Finite Group Databasebeta

← #22592618

Subgroups — #22592618

Order 9088.

700 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×C22⋊Dic71C211yesmaximal, characteristic
2#17985927C214yesmaximal, complemented
3#17985872C212yesmaximal, complemented
4#17985871C212yesmaximal, complemented
5#17985862C212yesmaximal, complemented
6#17985860C212yesmaximal, complemented
7#17985855C211yesmaximal, characteristic
8C22×C22⋊C4×C71C211yesmaximal, characteristic
9C23×Dic71C2211yescharacteristic
10C23×Dic71C2211yescharacteristic
11C2×Dic71⋊C4C2214yescomplemented
12C2×C22⋊Dic71—24no—
13C2×C22⋊Dic71C2212yes—
14C2×C22⋊Dic71C2212yes—
15C2×C22⋊Dic71C2212yes—
16C2×C22⋊Dic71C2212yes—
17C2×C22⋊Dic71C2212yes—
18C71⋊#95C2214yescomplemented
19C71⋊#95C2214yescomplemented
20C71⋊#95C2214yescomplemented
21C25×C71C2211yescharacteristic
22C23×C4×C71C2211yescharacteristic
23C23×C4×C71C2211yescharacteristic
24C2×C22⋊C4×C71—24no—
25C2×C22⋊C4×C71C2212yes—
26C2×C22⋊C4×C71C2212yes—
27C22×Dic71—22no—
28C22×Dic71—22no—
29C22×Dic71—24no—
30C22×Dic71—42no—
31C22×Dic71C2312yes—
32C22×Dic71C2312yes—
33C22×Dic71C2314yes—
34Dic71⋊C4—44no—
35C22⋊Dic71—24no—
36C22⋊Dic71—24no—
37C22⋊Dic71—24no—
38C22⋊Dic71—24no—
39C22⋊Dic71—42no—
40C24×C71—24no—
41C24×C71—24no—
42C24×C71—42no—
43C24×C71D414yes—
44C24×C71C2311yescharacteristic
45C24×C71C2311yescharacteristic
46C24×C71C2311yescharacteristic
47C22×C4×C71—22no—
48C22×C4×C71—22no—
49C22×C4×C71—24no—
50C22×C4×C71—42no—
51C22×C4×C71C2312yes—
52C22×C4×C71C2312yes—
53C22⋊C4×C71—24no—
54C22⋊C4×C71—24no—
55C23×C71—22no—
56C23×C71—22no—
57C23×C71—22no—
58C23×C71—22no—
59C23×C71—28no—
60C23×C71—28no—
61C23×C71—22no—
62C23×C71—22no—
63C23×C71—22no—
64C23×C71—44no—
65C23×C71—416no—
66C23×C71C2×D412yes—
67C23×C71C2×D412yes—
68C23×C71C2×D412yes—
69C23×C71#4114yes—
70C23×C71#4114yes—
71C23×C71C2411yescharacteristic
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—24no—
75C2×C4×C71—24no—
76C2×C4×C71—44no—
77C2×C4×C71—44no—
78C2×Dic71—22no—
79C2×Dic71—22no—
80C2×Dic71—24no—
81C2×Dic71—24no—
82C2×Dic71—24no—
83C2×Dic71—24no—
84C2×Dic71—24no—
85C2×Dic71—44no—
86C2×Dic71—44no—
87C22×C71—22no—
88C22×C71—22no—
89C22×C71—22no—
90C22×C71—22no—
91C22×C71—28no—
92C22×C71—28no—
93C22×C71—22no—
94C22×C71—22no—
95C22×C71—22no—
96C22×C71—416no—
97C22×C71—44no—
98C22×C71#12314yes—
99C22×C71#12314yes—
100C22×C71C22×D411yescharacteristic
101C22×C71C2×#4112yes—
102C22×C71C2×#4112yes—
103C22×C71H(5,𝔽2)12yes—
104C4×C71—42no—
105C4×C71—42no—
106Dic71—42no—
107Dic71—42no—
108Dic71—44no—
109C2×C71—24no—
110C2×C71—24no—
111C2×C71—42no—
112C2×C71C2×#12311yescharacteristic
113C2×C71C2×#12311yescharacteristic
114C2×C71#53411yescharacteristic
115C2×C71#55114yes—
116#2483—711nomaximal, Sylow
117C71#248311yescharacteristic, complemented, Sylow
118C22×C22⋊C4—711no—
119C22×C22⋊C4D7111yescharacteristic
120#397—714no—
121#385—712no—
122#385—712no—
123#381—712no—
124#381—712no—
125#379—711no—
126C25C2×D7111yescharacteristic
127C23×C4—711no—
128C23×C4—711no—
129C23×C4C2×D7111yescharacteristic
130C23×C4C2×D7111yescharacteristic
131C2×C4⋊C4—714no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—712no—
134C2×C22⋊C4—712no—
135C2×C22⋊C4—712no—
136C2×C22⋊C4—712no—
137C2×C22⋊C4—712no—
138C2×C22⋊C4—1424no—
139C2×C22⋊C4C2×D7112yes—
140C2×C22⋊C4C2×D7112yes—
141#95—714no—
142#95—714no—
143#95—714no—
144C24—24no—
145C24—24no—
146C24—42no—
147C24C22⋊D7114yes—
148C24C22×D7111yescharacteristic
149C24C22×D7111yescharacteristic
150C24C22×D7111yescharacteristic
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—42no—
155C22×C4—2842no—
156C22×C4—712no—
157C22×C4—714no—
158C22×C4—712no—
159C22×C4—1422no—
160C22×C4—1422no—
161C22×C4—1424no—
162C22×C4C22×D7112yes—
163C22×C4C22×D7112yes—
164C4⋊C4—2844no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—2842no—
168C22⋊C4—1424no—
169C22⋊C4—1424no—
170C22⋊C4—1424no—
171C22⋊C4—1424no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—28no—
180C23—28no—
181C23—44no—
182C23—416no—
183C23C2×C22⋊D7112yes—
184C23C2×C22⋊D7112yes—
185C23C2×C22⋊D7112yes—
186C23#1176586414yes—
187C23#1176586514yes—
188C23C23×D7111yescharacteristic
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—2844no—
196C2×C4—2844no—
197C2×C4—1422no—
198C2×C4—1422no—
199C2×C4—1424no—
200C2×C4—1424no—
201C2×C4—1424no—
202C2×C4—1424no—
203C2×C4—1424no—
204C22—28no—
205C22—28no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—416no—
214C22—44no—
215C22#1455610914yes—
216C22#1455611114yes—
217C22C22×C22⋊D7111yescharacteristic
218C22C2×#1176586412yes—
219C22C2×#1176586512yes—
220C22#1455617312yes—
221C4—42no—
222C4—42no—
223C4—2844no—
224C4—2842no—
225C4—2842no—
226C2—24no—
227C2—24no—
228C2—42no—
229C2C2×#1455610911yescharacteristic
230C2C2×#1455611111yescharacteristic
231C2#1798655811yescharacteristic
232C2#1798667514yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC22×C22⋊C4×C71C2
hypercenterC22×C22⋊C4D71
hypocenterC71#2483
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #17985927C2semidirect
3. #17985872C2semidirect
4. #17985871C2semidirect
5. #17985862C2semidirect
6. #17985860C2semidirect
11. C2×Dic71⋊C4C22semidirect
18. C71⋊#95C22semidirect
19. C71⋊#95C22semidirect
20. C71⋊#95C22semidirect
117. C71#2483semidirect