Finite Group Databasebeta

← #22596040

Subgroups — #22596040

Order 9088.

472 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 129 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#17985921C212yesmaximal, complemented
2#17985919C211yesmaximal, characteristic, complemented
3#17985912C212yesmaximal, complemented
4#17985900C211yesmaximal, characteristic
5#17985882C212yesmaximal, complemented
6#17985877C212yesmaximal
7#17985873C212yesmaximal, complemented
8#17985858C212yesmaximal, complemented
9#400×C71C211yesmaximal, characteristic, complemented
10C22×Dic142C2211yescharacteristic
11C22×D284C2211yescharacteristic
12C22×C4×D71C2212yes—
13C2×C4⋊Dic71C2212yes—
14C2×Dic71⋊C4C2212yes—
15C2×Dic71⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×C4×Dic71C2211yescharacteristic
22C71⋊#95C2212yes—
23C71⋊#95C2211yescharacteristic
24C71⋊#95C2212yes—
25C71⋊#95C2212yes—
26C2×C4⋊C4×C71C2211yescharacteristic
27C2×C4⋊C4×C71C2211yescharacteristic
28#95×C71C2211yescharacteristic
29#95×C71C2212yes—
30#95×C71C2212yes—
31C23×D71C2312yes—
32C2×Dic142—41no—
33C2×Dic142—42no—
34C2×D284—41no—
35C2×D284—41no—
36C2×D284Q814yescomplemented
37C22×Dic71C2312yes—
38C22×Dic71C2312yes—
39C22×Dic71C2312yes—
40C2×C4×D71—24no—
41C2×C4×D71—42no—
42C2×C4×D71—42no—
43C4⋊Dic71—24no—
44Dic71⋊C4—42no—
45Dic71⋊C4—42no—
46(C2×D71)⋊C4—24no—
47(C2×D71)⋊C4—42no—
48(C2×D71)⋊C4—42no—
49(C2×D71)⋊C4—42no—
50(C2×D71)⋊C4—42no—
51C4×Dic71—41no—
52C22×C4×C71C2312yes—
53C22×C4×C71C2311yescharacteristic
54C22×C4×C71C2312yes—
55C22×C4×C71C2311yescharacteristic
56C22×C4×C71C2311yescharacteristic
57C4⋊C4×C71—41no—
58C4⋊C4×C71—41no—
59C23×C71C2411yescharacteristic
60C22×D71—22no—
61C22×D71—22no—
62C22×D71—22no—
63C22×D71—22no—
64C22×D71—22no—
65C22×D71—22no—
66C22×D71C2×Q814yes—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—22no—
79C2×C4×C71C2×Q812yes—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71—22no—
83C2×Dic71—22no—
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71—22no—
87C2×Dic71—22no—
88C2×Dic71#4114yes—
89D284—44no—
90C4×D71—48no—
91Dic142—44no—
92C22×C71C22×Q811yescharacteristic
93C22×C71C2×#4112yes—
94C22×C71H(5,𝔽2)11yescharacteristic
95C22×C71H(5,𝔽2)11yescharacteristic
96C22×C71H(5,𝔽2)12yes—
97C2×D71—24no—
98C2×D71—24no—
99C2×D71—24no—
100C2×D71—42no—
101C2×D71—42no—
102C2×D71—42no—
103C2×D71—42no—
104C4×C71—22no—
105C4×C71—42no—
106C4×C71—42no—
107C4×C71—41no—
108C4×C71—41no—
109Dic71—24no—
110Dic71—42no—
111Dic71—42no—
112C2×C71#53812yes—
113C2×C71#54311yescharacteristic
114C2×C71#55412yes—
115C2×C71#55511yescharacteristic
116C2×C71#56011yescharacteristic
117D71—44no—
118#2784—711nomaximal, Sylow
119C71#278411yescharacteristic, complemented, Sylow
120#400D7111yescharacteristic, complemented
121#396—712no—
122#395—711no—
123#393—712no—
124#390—711no—
125#387—712no—
126#386—712no—
127#385—712no—
128#380—712no—
129C22×Q8—711no—
130C22×D4—711no—
131C23×C4—712no—
132C2×C4⋊C4—712no—
133C2×C4⋊C4—712no—
134C2×C4⋊C4—712no—
135C2×C4⋊C4C2×D7111yescharacteristic
136C2×C4⋊C4C2×D7111yescharacteristic
137C2×C22⋊C4—712no—
138C2×C22⋊C4—712no—
139C2×C22⋊C4—712no—
140C2×C22⋊C4—712no—
141C2×C22⋊C4—712no—
142C2×C42—711no—
143#95—712no—
144#95—711no—
145#95—712no—
146#95—712no—
147#95C2×D7111yescharacteristic
148#95C2×D7112yes—
149#95C2×D7112yes—
150C24—712no—
151C2×Q8—2841no—
152C2×Q8—2842no—
153C2×D4—2841no—
154C2×D4—2841no—
155C2×D4—714no—
156C22×C4—2842no—
157C22×C4—2842no—
158C22×C4—712no—
159C22×C4—712no—
160C22×C4—712no—
161C22×C4—1424no—
162C22×C4C22×D7112yes—
163C22×C4C22×D7112yes—
164C22×C4C22×D7111yescharacteristic
165C22×C4C22×D7111yescharacteristic
166C22×C4C22×D7111yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—2842no—
170C4⋊C4—2842no—
171C4⋊C4—1424no—
172C22⋊C4—2842no—
173C22⋊C4—2842no—
174C22⋊C4—2842no—
175C22⋊C4—2842no—
176C22⋊C4—1424no—
177C42—2841no—
178C23—714no—
179C23—1422no—
180C23—1422no—
181C23—1422no—
182C23—1422no—
183C23—1422no—
184C23—1422no—
185C23C23×D7111yescharacteristic
186Q8—2844no—
187D4—2844no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—2848no—
201C2×C4—714no—
202C2×C4—1422no—
203C2×C4—1422no—
204C2×C4—1422no—
205C2×C4—1422no—
206C2×C4—1422no—
207C2×C4—1422no—
208C2×C4—1422no—
209C2×C4—1422no—
210C2×C4Q8×D7112yes—
211C22—2842no—
212C22—2842no—
213C22—2842no—
214C22—2842no—
215C22—1424no—
216C22—1424no—
217C22—1424no—
218C22C2×Q8×D7111yescharacteristic
219C22#41×D7112yes—
220C22#1455617311yescharacteristic
221C22#1455617411yescharacteristic
222C22#1455617412yes—
223C4—22no—
224C4—42no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—2842no—
229C4—2842no—
230C4—1424no—
231C2—2844no—
232C2#1798658812yes—
233C2#1798662411yescharacteristic
234C2#1798670512yes—
235C2#1798671111yescharacteristic
236C2#1798673911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2784
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #17985921C2semidirect
2. #17985919C2semidirect
3. #17985912C2semidirect
5. #17985882C2semidirect
7. #17985873C2semidirect
8. #17985858C2semidirect
9. #400×C71C2semidirect
36. C2×D284Q8semidirect
119. C71#2784semidirect
120. #400D71semidirect