Finite Group Databasebeta

← #22592813

Subgroups — #22592813

Order 9088.

544 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#14556106C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic
3#17985931C211yesmaximal, characteristic, complemented
4#17985930C211yesmaximal, characteristic, complemented
5#17985929C211yesmaximal, characteristic, complemented
6#17985903C212yesmaximal, complemented
7#17985873C212yesmaximal
8#17985870C212yesmaximal
9#17985868C212yesmaximal, complemented
10#17985846C211yesmaximal, characteristic
11#398×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C22×C4×D71C2212yes—
15#14556106—24no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2212yes—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2212yes—
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2212yes—
22C2×Dic71⋊C4C2212yes—
23C4⋊C4×D71—24no—
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×C4×Dic71C2212yes—
28C71⋊#95C2211yescharacteristic
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C4⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—24no—
40C22×Dic71C2312yes—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2312yes—
44C22×Dic71C2311yescharacteristic
45C2×C4×D71—24no—
46C2×C4×D71—24no—
47C2×C4×D71—42no—
48C2×C4×D71—42no—
49C2×C4×D71D414yes—
50C4⋊Dic71—22no—
51C4⋊Dic71—42no—
52Dic71⋊C4—22no—
53Dic71⋊C4—24no—
54Dic71⋊C4—22no—
55Dic71⋊C4—42no—
56Dic71⋊C4—42no—
57(C2×D71)⋊C4—22no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4Q818yescomplemented
60C4×Dic71—24no—
61C22×C4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2311yescharacteristic
65C22×C4×C71C2312yes—
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C4⋊C4×C71—42no—
69C23×C71C2411yescharacteristic
70C22×D71—22no—
71C22×D71—21no—
72C22×D71—21no—
73C22×D71C2×D412yes—
74C22×D71C2×Q814yes—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71C2×D412yes—
87C2×C4×C71C2×Q814yes—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—21no—
95C2×Dic71—22no—
96C2×Dic71—21no—
97C2×Dic71—22no—
98C2×Dic71C2×D412yes—
99C2×Dic71C2×Q814yes—
100C2×Dic71#4114yes—
101C4×D71—28no—
102C4×D71—48no—
103Dic142—44no—
104C22×C71C22×D411yescharacteristic
105C22×C71C22×Q812yes—
106C22×C71C2×#4112yes—
107C22×C71H(5,𝔽2)11yescharacteristic
108C22×C71H(5,𝔽2)11yescharacteristic
109C2×D71—22no—
110C2×D71—22no—
111C2×D71—24no—
112C2×D71—24no—
113C2×D71C4⋊Q814yes—
114C4×C71—22no—
115C4×C71—24no—
116C4×C71—41no—
117C4×C71—42no—
118C4×C71—41no—
119Dic71—24no—
120Dic71—41no—
121Dic71—42no—
122Dic71—41no—
123Dic71#12314yes—
124C2×C71C2×#12311yescharacteristic
125C2×C71C2×C4⋊Q811yescharacteristic
126C2×C71#53511yescharacteristic
127C2×C71#55412yes—
128C2×C71#55412yes—
129D71—24no—
130#2502—711nomaximal, Sylow
131C71#250211yescharacteristic, complemented, Sylow
132C2×C22⋊Q8—711no—
133C22×C4⋊C4—711no—
134#398—711no—
135#398—711no—
136#398—711no—
137#398D7111yescharacteristic, complemented
138#391—712no—
139#385—712no—
140#385—712no—
141#384—712no—
142C4×C22⋊C4—711no—
143C22×Q8—711no—
144C23×C4—712no—
145C23×C4—711no—
146C22⋊Q8—1424no—
147C2×C4⋊C4—712no—
148C2×C4⋊C4—712no—
149C2×C4⋊C4—712no—
150C2×C4⋊C4—712no—
151C2×C4⋊C4—711no—
152C2×C4⋊C4—711no—
153C2×C4⋊C4—711no—
154C2×C4⋊C4—1424no—
155C2×C4⋊C4C2×D7111yescharacteristic
156C2×C4⋊C4C2×D7111yescharacteristic
157C2×C4⋊C4C2×D7112yes—
158C2×C22⋊C4—712no—
159C2×C22⋊C4—711no—
160C2×C22⋊C4—711no—
161C2×C42—712no—
162#95—711no—
163#95—712no—
164#95—712no—
165#95—712no—
166#95C2×D7111yescharacteristic
167#95C2×D7112yes—
168C24—711no—
169C2×Q8—1422no—
170C2×Q8—1424no—
171C22×C4—2842no—
172C22×C4—2842no—
173C22×C4—711no—
174C22×C4—712no—
175C22×C4—714no—
176C22×C4—711no—
177C22×C4—712no—
178C22×C4—711no—
179C22×C4—1424no—
180C22×C4—1424no—
181C22×C4C22×D7111yescharacteristic
182C22×C4C22×D7112yes—
183C22×C4C22×D7111yescharacteristic
184C22×C4C22×D7111yescharacteristic
185C22×C4C22×D7112yes—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—2842no—
190C4⋊C4—2842no—
191C4⋊C4—2842no—
192C4⋊C4—1424no—
193C4⋊C4—1422no—
194C4⋊C4—1422no—
195C4⋊C4—1422no—
196C22⋊C4—718no—
197C22⋊C4—1422no—
198C22⋊C4—1422no—
199C42—1424no—
200C23—712no—
201C23—714no—
202C23—1422no—
203C23—1421no—
204C23—1421no—
205C23C23×D7111yescharacteristic
206Q8—2844no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—2848no—
219C2×C4—712no—
220C2×C4—714no—
221C2×C4—714no—
222C2×C4—1428no—
223C2×C4—1421no—
224C2×C4—1421no—
225C2×C4—1422no—
226C2×C4—1422no—
227C2×C4—1421no—
228C2×C4—1422no—
229C2×C4—1421no—
230C2×C4—1422no—
231C2×C4—1422no—
232C2×C4—1422no—
233C2×C4D4×D7112yes—
234C2×C4Q8×D7114yes—
235C22—714no—
236C22—1422no—
237C22—1424no—
238C22—1424no—
239C22—1422no—
240C22C2×D4×D7111yescharacteristic
241C22C2×Q8×D7112yes—
242C22#41×D7112yes—
243C22#1455617311yescharacteristic
244C22#1455617411yescharacteristic
245C4—24no—
246C4—22no—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—2842no—
251C4—2841no—
252C4—2841no—
253C4—714no—
254C4—1424no—
255C2—1424no—
256C2#123×D7111yescharacteristic
257C2C4⋊Q8×D7111yescharacteristic
258C2#1798656511yescharacteristic
259C2#1798670312yes—
260C2#1798670512yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#398×C71C2
hypercenter#398D71
hypocenterC71#2502
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #17985931C2semidirect
4. #17985930C2semidirect
5. #17985929C2semidirect
6. #17985903C2semidirect
9. #17985868C2semidirect
11. #398×C71C2semidirect
59. (C2×D71)⋊C4Q8semidirect
131. C71#2502semidirect
137. #398D71semidirect