Finite Group Databasebeta

← #22596404

Subgroups — #22596404

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×#14556123C211yesmaximal, characteristic
3#17985940C212yesmaximal, complemented
4#17985934C212yesmaximal
5#17985932C212yesmaximal, complemented
6#17985928C212yesmaximal
7#17985919C211yesmaximal, characteristic, complemented
8#17985912C211yesmaximal, characteristic
9#17985910C211yesmaximal, characteristic
10#17985866C211yesmaximal, characteristic, complemented
11#395×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13#14556126—24no—
14#14556123—24no—
15C2×C4⋊Dic71C2211yescharacteristic
16C2×Dic71⋊C4C2212yes—
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2211yescharacteristic
21C2×(C2×D71)⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×C4×Dic71C2211yescharacteristic
26C71⋊#95C2211yescharacteristic
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C22×Q8×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34C2×C4⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—42no—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C4⋊Dic71—22no—
46Dic71⋊C4—22no—
47Dic71⋊C4—24no—
48Dic71⋊C4—24no—
49Dic71⋊C4—42no—
50(C2×D71)⋊C4—22no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—42no—
54(C2×D71)⋊C4—42no—
55C4×Dic71D414yes—
56C2×Q8×C71—22no—
57C2×Q8×C71—42no—
58C22×C4×C71C2312yes—
59C22×C4×C71C2311yescharacteristic
60C22×C4×C71C2312yes—
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C4⋊C4×C71—22no—
64C4⋊C4×C71—42no—
65C23×C71C2411yescharacteristic
66C22×D71—21no—
67C22×D71—22no—
68C22×D71—21no—
69C22×D71—21no—
70C22×D71—22no—
71Q8×C71—44no—
72C2×C4×C71—22no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D412yes—
85C2×Dic71—21no—
86C2×Dic71—21no—
87C2×Dic71—22no—
88C2×Dic71—21no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71C2×D412yes—
97C2×Dic71C2×D412yes—
98Dic142—44no—
99C22×C71C22×D411yescharacteristic
100C22×C71H(5,𝔽2)12yes—
101C22×C712−1+411yescharacteristic
102C22×C712−1+411yescharacteristic
103C22×C712−1+412yes—
104C2×D71—41no—
105C2×D71—41no—
106C2×D71—41no—
107C2×D71—42no—
108C2×D71—42no—
109C4×C71—22no—
110C4×C71—41no—
111C4×C71—42no—
112C4×C71—41no—
113C4×C71—42no—
114Dic71—22no—
115Dic71—22no—
116Dic71—42no—
117Dic71—41no—
118Dic71—42no—
119C2×C71#53511yescharacteristic
120C2×C71#53611yescharacteristic
121C2×C71#53611yescharacteristic
122C2×C71#56312yes—
123C2×C71#56312yes—
124D71—81no—
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#395D7111yescharacteristic, complemented
135#393—711no—
136#393—711no—
137#383—711no—
138C22×Q8—711no—
139C22×Q8C2×D7111yescharacteristic
140C4⋊Q8—1424no—
141#126—1424no—
142C2×C4⋊C4—712no—
143C2×C4⋊C4—711no—
144C2×C4⋊C4—711no—
145C2×C4⋊C4—712no—
146C2×C4⋊C4—712no—
147C2×C4⋊C4C2×D7111yescharacteristic
148C2×C4⋊C4C2×D7112yes—
149C2×C22⋊C4—712no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4—711no—
152C2×C22⋊C4—711no—
153C2×C22⋊C4—711no—
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—
163C24—711no—
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×C4C22×D7112yes—
174C22×C4C22×D7111yescharacteristic
175C22×C4C22×D7112yes—
176C22×C4C22×D7111yescharacteristic
177C22×C4C22×D7111yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—2842no—
181C4⋊C4—1422no—
182C4⋊C4—1424no—
183C4⋊C4—1424no—
184C4⋊C4—1422no—
185C22⋊C4—2842no—
186C22⋊C4—2842no—
187C22⋊C4—1422no—
188C22⋊C4—1422no—
189C22⋊C4—1422no—
190C42—714no—
191C23—1421no—
192C23—1422no—
193C23—1421no—
194C23—1421no—
195C23—1422no—
196C23C23×D7111yescharacteristic
197Q8—44no—
198Q8—2844no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—712no—
212C2×C4—712no—
213C2×C4—1421no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1422no—
220C2×C4—1422no—
221C2×C4—1421no—
222C2×C4—1421no—
223C2×C4—1422no—
224C2×C4D4×D7112yes—
225C22—2841no—
226C22—2841no—
227C22—2841no—
228C22—2842no—
229C22—2842no—
230C22C2×D4×D7111yescharacteristic
231C22#1455617312yes—
232C22#1455617511yescharacteristic
233C22#1455617611yescharacteristic
234C22#1455617612yes—
235C4—22no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—42no—
240C4—2842no—
241C4—2841no—
242C4—2842no—
243C4—1422no—
244C4—1422no—
245C2—5681no—
246C2#1798656411yescharacteristic
247C2#1798657211yescharacteristic
248C2#1798657411yescharacteristic
249C2#1798674812yes—
250C2#1798675012yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#395×C71C2
hypercenter#395D71
hypocenterC71#2821
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#14556126C2semidirect
3. #17985940C2semidirect
5. #17985932C2semidirect
7. #17985919C2semidirect
10. #17985866C2semidirect
11. #395×C71C2semidirect
126. C71#2821semidirect
134. #395D71semidirect