Finite Group Databasebeta

← #22598980

Subgroups — #22598980

Order 9088.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986622C212yesmaximal, complemented
2#17986403C211yesmaximal, characteristic, complemented
3#17986384C211yesmaximal, characteristic, complemented
4#17986360C212yesmaximal, complemented
5#17986247C212yesmaximal, complemented
6#17986235C211yesmaximal, characteristic
7#17986102C212yesmaximal, complemented
8Q16×Dic71C211yesmaximal, characteristic, complemented
9#17986081C211yesmaximal, characteristic, complemented
10#17985954C211yesmaximal, characteristic, complemented
11#467×C71C211yesmaximal, characteristic, complemented
12C2×#11765852C2212yes—
13C2×D568C2211yescharacteristic
14C2×#11765849C2211yescharacteristic
15#14556127C2212yes—
16#14556123—22no—
17#14556117C2212yes—
18#14556114—22no—
19#14556110—22no—
20#14556108—22no—
21#14556102C2212yes—
22Q8×Dic71C2212yes—
23#14556090C2212yes—
24#14556087C2211yescharacteristic
25#14556062C2211yescharacteristic
26#14556061C2212yes—
27#14556052C2212yes—
28#14556050C2212yes—
29#14556048C2212yes—
30#14556038C2211yescharacteristic
31#14556038C2211yescharacteristic
32C8×Dic71C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C2×Q16×C71C2211yescharacteristic
35C22×C8×C71C2211yescharacteristic
36C22⋊Q8×C71C2212yes—
37#107×C71C2211yescharacteristic
38#103×C71C2212yes—
39C2×Dic142—22no—
40C2×D284C2312yes—
41C2×C22⋊D71—22no—
42C2×C4×D71—22no—
43#11765852—24no—
44D568—22no—
45#11765849—22no—
46C2×C71⋊C8C2311yescharacteristic
47C2×C71⋊C8C2311yescharacteristic
48C4⋊Dic71—22no—
49C4⋊Dic71D411yescharacteristic
50C4⋊Dic71D411yescharacteristic
51C4⋊Dic71C2312yes—
52Dic71⋊C4—22no—
53C22⋊Dic71D411yescharacteristic
54C22⋊Dic71D411yescharacteristic
55(C2×D71)⋊C4—22no—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—22no—
59C4×Dic71—22no—
60C4×Dic71C2311yescharacteristic
61C4×Dic71C2311yescharacteristic
62C2×Q8×C71C2312yes—
63C22×C4×C71C2311yescharacteristic
64Q16×C71—22no—
65C2×C8×C71—22no—
66C2×C8×C71C2311yescharacteristic
67C2×C8×C71C2311yescharacteristic
68C4⋊C4×C71—22no—
69C4⋊C4×C71C2312yes—
70C22⋊C4×C71—22no—
71C23×C71C2×D411yescharacteristic
72C22×D71—22no—
73Q8×C71—22no—
74Q8×C71—22no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D411yescharacteristic
80C2×C4×C71C2411yescharacteristic
81C22⋊D71—82no—
82C2×Dic71—22no—
83C2×Dic71C2×D411yescharacteristic
84C2×Dic71C2×D411yescharacteristic
85C2×Dic71C2×D411yescharacteristic
86C2×Dic71C2×D411yescharacteristic
87D284—22no—
88D284—22no—
89C4×D71—82no—
90Dic142—82no—
91C8×C71—21no—
92C8×C71#4112yes—
93C71⋊C8—21no—
94C71⋊C8#4112yes—
95C22×C71—21no—
96C22×C71—21no—
97C22×C71—21no—
98C22×C71C22×D411yescharacteristic
99C2×D71—42no—
100C2×D71—42no—
101C2×D71—42no—
102C4×C71—21no—
103C4×C71—42no—
104C4×C71—42no—
105C4×C71C2×#4111yescharacteristic
106C4×C71C2×#4111yescharacteristic
107Dic71—21no—
108Dic71—21no—
109Dic71—21no—
110Dic71—21no—
111Dic71—42no—
112C2×C71—41no—
113C2×C71#53211yescharacteristic
114C2×C71#57611yescharacteristic
115C2×C71#57811yescharacteristic
116D71—82no—
117#3175—711nomaximal, Sylow
118C71#317511yescharacteristic, complemented, Sylow
119#542—712no—
120#500—711no—
121#495—711no—
122#489—712no—
123#468—712no—
124#467D7111yescharacteristic, complemented
125#466—711no—
126#440—712no—
127C4×Q16—711no—
128C4×D8—711no—
129#405—711no—
130C2×Q16C2×D7111yescharacteristic
131C2×SD16—712no—
132C2×D8—711no—
133C2×M16—711no—
134C22×C8C2×D7111yescharacteristic
135C4⋊Q8—712no—
136#126—1422no—
137#124—712no—
138#124—1422no—
139#123—1422no—
140C22⋊Q8—1422no—
141C22⋊Q8C2×D7112yes—
142#121—712no—
143C4×Q8—712no—
144C4×D4—712no—
145#117—711no—
146#107—711no—
147#107—712no—
148#107C2×D7111yescharacteristic
149#103—712no—
150#103C2×D7112yes—
151#102—712no—
152#102—712no—
153#97—711no—
154#97—711no—
155C4×C8—711no—
156C4×C8—711no—
157C2×Q8—1422no—
158C2×Q8C22×D7112yes—
159C2×D4—712no—
160C2×D4—1422no—
161C22×C4—1422no—
162C22×C4C22×D7111yescharacteristic
163Q16—22no—
164SD16—1424no—
165D8—1422no—
166M16—1422no—
167C2×C8—22no—
168C2×C8—711no—
169C2×C8—711no—
170C2×C8C22×D7111yescharacteristic
171C2×C8C22×D7111yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—711no—
174C4⋊C4—711no—
175C4⋊C4—712no—
176C4⋊C4—1422no—
177C4⋊C4—1422no—
178C4⋊C4C22×D7112yes—
179C22⋊C4—22no—
180C22⋊C4—711no—
181C22⋊C4—711no—
182C22⋊C4—1422no—
183C22⋊C4—1422no—
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C42—711no—
187C42—711no—
188C42—1422no—
189C23—1422no—
190C23D4×D7111yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—5682no—
194D4—5682no—
195D4—1422no—
196D4—1422no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—5682no—
202C2×C4—711no—
203C2×C4—711no—
204C2×C4—711no—
205C2×C4—711no—
206C2×C4—1422no—
207C2×C4D4×D7111yescharacteristic
208C2×C4C23×D7111yescharacteristic
209C8—21no—
210C8—712no—
211C8—1421no—
212C8#1176586412yes—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—2842no—
217C22—2842no—
218C22—2842no—
219C22C2×D4×D7111yescharacteristic
220C4—21no—
221C4—42no—
222C4—42no—
223C4—2842no—
224C4—1421no—
225C4—1421no—
226C4—1421no—
227C4—1421no—
228C4#41×D7111yescharacteristic
229C4C2×#1176586411yescharacteristic
230C2—41no—
231C2—5682no—
232C2#1798654411yescharacteristic
233C2#1798680111yescharacteristic
234C2#1798681211yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#467×C71C2
socleC22×C71C22×D4
hypercenter#467D71
hypocenterC71#3175
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #17986622C2semidirect
2. #17986403C2semidirect
3. #17986384C2semidirect
4. #17986360C2semidirect
5. #17986247C2semidirect
7. #17986102C2semidirect
8. Q16×Dic71C2semidirect
9. #17986081C2semidirect
10. #17985954C2semidirect
11. #467×C71C2semidirect
118. C71#3175semidirect
124. #467D71semidirect