Finite Group Databasebeta

← #31894893

Subgroups — #31894893

Order 18176.

618 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 115 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#22590697C211yesmaximal, characteristic
2#22586860C212yesmaximal
3#22586670C211yesmaximal, characteristic
4#22586226C212yesmaximal
5C71×#2583C211yesmaximal, characteristic
6C2×#14556087—21no—
7#17985963C2212yes—
8C4⋊C4×Dic71—24no—
9C4×C22⋊Dic71—21no—
10C4×C22⋊Dic71C2212yes—
11#17985736—24no—
12#17985725—22no—
13C2×C22⋊Q8×C71C2211yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#398×C71C412yescomplemented
16#395×C71C412yescomplemented
17#389×C71—22no—
18#385×C71—21no—
19#385×C71—21no—
20C4×C22⋊C4×C71C2211yescharacteristic
21C2×#11765849—22no—
22#14556087—28no—
23C2×C4⋊Dic71—22no—
24C2×C22⋊Dic71—22no—
25C2×C4×Dic71—22no—
26C2×C4×Dic71—22no—
27C2×C4×Dic71—22no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C71⋊#95—22no—
31C71⋊#95D414yes—
32C22×Q8×C71C2×C411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C23×C4×C71C2311yescharacteristic
36C22⋊Q8×C71—22no—
37C22⋊Q8×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2×C412yes—
45C2×C4⋊C4×C71D412yes—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C42×C71—21no—
49#95×C71—22no—
50#95×C71—22no—
51#95×C71C2×C412yes—
52C22×Dic71—24no—
53#11765849—82no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—28no—
56C22⋊Dic71—28no—
57C4×Dic71—28no—
58C4×Dic71—82no—
59C4×Dic71—82no—
60C24×C71C2×D411yescharacteristic
61C2×Q8×C71—22no—
62C2×Q8×C71—81no—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—24no—
66C22×C4×C71—22no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—81no—
72C22×C4×C71—81no—
73C22×C4×C71C22⋊C412yes—
74C22×C4×C71C22⋊C412yes—
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D412yes—
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71#4112yes—
79C4⋊C4×C71—24no—
80C4⋊C4×C71—24no—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—44no—
83C4⋊C4×C71—42no—
84C4⋊C4×C71—82no—
85C22⋊C4×C71—24no—
86C22⋊C4×C71—41no—
87C22⋊C4×C71—41no—
88C42×C71—81no—
89C23×C71—22no—
90C23×C71—22no—
91C23×C71—41no—
92C23×C71C2×C22⋊C411yescharacteristic
93C23×C71#12011yescharacteristic
94C23×C71#12311yescharacteristic
95Q8×C71—82no—
96C2×C4×C71—24no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—24no—
102C2×C4×C71—22no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—42no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—42no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—42no—
115C2×C4×C71—81no—
116C2×C4×C71#10414yes—
117C2×C4×C71C4×D412yes—
118C2×C4×C71#12112yes—
119C2×Dic71—28no—
120C2×Dic71—44no—
121C2×Dic71—44no—
122C71⋊C8—82no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—44no—
131C22×C71#38611yescharacteristic
132C22×C71C2×#10412yes—
133C22×C71#45612yes—
134C4×C71—24no—
135C4×C71—22no—
136C4×C71—22no—
137C4×C71—44no—
138C4×C71—42no—
139C4×C71—81no—
140C4×C71—81no—
141Dic71—48no—
142#9408—711nomaximal, Sylow
143C2×C71—22no—
144C2×C71—22no—
145C2×C71—41no—
146C2×C71#188611yescharacteristic
147C2×C71#189611yescharacteristic
148C2×C71#191311yescharacteristic
149#2583D7111yescharacteristic
150#2274—711no—
151#1508—712no—
152#1463—711no—
153#1297—712no—
154C71#940811yescharacteristic, complemented, Sylow
155C2×C22⋊Q8C2×D7111yescharacteristic
156C2×#117—1421no—
157C22×C4⋊C4C2×D7111yescharacteristic
158#407—712no—
159#398Dic7112yescomplemented
160#395Dic7112yescomplemented
161#389—22no—
162#385—21no—
163#385—21no—
164C4×C4⋊C4—1424no—
165C4×C22⋊C4—712no—
166C4×C22⋊C4—1421no—
167C4×C22⋊C4C2×D7111yescharacteristic
168#328—1424no—
169#324—1422no—
170C22×Q8C2×Dic7111yescharacteristic
171C23×C4C22⋊D7111yescharacteristic
172C23×C4C22⋊D7111yescharacteristic
173C23×C4C22×D7111yescharacteristic
174C2×M16—1422no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177#117—1428no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—1422no—
184C2×C4⋊C4C2×Dic7111yescharacteristic
185C2×C4⋊C4C2×Dic7112yes—
186C2×C4⋊C4C22⋊D7112yes—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—1422no—
190C2×C42—21no—
191C2×C42—1422no—
192C2×C42—1422no—
193C2×C42—1422no—
194C22⋊C8—1422no—
195C22⋊C8—1422no—
196#95—22no—
197#95—22no—
198#95—714no—
199#95—1422no—
200#95C2×Dic7112yes—
201C24C2×C22⋊D7111yescharacteristic
202C2×Q8—22no—
203C2×Q8—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—1424no—
215C22×C4C22⋊Dic7112yes—
216C22×C4C22⋊Dic7112yes—
217C22×C4C22×Dic7111yescharacteristic
218C22×C4D4×D7112yes—
219C22×C4C2×C22⋊D7111yescharacteristic
220C22×C4#1176586412yes—
221M16—5682no—
222C2×C8—2842no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—82no—
229C4⋊C4—1428no—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—1428no—
234C42—81no—
235C42—5682no—
236C42—5682no—
237C42—1428no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23C2×C22⋊Dic7111yescharacteristic
242C23#1455609611yescharacteristic
243C23#1455610911yescharacteristic
244Q8—82no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—2844no—
266C2×C4—2844no—
267C2×C4—1428no—
268C2×C4#1455605514yes—
269C2×C4D4×Dic7112yes—
270C2×C4#1455610012yes—
271C8—5682no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1798587911yescharacteristic
281C22C2×#1455605512yes—
282C22#1798618112yes—
283C4—24no—
284C4—22no—
285C4—22no—
286C4—44no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—2848no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#2258873411yescharacteristic
295C2#2258878411yescharacteristic
296C2#2258888311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2583C2
socleC22×C71#386
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2583D71
hypocenterC71#9408
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
15. #398×C71C4semidirect
16. #395×C71C4semidirect
154. C71#9408semidirect
159. #398Dic71semidirect
160. #395Dic71semidirect

Maximal subgroups

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

subgroupindex
#225906972
#225868602
#225866702
#225862262
C71×#25832
#940871