Finite Group Databasebeta

← #22596459

Subgroups — #22596459

Order 9088.

508 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#17985938C211yesmaximal, characteristic, complemented
2#17985927C212yesmaximal, complemented
3#17985916C211yesmaximal, characteristic, complemented
4#17985915C212yesmaximal, complemented
5#17985914C212yesmaximal, complemented
6#17985908C212yesmaximal, complemented
7#17985907C212yesmaximal, complemented
8#17985890C212yesmaximal, complemented
9#390×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C22⋊D71C2212yes—
12C22×C22⋊D71C2212yes—
13C2×C4⋊Dic71C2211yescharacteristic
14C2×Dic71⋊C4C2212yes—
15C2×C22⋊Dic71C2212yes—
16C2×C22⋊Dic71C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C71⋊#95C2212yes—
22C71⋊#95C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2211yescharacteristic
25C71⋊#95C2211yescharacteristic
26C22×D4×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28C2×C22⋊C4×C71C2212yes—
29C2×C22⋊C4×C71C2212yes—
30C2×C42×C71C2211yescharacteristic
31C23×D71C2312yes—
32C2×D284—42no—
33C2×D284—41no—
34C2×C22⋊D71—42no—
35C2×C22⋊D71—42no—
36C2×C22⋊D71—42no—
37C2×C22⋊D71—42no—
38C2×C22⋊D71—42no—
39C2×C22⋊D71—42no—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C22×Dic71C2312yes—
43C4⋊Dic71—41no—
44Dic71⋊C4—42no—
45C22⋊Dic71—42no—
46C22⋊Dic71—42no—
47(C2×D71)⋊C4—24no—
48(C2×D71)⋊C4—42no—
49(C2×D71)⋊C4—42no—
50(C2×D71)⋊C4—42no—
51C24×C71C2312yes—
52C2×D4×C71—22no—
53C2×D4×C71—41no—
54C2×D4×C71—41no—
55C22×C4×C71C2311yescharacteristic
56C22×C4×C71C2312yes—
57C22×C4×C71C2311yescharacteristic
58C22×C4×C71C2311yescharacteristic
59C4⋊C4×C71—41no—
60C22⋊C4×C71—42no—
61C22⋊C4×C71—42no—
62C42×C71—41no—
63C23×C71—22no—
64C23×C71—22no—
65C23×C71—22no—
66C23×C71—22no—
67C23×C71—22no—
68C23×C71—22no—
69C23×C71—22no—
70C23×C71C2411yescharacteristic
71C22×D71—22no—
72C22×D71—22no—
73C22×D71—22no—
74C22×D71—22no—
75C22×D71—22no—
76C22×D71—22no—
77C22×D71—22no—
78D4×C71—44no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71#4112yes—
89C22⋊D71—48no—
90C22⋊D71—48no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100D284—44no—
101C22×C71—42no—
102C22×C71—42no—
103C22×C71—42no—
104C22×C71—42no—
105C22×C71—42no—
106C22×C71—42no—
107C22×C71—42no—
108C22×C71C2×#4111yescharacteristic
109C22×C71H(5,𝔽2)11yescharacteristic
110C22×C71H(5,𝔽2)11yescharacteristic
111C22×C71H(5,𝔽2)12yes—
112C22×C71H(5,𝔽2)12yes—
113C2×D71—42no—
114C2×D71—42no—
115C2×D71—42no—
116C2×D71—42no—
117C2×D71—42no—
118C2×D71—42no—
119C2×D71—42no—
120C4×C71—22no—
121C4×C71—41no—
122C4×C71—41no—
123C4×C71—42no—
124Dic71—42no—
125Dic71—42no—
126Dic71—42no—
127C2×C71—82no—
128C2×C71#53811yescharacteristic
129C2×C71#54012yes—
130C2×C71#56011yescharacteristic
131C2×C71#56012yes—
132C2×C71H(3,𝔽4)11yescharacteristic
133D71—82no—
134#2828—711nomaximal, Sylow
135C71#282811yescharacteristic, complemented, Sylow
136#400—711no—
137#397—712no—
138#394—711no—
139#394—712no—
140#394—712no—
141#392—712no—
142#392—712no—
143#390D7111yescharacteristic, complemented
144#388—712no—
145C22×D4—712no—
146C22×D4—712no—
147C22×D4—711no—
148C22×D4C2×D7111yescharacteristic
149C2×C4⋊C4—712no—
150C2×C4⋊C4—711no—
151C2×C4⋊C4C2×D7111yescharacteristic
152C2×C22⋊C4—712no—
153C2×C22⋊C4—712no—
154C2×C22⋊C4—712no—
155C2×C22⋊C4—712no—
156C2×C22⋊C4—712no—
157C2×C22⋊C4—712no—
158C2×C22⋊C4C2×D7112yes—
159C2×C22⋊C4C2×D7112yes—
160C2×C42C2×D7111yescharacteristic
161#95—712no—
162#95—712no—
163#95—712no—
164#95—711no—
165#95—711no—
166C24—712no—
167C24C22×D7112yes—
168C2×D4—22no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—2842no—
172C2×D4—2841no—
173C2×D4—2842no—
174C2×D4—2842no—
175C2×D4—2842no—
176C2×D4—2842no—
177C2×D4—2842no—
178C2×D4—2842no—
179C22×C4—712no—
180C22×C4—712no—
181C22×C4—712no—
182C22×C4C22×D7111yescharacteristic
183C22×C4C22×D7112yes—
184C22×C4C22×D7111yescharacteristic
185C22×C4C22×D7111yescharacteristic
186C4⋊C4—41no—
187C4⋊C4—2842no—
188C4⋊C4—2841no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—2842no—
192C22⋊C4—2842no—
193C22⋊C4—2842no—
194C22⋊C4—2842no—
195C22⋊C4—2842no—
196C22⋊C4—1424no—
197C42—41no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—1422no—
206C23—1422no—
207C23—1422no—
208C23—1422no—
209C23—1422no—
210C23—1422no—
211C23—1422no—
212C23C23×D7111yescharacteristic
213D4—44no—
214D4—2844no—
215D4—2848no—
216D4—2848no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—1422no—
227C2×C4—1422no—
228C2×C4—1422no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4—1422no—
232C2×C4—1422no—
233C2×C4—1422no—
234C2×C4—1422no—
235C2×C4#1176586512yes—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—2842no—
244C22—2842no—
245C22—2842no—
246C22—2842no—
247C22—2842no—
248C22—2842no—
249C22—2842no—
250C22C2×#1176586511yescharacteristic
251C22#1455617311yescharacteristic
252C22#1455617311yescharacteristic
253C22#1455617312yes—
254C22#1455617412yes—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—2842no—
260C4—2842no—
261C4—2842no—
262C2—82no—
263C2—5682no—
264C2#1798658711yescharacteristic
265C2#1798660012yes—
266C2#1798673611yescharacteristic
267C2#1798673712yes—
268C2#1798674111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#390×C71C2
hypercenter#390D71
hypocenterC71#2828
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #17985938C2semidirect
2. #17985927C2semidirect
3. #17985916C2semidirect
4. #17985915C2semidirect
5. #17985914C2semidirect
6. #17985908C2semidirect
7. #17985907C2semidirect
8. #17985890C2semidirect
9. #390×C71C2semidirect
135. C71#2828semidirect
143. #390D71semidirect