Finite Group Databasebeta

← #31895137

Subgroups — #31895137

Order 18176.

468 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 111 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#22587066C212yesmaximal
3#22586684C211yesmaximal, characteristic
4#22586196C212yesmaximal
5C71×#2546C211yesmaximal, characteristic
6#17986130—24no—
7#17986056—21no—
8#17986056C2212yes—
9#17986056C2212yes—
10#17986052—21no—
11#17985748—24no—
12#17985725—21no—
13#17985725—21no—
14C2×C4⋊Q8×C71C412yes—
15C2×C4⋊Q8×C71C2211yescharacteristic
16C2×C4×Q8×C71C2211yescharacteristic
17#398×C71—21no—
18#398×C71—21no—
19#391×C71C412yes—
20#389×C71—21no—
21#389×C71—21no—
22C4×C4⋊C4×C71C2211yescharacteristic
23C4⋊(C71⋊C8)—22no—
24C4⋊(C71⋊C8)—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)D414yes—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29#14556039—22no—
30#14556039—22no—
31#14556039—22no—
32C4×C71⋊C8—22no—
33C22×Q8×C71C2×C412yes—
34C22×Q8×C71C2×C411yescharacteristic
35C4⋊Q8×C71—22no—
36C4⋊Q8×C71—22no—
37C4⋊Q8×C71—44no—
38C4×Q8×C71—22no—
39C4×Q8×C71—22no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D411yescharacteristic
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C42×C71C2×C412yes—
49C2×C42×C71D411yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×C42×C71C2311yescharacteristic
52#95×C71—22no—
53#95×C71—22no—
54#95×C71—21no—
55C2×C71⋊C8—24no—
56C2×C71⋊C8—42no—
57C2×Q8×C71—22no—
58C2×Q8×C71—22no—
59C2×Q8×C71—24no—
60C2×Q8×C71—41no—
61C2×Q8×C71—42no—
62C2×Q8×C71—42no—
63C22×C4×C71—22no—
64C22×C4×C71—22no—
65C22×C4×C71C22⋊C411yescharacteristic
66C22×C4×C71C22⋊C411yescharacteristic
67C22×C4×C71C22⋊C411yescharacteristic
68C22×C4×C71C22⋊C411yescharacteristic
69C22×C4×C71C22×C411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C22×C4×C71C2×D411yescharacteristic
72C4⋊C4×C71—24no—
73C4⋊C4×C71—22no—
74C4⋊C4×C71—22no—
75C4⋊C4×C71—22no—
76C4⋊C4×C71—44no—
77C4⋊C4×C71—44no—
78C4⋊C4×C71—41no—
79C4⋊C4×C71—42no—
80C4⋊C4×C71—41no—
81C42×C71—21no—
82C42×C71—21no—
83C42×C71—21no—
84C42×C71—21no—
85C42×C71—42no—
86C42×C71C2×D412yes—
87C42×C71#4112yes—
88C23×C71C2×C22⋊C411yescharacteristic
89Q8×C71—44no—
90Q8×C71—84no—
91C2×C4×C71—21no—
92C2×C4×C71—21no—
93C2×C4×C71—21no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—24no—
97C2×C4×C71—21no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—41no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71#10012yes—
111C2×C4×C71C4×D412yes—
112C2×C4×C71#12011yescharacteristic
113C2×C4×C71#12112yes—
114C2×C4×C71#12311yescharacteristic
115C71⋊C8—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—44no—
127C4×C71—41no—
128C4×C71—41no—
129C4×C71—41no—
130C4×C71—82no—
131C4×C71#45612yes—
132#9452—711nomaximal, Sylow
133C2×C71—21no—
134C2×C71—21no—
135C2×C71#188611yescharacteristic
136C2×C71#190011yescharacteristic
137C2×C71#190911yescharacteristic
138#2983—711no—
139#2546D7111yescharacteristic
140#1558—712no—
141#1467—711no—
142#1285—712no—
143C71#945211yescharacteristic, complemented, Sylow
144C2×C4⋊Q8Dic7112yes—
145C2×C4⋊Q8C2×D7111yescharacteristic
146C2×C4×Q8C2×D7111yescharacteristic
147#446—1424no—
148#432—712no—
149#432—712no—
150#432—1421no—
151#431—1421no—
152#398—21no—
153#398—21no—
154#391Dic7112yes—
155#389—21no—
156#389—21no—
157C4×C4⋊C4C2×D7111yescharacteristic
158#333—1424no—
159#324—1421no—
160#324—1421no—
161C22×Q8C2×Dic7112yes—
162C22×Q8C2×Dic7111yescharacteristic
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—44no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C2×Dic7111yescharacteristic
174C2×C4⋊C4C22⋊D7111yescharacteristic
175C2×C4⋊C4C22⋊D7111yescharacteristic
176C2×C42C2×Dic7112yes—
177C2×C42C22⋊D7111yescharacteristic
178C2×C42C22⋊D7111yescharacteristic
179C2×C42C22×D7111yescharacteristic
180C4⋊C8—714no—
181C4⋊C8—1422no—
182C4⋊C8—1422no—
183C4⋊C8—1422no—
184C22⋊C8—1422no—
185C22⋊C8—1422no—
186#97—1422no—
187#97—1422no—
188#97—1422no—
189C4×C8—1422no—
190#95—22no—
191#95—22no—
192#95—21no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—24no—
196C2×Q8—41no—
197C2×Q8—42no—
198C2×Q8—42no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4C22⋊Dic7111yescharacteristic
202C22×C4C22⋊Dic7111yescharacteristic
203C22×C4C22⋊Dic7111yescharacteristic
204C22×C4C22⋊Dic7111yescharacteristic
205C22×C4C22×Dic7111yescharacteristic
206C22×C4C2×C22⋊D7111yescharacteristic
207C22×C4C2×C22⋊D7111yescharacteristic
208C2×C8—2842no—
209C2×C8—1424no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—42no—
224C42D4×D7112yes—
225C42#1176586412yes—
226C23C2×C22⋊Dic7111yescharacteristic
227Q8—44no—
228Q8—84no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4#1455604512yes—
249C2×C4D4×Dic7112yes—
250C2×C4#1455609611yescharacteristic
251C2×C4#1455610012yes—
252C2×C4#1455610911yescharacteristic
253C8—2844no—
254C8—5682no—
255C22—21no—
256C22—21no—
257C22#1798587911yescharacteristic
258C22C2×#1455604511yescharacteristic
259C22#1798598311yescharacteristic
260C4—21no—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—44no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—82no—
269C4#1798618112yes—
270C2—21no—
271C2—21no—
272C2#2258873411yescharacteristic
273C2#2258880411yescharacteristic
274C2#2258886311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2546C2
socleC22×C71#386
prime radicalC23×C71C2×C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2546D71
hypocenterC71#9452
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

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
#225974252
#225870662
#225866842
#225861962
C71×#25462
#945271