Finite Group Databasebeta

← #22596019

Subgroups — #22596019

Order 9088.

448 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 121 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#17985941C212yesmaximal, complemented
2#17985926C212yesmaximal, complemented
3#17985916C211yesmaximal, characteristic
4#17985891C212yesmaximal, complemented
5#17985877C212yesmaximal
6#17985873C212yesmaximal, complemented
7#17985870C212yesmaximal, complemented
8#17985864C211yesmaximal, characteristic, complemented
9#400×C71C211yesmaximal, characteristic, complemented
10C22×D284C2211yescharacteristic
11C22×C4×D71C2212yes—
12C2×C4⋊Dic71C2211yescharacteristic
13C2×C4⋊Dic71C2212yes—
14C2×Dic71⋊C4C2212yes—
15C2×(C2×D71)⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×C4×Dic71C2211yescharacteristic
21C71⋊#95C2212yes—
22C71⋊#95C2212yes—
23C71⋊#95C2212yes—
24C71⋊#95C2212yes—
25C71⋊#95C2211yescharacteristic
26C2×C4⋊C4×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28#95×C71C2212yes—
29#95×C71C2212yes—
30#95×C71C2211yescharacteristic
31C23×D71C2312yes—
32C2×D284—24no—
33C2×D284—41no—
34C22×Dic71C2312yes—
35C22×Dic71C2312yes—
36C22×Dic71C2312yes—
37C2×C4×D71—42no—
38C2×C4×D71—42no—
39C2×C4×D71—42no—
40C4⋊Dic71—42no—
41C4⋊Dic71—41no—
42Dic71⋊C4—42no—
43(C2×D71)⋊C4—24no—
44(C2×D71)⋊C4—24no—
45(C2×D71)⋊C4—24no—
46(C2×D71)⋊C4—42no—
47(C2×D71)⋊C4—42no—
48C4×Dic71—22no—
49C22×C4×C71C2312yes—
50C22×C4×C71C2311yescharacteristic
51C22×C4×C71C2312yes—
52C22×C4×C71C2311yescharacteristic
53C22×C4×C71C2311yescharacteristic
54C4⋊C4×C71—41no—
55C4⋊C4×C71—41no—
56C23×C71C2411yescharacteristic
57C22×D71—22no—
58C22×D71—22no—
59C22×D71—22no—
60C22×D71—22no—
61C22×D71—22no—
62C22×D71—22no—
63C22×D71#4114yes—
64C2×C4×C71—21no—
65C2×C4×C71—21no—
66C2×C4×C71—22no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71—21no—
70C2×C4×C71—22no—
71C2×C4×C71—21no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71—22no—
76C2×C4×C71#4112yes—
77C2×Dic71—22no—
78C2×Dic71—22no—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86D284—44no—
87C4×D71—48no—
88C22×C71C2×#4112yes—
89C22×C71C2×#4111yescharacteristic
90C22×C71H(5,𝔽2)11yescharacteristic
91C22×C71H(5,𝔽2)12yes—
92C22×C712−1+411yescharacteristic
93C2×D71—24no—
94C2×D71—24no—
95C2×D71—24no—
96C2×D71—42no—
97C2×D71—42no—
98C2×D71—42no—
99C2×D71—42no—
100C4×C71—22no—
101C4×C71—42no—
102C4×C71—41no—
103C4×C71—41no—
104C4×C71—42no—
105Dic71—42no—
106Dic71—42no—
107Dic71—42no—
108C2×C71#53912yes—
109C2×C71#53911yescharacteristic
110C2×C71#55111yescharacteristic
111C2×C71#55312yes—
112C2×C71H(3,𝔽4)11yescharacteristic
113D71—44no—
114#2782—711nomaximal, Sylow
115C71#278211yescharacteristic, complemented, Sylow
116#400—712no—
117#400D7111yescharacteristic, complemented
118#397—712no—
119#394—711no—
120#388—712no—
121#386—712no—
122#385—712no—
123#385—712no—
124#382—711no—
125C22×D4—711no—
126C23×C4—712no—
127C2×C4⋊C4—711no—
128C2×C4⋊C4—712no—
129C2×C4⋊C4—712no—
130C2×C4⋊C4C2×D7111yescharacteristic
131C2×C4⋊C4C2×D7111yescharacteristic
132C2×C22⋊C4—712no—
133C2×C22⋊C4—712no—
134C2×C22⋊C4—712no—
135C2×C22⋊C4—712no—
136C2×C22⋊C4—712no—
137C2×C42—711no—
138#95—712no—
139#95—712no—
140#95—712no—
141#95—712no—
142#95—711no—
143#95C2×D7112yes—
144#95C2×D7112yes—
145#95C2×D7111yescharacteristic
146C24—712no—
147C2×D4—2841no—
148C2×D4—1424no—
149C22×C4—2842no—
150C22×C4—2842no—
151C22×C4—2842no—
152C22×C4—712no—
153C22×C4—712no—
154C22×C4—712no—
155C22×C4C22×D7111yescharacteristic
156C22×C4C22×D7111yescharacteristic
157C22×C4C22×D7112yes—
158C22×C4C22×D7112yes—
159C22×C4C22×D7111yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—2842no—
163C4⋊C4—2841no—
164C4⋊C4—2842no—
165C22⋊C4—2842no—
166C22⋊C4—2842no—
167C22⋊C4—1424no—
168C22⋊C4—1424no—
169C22⋊C4—1424no—
170C42—1422no—
171C23—714no—
172C23—1422no—
173C23—1422no—
174C23—1422no—
175C23—1422no—
176C23—1422no—
177C23—1422no—
178C23C23×D7111yescharacteristic
179D4—2844no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—2848no—
193C2×C4—1422no—
194C2×C4—1422no—
195C2×C4—1422no—
196C2×C4—1422no—
197C2×C4—1422no—
198C2×C4—1422no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4—1422no—
202C2×C4#1176586612yes—
203C22—2842no—
204C22—2842no—
205C22—2842no—
206C22—2842no—
207C22—1424no—
208C22—1424no—
209C22—1424no—
210C22#41×D7112yes—
211C22C2×#1176586611yescharacteristic
212C22#1455617311yescharacteristic
213C22#1455617412yes—
214C22#1455617511yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—42no—
220C4—2842no—
221C4—2842no—
222C4—2842no—
223C2—2844no—
224C2#1798659012yes—
225C2#1798659811yescharacteristic
226C2#1798667711yescharacteristic
227C2#1798670212yes—
228C2#1798674111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2782
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #17985941C2semidirect
2. #17985926C2semidirect
4. #17985891C2semidirect
6. #17985873C2semidirect
7. #17985870C2semidirect
8. #17985864C2semidirect
9. #400×C71C2semidirect
115. C71#2782semidirect
117. #400D71semidirect