Finite Group Databasebeta

← #22594504

Subgroups — #22594504

Order 9088.

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

normal subgroups only

This group has 434 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×#14556121C211yesmaximal, characteristic
2C2×#14556119C211yesmaximal, characteristic, complemented
3#17985937C211yesmaximal, characteristic, complemented
4#17985931C211yesmaximal, characteristic, complemented
5#17985930C212yesmaximal, complemented
6#17985924C211yesmaximal, characteristic
7#17985922C212yesmaximal
8#17985898C212yesmaximal, complemented
9#17985892C212yesmaximal
10#17985850C211yesmaximal, characteristic, complemented
11#399×C71C211yesmaximal, characteristic
12#14556121—24no—
13#14556119—24no—
14C2×C4⋊Dic71C2212yes—
15C2×C4⋊Dic71C2211yescharacteristic
16C2×C4⋊Dic71C2211yescharacteristic
17C2×C4⋊Dic71C2212yes—
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2212yes—
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2211yescharacteristic
22C2×C22⋊Dic71C2211yescharacteristic
23C2×C22⋊Dic71C2211yescharacteristic
24C2×C22⋊Dic71C2212yes—
25C2×C4×Dic71C2212yes—
26C2×C4×Dic71C2211yescharacteristic
27C71⋊#95C2211yescharacteristic
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2212yes—
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35C2×C22⋊C4×C71C2211yescharacteristic
36#95×C71C2211yescharacteristic
37C22×Dic71C2312yes—
38C22×Dic71C2311yescharacteristic
39C22×Dic71C2311yescharacteristic
40C22×Dic71C2312yes—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C4⋊Dic71—22no—
44C4⋊Dic71—24no—
45C4⋊Dic71—22no—
46C4⋊Dic71—42no—
47Dic71⋊C4—22no—
48Dic71⋊C4—24no—
49Dic71⋊C4—24no—
50Dic71⋊C4—22no—
51C22⋊Dic71—22no—
52C22⋊Dic71—22no—
53C22⋊Dic71—42no—
54C4×Dic71—42no—
55C4×Dic71D414yes—
56C24×C71C2311yescharacteristic
57C22×C4×C71C2311yescharacteristic
58C22×C4×C71C2312yes—
59C22×C4×C71C2311yescharacteristic
60C22×C4×C71C2312yes—
61C4⋊C4×C71—22no—
62C4⋊C4×C71—24no—
63C22⋊C4×C71—22no—
64C22⋊C4×C71—24no—
65C23×C71—21no—
66C23×C71—22no—
67C23×C71—21no—
68C23×C71—22no—
69C23×C71—21no—
70C23×C71C2411yescharacteristic
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71C2×D412yes—
80C2×C4×C71#4114yes—
81C2×Dic71—22no—
82C2×Dic71—21no—
83C2×Dic71—21no—
84C2×Dic71—21no—
85C2×Dic71—21no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—21no—
94C2×Dic71—22no—
95C2×Dic71C2×D412yes—
96C2×Dic71C2×D412yes—
97C22×C71—41no—
98C22×C71—41no—
99C22×C71—41no—
100C22×C71—42no—
101C22×C71—42no—
102C22×C71C22×D411yescharacteristic
103C22×C71C2×#4112yes—
104C22×C71H(5,𝔽2)11yescharacteristic
105C22×C712−1+412yes—
106C22×C712−1+411yescharacteristic
107C4×C71—22no—
108C4×C71—24no—
109C4×C71—41no—
110C4×C71—42no—
111Dic71—22no—
112Dic71—22no—
113Dic71—41no—
114Dic71—41no—
115Dic71—42no—
116Dic71—42no—
117C2×C71—81no—
118C2×C71#53211yescharacteristic
119C2×C71#53611yescharacteristic
120C2×C71#53711yescharacteristic
121C2×C71#53912yes—
122C2×C71#54112yes—
123#2664—711nomaximal, Sylow
124C71#266411yescharacteristic, complemented, Sylow
125C2×#126—711no—
126C2×#125—711no—
127#400—711no—
128#399D7111yescharacteristic
129#398—712no—
130#398—711no—
131#397—711no—
132#396—712no—
133#389—712no—
134#388—712no—
135C4×C4⋊C4—711no—
136#126—1424no—
137#125—1424no—
138C2×C4⋊C4—712no—
139C2×C4⋊C4—712no—
140C2×C4⋊C4—712no—
141C2×C4⋊C4—711no—
142C2×C4⋊C4—711no—
143C2×C4⋊C4—711no—
144C2×C4⋊C4—711no—
145C2×C4⋊C4—712no—
146C2×C4⋊C4C2×D7111yescharacteristic
147C2×C4⋊C4C2×D7112yes—
148C2×C22⋊C4—712no—
149C2×C22⋊C4—711no—
150C2×C22⋊C4—711no—
151C2×C22⋊C4C2×D7111yescharacteristic
152C2×C22⋊C4C2×D7112yes—
153C2×C42—712no—
154C2×C42—711no—
155#95—711no—
156#95—712no—
157#95—712no—
158#95—712no—
159#95—712no—
160#95C2×D7111yescharacteristic
161C24C22×D7111yescharacteristic
162C22×C4—711no—
163C22×C4—711no—
164C22×C4—712no—
165C22×C4—711no—
166C22×C4—711no—
167C22×C4—712no—
168C22×C4C22×D7111yescharacteristic
169C22×C4C22×D7112yes—
170C22×C4C22×D7111yescharacteristic
171C22×C4C22×D7112yes—
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—2842no—
175C4⋊C4—1424no—
176C4⋊C4—1422no—
177C4⋊C4—1424no—
178C4⋊C4—1422no—
179C4⋊C4—1422no—
180C4⋊C4—1422no—
181C4⋊C4—1424no—
182C22⋊C4—22no—
183C22⋊C4—24no—
184C22⋊C4—2842no—
185C22⋊C4—1422no—
186C22⋊C4—1422no—
187C42—2842no—
188C42—714no—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—22no—
193C23—21no—
194C23C23×D7111yescharacteristic
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—712no—
204C2×C4—712no—
205C2×C4—1421no—
206C2×C4—1421no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4—1422no—
210C2×C4—1422no—
211C2×C4—1421no—
212C2×C4—1422no—
213C2×C4—1422no—
214C2×C4—1421no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1421no—
218C2×C4—1421no—
219C2×C4D4×D7112yes—
220C2×C4#1176586414yes—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22C2×D4×D7111yescharacteristic
227C22C2×#1176586412yes—
228C22#1455617411yescharacteristic
229C22#1455617512yes—
230C22#1455617611yescharacteristic
231C4—22no—
232C4—24no—
233C4—41no—
234C4—42no—
235C4—2841no—
236C4—2841no—
237C4—2842no—
238C4—2842no—
239C4—1422no—
240C4—1422no—
241C2—81no—
242C2#1798654711yescharacteristic
243C2#1798657111yescharacteristic
244C2#1798657911yescharacteristic
245C2#1798659312yes—
246C2#1798660912yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#399×C71C2
hypercenter#399D71
hypocenterC71#2664
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#14556119C2semidirect
3. #17985937C2semidirect
4. #17985931C2semidirect
5. #17985930C2semidirect
8. #17985898C2semidirect
10. #17985850C2semidirect
124. C71#2664semidirect