Finite Group Databasebeta

← #22601137

Subgroups — #22601137

Order 9088.

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

normal subgroups only

This group has 384 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#17986735C211yesmaximal, characteristic, complemented
2#17986664C211yesmaximal, characteristic, complemented
3#17986357C212yesmaximal, complemented
4#17986287C212yesmaximal, complemented
5#17986281C211yesmaximal, characteristic, complemented
6#17986213C212yesmaximal, complemented
7#17986191C211yesmaximal, characteristic
8#17986129C211yesmaximal, characteristic, complemented
9#17986108C212yesmaximal, complemented
10C4×#11765854C211yesmaximal, characteristic, complemented
11#486×C71C211yesmaximal, characteristic, complemented
12C2×Q8×D71—21no—
13C2×#11765854C2211yescharacteristic
14C2×#11765850C2212yes—
15#14556127C2212yes—
16#14556115—21no—
17#14556115C2212yes—
18#14556109—22no—
19#14556106—22no—
20#14556104—21no—
21C4×Dic142C2211yescharacteristic
22C4×Dic142C2211yescharacteristic
23C4×D284C2211yescharacteristic
24D4×Dic71—21no—
25D4×Dic71C2212yes—
26#14556090C2212yes—
27#14556060C2211yescharacteristic
28#14556059C2212yes—
29C4⋊(C71⋊C8)C2211yescharacteristic
30Dic71⋊C8C2212yes—
31#14556053C2211yescharacteristic
32#14556051C2212yes—
33#14556050C2211yescharacteristic
34#14556049C2212yes—
35#14556038C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37#127×C71C2211yescharacteristic
38#102×C71C2212yes—
39#102×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41C2×Dic142—22no—
42C2×Dic142D411yescharacteristic
43C2×Dic142C2311yescharacteristic
44Q8×D71—44no—
45C2×D284C2311yescharacteristic
46C22×Dic71—22no—
47C2×C4×D71—21no—
48C2×C4×D71—22no—
49#11765854—22no—
50#11765850—24no—
51C2×C71⋊C8C2312yes—
52C4⋊Dic71D411yescharacteristic
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—21no—
56Dic71⋊C4—22no—
57Dic71⋊C4D412yes—
58C22⋊Dic71—22no—
59C22⋊Dic71—22no—
60C22⋊Dic71—22no—
61(C2×D71)⋊C4—21no—
62(C2×D71)⋊C4—22no—
63C4×Dic71—21no—
64C4×Dic71C2312yes—
65C2×Q8×C71—21no—
66C2×D4×C71—21no—
67C2×D4×C71—21no—
68C2×D4×C71C2312yes—
69C2×C8×C71C2311yescharacteristic
70C2×C8×C71C2311yescharacteristic
71C4⋊C4×C71—21no—
72C4⋊C4×C71C2312yes—
73C42×C71C2311yescharacteristic
74C23×C71—22no—
75C22×D71—21no—
76Q8×C71—42no—
77D4×C71—22no—
78D4×C71—22no—
79D4×C71—42no—
80D4×C71—81no—
81C2×C4×C71—22no—
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2411yescharacteristic
85C2×Dic71—21no—
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—82no—
89C2×Dic71C2×D412yes—
90C2×Dic71C2×D412yes—
91D284—21no—
92D284#4112yes—
93C4×D71—44no—
94C4×D71—42no—
95Dic142—22no—
96Dic142—21no—
97Dic142—21no—
98Dic142—44no—
99C8×C71—21no—
100C8×C71—21no—
101C71⋊C8—22no—
102C22×C71—42no—
103C22×C71—42no—
104C22×C71—42no—
105C22×C71C22×D411yescharacteristic
106C2×D71—22no—
107C2×D71—41no—
108C2×D71—41no—
109C4×C71—21no—
110C4×C71—42no—
111C4×C71AGL(1,ℤ/8)12yes—
112C4×C71C2×#4111yescharacteristic
113C4×C712−1+411yescharacteristic
114Dic71—22no—
115Dic71—22no—
116Dic71—41no—
117Dic71—41no—
118Dic71—41no—
119C2×C71—82no—
120C2×C71#54811yescharacteristic
121C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
122C2×C71#57711yescharacteristic
123D71—42no—
124#3373—711nomaximal, Sylow
125C71#337311yescharacteristic, complemented, Sylow
126#559—711no—
127D4×Q8—711no—
128#489—712no—
129#486D7111yescharacteristic, complemented
130#476—712no—
131#475—711no—
132#463—712no—
133#459—711no—
134#446—711no—
135#442—712no—
136C4×SD16—711no—
137C22×Q8—1421no—
138C2×SD16—711no—
139C2×D8—712no—
140C4⋊Q8—712no—
141C4⋊Q8C2×D7111yescharacteristic
142#127C2×D7111yescharacteristic
143#124—712no—
144#124—1421no—
145#123—1422no—
146C22⋊Q8—1422no—
147C22⋊Q8—1421no—
148C4×Q8—711no—
149C4×Q8—711no—
150C4×D4—712no—
151C4×D4—712no—
152C4×D4—711no—
153C4×D4—1421no—
154#106—711no—
155#106—712no—
156C4⋊C8—712no—
157C4⋊C8—711no—
158#103—711no—
159#103—712no—
160#102—712no—
161#102—711no—
162#102C2×D7112yes—
163#102C2×D7112yes—
164#97—712no—
165C4×C8C2×D7111yescharacteristic
166C2×Q8—21no—
167C2×Q8—2844no—
168C2×Q8—711no—
169C2×Q8—711no—
170C2×Q8—1422no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—711no—
174C2×D4C22×D7112yes—
175C22×C4—1422no—
176C22×C4—1421no—
177C22×C4—1422no—
178SD16—1422no—
179D8—1424no—
180C2×C8—712no—
181C2×C8C22×D7111yescharacteristic
182C2×C8C22×D7111yescharacteristic
183C4⋊C4—21no—
184C4⋊C4—712no—
185C4⋊C4—711no—
186C4⋊C4—711no—
187C4⋊C4—711no—
188C4⋊C4—1422no—
189C4⋊C4—1421no—
190C4⋊C4C22×D7112yes—
191C22⋊C4—1422no—
192C22⋊C4—1422no—
193C22⋊C4—1422no—
194C22⋊C4—1422no—
195C22⋊C4—1421no—
196C42—712no—
197C42—1421no—
198C42C22×D7111yescharacteristic
199C23—22no—
200C23—1421no—
201Q8—42no—
202Q8—2844no—
203Q8—1421no—
204Q8—1421no—
205Q8—1422no—
206D4—22no—
207D4—22no—
208D4—42no—
209D4—81no—
210D4—712no—
211D4—1421no—
212C2×C4—22no—
213C2×C4—2844no—
214C2×C4—2842no—
215C2×C4—5682no—
216C2×C4—712no—
217C2×C4—712no—
218C2×C4—1421no—
219C2×C4—1421no—
220C2×C4—1421no—
221C2×C4D4×D7111yescharacteristic
222C2×C4D4×D7111yescharacteristic
223C2×C4C23×D7111yescharacteristic
224C8—21no—
225C8—21no—
226C8—1422no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—2841no—
231C22—2841no—
232C22—1422no—
233C22C2×D4×D7111yescharacteristic
234C4—21no—
235C4—42no—
236C4—2841no—
237C4—2841no—
238C4—2841no—
239C4—1422no—
240C4—1422no—
241C4#1455615212yes—
242C4#41×D7111yescharacteristic
243C4#1455617511yescharacteristic
244C2—82no—
245C2—2842no—
246C2#1798665711yescharacteristic
247C2C2×#1455615211yescharacteristic
248C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#486×C71C2
socleC22×C71C22×D4
hypercenter#486D71
hypocenterC71#3373
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #17986735C2semidirect
2. #17986664C2semidirect
3. #17986357C2semidirect
4. #17986287C2semidirect
5. #17986281C2semidirect
6. #17986213C2semidirect
8. #17986129C2semidirect
9. #17986108C2semidirect
10. C4×#11765854C2semidirect
11. #486×C71C2semidirect
125. C71#3373semidirect
129. #486D71semidirect