Finite Group Databasebeta

← #22598265

Subgroups — #22598265

Order 9088.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#17986510C211yesmaximal, characteristic
2C22×C4⋊Dic71C211yesmaximal, characteristic, complemented
3#17986335C212yesmaximal, complemented
4#17986330C212yesmaximal, complemented
5#17986318C212yesmaximal, complemented
6#17986313C212yesmaximal, complemented
7#17986041C212yesmaximal, complemented
8#17986001C212yesmaximal, complemented
9#407×C71C211yesmaximal, characteristic, complemented
10C2×#11765865C2211yescharacteristic
11C23×Dic71—21no—
12#14556111—21no—
13#14556107C2211yescharacteristic
14#14556107C2211yescharacteristic
15#14556102C2211yescharacteristic
16#14556102C2211yescharacteristic
17#14556097—21no—
18C4×C22⋊D71—21no—
19C4×C22⋊D71—21no—
20#14556087C2211yescharacteristic
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71C2212yes—
25C2×C4⋊Dic71C2212yes—
26C2×C4⋊Dic71C2212yes—
27#14556062C2214yescomplemented
28#14556060C2214yescomplemented
29#14556053C2214yescomplemented
30#14556050C2214yescomplemented
31C23×C4×C71C2211yescharacteristic
32C2×M16×C71C2212yes—
33C22⋊C8×C71C2212yes—
34C22⋊C8×C71C2212yes—
35#11765865—24no—
36C2×Dic142C2311yescharacteristic
37C2×D284C2311yescharacteristic
38C2×C22⋊D71—21no—
39C2×C22⋊D71—21no—
40C2×C22⋊D71—21no—
41C22×Dic71—22no—
42C22×Dic71—22no—
43C22×Dic71—22no—
44C22×Dic71—44no—
45C2×C4×D71—21no—
46C4⋊Dic71—22no—
47C4⋊Dic71—22no—
48C4⋊Dic71—22no—
49C4⋊Dic71C2314yes—
50C4⋊Dic71C2311yescharacteristic
51C4⋊Dic71C2311yescharacteristic
52Dic71⋊C4—21no—
53Dic71⋊C4—21no—
54C22⋊Dic71—21no—
55C22⋊Dic71—21no—
56C22⋊Dic71—21no—
57(C2×D71)⋊C4—21no—
58(C2×D71)⋊C4—21no—
59C4×Dic71—21no—
60C24×C71D411yescharacteristic
61C22×C4×C71—24no—
62C22×C4×C71D411yescharacteristic
63C22×C4×C71D411yescharacteristic
64C22×C4×C71D411yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C22×C4×C71C2311yescharacteristic
67C22×C4×C71C2311yescharacteristic
68M16×C71—24no—
69C2×C8×C71C2314yes—
70C23×C71—22no—
71C23×C71—22no—
72C23×C71—22no—
73C23×C71C2×D411yescharacteristic
74C23×C71C2×D411yescharacteristic
75C23×C71C2×D411yescharacteristic
76C22×D71—21no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71#4114yes—
86C2×C4×C71C2411yescharacteristic
87C22⋊D71—42no—
88C22⋊D71—42no—
89C22⋊D71—42no—
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—24no—
94C2×Dic71—44no—
95C2×Dic71—44no—
96C2×Dic71—44no—
97D284—22no—
98C4×D71—42no—
99Dic142—22no—
100C8×C71—24no—
101C22×C71—28no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71—21no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71—21no—
108C22×C71#12312yes—
109C22×C71AGL(1,ℤ/8)12yes—
110C22×C71#13712yes—
111C22×C71C22×D411yescharacteristic
112C2×D71—41no—
113C2×D71—41no—
114C2×D71—41no—
115C4×C71—21no—
116C4×C71—21no—
117C4×C71#12312yes—
118C4×C71C2×#4112yes—
119Dic71—41no—
120Dic71—44no—
121Dic71—41no—
122Dic71—41no—
123C2×C71—22no—
124C2×C71—22no—
125C2×C71—22no—
126C2×C71C2×#12311yescharacteristic
127C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C71C2×#13711yescharacteristic
129D71—81no—
130#3092—711nomaximal, Sylow
131C71#309211yescharacteristic, complemented, Sylow
132#525—711no—
133C22×C4⋊C4—711no—
134#484—712no—
135#483—712no—
136#481—712no—
137#480—712no—
138#428—712no—
139#417—712no—
140#407D7111yescharacteristic, complemented
141C2×#41—711no—
142C23×C4—1421no—
143C23×C4C2×D7111yescharacteristic
144C2×M16C2×D7112yes—
145#123—1421no—
146C22⋊Q8—711no—
147C22⋊Q8—711no—
148#121—711no—
149#121—711no—
150#120—1421no—
151C4×D4—1421no—
152C4×D4—1421no—
153#117—711no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4—712no—
156C2×C4⋊C4—712no—
157C2×C4⋊C4—1421no—
158C2×C4⋊C4—1421no—
159C2×C4⋊C4—1421no—
160#107—714no—
161#106—714no—
162#103—714no—
163#102—714no—
164C22⋊C8C2×D7112yes—
165C22⋊C8C2×D7112yes—
166C24D28411yescharacteristic
167#41—1424no—
168C2×Q8—711no—
169C2×D4—711no—
170C2×D4—1421no—
171C2×D4—1421no—
172C2×D4—1421no—
173C22×C4—24no—
174C22×C4—2844no—
175C22×C4—1422no—
176C22×C4—1422no—
177C22×C4—1422no—
178C22×C4—1421no—
179C22×C4D28411yescharacteristic
180C22×C4D28411yescharacteristic
181C22×C4D28411yescharacteristic
182C22×C4C22×D7111yescharacteristic
183C22×C4C22×D7111yescharacteristic
184C22×C4C22×D7111yescharacteristic
185M16—24no—
186C2×C8C22×D7114yes—
187C4⋊C4—714no—
188C4⋊C4—711no—
189C4⋊C4—711no—
190C4⋊C4—1421no—
191C4⋊C4—1421no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C22⋊C4—1421no—
196C22⋊C4—1421no—
197C22⋊C4—1421no—
198C22⋊C4—1421no—
199C22⋊C4—1421no—
200C42—1421no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—1421no—
205C23C2×D28411yescharacteristic
206C23C2×D28411yescharacteristic
207C23C2×D28411yescharacteristic
208Q8—1422no—
209D4—2842no—
210D4—2842no—
211D4—2842no—
212D4—1422no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—2844no—
219C2×C4—2844no—
220C2×C4—2844no—
221C2×C4—2842no—
222C2×C4—1421no—
223C2×C4—1421no—
224C2×C4—1421no—
225C2×C4—1424no—
226C2×C4C2×D28411yescharacteristic
227C2×C4C2×D28411yescharacteristic
228C2×C4C2×D28411yescharacteristic
229C2×C4#1176586414yes—
230C2×C4C23×D7111yescharacteristic
231C8—24no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—2841no—
240C22—2841no—
241C22—2841no—
242C22#1455611212yes—
243C22#1455615612yes—
244C22#1455616112yes—
245C22C22×D28411yescharacteristic
246C4—21no—
247C4—21no—
248C4—2841no—
249C4—2844no—
250C4—2841no—
251C4—2841no—
252C4#1455611212yes—
253C4C2×#1176586412yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—5681no—
258C2C2×#1455611211yescharacteristic
259C2C2×#1455615611yescharacteristic
260C2C2×#1455616111yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#407×C71C2
socleC22×C71C22×D4
hypercenter#407D71
hypocenterC71#3092
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic71C2semidirect
3. #17986335C2semidirect
4. #17986330C2semidirect
5. #17986318C2semidirect
6. #17986313C2semidirect
7. #17986041C2semidirect
8. #17986001C2semidirect
9. #407×C71C2semidirect
27. #14556062C22semidirect
28. #14556060C22semidirect
29. #14556053C22semidirect
30. #14556050C22semidirect
131. C71#3092semidirect
140. #407D71semidirect