Finite Group Databasebeta

← #22596440

Subgroups — #22596440

Order 9088.

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

normal subgroups only

This group has 464 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#17985938C212yesmaximal, complemented
2#17985928C212yesmaximal, complemented
3#17985921C212yesmaximal, complemented
4#17985916C212yesmaximal, complemented
5#17985915C212yesmaximal, complemented
6#17985908C211yesmaximal, characteristic
7#17985876C212yesmaximal, complemented
8#17985871C211yesmaximal, characteristic, complemented
9#399×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C22⋊D71C2212yes—
12C23×Dic71C2211yescharacteristic
13C2×C4⋊Dic71C2212yes—
14C2×Dic71⋊C4C2212yes—
15C2×C22⋊Dic71C2211yescharacteristic
16C2×(C2×D71)⋊C4C2212yes—
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⋊#95C2212yes—
26C2×C4⋊C4×C71C2212yes—
27C2×C4⋊C4×C71C2211yescharacteristic
28C2×C22⋊C4×C71C2211yescharacteristic
29C2×C22⋊C4×C71C2212yes—
30#95×C71C2211yescharacteristic
31C23×D71C2312yes—
32C2×D284—41no—
33C2×D284—42no—
34C2×C22⋊D71—24no—
35C2×C22⋊D71—42no—
36C2×C22⋊D71—42no—
37C22×Dic71—41no—
38C22×Dic71—42no—
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C4⋊Dic71—42no—
43Dic71⋊C4—42no—
44C22⋊Dic71—22no—
45(C2×D71)⋊C4—42no—
46(C2×D71)⋊C4—42no—
47(C2×D71)⋊C4—42no—
48(C2×D71)⋊C4—42no—
49(C2×D71)⋊C4—42no—
50C24×C71C2311yescharacteristic
51C22×C4×C71C2311yescharacteristic
52C22×C4×C71C2312yes—
53C22×C4×C71C2311yescharacteristic
54C22×C4×C71C2312yes—
55C4⋊C4×C71—41no—
56C4⋊C4×C71—42no—
57C22⋊C4×C71—42no—
58C22⋊C4×C71—41no—
59C23×C71—21no—
60C23×C71—22no—
61C23×C71—22no—
62C23×C71—21no—
63C23×C71#4112yes—
64C23×C71C2411yescharacteristic
65C22×D71—22no—
66C22×D71—22no—
67C22×D71—22no—
68C22×D71—22no—
69C22×D71—22no—
70C22×D71—22no—
71C22×D71—22no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C22⋊D71—48no—
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—
92C2×Dic71—44no—
93D284—44no—
94C22×C71—24no—
95C22×C71—22no—
96C22×C71—41no—
97C22×C71—41no—
98C22×C71—42no—
99C22×C71C2×#4111yescharacteristic
100C22×C71H(5,𝔽2)12yes—
101C22×C71H(5,𝔽2)11yescharacteristic
102C22×C71H(5,𝔽2)12yes—
103C22×C712−1+411yescharacteristic
104C2×D71—42no—
105C2×D71—42no—
106C2×D71—42no—
107C2×D71—42no—
108C2×D71—42no—
109C2×D71—42no—
110C2×D71—42no—
111C4×C71—41no—
112C4×C71—42no—
113C4×C71—41no—
114C4×C71—42no—
115Dic71—42no—
116Dic71—42no—
117Dic71—42no—
118C2×C71—42no—
119C2×C71#53812yes—
120C2×C71#53911yescharacteristic
121C2×C71#56011yescharacteristic
122C2×C71#56011yescharacteristic
123C2×C71#56212yes—
124D71—82no—
125#2826—711nomaximal, Sylow
126C71#282611yescharacteristic, complemented, Sylow
127#400—712no—
128#399D7111yescharacteristic, complemented
129#397—712no—
130#396—712no—
131#394—712no—
132#394—712no—
133#392—711no—
134#386—712no—
135#385—711no—
136C22×D4—712no—
137C22×D4—711no—
138C23×C4—711no—
139C2×C4⋊C4—712no—
140C2×C4⋊C4—712no—
141C2×C4⋊C4C2×D7112yes—
142C2×C4⋊C4C2×D7111yescharacteristic
143C2×C22⋊C4—712no—
144C2×C22⋊C4—712no—
145C2×C22⋊C4—712no—
146C2×C22⋊C4—712no—
147C2×C22⋊C4—712no—
148C2×C22⋊C4—711no—
149C2×C22⋊C4C2×D7111yescharacteristic
150C2×C22⋊C4C2×D7112yes—
151#95—712no—
152#95—712no—
153#95—712no—
154#95—711no—
155#95—712no—
156#95C2×D7111yescharacteristic
157C24—712no—
158C24C22×D7111yescharacteristic
159C2×D4—2841no—
160C2×D4—2842no—
161C2×D4—2842no—
162C2×D4—2842no—
163C2×D4—1424no—
164C22×C4—2841no—
165C22×C4—2842no—
166C22×C4—712no—
167C22×C4—712no—
168C22×C4—712no—
169C22×C4C22×D7111yescharacteristic
170C22×C4C22×D7112yes—
171C22×C4C22×D7112yes—
172C22×C4C22×D7111yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—2842no—
176C4⋊C4—2842no—
177C22⋊C4—42no—
178C22⋊C4—41no—
179C22⋊C4—2842no—
180C22⋊C4—2842no—
181C22⋊C4—2842no—
182C22⋊C4—2842no—
183C22⋊C4—2842no—
184C22⋊C4—1422no—
185C23—21no—
186C23—22no—
187C23—22no—
188C23—21no—
189C23—1422no—
190C23—1422no—
191C23—1422no—
192C23—1422no—
193C23—1422no—
194C23—1422no—
195C23—1422no—
196C23#1176586612yes—
197C23C23×D7111yescharacteristic
198D4—2844no—
199D4—2848no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—2844no—
211C2×C4—1422no—
212C2×C4—1422no—
213C2×C4—1422no—
214C2×C4—1422no—
215C2×C4—1422no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4—1422no—
219C2×C4—1422no—
220C22—24no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—2842no—
226C22—2842no—
227C22—2842no—
228C22—2842no—
229C22—2842no—
230C22—2842no—
231C22—2842no—
232C22C2×#1176586611yescharacteristic
233C22#1455617312yes—
234C22#1455617311yescharacteristic
235C22#1455617412yes—
236C22#1455617511yescharacteristic
237C4—41no—
238C4—42no—
239C4—41no—
240C4—42no—
241C4—2842no—
242C4—2842no—
243C4—2842no—
244C2—42no—
245C2—5682no—
246C2#1798658912yes—
247C2#1798659811yescharacteristic
248C2#1798673611yescharacteristic
249C2#1798673911yescharacteristic
250C2#1798674612yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#399×C71C2
hypercenter#399D71
hypocenterC71#2826
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #17985938C2semidirect
2. #17985928C2semidirect
3. #17985921C2semidirect
4. #17985916C2semidirect
5. #17985915C2semidirect
7. #17985876C2semidirect
8. #17985871C2semidirect
9. #399×C71C2semidirect
126. C71#2826semidirect
128. #399D71semidirect