Finite Group Databasebeta

← #22594154

Subgroups — #22594154

Order 9088.

508 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14556108C211yesmaximal, characteristic
2C2×C4⋊C4×D71C211yesmaximal, characteristic
3#17985937C211yesmaximal, characteristic, complemented
4#17985933C212yesmaximal, complemented
5#17985931C212yesmaximal, complemented
6#17985920C211yesmaximal, characteristic, complemented
7#17985919C211yesmaximal, characteristic, complemented
8#17985873C212yesmaximal
9#17985861C212yesmaximal
10#17985858C211yesmaximal, characteristic
11#400×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×C4×D71C2211yescharacteristic
14C22×C4×D71C2211yescharacteristic
15C22×C4×D71C2211yescharacteristic
16#14556108—24no—
17C2×C4⋊Dic71C2211yescharacteristic
18C2×C4⋊Dic71C2212yes—
19C2×C4⋊Dic71C2212yes—
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2211yescharacteristic
22C2×Dic71⋊C4C2212yes—
23C2×Dic71⋊C4C2212yes—
24C4⋊C4×D71—24no—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C71⋊#95C2211yescharacteristic
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—42no—
40C22×Dic71C2311yescharacteristic
41C22×Dic71C2312yes—
42C22×Dic71C2312yes—
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C2×C4×D71—22no—
46C2×C4×D71—22no—
47C2×C4×D71—22no—
48C2×C4×D71—42no—
49C2×C4×D71—42no—
50C2×C4×D71D414yes—
51C2×C4×D71Q814yescomplemented
52C4⋊Dic71—22no—
53C4⋊Dic71—24no—
54C4⋊Dic71—42no—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—42no—
58Dic71⋊C4—42no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61C22×C4×C71C2311yescharacteristic
62C22×C4×C71C2312yes—
63C22×C4×C71C2312yes—
64C22×C4×C71C2311yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C23×C71C2411yescharacteristic
69C22×D71—22no—
70C22×D71—22no—
71C22×D71C2×D412yes—
72C22×D71C2×Q812yes—
73C22×D71#4112yes—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—24no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D412yes—
85C2×C4×C71C2×Q812yes—
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71C2×D412yes—
98C2×Dic71C2×Q812yes—
99C4×D71—28no—
100C4×D71—44no—
101C4×D71—44no—
102Dic142—44no—
103C22×C71C22×D411yescharacteristic
104C22×C71C22×Q811yescharacteristic
105C22×C71C2×#4111yescharacteristic
106C22×C71H(5,𝔽2)12yes—
107C22×C712−1+412yes—
108C2×D71—22no—
109C2×D71—22no—
110C2×D71—24no—
111C2×D71—24no—
112C2×D71C22⋊Q814yes—
113C4×C71—22no—
114C4×C71—22no—
115C4×C71—41no—
116C4×C71—42no—
117C4×C71—42no—
118Dic71—22no—
119Dic71—22no—
120Dic71—42no—
121Dic71—41no—
122Dic71—42no—
123C2×C71C2×C22⋊Q811yescharacteristic
124C2×C71#53411yescharacteristic
125C2×C71#53611yescharacteristic
126C2×C71#53912yes—
127C2×C71#54412yes—
128D71—24no—
129#2627—711nomaximal, Sylow
130C71#262711yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—711no—
132C22×C4⋊C4—711no—
133#400—711no—
134#400D7111yescharacteristic, complemented
135#398—712no—
136#398—712no—
137#395—711no—
138#395—711no—
139#385—712no—
140#381—712no—
141#380—711no—
142C22×Q8—711no—
143C23×C4—711no—
144C23×C4—711no—
145C23×C4—711no—
146C22⋊Q8—1424no—
147C2×C4⋊C4—712no—
148C2×C4⋊C4—712no—
149C2×C4⋊C4—712no—
150C2×C4⋊C4—711no—
151C2×C4⋊C4—711no—
152C2×C4⋊C4—711no—
153C2×C4⋊C4—712no—
154C2×C4⋊C4—1424no—
155C2×C4⋊C4C2×D7111yescharacteristic
156C2×C4⋊C4C2×D7111yescharacteristic
157C2×C22⋊C4—712no—
158C2×C22⋊C4—712no—
159#95—711no—
160#95—712no—
161#95—712no—
162#95—712no—
163#95—712no—
164#95C2×D7111yescharacteristic
165#95C2×D7112yes—
166#95C2×D7112yes—
167C24—711no—
168C2×Q8—2842no—
169C2×Q8—1422no—
170C22×C4—2842no—
171C22×C4—2842no—
172C22×C4—714no—
173C22×C4—711no—
174C22×C4—711no—
175C22×C4—714no—
176C22×C4—712no—
177C22×C4—712no—
178C22×C4—711no—
179C22×C4—1422no—
180C22×C4—1422no—
181C22×C4—1422no—
182C22×C4C22×D7112yes—
183C22×C4C22×D7112yes—
184C22×C4C22×D7111yescharacteristic
185C22×C4C22×D7111yescharacteristic
186C22×C4C22×D7111yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—2842no—
190C4⋊C4—2842no—
191C4⋊C4—2842no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C4⋊C4—1424no—
195C4⋊C4—1422no—
196C22⋊C4—1424no—
197C22⋊C4—1424no—
198C23—712no—
199C23—712no—
200C23—712no—
201C23—1422no—
202C23—1422no—
203C23C23×D7111yescharacteristic
204Q8—2844no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—2844no—
216C2×C4—2844no—
217C2×C4—712no—
218C2×C4—712no—
219C2×C4—1428no—
220C2×C4—1422no—
221C2×C4—1421no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4—1422no—
227C2×C4—1422no—
228C2×C4—1421no—
229C2×C4—1421no—
230C2×C4—1422no—
231C2×C4D4×D7112yes—
232C2×C4Q8×D7112yes—
233C22—714no—
234C22—1422no—
235C22—1422no—
236C22—1424no—
237C22—1424no—
238C22C2×D4×D7111yescharacteristic
239C22C2×Q8×D7111yescharacteristic
240C22#41×D7111yescharacteristic
241C22#1455617412yes—
242C22#1455617512yes—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—42no—
247C4—42no—
248C4—2841no—
249C4—2842no—
250C4—2842no—
251C4—1422no—
252C4—1422no—
253C2—1424no—
254C2C22⋊Q8×D7111yescharacteristic
255C2#1798656011yescharacteristic
256C2#1798656911yescharacteristic
257C2#1798659012yes—
258C2#1798662912yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2627
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #17985937C2semidirect
4. #17985933C2semidirect
5. #17985931C2semidirect
6. #17985920C2semidirect
7. #17985919C2semidirect
11. #400×C71C2semidirect
51. C2×C4×D71Q8semidirect
130. C71#2627semidirect
134. #400D71semidirect