Finite Group Databasebeta

← #31895287

Subgroups — #31895287

Order 18176.

518 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 107 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#22590943C211yesmaximal, characteristic
2#22588439C212yesmaximal
3#22586670C211yesmaximal, characteristic
4#22586192C212yesmaximal
5C71×#2484C211yesmaximal, characteristic
6#17985963C2212yes—
7#17985885—24no—
8#17985862—21no—
9#17985859—21no—
10C4×C22⋊Dic71C2212yes—
11#17985758—24no—
12#17985725—22no—
13C2×#117×C71C2211yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#400×C71C412yescomplemented
16#398×C71C412yescomplemented
17#387×C71—22no—
18#385×C71—21no—
19#381×C71—21no—
20#381×C71C2211yescharacteristic
21C2×#11765849—22no—
22C2×C4⋊Dic71—22no—
23C2×C22⋊Dic71—22no—
24C2×C4×Dic71D414yes—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C71⋊#95—22no—
28C71⋊#95—24no—
29C71⋊#95—22no—
30C23×C4×C71D411yescharacteristic
31C23×C4×C71D411yescharacteristic
32C23×C4×C71C2311yescharacteristic
33#117×C71—22no—
34#117×C71—22no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C412yes—
41C2×C4⋊C4×C71D412yes—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C42×C71—21no—
45#95×C71—22no—
46#95×C71—22no—
47#95×C71—22no—
48#95×C71—21no—
49#95×C71C2×C412yes—
50C22×Dic71—24no—
51#11765849—44no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—82no—
54C22⋊Dic71—44no—
55C4×Dic71—44no—
56C24×C71C2×D411yescharacteristic
57C22×C4×C71—21no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71—21no—
61C22×C4×C71—22no—
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—42no—
65C22×C4×C71—42no—
66C22×C4×C71—42no—
67C22×C4×C71C22⋊C412yes—
68C22×C4×C71C22⋊C412yes—
69C22×C4×C71C22×C411yescharacteristic
70C22×C4×C71C2×D412yes—
71C22×C4×C71C2×D411yescharacteristic
72C22×C4×C71#4112yes—
73C4⋊C4×C71—24no—
74C4⋊C4×C71—24no—
75C4⋊C4×C71—22no—
76C4⋊C4×C71—22no—
77C4⋊C4×C71—44no—
78C22⋊C4×C71—42no—
79C22⋊C4×C71—41no—
80C22⋊C4×C71—41no—
81C42×C71—42no—
82C23×C71—22no—
83C23×C71—22no—
84C23×C71—41no—
85C23×C71C2×C22⋊C411yescharacteristic
86C23×C71#12011yescharacteristic
87C23×C71#12311yescharacteristic
88C2×C4×C71—24no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—41no—
94C2×C4×C71—41no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—42no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—81no—
108C2×C4×C71C4×D412yes—
109C2×C4×C71#12112yes—
110C2×Dic71—44no—
111C2×Dic71—44no—
112C2×Dic71—44no—
113C71⋊C8—82no—
114C22×C71—21no—
115C22×C71—21no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—44no—
122C22×C71#38611yescharacteristic
123C22×C71#42112yes—
124C22×C71#45812yes—
125C4×C71—22no—
126C4×C71—22no—
127C4×C71—42no—
128C4×C71—42no—
129C4×C71—82no—
130C4×C71—81no—
131C4×C71—81no—
132Dic71—84no—
133#9482—711nomaximal, Sylow
134C2×C71—22no—
135C2×C71—22no—
136C2×C71—41no—
137C2×C71#189011yescharacteristic
138C2×C71#189711yescharacteristic
139C2×C71#191211yescharacteristic
140#2484D7111yescharacteristic
141#2313—711no—
142#1826—712no—
143#1463—711no—
144#1284—712no—
145C71#948211yescharacteristic, complemented, Sylow
146C2×#117C2×D7111yescharacteristic
147C22×C4⋊C4C2×D7111yescharacteristic
148#407—712no—
149#400Dic7112yescomplemented
150#398Dic7112yescomplemented
151#387—22no—
152#387—1424no—
153#385—21no—
154#381—21no—
155#381—1421no—
156#381C2×D7111yescharacteristic
157#380—1421no—
158C4×C22⋊C4—712no—
159#339—1424no—
160#324—1422no—
161C23×C4C22⋊D7111yescharacteristic
162C23×C4C22⋊D7111yescharacteristic
163C23×C4C22×D7111yescharacteristic
164C2×M16—1422no—
165#117—22no—
166#117—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—1422no—
171C2×C4⋊C4C2×Dic7111yescharacteristic
172C2×C4⋊C4C2×Dic7111yescharacteristic
173C2×C4⋊C4C2×Dic7112yes—
174C2×C4⋊C4C22⋊D7112yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—1422no—
178C2×C42—21no—
179C2×C42—714no—
180C22⋊C8—1422no—
181C22⋊C8—1422no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—21no—
186#95—1422no—
187#95—1424no—
188#95—1422no—
189#95C2×Dic7112yes—
190C24C2×C22⋊D7111yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—1424no—
202C22×C4C22⋊Dic7112yes—
203C22×C4C22⋊Dic7112yes—
204C22×C4C22×Dic7111yescharacteristic
205C22×C4D4×D7112yes—
206C22×C4C2×C22⋊D7111yescharacteristic
207C22×C4#1176586412yes—
208M16—2844no—
209C2×C8—2842no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—44no—
215C4⋊C4—5682no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—2844no—
220C42—42no—
221C42—2844no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23C2×C22⋊Dic7111yescharacteristic
226C23#1455609611yescharacteristic
227C23#1455610911yescharacteristic
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—2844no—
249C2×C4—2844no—
250C2×C4—2844no—
251C2×C4D4×Dic7112yes—
252C2×C4#1455610012yes—
253C8—5682no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22#1798587911yescharacteristic
263C22#1798601612yes—
264C22#1798618812yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—82no—
270C4—81no—
271C4—81no—
272C4—5684no—
273C2—22no—
274C2—22no—
275C2—41no—
276C2#2258875411yescharacteristic
277C2#2258878911yescharacteristic
278C2#2258887811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2484C2
socleC22×C71#386
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2484D71
hypocenterC71#9482
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
15. #400×C71C4semidirect
16. #398×C71C4semidirect
145. C71#9482semidirect
149. #400Dic71semidirect
150. #398Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909432
#225884392
#225866702
#225861922
C71×#24842
#948271