Finite Group Databasebeta

← #22594278

Subgroups — #22594278

Order 9088.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556126C211yesmaximal, characteristic, complemented
2C2×#14556114C211yesmaximal, characteristic
3#17985928C212yesmaximal, complemented
4#17985927C212yesmaximal, complemented
5#17985912C211yesmaximal, characteristic, complemented
6#17985911C211yesmaximal, characteristic, complemented
7#17985910C211yesmaximal, characteristic, complemented
8#17985903C212yesmaximal, complemented
9#17985899C212yesmaximal
10#17985844C211yesmaximal, characteristic, complemented
11#393×C71C211yesmaximal, characteristic, complemented
12C22×Dic142C2211yescharacteristic
13C22×Dic142C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15#14556126—24no—
16#14556114—24no—
17C2×Dic71⋊C4C2212yes—
18C2×Dic71⋊C4C2212yes—
19C2×C22⋊Dic71C2211yescharacteristic
20C2×C22⋊Dic71C2212yes—
21C2×(C2×D71)⋊C4C2211yescharacteristic
22C2×(C2×D71)⋊C4C2211yescharacteristic
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2212yes—
25C2×C4×Dic71C2212yes—
26C2×C4×Dic71C2211yescharacteristic
27C71⋊#95C2211yescharacteristic
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C22×Q8×C71C2211yescharacteristic
33C2×C22⋊C4×C71C2211yescharacteristic
34C2×C22⋊C4×C71C2212yes—
35#95×C71C2211yescharacteristic
36#95×C71C2212yes—
37C23×D71C2311yescharacteristic
38C2×Dic142—22no—
39C2×Dic142—22no—
40C2×Dic142—24no—
41C2×Dic142—42no—
42C2×C22⋊D71—22no—
43C2×C22⋊D71—24no—
44C22×Dic71C2311yescharacteristic
45C22×Dic71C2312yes—
46C22×Dic71C2312yes—
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2311yescharacteristic
49Dic71⋊C4—24no—
50Dic71⋊C4—24no—
51C22⋊Dic71—22no—
52C22⋊Dic71—42no—
53(C2×D71)⋊C4—22no—
54(C2×D71)⋊C4—22no—
55(C2×D71)⋊C4—42no—
56(C2×D71)⋊C4—42no—
57C4×Dic71—24no—
58C4×Dic71D414yes—
59C24×C71C2311yescharacteristic
60C2×Q8×C71—22no—
61C2×Q8×C71—42no—
62C22×C4×C71C2311yescharacteristic
63C22×C4×C71C2312yes—
64C22×C4×C71C2312yes—
65C22×C4×C71C2311yescharacteristic
66C22⋊C4×C71—22no—
67C22⋊C4×C71—42no—
68C23×C71—21no—
69C23×C71—22no—
70C23×C71—22no—
71C23×C71—21no—
72C23×C71—21no—
73C23×C71C2411yescharacteristic
74C22×D71—21no—
75C22×D71—22no—
76C22×D71—22no—
77C22×D71—21no—
78C22×D71—21no—
79Q8×C71—44no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71C2×D412yes—
90C22⋊D71—44no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—21no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—22no—
101C2×Dic71C2×D412yes—
102C2×Dic71C2×D412yes—
103C2×Dic71#4114yes—
104Dic142—44no—
105Dic142—44no—
106C22×C71—41no—
107C22×C71—41no—
108C22×C71—41no—
109C22×C71—42no—
110C22×C71—42no—
111C22×C71C22×D411yescharacteristic
112C22×C71C2×#4112yes—
113C22×C71H(5,𝔽2)12yes—
114C22×C712−1+411yescharacteristic
115C22×C712−1+411yescharacteristic
116C2×D71—41no—
117C2×D71—41no—
118C2×D71—41no—
119C2×D71—42no—
120C2×D71—42no—
121C4×C71—22no—
122C4×C71—41no—
123C4×C71—42no—
124C4×C71—42no—
125Dic71—22no—
126Dic71—41no—
127Dic71—42no—
128Dic71—42no—
129Dic71#12414yes—
130C2×C71—81no—
131C2×C71C2×#12411yescharacteristic
132C2×C71#53511yescharacteristic
133C2×C71#53611yescharacteristic
134C2×C71#54212yes—
135C2×C71#54212yes—
136D71—81no—
137#2640—711nomaximal, Sylow
138C71#264011yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—711no—
140C2×#124—711no—
141#397—712no—
142#397—712no—
143#393—711no—
144#393—711no—
145#393—711no—
146#393D7111yescharacteristic, complemented
147#391—712no—
148#390—712no—
149#376—711no—
150C22×Q8—711no—
151C22×Q8—711no—
152C22×Q8C2×D7111yescharacteristic
153C22×D4—711no—
154C4⋊Q8—1424no—
155#124—1424no—
156C2×C4⋊C4—712no—
157C2×C4⋊C4—712no—
158C2×C22⋊C4—712no—
159C2×C22⋊C4—712no—
160C2×C22⋊C4—712no—
161C2×C22⋊C4—711no—
162C2×C22⋊C4—711no—
163C2×C22⋊C4—711no—
164C2×C22⋊C4C2×D7111yescharacteristic
165C2×C22⋊C4C2×D7112yes—
166C2×C42—711no—
167C2×C42—712no—
168#95—711no—
169#95—712no—
170#95—712no—
171#95—712no—
172#95—712no—
173#95C2×D7111yescharacteristic
174#95C2×D7112yes—
175C24—711no—
176C24C22×D7111yescharacteristic
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—2842no—
180C2×Q8—1422no—
181C2×Q8—1424no—
182C2×Q8—1422no—
183C2×D4—1422no—
184C2×D4—1424no—
185C22×C4—711no—
186C22×C4—711no—
187C22×C4—712no—
188C22×C4—711no—
189C22×C4—712no—
190C22×C4C22×D7112yes—
191C22×C4C22×D7111yescharacteristic
192C22×C4C22×D7112yes—
193C22×C4C22×D7111yescharacteristic
194C4⋊C4—1424no—
195C4⋊C4—1424no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—2842no—
199C22⋊C4—2842no—
200C22⋊C4—2842no—
201C22⋊C4—1422no—
202C22⋊C4—1422no—
203C22⋊C4—1422no—
204C42—714no—
205C42—1424no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—1421no—
212C23—1422no—
213C23—1422no—
214C23—1421no—
215C23—1421no—
216C23C23×D7111yescharacteristic
217Q8—44no—
218Q8—2844no—
219Q8—2844no—
220D4—2844no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—712no—
231C2×C4—714no—
232C2×C4—712no—
233C2×C4—1421no—
234C2×C4—1422no—
235C2×C4—1422no—
236C2×C4—1422no—
237C2×C4—1422no—
238C2×C4—1422no—
239C2×C4—1422no—
240C2×C4—1422no—
241C2×C4—1421no—
242C2×C4—1421no—
243C2×C4D4×D7112yes—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—2841no—
250C22—2841no—
251C22—2842no—
252C22—2842no—
253C22—2841no—
254C22C2×D4×D7111yescharacteristic
255C22#41×D7112yes—
256C22#1455617312yes—
257C22#1455617511yescharacteristic
258C22#1455617611yescharacteristic
259C4—22no—
260C4—42no—
261C4—42no—
262C4—41no—
263C4—2842no—
264C4—2842no—
265C4—2841no—
266C4—714no—
267C4—1422no—
268C2—81no—
269C2—5681no—
270C2#124×D7111yescharacteristic
271C2#1798656411yescharacteristic
272C2#1798657211yescharacteristic
273C2#1798661312yes—
274C2#1798661912yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#393×C71C2
hypercenter#393D71
hypocenterC71#2640
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14556126C2semidirect
3. #17985928C2semidirect
4. #17985927C2semidirect
5. #17985912C2semidirect
6. #17985911C2semidirect
7. #17985910C2semidirect
8. #17985903C2semidirect
10. #17985844C2semidirect
11. #393×C71C2semidirect
138. C71#2640semidirect
146. #393D71semidirect