no—
198C42—711no—
199C42—711no—
200C42—711no—
201C42—1422no—
202C42—1421no—
203C42C22×D7111yescharacteristic
204C23—21no—
205C23—711no—
206Q8—22no—
207Q8—42no—
208Q8—2842no—
209D4—22no—
210D4—2842no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—2841no—
214C2×C4—2841no—
215C2×C4—2842no—
216C2×C4—711no—
217C2×C4—711no—
218C2×C4—711no—
219C2×C4—1421no—
220C2×C4—1422no—
221C2×C4—1421no—
222C2×C4—1422no—
223C2×C4D4×D7111yescharacteristic
224C2×C4D4×D7111yescharacteristic
225C2×C4C23×D7111yescharacteristic
226C8—22no—
227C8—1422no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—1421no—
232C22—1421no—
233C22—1421no—
234C22C2×D4×D7111yescharacteristic
235C4—21no—
236C4—21no—
237C4—41no—
238C4—42no—
239C4—2841no—
240C4—2841no—
241C4—2842no—
242C4—1421no—
243C4—1421no—
244C4—1421no—
245C4#1455617512yes—
246C2—81no—
247C2—2841no—
248C2#1798656911yescharacteristic
249C2#1798679511yescharacteristic
250C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#489×C71C2
socleC22×C71C22×D4
hypercenter#489D71
hypocenterC71#3240
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #17986554C2semidirect
3. #17986408C2semidirect
4. #17986405C2semidirect
5. #17986358C2semidirect
6. #17986339C2semidirect
8. #17986304C2semidirect
9. #17986297C2semidirect
10. #17986054C2semidirect
11. #489×C71C2semidirect
126. C71#3240semidirect
132. #489D71semidirect