Finite Group Databasebeta

← #31907096

Subgroups — #31907096

Order 18176.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 87 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#22586664C211yesmaximal, characteristic
3#22586273C214yesmaximal
4C71×#1467C211yesmaximal, characteristic
5C2×#14556128C411yescharacteristic
6C2×#14556126C412yes—
7#17986052—21no—
8#17985929—21no—
9#17985929—21no—
10#17985904C2212yes—
11#17985897—21no—
12#17985897—21no—
13C4×C4⋊Dic71C411yescharacteristic
14#17985725—21no—
15#17985725C2212yes—
16C2×C4×Q8×C71C2211yescharacteristic
17#432×C71—21no—
18#324×C71—21no—
19#324×C71C2212yes—
20C22×Dic142C2×C412yes—
21#14556128—41no—
22#14556128—41no—
23#14556126—44no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71—21no—
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—22no—
29C2×C4×Dic71C2×C412yes—
30C4⋊(C71⋊C8)—41no—
31C22⋊(C71⋊C8)—22no—
32C4×C71⋊C8—41no—
33C71⋊#95—21no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C22×Q8×C71C2×C411yescharacteristic
37C4×Q8×C71—41no—
38C4×Q8×C71—41no—
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C4⋊C8×C71—41no—
46C22⋊C8×C71—22no—
47#97×C71—41no—
48C2×Dic142—24no—
49C2×Dic142—42no—
50C2×Dic142—42no—
51C22×Dic71—22no—
52C22×Dic71—22no—
53C2×C71⋊C8—42no—
54C4⋊Dic71—24no—
55C4⋊Dic71—42no—
56C4⋊Dic71—41no—
57C4⋊Dic71—41no—
58Dic71⋊C4—44no—
59Dic71⋊C4—44no—
60C4×Dic71—42no—
61C2×Q8×C71—22no—
62C2×Q8×C71—22no—
63C2×Q8×C71—41no—
64C22×C4×C71C22⋊C411yescharacteristic
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22×C411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C2×C8×C71—42no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—41no—
75C42×C71—21no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C23×C71C2×C22⋊C411yescharacteristic
82Q8×C71—44no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—41no—
95C2×C4×C71—41no—
96C2×C4×C71—41no—
97C2×C4×C71#9911yescharacteristic
98C2×C4×C71#9911yescharacteristic
99C2×C4×C71#10012yes—
100C2×C4×C71#13112yes—
101C2×Dic71—24no—
102C2×Dic71—42no—
103C2×Dic71—42no—
104C2×Dic71—42no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107Dic142—84no—
108C8×C71—82no—
109C71⋊C8—82no—
110C22×C71—21no—
111C22×C71—21no—
112C22×C71#40811yescharacteristic
113C22×C71C2×#9911yescharacteristic
114C22×C71C2×#10011yescharacteristic
115C4×C71—21no—
116C4×C71—22no—
117C4×C71—22no—
118C4×C71—41no—
119C4×C71—41no—
120C4×C71—41no—
121C4×C71—41no—
122C4×C71#35612yes—
123Dic71—44no—
124Dic71—82no—
125#11568—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71#147111yescharacteristic
129C2×C71#212911yescharacteristic
130C2×C71C2×#35611yescharacteristic
131#2546—711no—
132#1467D7111yescharacteristic
133#1461—711no—
134#1319—714no—
135C71#1156811yescharacteristic, complemented, Sylow
136C2×C4⋊Q8—712no—
137C2×C4⋊Q8—711no—
138C2×C4×Q8C2×D7111yescharacteristic
139#432—21no—
140#431—1421no—
141#398—1421no—
142#398—1421no—
143#391—712no—
144#389—1421no—
145#389—1421no—
146C4×C4⋊C4—711no—
147#324—21no—
148#324—712no—
149#324—1421no—
150#324C2×D7112yes—
151C22×Q8—712no—
152C22×Q8C4×D7111yescharacteristic
153C4⋊Q8—2841no—
154C4⋊Q8—2841no—
155C4⋊Q8—2844no—
156C4×Q8—41no—
157C4×Q8—41no—
158C2×C4⋊C4—1422no—
159C2×C4⋊C4—1422no—
160C2×C4⋊C4—1421no—
161C2×C4⋊C4—1421no—
162C2×C4⋊C4—1421no—
163C2×C4⋊C4D28411yescharacteristic
164C2×C4⋊C4C22⋊D7111yescharacteristic
165C2×C4⋊C4C22×D7111yescharacteristic
166C2×C42—712no—
167C2×C42C4×D7111yescharacteristic
168C2×C42D28411yescharacteristic
169C2×C42C22⋊D7111yescharacteristic
170C4⋊C8—41no—
171C4⋊C8—2841no—
172C22⋊C8—22no—
173C22⋊C8—1422no—
174#97—41no—
175C4×C8—2841no—
176#95—1421no—
177#95—1422no—
178#95—1422no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×Q8—2842no—
183C2×Q8—2842no—
184C2×Q8—1424no—
185C22×C4—1422no—
186C22×C4—1422no—
187C22×C4(C2×D71)⋊C411yescharacteristic
188C22×C4(C2×D71)⋊C411yescharacteristic
189C22×C4(C2×D71)⋊C411yescharacteristic
190C22×C4(C2×D71)⋊C411yescharacteristic
191C22×C4C2×C4×D7111yescharacteristic
192C22×C4C2×C22⋊D7111yescharacteristic
193C22×C4C2×D28411yescharacteristic
194C2×C8—42no—
195C2×C8—2842no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—2844no—
200C4⋊C4—2842no—
201C4⋊C4—2841no—
202C4⋊C4—2841no—
203C4⋊C4—2844no—
204C4⋊C4—1424no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—2842no—
212C23C2×(C2×D71)⋊C411yescharacteristic
213Q8—44no—
214Q8—5684no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—2842no—
230C2×C4—2842no—
231C2×C4—2842no—
232C2×C4—2842no—
233C2×C4—2842no—
234C2×C4—1424no—
235C2×C4#1455604211yescharacteristic
236C2×C4#1455604211yescharacteristic
237C2×C4#1455604412yes—
238C2×C4#1455613612yes—
239C8—82no—
240C8—5682no—
241C22—21no—
242C22—21no—
243C22#1798596711yescharacteristic
244C22C2×#1455604211yescharacteristic
245C22C2×#1455604411yescharacteristic
246C4—21no—
247C4—22no—
248C4—22no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—2844no—
254C4—5682no—
255C4#1798579712yes—
256C2—21no—
257C2—21no—
258C2#2258670111yescharacteristic
259C2#2259001411yescharacteristic
260C2C2×#1798579711yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1467C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1467D71
hypocenterC71#11568
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
135. C71#11568semidirect

Maximal subgroups

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

subgroupindex
#225933252
#225866642
#225862732
C71×#14672
#1156871