Finite Group Databasebeta

← #22596322

Subgroups — #22596322

Order 9088.

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

normal subgroups only

This group has 428 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
1#17985926C212yesmaximal, complemented
2#17985917C212yesmaximal
3#17985910C212yesmaximal, complemented
4#17985901C211yesmaximal, characteristic
5#17985897C211yesmaximal, characteristic, complemented
6#17985891C212yesmaximal, complemented
7#17985889C212yesmaximal, complemented
8#17985886C212yesmaximal, complemented
9#391×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C2×C4⋊Dic71C2211yescharacteristic
12C2×C4⋊Dic71C2211yescharacteristic
13C2×C4⋊Dic71C2211yescharacteristic
14C2×Dic71⋊C4C2212yes—
15C2×(C2×D71)⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×C4×Dic71C2212yes—
22C71⋊#95C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2212yes—
26C22×Q8×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28C2×C42×C71C2211yescharacteristic
29#95×C71C2212yes—
30#95×C71C2212yes—
31C23×D71C2312yes—
32C2×D284—22no—
33C2×D284—41no—
34C2×D284—41no—
35C22×Dic71C2312yes—
36C22×Dic71C2312yes—
37C22×Dic71C2312yes—
38C4⋊Dic71—22no—
39C4⋊Dic71—41no—
40C4⋊Dic71—41no—
41Dic71⋊C4—24no—
42(C2×D71)⋊C4—24no—
43(C2×D71)⋊C4—24no—
44(C2×D71)⋊C4—42no—
45(C2×D71)⋊C4—42no—
46(C2×D71)⋊C4—42no—
47(C2×D71)⋊C4—42no—
48C4×Dic71—42no—
49C2×Q8×C71—22no—
50C2×Q8×C71—41no—
51C2×Q8×C71—41no—
52C22×C4×C71C2312yes—
53C22×C4×C71C2312yes—
54C22×C4×C71C2311yescharacteristic
55C22×C4×C71C2311yescharacteristic
56C22×C4×C71C2311yescharacteristic
57C4⋊C4×C71—41no—
58C42×C71—41no—
59C23×C71C2411yescharacteristic
60C22×D71—22no—
61C22×D71—22no—
62C22×D71—22no—
63C22×D71—22no—
64C22×D71—22no—
65C22×D71—22no—
66C22×D71—22no—
67Q8×C71—44no—
68C2×C4×C71—21no—
69C2×C4×C71—21no—
70C2×C4×C71—21no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
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#4112yes—
81C2×C4×C71#4112yes—
82C2×C4×C71#4112yes—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92D284—44no—
93C22×C71C2×#4111yescharacteristic
94C22×C71C2×#4111yescharacteristic
95C22×C71C2×#4111yescharacteristic
96C22×C71H(5,𝔽2)11yescharacteristic
97C22×C71H(5,𝔽2)11yescharacteristic
98C22×C71H(5,𝔽2)11yescharacteristic
99C22×C712−1+411yescharacteristic
100C2×D71—42no—
101C2×D71—42no—
102C2×D71—42no—
103C2×D71—42no—
104C2×D71—42no—
105C2×D71—42no—
106C2×D71—42no—
107C4×C71—22no—
108C4×C71—22no—
109C4×C71—22no—
110C4×C71—42no—
111C4×C71—42no—
112Dic71—42no—
113Dic71—42no—
114Dic71—42no—
115C2×C71#53811yescharacteristic
116C2×C71#53811yescharacteristic
117C2×C71#53911yescharacteristic
118C2×C71#55511yescharacteristic
119C2×C71#55611yescharacteristic
120C2×C71#55911yescharacteristic
121C2×C71#56211yescharacteristic
122D71—82no—
123#2808—711nomaximal, Sylow
124C71#280811yescharacteristic, complemented, Sylow
125#397—712no—
126#394—712no—
127#393—712no—
128#391D7111yescharacteristic, complemented
129#390—711no—
130#389—711no—
131#388—712no—
132#388—712no—
133#387—712no—
134C22×Q8C2×D7111yescharacteristic
135C22×D4—711no—
136C2×C4⋊C4—711no—
137C2×C4⋊C4—711no—
138C2×C4⋊C4—712no—
139C2×C4⋊C4—711no—
140C2×C4⋊C4C2×D7111yescharacteristic
141C2×C22⋊C4—712no—
142C2×C22⋊C4—712no—
143C2×C22⋊C4—712no—
144C2×C22⋊C4—712no—
145C2×C22⋊C4—712no—
146C2×C22⋊C4—712no—
147C2×C42—712no—
148C2×C42C2×D7111yescharacteristic
149#95—712no—
150#95—712no—
151#95—712no—
152#95—712no—
153#95C2×D7112yes—
154#95C2×D7112yes—
155C24—712no—
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×D4—2841no—
160C2×D4—2841no—
161C2×D4—1422no—
162C22×C4—712no—
163C22×C4—712no—
164C22×C4—712no—
165C22×C4C22×D7112yes—
166C22×C4C22×D7112yes—
167C22×C4C22×D7111yescharacteristic
168C22×C4C22×D7111yescharacteristic
169C22×C4C22×D7111yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—2841no—
172C4⋊C4—2841no—
173C4⋊C4—1424no—
174C4⋊C4—1422no—
175C22⋊C4—2842no—
176C22⋊C4—2842no—
177C22⋊C4—2842no—
178C22⋊C4—2842no—
179C22⋊C4—1424no—
180C22⋊C4—1424no—
181C42—41no—
182C42—2842no—
183C23—1422no—
184C23—1422no—
185C23—1422no—
186C23—1422no—
187C23—1422no—
188C23—1422no—
189C23—1422no—
190C23C23×D7111yescharacteristic
191Q8—44no—
192D4—2844no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—1422no—
206C2×C4—1422no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4—1422no—
210C2×C4—1422no—
211C2×C4—1422no—
212C2×C4—1422no—
213C2×C4—1422no—
214C2×C4#1176586512yes—
215C2×C4#1176586612yes—
216C2×C4#1176586612yes—
217C22—2842no—
218C22—2842no—
219C22—2842no—
220C22—2842no—
221C22—2842no—
222C22—2842no—
223C22—2842no—
224C22C2×#1176586511yescharacteristic
225C22C2×#1176586611yescharacteristic
226C22C2×#1176586611yescharacteristic
227C22#1455617311yescharacteristic
228C22#1455617311yescharacteristic
229C22#1455617411yescharacteristic
230C22#1455617511yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—2842no—
237C4—2842no—
238C4—2842no—
239C2—5682no—
240C2#1798658711yescharacteristic
241C2#1798658911yescharacteristic
242C2#1798659811yescharacteristic
243C2#1798671511yescharacteristic
244C2#1798672211yescharacteristic
245C2#1798673411yescharacteristic
246C2#1798674311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#391×C71C2
hypercenter#391D71
hypocenterC71#2808
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #17985926C2semidirect
3. #17985910C2semidirect
5. #17985897C2semidirect
6. #17985891C2semidirect
7. #17985889C2semidirect
8. #17985886C2semidirect
9. #391×C71C2semidirect
124. C71#2808semidirect
128. #391D71semidirect