Finite Group Databasebeta

← #22594119

Subgroups — #22594119

Order 9088.

526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556110C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic, complemented
3#17985941C212yesmaximal, complemented
4#17985935C212yesmaximal, complemented
5#17985931C211yesmaximal, characteristic, complemented
6#17985930C211yesmaximal, characteristic, complemented
7#17985924C211yesmaximal, characteristic, complemented
8#17985875C212yesmaximal
9#17985873C212yesmaximal, complemented
10#17985861C211yesmaximal, characteristic, complemented
11#397×C71C211yesmaximal, characteristic, complemented
12C22×C22⋊D71C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C22×C4×D71C2212yes—
15#14556110—24no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2211yescharacteristic
18C2×C4⋊Dic71C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2212yes—
22C4⋊C4×D71—24no—
23C2×C22⋊Dic71C2211yescharacteristic
24C2×C22⋊Dic71C2212yes—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2211yescharacteristic
27C71⋊#95C2211yescharacteristic
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×C22⋊D71—22no—
39C2×C22⋊D71—24no—
40C22×Dic71C2312yes—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2312yes—
43C22×Dic71C2311yescharacteristic
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—22no—
52C4⋊Dic71—42no—
53Dic71⋊C4—22no—
54Dic71⋊C4—24no—
55Dic71⋊C4—42no—
56C22⋊Dic71—22no—
57C22⋊Dic71—42no—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—24no—
60C24×C71C2311yescharacteristic
61C22×C4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2312yes—
64C22×C4×C71C2311yescharacteristic
65C4⋊C4×C71—22no—
66C22⋊C4×C71—22no—
67C22⋊C4×C71—42no—
68C23×C71—21no—
69C23×C71—22no—
70C23×C71—21no—
71C23×C71—21no—
72C23×C71—22no—
73C23×C71C2411yescharacteristic
74C22×D71—21no—
75C22×D71—21no—
76C22×D71—22no—
77C22×D71C2×D412yes—
78C22×D71#4114yes—
79C2×C4×C71—22no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71C2×D412yes—
89C22⋊D71—44no—
90C2×Dic71—22no—
91C2×Dic71—21no—
92C2×Dic71—22no—
93C2×Dic71—21no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—21no—
99C2×Dic71—21no—
100C2×Dic71—22no—
101C2×Dic71—22no—
102C2×Dic71C2×D412yes—
103C4×D71—28no—
104C4×D71—48no—
105C22×C71—41no—
106C22×C71—41no—
107C22×C71—41no—
108C22×C71—42no—
109C22×C71—42no—
110C22×C71C22×D411yescharacteristic
111C22×C71C2×#4112yes—
112C22×C71H(5,𝔽2)11yescharacteristic
113C22×C71H(5,𝔽2)11yescharacteristic
114C22×C712−1+412yes—
115C2×D71—22no—
116C2×D71—22no—
117C2×D71—24no—
118C2×D71—24no—
119C2×D71#12314yes—
120C4×C71—22no—
121C4×C71—42no—
122C4×C71—41no—
123C4×C71—42no—
124Dic71—22no—
125Dic71—41no—
126Dic71—42no—
127Dic71—41no—
128Dic71—42no—
129C2×C71—81no—
130C2×C71C2×#12311yescharacteristic
131C2×C71#53511yescharacteristic
132C2×C71#53611yescharacteristic
133C2×C71#53912yes—
134C2×C71#53912yes—
135D71—24no—
136#2623—711nomaximal, Sylow
137C71#262311yescharacteristic, complemented, Sylow
138C2×#123—711no—
139C22×C4⋊C4—711no—
140#400—712no—
141#399—712no—
142#398—711no—
143#398—711no—
144#397—711no—
145#397D7111yescharacteristic, complemented
146#386—712no—
147#385—712no—
148#381—711no—
149C22×D4—711no—
150C23×C4—712no—
151C23×C4—711no—
152#123—1424no—
153C2×C4⋊C4—712no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4—711no—
156C2×C4⋊C4—712no—
157C2×C4⋊C4—711no—
158C2×C4⋊C4—711no—
159C2×C4⋊C4—1424no—
160C2×C4⋊C4C2×D7111yescharacteristic
161C2×C22⋊C4—712no—
162C2×C22⋊C4—711no—
163C2×C22⋊C4—711no—
164C2×C22⋊C4—712no—
165C2×C22⋊C4C2×D7111yescharacteristic
166C2×C22⋊C4C2×D7112yes—
167#95—711no—
168#95—712no—
169#95—712no—
170#95—712no—
171#95—712no—
172#95C2×D7111yescharacteristic
173#95C2×D7112yes—
174C24—711no—
175C24C22×D7111yescharacteristic
176C2×D4—1422no—
177C2×D4—1424no—
178C22×C4—2842no—
179C22×C4—2842no—
180C22×C4—711no—
181C22×C4—714no—
182C22×C4—712no—
183C22×C4—711no—
184C22×C4—712no—
185C22×C4—711no—
186C22×C4—1424no—
187C22×C4—1424no—
188C22×C4C22×D7111yescharacteristic
189C22×C4C22×D7112yes—
190C22×C4C22×D7112yes—
191C22×C4C22×D7111yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—2842no—
194C4⋊C4—2842no—
195C4⋊C4—1424no—
196C4⋊C4—1422no—
197C4⋊C4—1422no—
198C4⋊C4—1422no—
199C22⋊C4—22no—
200C22⋊C4—42no—
201C22⋊C4—2842no—
202C22⋊C4—1424no—
203C22⋊C4—1422no—
204C22⋊C4—1422no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—712no—
211C23—714no—
212C23—1421no—
213C23—1421no—
214C23—1422no—
215C23C23×D7111yescharacteristic
216D4—2844no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—2848no—
227C2×C4—712no—
228C2×C4—1428no—
229C2×C4—1421no—
230C2×C4—1422no—
231C2×C4—1421no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4—1421no—
235C2×C4—1422no—
236C2×C4—1422no—
237C2×C4—1422no—
238C2×C4—1421no—
239C2×C4—1422no—
240C2×C4—1422no—
241C2×C4D4×D7112yes—
242C22—41no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22—41no—
247C22—714no—
248C22—1422no—
249C22—1422no—
250C22—1424no—
251C22—1424no—
252C22C2×D4×D7111yescharacteristic
253C22#41×D7112yes—
254C22#1455617311yescharacteristic
255C22#1455617411yescharacteristic
256C22#1455617512yes—
257C4—22no—
258C4—42no—
259C4—41no—
260C4—42no—
261C4—2842no—
262C4—2841no—
263C4—2842no—
264C4—2841no—
265C4—1422no—
266C2—81no—
267C2—1424no—
268C2#123×D7111yescharacteristic
269C2#1798656511yescharacteristic
270C2#1798656911yescharacteristic
271C2#1798659012yes—
272C2#1798659212yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#397×C71C2
hypercenter#397D71
hypocenterC71#2623
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D71C2semidirect
3. #17985941C2semidirect
4. #17985935C2semidirect
5. #17985931C2semidirect
6. #17985930C2semidirect
7. #17985924C2semidirect
9. #17985873C2semidirect
10. #17985861C2semidirect
11. #397×C71C2semidirect
137. C71#2623semidirect
145. #397D71semidirect