Finite Group Databasebeta

← #22595349

Subgroups — #22595349

Order 9088.

492 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14556117C211yesmaximal, characteristic
3#17985921C212yesmaximal, complemented
4#17985917C212yesmaximal
5#17985912C212yesmaximal, complemented
6#17985891C212yesmaximal, complemented
7#17985890C212yesmaximal, complemented
8#17985884C212yesmaximal, complemented
9#391×C71C211yesmaximal, characteristic, complemented
10C22×Dic142C2211yescharacteristic
11C22×D284C2211yescharacteristic
12#14556126—24no—
13#14556117—24no—
14C2×C4⋊Dic71C2212yes—
15C2×Dic71⋊C4C2212yes—
16C2×Dic71⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×(C2×D71)⋊C4C2212yes—
22C2×(C2×D71)⋊C4C2212yes—
23C2×C4×Dic71C2211yescharacteristic
24C2×C4×Dic71C2211yescharacteristic
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C71⋊#95C2212yes—
28C22×Q8×C71C2211yescharacteristic
29C2×C4⋊C4×C71C2211yescharacteristic
30C2×C42×C71C2211yescharacteristic
31#95×C71C2212yes—
32#95×C71C2212yes—
33C23×D71C2312yes—
34C2×Dic142—22no—
35C2×Dic142—41no—
36C2×Dic142—41no—
37C2×D284—22no—
38C2×D284—41no—
39C2×D284—41no—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C22×Dic71C2312yes—
43C4⋊Dic71—24no—
44Dic71⋊C4—24no—
45Dic71⋊C4—24no—
46(C2×D71)⋊C4—24no—
47(C2×D71)⋊C4—24no—
48(C2×D71)⋊C4—24no—
49(C2×D71)⋊C4—24no—
50(C2×D71)⋊C4—42no—
51(C2×D71)⋊C4—42no—
52C4×Dic71—22no—
53C4×Dic71—22no—
54C2×Q8×C71—41no—
55C2×Q8×C71—41no—
56C2×Q8×C71D414yes—
57C22×C4×C71C2312yes—
58C22×C4×C71C2312yes—
59C22×C4×C71C2311yescharacteristic
60C22×C4×C71C2311yescharacteristic
61C22×C4×C71C2311yescharacteristic
62C4⋊C4×C71—41no—
63C42×C71—41no—
64C23×C71C2411yescharacteristic
65C22×D71—22no—
66C22×D71—22no—
67C22×D71—22no—
68C22×D71—22no—
69C22×D71—22no—
70C22×D71—22no—
71C22×D71—22no—
72Q8×C71—44no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71C2×D412yes—
86C2×C4×C71C2×D412yes—
87C2×C4×C71C2×D412yes—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71#4114yes—
97D284—44no—
98Dic142—44no—
99C22×C71C22×D411yescharacteristic
100C22×C71C2×#4112yes—
101C22×C71H(5,𝔽2)11yescharacteristic
102C22×C71H(5,𝔽2)11yescharacteristic
103C22×C712−1+412yes—
104C2×D71—42no—
105C2×D71—42no—
106C2×D71—42no—
107C2×D71—42no—
108C2×D71—42no—
109C2×D71—42no—
110C2×D71—42no—
111C4×C71—22no—
112C4×C71—22no—
113C4×C71—22no—
114C4×C71—42no—
115C4×C71—42no—
116Dic71—24no—
117Dic71—42no—
118Dic71—42no—
119C2×C71#53411yescharacteristic
120C2×C71#54212yes—
121C2×C71#54812yes—
122C2×C71#55511yescharacteristic
123C2×C71#56311yescharacteristic
124D71—82no—
125#2731—711nomaximal, Sylow
126C71#273111yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—711no—
128C2×#124—711no—
129#396—712no—
130#394—712no—
131#393—712no—
132#391D7111yescharacteristic, complemented
133#388—712no—
134#388—712no—
135#387—712no—
136C22×Q8—711no—
137C22×Q8C2×D7111yescharacteristic
138C22×D4—711no—
139C4⋊Q8—1424no—
140#124—1424no—
141C2×C4⋊C4—712no—
142C2×C4⋊C4—712no—
143C2×C4⋊C4—712no—
144C2×C4⋊C4C2×D7111yescharacteristic
145C2×C22⋊C4—712no—
146C2×C22⋊C4—712no—
147C2×C22⋊C4—712no—
148C2×C22⋊C4—712no—
149C2×C22⋊C4—712no—
150C2×C22⋊C4—712no—
151C2×C42—711no—
152C2×C42—711no—
153C2×C42C2×D7111yescharacteristic
154#95—712no—
155#95—712no—
156#95—712no—
157#95C2×D7112yes—
158#95C2×D7112yes—
159C24—712no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—2841no—
163C2×Q8—2841no—
164C2×Q8—1422no—
165C2×Q8D28414yes—
166C2×D4—2841no—
167C2×D4—2841no—
168C2×D4—1422no—
169C22×C4—712no—
170C22×C4—712no—
171C22×C4—712no—
172C22×C4C22×D7112yes—
173C22×C4C22×D7112yes—
174C22×C4C22×D7111yescharacteristic
175C22×C4C22×D7111yescharacteristic
176C22×C4C22×D7111yescharacteristic
177C4⋊C4—41no—
178C4⋊C4—1424no—
179C4⋊C4—1424no—
180C4⋊C4—1424no—
181C22⋊C4—2842no—
182C22⋊C4—2842no—
183C22⋊C4—1424no—
184C22⋊C4—1424no—
185C22⋊C4—1424no—
186C22⋊C4—1424no—
187C42—41no—
188C42—1422no—
189C42—1422no—
190C23—1422no—
191C23—1422no—
192C23—1422no—
193C23—1422no—
194C23—1422no—
195C23—1422no—
196C23—1422no—
197C23C23×D7111yescharacteristic
198Q8—44no—
199Q8—2844no—
200D4—2844no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—714no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1422no—
220C2×C4—1422no—
221C2×C4—1422no—
222C2×C4C2×D28412yes—
223C2×C4C2×D28412yes—
224C2×C4C2×D28412yes—
225C22—2842no—
226C22—2842no—
227C22—2842no—
228C22—2842no—
229C22—2842no—
230C22—2842no—
231C22—2842no—
232C22C22×D28411yescharacteristic
233C22#41×D7112yes—
234C22#1455617311yescharacteristic
235C22#1455617311yescharacteristic
236C22#1455617612yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—2842no—
243C4—2842no—
244C4—1424no—
245C2—5682no—
246C2#1798656111yescharacteristic
247C2#1798661912yes—
248C2#1798666112yes—
249C2#1798671111yescharacteristic
250C2#1798674811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#391×C71C2
hypercenter#391D71
hypocenterC71#2731
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14556126C2semidirect
3. #17985921C2semidirect
5. #17985912C2semidirect
6. #17985891C2semidirect
7. #17985890C2semidirect
8. #17985884C2semidirect
9. #391×C71C2semidirect
126. C71#2731semidirect
132. #391D71semidirect