Finite Group Databasebeta

← #31895136

Subgroups — #31895136

Order 18176.

468 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 107 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#22593325C211yesmaximal, characteristic
2#22587065C212yesmaximal
3#22586684C211yesmaximal, characteristic
4#22586197C212yesmaximal
5C71×#2983C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7C2×#14556126C412yes—
8#17986056C2212yes—
9#17985929—21no—
10#17985929—21no—
11#17985904C412yes—
12#17985897—21no—
13#17985897—21no—
14C4×C4⋊Dic71C2211yescharacteristic
15#17985747—24no—
16#17985725—21no—
17#17985725—21no—
18C2×C4×Q8×C71C2211yescharacteristic
19#446×C71—24no—
20#432×C71—21no—
21#432×C71C2212yes—
22#431×C71—21no—
23C22×Dic142C2×C412yes—
24#14556128—22no—
25#14556128—22no—
26#14556126—44no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71—21no—
29C2×C4⋊Dic71—21no—
30C2×Dic71⋊C4—22no—
31C2×Dic71⋊C4—22no—
32C2×C4×Dic71C2×C412yes—
33C4⋊(C71⋊C8)—22no—
34C22⋊(C71⋊C8)—22no—
35#14556039—22no—
36C71⋊#95—22no—
37C71⋊#95—22no—
38C71⋊#95—21no—
39C22×Q8×C71C2×C411yescharacteristic
40C4×Q8×C71—22no—
41C4×Q8×C71—22no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71D411yescharacteristic
44C2×C4⋊C4×C71D411yescharacteristic
45C2×C42×C71D411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71C2311yescharacteristic
48C4⋊C8×C71—22no—
49C4⋊C8×C71—22no—
50C4⋊C8×C71D414yes—
51C22⋊C8×C71—22no—
52#97×C71—22no—
53#97×C71—22no—
54C4×C8×C71—22no—
55C2×Dic142—24no—
56C2×Dic142—42no—
57C2×Dic142—42no—
58C22×Dic71—22no—
59C22×Dic71—22no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—24no—
62C4⋊Dic71—41no—
63C4⋊Dic71—42no—
64C4⋊Dic71—41no—
65Dic71⋊C4—44no—
66Dic71⋊C4—44no—
67C4×Dic71—42no—
68C2×Q8×C71—22no—
69C2×Q8×C71—22no—
70C2×Q8×C71—41no—
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C2×C8×C71—24no—
79C4⋊C4×C71—22no—
80C4⋊C4×C71—22no—
81C4⋊C4×C71—22no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—21no—
85C42×C71—21no—
86C42×C71C2×D412yes—
87C42×C71#4112yes—
88C23×C71C2×C22⋊C411yescharacteristic
89Q8×C71—44no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—22no—
95C2×C4×C71—21no—
96C2×C4×C71—21no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—41no—
103C2×C4×C71#10012yes—
104C2×C4×C71C4×D412yes—
105C2×C4×C71#12011yescharacteristic
106C2×C4×C71#12112yes—
107C2×C4×C71#12311yescharacteristic
108C2×Dic71—24no—
109C2×Dic71—42no—
110C2×Dic71—42no—
111C2×Dic71—42no—
112C2×Dic71—42no—
113C2×Dic71—42no—
114Dic142—84no—
115C8×C71—44no—
116C71⋊C8—82no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71#38611yescharacteristic
120C22×C71C2×#10011yescharacteristic
121C22×C71#41311yescharacteristic
122C4×C71—21no—
123C4×C71—22no—
124C4×C71—22no—
125C4×C71—22no—
126C4×C71—41no—
127C4×C71—41no—
128C4×C71—41no—
129C4×C71#45612yes—
130Dic71—44no—
131Dic71—82no—
132#9452—711nomaximal, Sylow
133C2×C71—21no—
134C2×C71—21no—
135C2×C71#188611yescharacteristic
136C2×C71#190011yescharacteristic
137C2×C71#190911yescharacteristic
138#2983D7111yescharacteristic
139#2546—711no—
140#1558—712no—
141#1467—711no—
142#1285—712no—
143C71#945211yescharacteristic, complemented, Sylow
144C2×C4⋊Q8—712no—
145C2×C4⋊Q8—711no—
146C2×C4×Q8C2×D7111yescharacteristic
147#446—24no—
148#432—21no—
149#432—712no—
150#432C2×D7112yes—
151#431—21no—
152#398—1421no—
153#398—1421no—
154#391—712no—
155#389—1421no—
156#389—1421no—
157C4×C4⋊C4—711no—
158#333—1424no—
159#324—1421no—
160#324—1421no—
161C22×Q8—712no—
162C22×Q8C4×D7111yescharacteristic
163C4⋊Q8—2844no—
164C4⋊Q8—1422no—
165C4⋊Q8—1422no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—1422no—
169C2×C4⋊C4—1422no—
170C2×C4⋊C4—1421no—
171C2×C4⋊C4—1421no—
172C2×C4⋊C4—1421no—
173C2×C4⋊C4C4×D7111yescharacteristic
174C2×C4⋊C4D28411yescharacteristic
175C2×C4⋊C4D28411yescharacteristic
176C2×C42—712no—
177C2×C42C22⋊D7111yescharacteristic
178C2×C42C22⋊D7111yescharacteristic
179C2×C42C22×D7111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—1422no—
183C4⋊C8C22⋊D7114yes—
184C22⋊C8—22no—
185C22⋊C8—1422no—
186#97—22no—
187#97—22no—
188#97—1422no—
189C4×C8—22no—
190#95—1422no—
191#95—1422no—
192#95—1421no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—2842no—
197C2×Q8—2842no—
198C2×Q8—1424no—
199C22×C4—1422no—
200C22×C4—1422no—
201C22×C4(C2×D71)⋊C411yescharacteristic
202C22×C4(C2×D71)⋊C411yescharacteristic
203C22×C4(C2×D71)⋊C411yescharacteristic
204C22×C4(C2×D71)⋊C411yescharacteristic
205C22×C4C2×C4×D7111yescharacteristic
206C22×C4C2×C22⋊D7111yescharacteristic
207C22×C4C2×D28411yescharacteristic
208C2×C8—24no—
209C2×C8—2842no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—2844no—
214C4⋊C4—2844no—
215C4⋊C4—2841no—
216C4⋊C4—2842no—
217C4⋊C4—2841no—
218C4⋊C4—1424no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—2842no—
224C42C2×C22⋊D7112yes—
225C42#1176586512yes—
226C23C2×(C2×D71)⋊C411yescharacteristic
227Q8—44no—
228Q8—5684no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—2842no—
243C2×C4—2842no—
244C2×C4—2842no—
245C2×C4—2842no—
246C2×C4—2842no—
247C2×C4—1424no—
248C2×C4#1455604412yes—
249C2×C4C4×C22⋊D7112yes—
250C2×C4#1455609711yescharacteristic
251C2×C4#1455610212yes—
252C2×C4#1455611111yescharacteristic
253C8—44no—
254C8—5682no—
255C22—21no—
256C22—21no—
257C22#1798587811yescharacteristic
258C22C2×#1455604411yescharacteristic
259C22#1798598111yescharacteristic
260C4—21no—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—2844no—
268C4—5682no—
269C4#1798618212yes—
270C2—21no—
271C2—21no—
272C2#2258873311yescharacteristic
273C2#2258880311yescharacteristic
274C2#2258886211yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2983C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2983D71
hypocenterC71#9452
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
143. C71#9452semidirect

Maximal subgroups

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

subgroupindex
#225933252
#225870652
#225866842
#225861972
C71×#29832
#945271