Finite Group Databasebeta

← #22600655

Subgroups — #22600655

Order 9088.

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

normal subgroups only

This group has 388 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#17986710C211yesmaximal, characteristic, complemented
2#17986652C211yesmaximal, characteristic
3#17986328C212yesmaximal, complemented
4#17986286C211yesmaximal, characteristic, complemented
5#17986256C212yesmaximal, complemented
6#17986206C211yesmaximal, characteristic, complemented
7#17986164C212yesmaximal, complemented
8C4×#11765854C211yesmaximal, characteristic, complemented
9#17986078C211yesmaximal, characteristic, complemented
10#17985999C212yesmaximal, complemented
11#487×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2212yes—
13C2×#11765855C2212yes—
14C2×#11765854C2211yescharacteristic
15C2×#11765848C2212yes—
16#14556119—21no—
17#14556113—22no—
18#14556108C2212yes—
19#14556104—22no—
20#14556104C2212yes—
21#14556103—21no—
22C4×Dic142C2211yescharacteristic
23C4×D284C2211yescharacteristic
24C4×D284C2211yescharacteristic
25#14556090—21no—
26C4⋊C4×D71—21no—
27C4⋊C4×D71C2212yes—
28#14556060C2211yescharacteristic
29#14556059C2212yes—
30C4⋊(C71⋊C8)C2211yescharacteristic
31#14556053C2211yescharacteristic
32#14556052C2212yes—
33#14556050C2211yescharacteristic
34#14556048C2212yes—
35(C2×D71)⋊C8C2212yes—
36C4⋊Q8×C71C2211yescharacteristic
37C4⋊Q8×C71C2211yescharacteristic
38#103×C71C2212yes—
39#103×C71C2212yes—
40C4×C8×C71C2211yescharacteristic
41#11765866—24no—
42#11765866—42no—
43C2×Dic142C2311yescharacteristic
44C2×D284—22no—
45C2×D284D411yescharacteristic
46C2×D284C2311yescharacteristic
47C2×C4×D71—21no—
48C2×C4×D71—22no—
49C2×C4×D71—22no—
50C2×C4×D71C2312yes—
51#11765855—42no—
52#11765854—22no—
53#11765848—42no—
54C2×C71⋊C8C2312yes—
55C4⋊Dic71—22no—
56C4⋊Dic71D411yescharacteristic
57C4⋊Dic71C2311yescharacteristic
58C4⋊Dic71C2311yescharacteristic
59Dic71⋊C4—21no—
60Dic71⋊C4—22no—
61Dic71⋊C4—22no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4—22no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4D412yes—
66C4×Dic71—21no—
67C2×Q8×C71—21no—
68C2×Q8×C71C2312yes—
69C2×C8×C71C2311yescharacteristic
70C2×C8×C71C2311yescharacteristic
71C4⋊C4×C71—21no—
72C4⋊C4×C71—21no—
73C4⋊C4×C71—21no—
74C4⋊C4×C71C2312yes—
75C42×C71C2311yescharacteristic
76C22×D71—21no—
77C22×D71C2×D412yes—
78Q8×C71—22no—
79Q8×C71—22no—
80Q8×C71—81no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2411yescharacteristic
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—21no—
89C2×Dic71C2×D412yes—
90D284—22no—
91D284—21no—
92D284—44no—
93D284#4112yes—
94C4×D71—22no—
95C4×D71—22no—
96C4×D71—44no—
97C4×D71—42no—
98C4×D71—82no—
99Dic142—21no—
100Dic142—21no—
101C8×C71—21no—
102C8×C71—21no—
103C71⋊C8—22no—
104C22×C71C22×D411yescharacteristic
105C2×D71—22no—
106C2×D71—22no—
107C2×D71—22no—
108C2×D71—22no—
109C2×D71—41no—
110C2×D71—41no—
111C4×C71—21no—
112C4×C71—42no—
113C4×C71—42no—
114C4×C71#13712yes—
115C4×C71C2×#4111yescharacteristic
116C4×C71H(5,𝔽2)11yescharacteristic
117Dic71—22no—
118Dic71—41no—
119Dic71—41no—
120Dic71—41no—
121C2×C71#54611yescharacteristic
122C2×C71C2×#13711yescharacteristic
123C2×C71#57711yescharacteristic
124D71—42no—
125D71—42no—
126#3331—711nomaximal, Sylow
127C71#333111yescharacteristic, complemented, Sylow
128#554—711no—
129#547—711no—
130#487D7111yescharacteristic, complemented
131#483—712no—
132#476—711no—
133#470—712no—
134#462—711no—
135#452—712no—
136C4×SD16—711no—
137#436—711no—
138#417—712no—
139C2×#41—712no—
140C2×Q16—712no—
141C2×SD16—711no—
142C2×M16—712no—
143C4⋊Q8C2×D7111yescharacteristic
144C4⋊Q8C2×D7111yescharacteristic
145#125—1421no—
146#123—1422no—
147C22⋊Q8—712no—
148C22⋊Q8—712no—
149C22⋊Q8—1422no—
150#121—1421no—
151C4×Q8—711no—
152C4×D4—711no—
153C4×D4—711no—
154C4×D4—1421no—
155C2×C4⋊C4—712no—
156C2×C4⋊C4—1421no—
157#106—711no—
158#106—712no—
159C4⋊C8—711no—
160#103—712no—
161#103—711no—
162#103C2×D7112yes—
163#103C2×D7112yes—
164#102—711no—
165#102—712no—
166C22⋊C8—712no—
167C4×C8C2×D7111yescharacteristic
168#41—2842no—
169#41—1424no—
170C2×Q8—21no—
171C2×Q8—711no—
172C2×Q8C22×D7112yes—
173C2×D4—711no—
174C2×D4—711no—
175C2×D4—1422no—
176C22×C4—712no—
177C22×C4—1422no—
178C22×C4—1422no—
179C22×C4—1421no—
180Q16—2842no—
181SD16—1422no—
182M16—2842no—
183C2×C8—712no—
184C2×C8C22×D7111yescharacteristic
185C2×C8C22×D7111yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—711no—
190C4⋊C4—711no—
191C4⋊C4—711no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C4⋊C4—1421no—
196C4⋊C4C22×D7112yes—
197C22⋊C4—712no—
198C22⋊C4—1422no—
199C22⋊C4—1422no—
200C22⋊C4—1421no—
201C42—1421no—
202C42C22×D7111yescharacteristic
203C23—712no—
204C23—1421no—
205Q8—22no—
206Q8—22no—
207Q8—81no—
208Q8—1421no—
209Q8—1421no—
210D4—2844no—
211D4—712no—
212D4—1422no—
213D4—1421no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—2844no—
217C2×C4—2842no—
218C2×C4—5682no—
219C2×C4—712no—
220C2×C4—1421no—
221C2×C4—1421no—
222C2×C4—1421no—
223C2×C4—1422no—
224C2×C4—1422no—
225C2×C4D4×D7111yescharacteristic
226C2×C4D4×D7111yescharacteristic
227C2×C4C23×D7111yescharacteristic
228C8—21no—
229C8—21no—
230C8—1422no—
231C22—2841no—
232C22—2841no—
233C22—1422no—
234C22—1422no—
235C22—1422no—
236C22—1422no—
237C22C2×D4×D7111yescharacteristic
238C4—21no—
239C4—42no—
240C4—42no—
241C4—2841no—
242C4—2841no—
243C4—2841no—
244C4—1422no—
245C4#1455616012yes—
246C4#41×D7111yescharacteristic
247C4#1455617411yescharacteristic
248C2—2842no—
249C2—2842no—
250C2#1798664211yescharacteristic
251C2C2×#1455616011yescharacteristic
252C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#487×C71C2
socleC22×C71C22×D4
hypercenter#487D71
hypocenterC71#3331
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #17986710C2semidirect
3. #17986328C2semidirect
4. #17986286C2semidirect
5. #17986256C2semidirect
6. #17986206C2semidirect
7. #17986164C2semidirect
8. C4×#11765854C2semidirect
9. #17986078C2semidirect
10. #17985999C2semidirect
11. #487×C71C2semidirect
127. C71#3331semidirect
130. #487D71semidirect