Finite Group Databasebeta

← #31895144

Subgroups — #31895144

Order 18176.

448 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 103 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#22597425C211yesmaximal, characteristic
2#22593545C211yesmaximal, characteristic
3#22587058C212yesmaximal
4#22586195C212yesmaximal
5C71×#1467C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986130—24no—
8#17986056—21no—
9#17986056C2212yes—
10#17986052—21no—
11#17985919—21no—
12#17985919—21no—
13#17985906C412yes—
14#17985897—21no—
15#17985897—21no—
16#17985868C412yes—
17C4×C4⋊Dic71C2211yescharacteristic
18#17985746—24no—
19C2×C4×Q8×C71C2211yescharacteristic
20#432×C71C2212yes—
21#324×C71—21no—
22#324×C71—21no—
23C22×Dic142—21no—
24#14556128—22no—
25#14556128—22no—
26C2×C4⋊Dic71—21no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71C2×C412yes—
29C2×Dic71⋊C4—22no—
30C2×Dic71⋊C4—22no—
31C2×C4×Dic71C2×C412yes—
32C4⋊(C71⋊C8)—22no—
33C4⋊(C71⋊C8)—22no—
34C4⋊(C71⋊C8)D414yes—
35C22⋊(C71⋊C8)—22no—
36#14556039—22no—
37#14556039—22no—
38C4×C71⋊C8—22no—
39C71⋊#95—22no—
40C71⋊#95—22no—
41C71⋊#95—21no—
42C22×Q8×C71C2×C411yescharacteristic
43C4×Q8×C71—22no—
44C4×Q8×C71—22no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50C2×C42×C71C2311yescharacteristic
51C4⋊C8×C71—22no—
52C22⋊C8×C71—22no—
53#97×C71—22no—
54C2×Dic142—42no—
55C2×Dic142—82no—
56C22×Dic71—22no—
57C22×Dic71—22no—
58C2×C71⋊C8—24no—
59C4⋊Dic71—24no—
60C4⋊Dic71—24no—
61C4⋊Dic71—41no—
62C4⋊Dic71—41no—
63Dic71⋊C4—82no—
64Dic71⋊C4—82no—
65C4×Dic71—42no—
66C2×Q8×C71—22no—
67C2×Q8×C71—22no—
68C2×Q8×C71—41no—
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C2×C8×C71—42no—
77C4⋊C4×C71—22no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—22no—
80C42×C71—21no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71C2×D412yes—
85C42×C71#4112yes—
86C23×C71C2×C22⋊C411yescharacteristic
87Q8×C71—44no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—22no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71#10112yes—
102C2×C4×C71C4×D412yes—
103C2×C4×C71#12011yescharacteristic
104C2×C4×C71#12112yes—
105C2×C4×C71#12311yescharacteristic
106C2×Dic71—24no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110C2×Dic71—42no—
111C2×Dic71—42no—
112Dic142—84no—
113C8×C71—82no—
114C71⋊C8—44no—
115C22×C71—21no—
116C22×C71—21no—
117C22×C71#38611yescharacteristic
118C22×C71C2×#10111yescharacteristic
119C22×C71#41311yescharacteristic
120C4×C71—21no—
121C4×C71—22no—
122C4×C71—22no—
123C4×C71—22no—
124C4×C71—41no—
125C4×C71—41no—
126C4×C71—41no—
127C4×C71#45612yes—
128Dic71—44no—
129Dic71—82no—
130#9454—711nomaximal, Sylow
131C2×C71—21no—
132C2×C71—21no—
133C2×C71#188611yescharacteristic
134C2×C71#190111yescharacteristic
135C2×C71#191011yescharacteristic
136#2983—711no—
137#2567—711no—
138#1556—712no—
139#1467D7111yescharacteristic
140#1285—712no—
141C71#945411yescharacteristic, complemented, Sylow
142C2×C4⋊Q8—711no—
143C2×C4×Q8C2×D7111yescharacteristic
144#446—1424no—
145#432—712no—
146#432—1421no—
147#432C2×D7112yes—
148#431—1421no—
149#395—1421no—
150#395—1421no—
151#391—712no—
152#389—1421no—
153#389—1421no—
154#384—712no—
155C4×C4⋊C4—711no—
156#332—1424no—
157#324—21no—
158#324—21no—
159C22×Q8—1421no—
160C22×Q8C4×D7111yescharacteristic
161C4⋊Q8—1422no—
162C4⋊Q8—1422no—
163C4×Q8—22no—
164C4×Q8—22no—
165C2×C4⋊C4—712no—
166C2×C4⋊C4—1421no—
167C2×C4⋊C4—1422no—
168C2×C4⋊C4—1422no—
169C2×C4⋊C4—1421no—
170C2×C4⋊C4C4×D7111yescharacteristic
171C2×C4⋊C4C22⋊D7111yescharacteristic
172C2×C4⋊C4C22⋊D7111yescharacteristic
173C2×C42—712no—
174C2×C42D28411yescharacteristic
175C2×C42D28411yescharacteristic
176C2×C42C22×D7111yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—714no—
179C4⋊C8—1422no—
180C4⋊C8—1422no—
181C22⋊C8—22no—
182C22⋊C8—1422no—
183#97—22no—
184#97—1422no—
185#97—1422no—
186C4×C8—1422no—
187#95—1422no—
188#95—1422no—
189#95—1421no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—2842no—
194C2×Q8—5682no—
195C22×C4—1422no—
196C22×C4—1422no—
197C22×C4(C2×D71)⋊C411yescharacteristic
198C22×C4(C2×D71)⋊C411yescharacteristic
199C22×C4(C2×D71)⋊C411yescharacteristic
200C22×C4(C2×D71)⋊C411yescharacteristic
201C22×C4C2×C4×D7111yescharacteristic
202C22×C4C2×C22⋊D7111yescharacteristic
203C22×C4C2×D28411yescharacteristic
204C2×C8—42no—
205C2×C8—1424no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—2841no—
210C4⋊C4—2841no—
211C4⋊C4—5682no—
212C4⋊C4—5682no—
213C4⋊C4—1424no—
214C4⋊C4—1424no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—2842no—
220C42D4×D7112yes—
221C42#1176586412yes—
222C23C2×(C2×D71)⋊C411yescharacteristic
223Q8—44no—
224Q8—5684no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—2842no—
239C2×C4—2842no—
240C2×C4—2842no—
241C2×C4—2842no—
242C2×C4—2842no—
243C2×C4—1424no—
244C2×C4#1455604612yes—
245C2×C4(C22⋊D71)⋊C412yes—
246C2×C4#1455609811yescharacteristic
247C2×C4#1455610012yes—
248C2×C4#1455611211yescharacteristic
249C8—82no—
250C8—2844no—
251C22—21no—
252C22—21no—
253C22#1798587611yescharacteristic
254C22C2×#1455604611yescharacteristic
255C22#1798598111yescharacteristic
256C4—21no—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—2844no—
264C4—5682no—
265C4#1798618412yes—
266C2—21no—
267C2—21no—
268C2#2258873111yescharacteristic
269C2#2258880611yescharacteristic
270C2#2258886511yescharacteristic
subgroupG/H
centerC22#17985876
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1467C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1467D71
hypocenterC71#9454
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
141. C71#9454semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225974252
#225935452
#225870582
#225861952
C71×#14672
#945471