Finite Group Databasebeta

← #31907095

Subgroups — #31907095

Order 18176.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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#22586684C211yesmaximal, characteristic
2#22586664C211yesmaximal, characteristic
3#22586274C214yesmaximal
4C71×#2546C211yesmaximal, characteristic
5#17986056—21no—
6#17986052—21no—
7#17985725—21no—
8#17985725—21no—
9#17985725C2212yes—
10#17985725C2212yes—
11C2×C4⋊Q8×C71C411yescharacteristic
12C2×C4⋊Q8×C71C412yes—
13C2×C4×Q8×C71C2211yescharacteristic
14#398×C71—21no—
15#398×C71—21no—
16#391×C71C2212yes—
17#389×C71—21no—
18#389×C71—21no—
19C4×C4⋊C4×C71C411yescharacteristic
20C4⋊(C71⋊C8)—41no—
21C4⋊(C71⋊C8)—41no—
22C22⋊(C71⋊C8)—22no—
23C22⋊(C71⋊C8)—22no—
24#14556039—41no—
25C4×C71⋊C8—41no—
26C22×Q8×C71C2×C412yes—
27C22×Q8×C71C2×C411yescharacteristic
28C4⋊Q8×C71—41no—
29C4⋊Q8×C71—41no—
30C4⋊Q8×C71—44no—
31C4×Q8×C71—41no—
32C4×Q8×C71—41no—
33C2×C4⋊C4×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71D411yescharacteristic
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71C2311yescharacteristic
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71C2×C412yes—
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45#95×C71—21no—
46#95×C71—22no—
47#95×C71—22no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C2×Q8×C71—22no—
51C2×Q8×C71—22no—
52C2×Q8×C71—24no—
53C2×Q8×C71—41no—
54C2×Q8×C71—42no—
55C2×Q8×C71—42no—
56C22×C4×C71—22no—
57C22×C4×C71—22no—
58C22×C4×C71C22⋊C411yescharacteristic
59C22×C4×C71C22⋊C411yescharacteristic
60C22×C4×C71C22⋊C411yescharacteristic
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22×C411yescharacteristic
63C22×C4×C71C2×D411yescharacteristic
64C22×C4×C71C2×D411yescharacteristic
65C4⋊C4×C71—24no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—44no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—41no—
73C4⋊C4×C71—44no—
74C42×C71—21no—
75C42×C71—21no—
76C42×C71—21no—
77C42×C71—21no—
78C42×C71—21no—
79C42×C71—21no—
80C42×C71—42no—
81C23×C71C2×C22⋊C411yescharacteristic
82Q8×C71—44no—
83Q8×C71—84no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
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—42no—
97C2×C4×C71—42no—
98C2×C4×C71—41no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71#9911yescharacteristic
105C2×C4×C71#9911yescharacteristic
106C2×C4×C71#10012yes—
107C2×C4×C71#13112yes—
108C71⋊C8—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—44no—
119C4×C71—41no—
120C4×C71—41no—
121C4×C71—41no—
122C4×C71—41no—
123C4×C71—82no—
124C4×C71#35612yes—
125#11568—711nomaximal, Sylow
126C2×C71—21no—
127C2×C71—21no—
128C2×C71#147111yescharacteristic
129C2×C71#212911yescharacteristic
130C2×C71C2×#35611yescharacteristic
131#2546D7111yescharacteristic
132#1467—711no—
133#1461—711no—
134#1319—714no—
135C71#1156811yescharacteristic, complemented, Sylow
136C2×C4⋊Q8Dic7111yescharacteristic
137C2×C4⋊Q8Dic7112yes—
138C2×C4×Q8C2×D7111yescharacteristic
139#432—1421no—
140#431—1421no—
141#398—21no—
142#398—21no—
143#391C2×D7112yes—
144#389—21no—
145#389—21no—
146C4×C4⋊C4Dic7111yescharacteristic
147#324—712no—
148#324—712no—
149#324—1421no—
150#324—1421no—
151C22×Q8C2×Dic7112yes—
152C22×Q8C2×Dic7111yescharacteristic
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8—44no—
156C4×Q8—41no—
157C4×Q8—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊D7111yescharacteristic
164C2×C4⋊C4C22⋊D7111yescharacteristic
165C2×C4⋊C4C22×D7111yescharacteristic
166C2×C42C2×Dic7111yescharacteristic
167C2×C42C2×Dic7112yes—
168C2×C42C22⋊D7111yescharacteristic
169C2×C42C22⋊D7111yescharacteristic
170C4⋊C8—2841no—
171C4⋊C8—2841no—
172C22⋊C8—1422no—
173C22⋊C8—1422no—
174#97—2841no—
175C4×C8—2841no—
176#95—21no—
177#95—22no—
178#95—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—42no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C22⋊Dic7111yescharacteristic
188C22×C4C22⋊Dic7111yescharacteristic
189C22×C4C22⋊Dic7111yescharacteristic
190C22×C4C22⋊Dic7111yescharacteristic
191C22×C4C22×Dic7111yescharacteristic
192C22×C4C2×C22⋊D7111yescharacteristic
193C22×C4C2×C22⋊D7111yescharacteristic
194C2×C8—2842no—
195C2×C8—2842no—
196C4⋊C4—24no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—42no—
212C23C2×C22⋊Dic7111yescharacteristic
213Q8—44no—
214Q8—84no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4#1455604311yescharacteristic
236C2×C4#1455604311yescharacteristic
237C2×C4#1455604512yes—
238C2×C4#1455613512yes—
239C8—5682no—
240C8—5682no—
241C22—21no—
242C22—21no—
243C22#1798596611yescharacteristic
244C22C2×#1455604311yescharacteristic
245C22C2×#1455604511yescharacteristic
246C4—21no—
247C4—22no—
248C4—22no—
249C4—44no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4#1798579812yes—
256C2—21no—
257C2—21no—
258C2#2258670011yescharacteristic
259C2#2259001511yescharacteristic
260C2C2×#1798579811yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2546C2
socleC22×C71C2×#99
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2546D71
hypocenterC71#11568
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2546

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
#225866842
#225866642
#225862742
C71×#25462
#1156871