Finite Group Databasebeta

← #31906880

Subgroups — #31906880

Order 18176.

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

normal subgroups only

This group has 418 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#22590960C211yesmaximal, characteristic
2#22586674C211yesmaximal, characteristic
3#22586289C214yesmaximal
4C71×#1464C211yesmaximal, characteristic
5C2×#14556087C411yescharacteristic
6C22×C4⋊Dic71C411yescharacteristic
7#17986056—21no—
8#17985897—21no—
9#17985897—21no—
10#17985871C412yes—
11#17985871C2212yes—
12#17985868—21no—
13#17985864—21no—
14#17985725—21no—
15#17985723C2212yes—
16C2×#117×C71C2211yescharacteristic
17#432×C71—21no—
18#324×C71—21no—
19#323×C71C2212yes—
20C23×Dic71C2×C412yes—
21#14556087—24no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71—21no—
25C2×C4⋊Dic71—41no—
26C2×C4⋊Dic71—41no—
27C2×Dic71⋊C4—22no—
28C2×Dic71⋊C4—22no—
29C2×C22⋊Dic71C2×C412yes—
30C2×C4×Dic71—21no—
31C4⋊(C71⋊C8)—41no—
32C22⋊(C71⋊C8)—22no—
33#14556039—41no—
34C71⋊#95—22no—
35C71⋊#95—22no—
36C23×C4×C71C2×C411yescharacteristic
37#117×C71—41no—
38#117×C71—41no—
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C22⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71C2311yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45C4⋊C8×C71—41no—
46C22⋊C8×C71—22no—
47#97×C71—41no—
48C22×Dic71—22no—
49C22×Dic71—22no—
50C22×Dic71—42no—
51C22×Dic71—82no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—24no—
54C4⋊Dic71—42no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57Dic71⋊C4—82no—
58Dic71⋊C4—82no—
59C22⋊Dic71—22no—
60C22⋊Dic71M1614yes—
61C4×Dic71—24no—
62C24×C71C22×C411yescharacteristic
63C22×C4×C71—22no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—22no—
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72C22×C4×C71C2×D411yescharacteristic
73C2×C8×C71—42no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—41no—
76C22⋊C4×C71—41no—
77C22⋊C4×C71—41no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—21no—
81C42×C71—21no—
82C23×C71—21no—
83C23×C71—21no—
84C23×C71—41no—
85C23×C71—41no—
86C23×C71C2×C22⋊C411yescharacteristic
87C23×C71C2×M1612yes—
88C2×C4×C71—24no—
89C2×C4×C71—24no—
90C2×C4×C71—21no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
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—41no—
102C2×C4×C71#9911yescharacteristic
103C2×C4×C71#9911yescharacteristic
104C2×C4×C71#10112yes—
105C2×Dic71—24no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—42no—
110C2×Dic71—42no—
111C2×Dic71—84no—
112C8×C71—82no—
113C71⋊C8—82no—
114C22×C71—21no—
115C22×C71—21no—
116C22×C71—22no—
117C22×C71—42no—
118C22×C71—42no—
119C22×C71—41no—
120C22×C71—41no—
121C22×C71#40711yescharacteristic
122C22×C71C2×#9911yescharacteristic
123C22×C71C2×#10111yescharacteristic
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—22no—
127C4×C71—41no—
128C4×C71—41no—
129C4×C71—41no—
130C4×C71—41no—
131Dic71—44no—
132Dic71—82no—
133#11520—711nomaximal, Sylow
134C2×C71—21no—
135C2×C71—21no—
136C2×C71—42no—
137C2×C71#146311yescharacteristic
138C2×C71#212211yescharacteristic
139C2×C71#212311yescharacteristic
140#2317—711no—
141#1464—711no—
142#1464D7111yescharacteristic
143#1326—714no—
144C71#1152011yescharacteristic, complemented, Sylow
145C2×#117—711no—
146C2×#117C2×D7111yescharacteristic
147C22×C4⋊C4—711no—
148#432—21no—
149#432—1421no—
150#389—1421no—
151#389—1421no—
152#385—712no—
153#385—712no—
154#384—1421no—
155#382—1421no—
156#324—21no—
157#324—1421no—
158#323—712no—
159#323C2×D7112yes—
160C23×C4—712no—
161C23×C4C4×D7111yescharacteristic
162#117—41no—
163#117—41no—
164#117—1424no—
165C2×C4⋊C4—2841no—
166C2×C4⋊C4—2841no—
167C2×C4⋊C4—1421no—
168C2×C4⋊C4—1421no—
169C2×C4⋊C4—1422no—
170C2×C4⋊C4—1422no—
171C2×C4⋊C4—1421no—
172C2×C4⋊C4D28411yescharacteristic
173C2×C4⋊C4C22⋊D7111yescharacteristic
174C2×C22⋊C4—712no—
175C2×C22⋊C4C4×D7111yescharacteristic
176C2×C22⋊C4C22×D7111yescharacteristic
177C2×C42—1421no—
178C2×C42D28411yescharacteristic
179C2×C42C22⋊D7111yescharacteristic
180C4⋊C8—41no—
181C4⋊C8—2841no—
182C22⋊C8—22no—
183C22⋊C8—1422no—
184#97—41no—
185#97—2841no—
186#95—1422no—
187#95—1422no—
188C24C2×C4×D7111yescharacteristic
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—2842no—
194C22×C4—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×C22⋊D7111yescharacteristic
202C22×C4C2×D28411yescharacteristic
203C2×C8—42no—
204C2×C8—2842no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—2842no—
208C4⋊C4—2841no—
209C4⋊C4—2841no—
210C4⋊C4—5682no—
211C4⋊C4—5682no—
212C4⋊C4—1424no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—714no—
216C22⋊C4—1422no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—1424no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23C2×(C2×D71)⋊C411yescharacteristic
227C23M16×D7112yes—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
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—5684no—
248C2×C4—1424no—
249C2×C4#1455604211yescharacteristic
250C2×C4#1455604211yescharacteristic
251C2×C4#1455604612yes—
252C8—82no—
253C8—5682no—
254C22—21no—
255C22—21no—
256C22—22no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22#1798596211yescharacteristic
262C22C2×#1455604211yescharacteristic
263C22C2×#1455604611yescharacteristic
264C4—21no—
265C4—21no—
266C4—22no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—2844no—
272C4—5682no—
273C2—21no—
274C2—21no—
275C2—42no—
276C2#2258666911yescharacteristic
277C2#2258999111yescharacteristic
278C2#2258999311yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutator, prime radicalC24×C71C22×C4
FrattiniC2×C22⋊C4C22×D71
FittingC71×#1464C2
socleC22×C71C2×#99
cosocleC2×C22⋊C4×C71C23
hypercenter#1464D71
hypocenterC71#11520
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
144. C71#11520semidirect

Maximal subgroups

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

subgroupindex
#225909602
#225866742
#225862892
C71×#14642
#1152071