Finite Group Databasebeta

← #22593502

Subgroups — #22593502

Order 9088.

532 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14556118C211yesmaximal, characteristic, complemented
2C2×#14556090C211yesmaximal, characteristic
3#17985937C211yesmaximal, characteristic, complemented
4#17985917C211yesmaximal, characteristic
5#17985916C211yesmaximal, characteristic
6#17985891C212yesmaximal, complemented
7#17985888C212yesmaximal, complemented
8#17985858C212yesmaximal, complemented
9#17985858C212yesmaximal, complemented
10#17985850C211yesmaximal, characteristic, complemented
11#400×C71C211yesmaximal, characteristic, complemented
12C22×D284C2211yescharacteristic
13C22×C4×D71C2212yes—
14#14556118—24no—
15#14556090—24no—
16C2×C4⋊Dic71C2211yescharacteristic
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yes—
27C2×C4×Dic71C2212yes—
28C2×C4×Dic71C2211yescharacteristic
29C71⋊#95C2211yescharacteristic
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C23×D71C2312yes—
38C2×D284—24no—
39C2×D284—22no—
40C22×Dic71C2312yes—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C2×C4×D71—24no—
46C2×C4×D71—24no—
47C2×C4×D71—24no—
48C4⋊Dic71—22no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53Dic71⋊C4—24no—
54(C2×D71)⋊C4—24no—
55(C2×D71)⋊C4—24no—
56(C2×D71)⋊C4—24no—
57(C2×D71)⋊C4—42no—
58(C2×D71)⋊C4—42no—
59C4×Dic71—42no—
60C4×Dic71D414yes—
61C22×C4×C71C2312yes—
62C22×C4×C71C2312yes—
63C22×C4×C71C2311yescharacteristic
64C22×C4×C71C2311yescharacteristic
65C22×C4×C71C2311yescharacteristic
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C23×C71C2411yescharacteristic
69C22×D71—22no—
70C22×D71—22no—
71C22×D71—22no—
72C22×D71—22no—
73C22×D71—22no—
74C22×D71—22no—
75C22×D71#4114yes—
76C2×C4×C71—22no—
77C2×C4×C71—22no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71C2×D412yes—
89C2×Dic71—21no—
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—21no—
98C2×Dic71—21no—
99C2×Dic71C2×D412yes—
100C2×Dic71C2×D412yes—
101C2×Dic71#4114yes—
102D284—28no—
103C4×D71—48no—
104C22×C71C22×D411yescharacteristic
105C22×C71C2×#4112yes—
106C22×C71C2×#4112yes—
107C22×C71H(5,𝔽2)11yescharacteristic
108C22×C712−1+411yescharacteristic
109C2×D71—24no—
110C2×D71—24no—
111C2×D71—24no—
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—42no—
115C2×D71—42no—
116C4×C71—22no—
117C4×C71—41no—
118C4×C71—42no—
119C4×C71—41no—
120C4×C71—42no—
121Dic71—22no—
122Dic71—41no—
123Dic71—41no—
124Dic71—42no—
125Dic71#12414yes—
126C2×C71#52511yescharacteristic
127C2×C71C2×#12411yescharacteristic
128C2×C71#53711yescharacteristic
129C2×C71#55212yes—
130C2×C71#55312yes—
131D71—44no—
132#2562—711nomaximal, Sylow
133C71#256211yescharacteristic, complemented, Sylow
134C2×#125—711no—
135C2×C4×D4—711no—
136#400—711no—
137#400D7111yescharacteristic, complemented
138#394—711no—
139#394—711no—
140#388—712no—
141#388—712no—
142#380—712no—
143#380—712no—
144C4×C4⋊C4—711no—
145C22×D4—711no—
146C23×C4—712no—
147#125—1424no—
148C4×D4—1424no—
149C2×C4⋊C4—711no—
150C2×C4⋊C4—711no—
151C2×C4⋊C4—711no—
152C2×C4⋊C4—711no—
153C2×C4⋊C4—711no—
154C2×C4⋊C4—712no—
155C2×C4⋊C4C2×D7111yescharacteristic
156C2×C4⋊C4C2×D7111yescharacteristic
157C2×C22⋊C4—712no—
158C2×C22⋊C4—712no—
159C2×C22⋊C4—712no—
160C2×C22⋊C4—712no—
161C2×C22⋊C4—712no—
162C2×C42—712no—
163C2×C42—711no—
164#95—711no—
165#95—712no—
166#95—712no—
167#95C2×D7111yescharacteristic
168#95C2×D7112yes—
169#95C2×D7112yes—
170C24—712no—
171C2×D4—1424no—
172C2×D4—1422no—
173C22×C4—711no—
174C22×C4—711no—
175C22×C4—712no—
176C22×C4—711no—
177C22×C4—711no—
178C22×C4—1424no—
179C22×C4—1424no—
180C22×C4—1424no—
181C22×C4C22×D7112yes—
182C22×C4C22×D7112yes—
183C22×C4C22×D7111yescharacteristic
184C22×C4C22×D7111yescharacteristic
185C22×C4C22×D7111yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—1424no—
189C4⋊C4—1422no—
190C4⋊C4—1422no—
191C4⋊C4—1422no—
192C4⋊C4—1422no—
193C4⋊C4—1422no—
194C22⋊C4—2842no—
195C22⋊C4—2842no—
196C22⋊C4—1424no—
197C22⋊C4—1424no—
198C22⋊C4—1424no—
199C42—2842no—
200C42—714no—
201C23—714no—
202C23—1422no—
203C23—1422no—
204C23—1422no—
205C23—1422no—
206C23—1422no—
207C23—1422no—
208C23C23×D7111yescharacteristic
209D4—1428no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—2848no—
223C2×C4—712no—
224C2×C4—714no—
225C2×C4—712no—
226C2×C4—1421no—
227C2×C4—1421no—
228C2×C4—1422no—
229C2×C4—1422no—
230C2×C4—1422no—
231C2×C4—1422no—
232C2×C4—1421no—
233C2×C4—1421no—
234C2×C4—1421no—
235C2×C4—1421no—
236C2×C4D4×D7112yes—
237C22—2842no—
238C22—2842no—
239C22—2842no—
240C22—2842no—
241C22—1424no—
242C22—1424no—
243C22—1424no—
244C22C2×D4×D7111yescharacteristic
245C22#41×D7112yes—
246C22#41×D7112yes—
247C22#1455617411yescharacteristic
248C22#1455617611yescharacteristic
249C4—22no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—42no—
254C4—2841no—
255C4—2842no—
256C4—2841no—
257C4—714no—
258C4—1422no—
259C2—2844no—
260C2#1798650711yescharacteristic
261C2#124×D7111yescharacteristic
262C2#1798657911yescharacteristic
263C2#1798668112yes—
264C2#1798670012yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2562
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#14556118C2semidirect
3. #17985937C2semidirect
6. #17985891C2semidirect
7. #17985888C2semidirect
8. #17985858C2semidirect
9. #17985858C2semidirect
10. #17985850C2semidirect
11. #400×C71C2semidirect
133. C71#2562semidirect
137. #400D71semidirect