Finite Group Databasebeta

← #22596402

Subgroups — #22596402

Order 9088.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14556126C211yesmaximal, characteristic, complemented
2C2×#14556121C211yesmaximal, characteristic
3#17985941C212yesmaximal, complemented
4#17985935C212yesmaximal
5#17985933C212yesmaximal, complemented
6#17985927C212yesmaximal
7#17985920C211yesmaximal, characteristic, complemented
8#17985918C211yesmaximal, characteristic, complemented
9#17985911C211yesmaximal, characteristic
10#17985866C211yesmaximal, characteristic, complemented
11#393×C71C211yesmaximal, characteristic
12C22×Dic142C2211yescharacteristic
13#14556126—24no—
14#14556121—24no—
15C2×C4⋊Dic71C2211yescharacteristic
16C2×C4⋊Dic71C2212yes—
17C2×Dic71⋊C4C2211yescharacteristic
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2212yes—
21C2×Dic71⋊C4C2212yes—
22C2×C22⋊Dic71C2211yescharacteristic
23C2×C22⋊Dic71C2211yescharacteristic
24C2×C22⋊Dic71C2212yes—
25C2×C4×Dic71C2211yescharacteristic
26C71⋊#95C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C22×Q8×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C2×Dic142—22no—
38C2×Dic142—42no—
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C4⋊Dic71—22no—
46C4⋊Dic71—42no—
47Dic71⋊C4—22no—
48Dic71⋊C4—22no—
49Dic71⋊C4—24no—
50Dic71⋊C4—24no—
51Dic71⋊C4—42no—
52C22⋊Dic71—22no—
53C22⋊Dic71—22no—
54C22⋊Dic71—42no—
55C4×Dic71D414yes—
56C24×C71C2311yescharacteristic
57C2×Q8×C71—22no—
58C2×Q8×C71—42no—
59C22×C4×C71C2312yes—
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63C22⋊C4×C71—22no—
64C22⋊C4×C71—42no—
65C23×C71—21no—
66C23×C71—22no—
67C23×C71—22no—
68C23×C71—21no—
69C23×C71—21no—
70C23×C71C2411yescharacteristic
71Q8×C71—44no—
72C2×C4×C71—22no—
73C2×C4×C71—21no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71C2×D412yes—
82C2×Dic71—22no—
83C2×Dic71—21no—
84C2×Dic71—21no—
85C2×Dic71—21no—
86C2×Dic71—21no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—21no—
95C2×Dic71—22no—
96C2×Dic71C2×D412yes—
97C2×Dic71C2×D412yes—
98Dic142—44no—
99C22×C71—41no—
100C22×C71—41no—
101C22×C71—41no—
102C22×C71—42no—
103C22×C71—42no—
104C22×C71C22×D411yescharacteristic
105C22×C71H(5,𝔽2)12yes—
106C22×C712−1+412yes—
107C22×C712−1+411yescharacteristic
108C22×C712−1+411yescharacteristic
109C4×C71—22no—
110C4×C71—42no—
111C4×C71—41no—
112C4×C71—42no—
113Dic71—22no—
114Dic71—22no—
115Dic71—42no—
116Dic71—41no—
117Dic71—41no—
118Dic71—42no—
119C2×C71—81no—
120C2×C71#53511yescharacteristic
121C2×C71#53611yescharacteristic
122C2×C71#53611yescharacteristic
123C2×C71#56312yes—
124C2×C71#56312yes—
125#2821—711nomaximal, Sylow
126C71#282111yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—711no—
128C2×#126—711no—
129#400—712no—
130#399—712no—
131#398—712no—
132#397—712no—
133#395—711no—
134#395—711no—
135#393—711no—
136#393D7111yescharacteristic
137#383—711no—
138C22×Q8—711no—
139C22×Q8C2×D7111yescharacteristic
140C4⋊Q8—1424no—
141#126—1424no—
142C2×C4⋊C4—712no—
143C2×C4⋊C4—712no—
144C2×C4⋊C4—711no—
145C2×C4⋊C4—711no—
146C2×C4⋊C4—711no—
147C2×C4⋊C4—712no—
148C2×C4⋊C4—712no—
149C2×C22⋊C4—712no—
150C2×C22⋊C4—711no—
151C2×C22⋊C4—711no—
152C2×C22⋊C4C2×D7111yescharacteristic
153C2×C22⋊C4C2×D7112yes—
154C2×C42—711no—
155#95—711no—
156#95—712no—
157#95—712no—
158#95—712no—
159#95—712no—
160#95—712no—
161#95C2×D7111yescharacteristic
162#95C2×D7112yes—
163C24C22×D7111yescharacteristic
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—2842no—
167C2×Q8—1422no—
168C22×C4—711no—
169C22×C4—711no—
170C22×C4—712no—
171C22×C4—711no—
172C22×C4—712no—
173C22×C4—711no—
174C22×C4C22×D7112yes—
175C22×C4C22×D7111yescharacteristic
176C22×C4C22×D7112yes—
177C22×C4C22×D7111yescharacteristic
178C4⋊C4—2842no—
179C4⋊C4—2842no—
180C4⋊C4—1422no—
181C4⋊C4—1424no—
182C4⋊C4—1424no—
183C4⋊C4—1422no—
184C4⋊C4—1422no—
185C22⋊C4—22no—
186C22⋊C4—42no—
187C22⋊C4—2842no—
188C22⋊C4—1422no—
189C22⋊C4—1422no—
190C42—714no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23—21no—
195C23—21no—
196C23C23×D7111yescharacteristic
197Q8—44no—
198Q8—2844no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—712no—
209C2×C4—712no—
210C2×C4—1421no—
211C2×C4—1421no—
212C2×C4—1422no—
213C2×C4—1422no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1421no—
218C2×C4—1422no—
219C2×C4—1421no—
220C2×C4—1422no—
221C2×C4—1422no—
222C2×C4—1421no—
223C2×C4—1421no—
224C2×C4D4×D7112yes—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22—42no—
230C22C2×D4×D7111yescharacteristic
231C22#1455617312yes—
232C22#1455617512yes—
233C22#1455617511yescharacteristic
234C22#1455617611yescharacteristic
235C4—22no—
236C4—42no—
237C4—41no—
238C4—42no—
239C4—2842no—
240C4—2841no—
241C4—2841no—
242C4—2842no—
243C4—1422no—
244C4—1422no—
245C2—81no—
246C2#1798656411yescharacteristic
247C2#1798656911yescharacteristic
248C2#1798657211yescharacteristic
249C2#1798674912yes—
250C2#1798675012yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#393×C71C2
hypercenter#393D71
hypocenterC71#2821
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14556126C2semidirect
3. #17985941C2semidirect
5. #17985933C2semidirect
7. #17985920C2semidirect
8. #17985918C2semidirect
10. #17985866C2semidirect
126. C71#2821semidirect