Finite Group Databasebeta

← #22593981

Subgroups — #22593981

Order 9088.

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

normal subgroups only

This group has 470 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×#14556123C211yesmaximal, characteristic
2C2×#14556093C211yesmaximal, characteristic, complemented
3#17985934C211yesmaximal, characteristic
4#17985919C211yesmaximal, characteristic, complemented
5#17985912C211yesmaximal, characteristic
6#17985896C212yesmaximal, complemented
7#17985888C212yesmaximal
8#17985888C212yesmaximal
9#17985881C212yesmaximal, complemented
10#17985844C211yesmaximal, characteristic, complemented
11#400×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13#14556123—24no—
14#14556093—24no—
15C2×C4⋊Dic71C2211yescharacteristic
16C2×Dic71⋊C4C2211yescharacteristic
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2212yes—
19C2×Dic71⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2211yescharacteristic
24C2×(C2×D71)⋊C4C2211yescharacteristic
25C2×C4×Dic71C2212yes—
26C2×C4×Dic71C2212yes—
27C2×C4×Dic71C2211yescharacteristic
28C71⋊#95C2212yes—
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—24no—
40C22×Dic71C2311yescharacteristic
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2312yes—
44C22×Dic71C2312yes—
45C4⋊Dic71—22no—
46Dic71⋊C4—24no—
47Dic71⋊C4—22no—
48Dic71⋊C4—24no—
49Dic71⋊C4—24no—
50(C2×D71)⋊C4—22no—
51(C2×D71)⋊C4—22no—
52(C2×D71)⋊C4—22no—
53(C2×D71)⋊C4—24no—
54(C2×D71)⋊C4—24no—
55C4×Dic71—24no—
56C4×Dic71—24no—
57C4×Dic71D414yes—
58C22×C4×C71C2312yes—
59C22×C4×C71C2311yescharacteristic
60C22×C4×C71C2312yes—
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2311yescharacteristic
63C4⋊C4×C71—22no—
64C4⋊C4×C71—22no—
65C23×C71C2411yescharacteristic
66C22×D71—21no—
67C22×D71—22no—
68C22×D71—22no—
69C22×D71—21no—
70C22×D71—21no—
71C2×C4×C71—22no—
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×C71C2×D412yes—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—21no—
89C2×Dic71—21no—
90C2×Dic71—22no—
91C2×Dic71—21no—
92C2×Dic71—22no—
93C2×Dic71C2×D412yes—
94C2×Dic71C2×D412yes—
95C2×Dic71#4114yes—
96C2×Dic71#4114yes—
97Dic142—28no—
98C22×C71C22×D411yescharacteristic
99C22×C71C2×#4112yes—
100C22×C71C2×#4112yes—
101C22×C712−1+411yescharacteristic
102C22×C712−1+411yescharacteristic
103C2×D71—41no—
104C2×D71—41no—
105C2×D71—41no—
106C2×D71—42no—
107C2×D71—42no—
108C4×C71—22no—
109C4×C71—41no—
110C4×C71—42no—
111C4×C71—41no—
112C4×C71—42no—
113Dic71—24no—
114Dic71—22no—
115Dic71—22no—
116Dic71—24no—
117Dic71—41no—
118C2×C71#52511yescharacteristic
119C2×C71#53211yescharacteristic
120C2×C71#53611yescharacteristic
121C2×C71#55212yes—
122C2×C71#55612yes—
123D71—81no—
124#2604—711nomaximal, Sylow
125C71#260411yescharacteristic, complemented, Sylow
126C2×#126—711no—
127C2×C4×Q8—711no—
128#400D7111yescharacteristic, complemented
129#399—711no—
130#395—711no—
131#393—711no—
132#389—712no—
133#388—712no—
134#388—712no—
135#387—712no—
136#376—711no—
137C22×Q8—711no—
138#126—1424no—
139C4×Q8—1424no—
140C2×C4⋊C4—712no—
141C2×C4⋊C4—711no—
142C2×C4⋊C4—711no—
143C2×C4⋊C4—712no—
144C2×C4⋊C4—712no—
145C2×C4⋊C4C2×D7111yescharacteristic
146C2×C4⋊C4C2×D7111yescharacteristic
147C2×C22⋊C4—711no—
148C2×C22⋊C4—711no—
149C2×C22⋊C4—711no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4—712no—
152C2×C42—711no—
153C2×C42—712no—
154C2×C42—712no—
155#95—712no—
156#95—711no—
157#95—712no—
158#95—712no—
159#95C2×D7111yescharacteristic
160#95C2×D7112yes—
161#95C2×D7112yes—
162C24—711no—
163C2×Q8—1424no—
164C2×Q8—1422no—
165C22×C4—712no—
166C22×C4—711no—
167C22×C4—712no—
168C22×C4—711no—
169C22×C4—711no—
170C22×C4C22×D7112yes—
171C22×C4C22×D7111yescharacteristic
172C22×C4C22×D7112yes—
173C22×C4C22×D7111yescharacteristic
174C22×C4C22×D7111yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—1424no—
178C4⋊C4—1424no—
179C4⋊C4—1424no—
180C4⋊C4—1422no—
181C4⋊C4—1422no—
182C22⋊C4—1424no—
183C22⋊C4—1424no—
184C22⋊C4—1422no—
185C22⋊C4—1422no—
186C22⋊C4—1422no—
187C42—714no—
188C42—1424no—
189C42—1424no—
190C23—1421no—
191C23—1422no—
192C23—1422no—
193C23—1421no—
194C23—1421no—
195C23C23×D7111yescharacteristic
196Q8—1428no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—712no—
210C2×C4—712no—
211C2×C4—714no—
212C2×C4—714no—
213C2×C4—1421no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1421no—
220C2×C4—1421no—
221C2×C4—1422no—
222C2×C4D4×D7112yes—
223C22—2841no—
224C22—2842no—
225C22—2842no—
226C22—2841no—
227C22—2841no—
228C22C2×D4×D7111yescharacteristic
229C22#41×D7112yes—
230C22#41×D7112yes—
231C22#1455617511yescharacteristic
232C22#1455617611yescharacteristic
233C4—22no—
234C4—41no—
235C4—42no—
236C4—41no—
237C4—42no—
238C4—2841no—
239C4—1422no—
240C4—1424no—
241C4—1424no—
242C4—1422no—
243C2—5681no—
244C2#1798650711yescharacteristic
245C2#1798654811yescharacteristic
246C2#1798657211yescharacteristic
247C2#1798668112yes—
248C2#1798671612yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2604
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#14556093C2semidirect
4. #17985919C2semidirect
6. #17985896C2semidirect
9. #17985881C2semidirect
10. #17985844C2semidirect
11. #400×C71C2semidirect
125. C71#2604semidirect
128. #400D71semidirect