Finite Group Databasebeta

← #31895286

Subgroups — #31895286

Order 18176.

518 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 103 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#22592627C211yesmaximal, characteristic
2#22588440C212yesmaximal
3#22586670C211yesmaximal, characteristic
4#22586193C212yesmaximal
5C71×#2313C211yesmaximal, characteristic
6C2×#14556087C2211yescharacteristic
7#17985963C2212yes—
8#17985938C412yescomplemented
9#17985929C412yescomplemented
10#17985880—22no—
11#17985872—21no—
12#17985862—21no—
13#17985862C2211yescharacteristic
14#17985759—24no—
15#17985725—22no—
16C22×C4⋊C4×C71C2211yescharacteristic
17#387×C71—24no—
18#381×C71—21no—
19#380×C71—21no—
20C4×C22⋊C4×C71C2212yes—
21C2×#11765849—22no—
22#14556087—22no—
23#14556087—22no—
24C2×C4⋊Dic71C2×C411yescharacteristic
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×Dic71⋊C4—22no—
27C2×C22⋊Dic71—21no—
28C2×C22⋊Dic71—21no—
29C2×C4×Dic71—21no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32C71⋊#95—22no—
33C71⋊#95—22no—
34C71⋊#95—22no—
35C71⋊#95—21no—
36C71⋊#95C2×C412yes—
37C23×C4×C71D411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—22no—
42C2×C4⋊C4×C71—22no—
43C2×C4⋊C4×C71C2×C412yes—
44C2×C4⋊C4×C71D412yes—
45C2×C22⋊C4×C71—22no—
46C2×C42×C71D414yes—
47#95×C71—22no—
48#95×C71—24no—
49#95×C71—22no—
50C22×Dic71—22no—
51C22×Dic71—21no—
52C22×Dic71—21no—
53#11765849—44no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—22no—
56C4⋊Dic71—22no—
57Dic71⋊C4—44no—
58C22⋊Dic71—42no—
59C22⋊Dic71—41no—
60C22⋊Dic71—41no—
61C4×Dic71—42no—
62C24×C71C2×D411yescharacteristic
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—24no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71C22⋊C412yes—
72C22×C4×C71C22⋊C412yes—
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D412yes—
75C22×C4×C71C2×D411yescharacteristic
76C22×C4×C71#4112yes—
77C4⋊C4×C71—24no—
78C4⋊C4×C71—24no—
79C4⋊C4×C71—82no—
80C22⋊C4×C71—44no—
81C42×C71—44no—
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—44no—
101C2×C4×C71—44no—
102C2×C4×C71—44no—
103C2×C4×C71—81no—
104C2×C4×C71C4×D412yes—
105C2×C4×C71#12112yes—
106C2×Dic71—42no—
107C2×Dic71—42no—
108C2×Dic71—42no—
109C2×Dic71—41no—
110C2×Dic71—41no—
111C2×Dic71—42no—
112C2×Dic71—42no—
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—84no—
130Dic71—82no—
131Dic71—81no—
132Dic71—81no—
133#9482—711nomaximal, Sylow
134C2×C71—22no—
135C2×C71—22no—
136C2×C71—41no—
137C2×C71#189011yescharacteristic
138C2×C71#189711yescharacteristic
139C2×C71#191211yescharacteristic
140#2484—711no—
141#2313D7111yescharacteristic
142#1826—712no—
143#1463—711no—
144#1284—712no—
145C71#948211yescharacteristic, complemented, Sylow
146C2×#117—711no—
147C22×C4⋊C4C2×D7111yescharacteristic
148#407—712no—
149#400—712no—
150#398—712no—
151#387—24no—
152#387—1422no—
153#385—1421no—
154#381—21no—
155#381—711no—
156#381—1421no—
157#380—21no—
158C4×C22⋊C4C2×D7112yes—
159#339—1424no—
160#324—1422no—
161C23×C4C22⋊D7111yescharacteristic
162C23×C4C22⋊D7111yescharacteristic
163C23×C4C22×D7111yescharacteristic
164C2×M16—1422no—
165#117—1422no—
166#117—1422no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—711no—
171C2×C4⋊C4—711no—
172C2×C4⋊C4—1422no—
173C2×C4⋊C4C4×D7112yes—
174C2×C4⋊C4D28412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—1421no—
177C2×C22⋊C4—1421no—
178C2×C42—1421no—
179C2×C42C22⋊D7114yes—
180C22⋊C8—1422no—
181C22⋊C8—1422no—
182#95—22no—
183#95—24no—
184#95—22no—
185#95—712no—
186#95—1422no—
187#95—1422no—
188#95—1422no—
189#95—1421no—
190C24C2×C22⋊D7111yescharacteristic
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—1422no—
200C22×C4—1421no—
201C22×C4—1421no—
202C22×C4(C2×D71)⋊C412yes—
203C22×C4(C2×D71)⋊C412yes—
204C22×C4C2×C4×D7111yescharacteristic
205C22×C4C2×C22⋊D7112yes—
206C22×C4C2×D28411yescharacteristic
207C22×C4#1176586512yes—
208M16—2844no—
209C2×C8—2842no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—82no—
213C4⋊C4—2844no—
214C4⋊C4—1422no—
215C4⋊C4—1422no—
216C22⋊C4—44no—
217C22⋊C4—2842no—
218C22⋊C4—2841no—
219C22⋊C4—2841no—
220C42—44no—
221C42—2842no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23C2×(C2×D71)⋊C411yescharacteristic
226C23#1455609711yescharacteristic
227C23#1455611111yescharacteristic
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—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—81no—
244C2×C4—2842no—
245C2×C4—2842no—
246C2×C4—2842no—
247C2×C4—2841no—
248C2×C4—2841no—
249C2×C4—2842no—
250C2×C4—2842no—
251C2×C4C4×C22⋊D7112yes—
252C2×C4#1455610212yes—
253C8—5682no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22#1798587811yescharacteristic
263C22#1798601712yes—
264C22#1798618912yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—84no—
270C4—5682no—
271C4—5681no—
272C4—5681no—
273C2—22no—
274C2—22no—
275C2—41no—
276C2#2258875311yescharacteristic
277C2#2258878811yescharacteristic
278C2#2258887711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2313C2
socleC22×C71#386
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2313D71
hypocenterC71#9482
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
8. #17985938C4semidirect
9. #17985929C4semidirect
145. C71#9482semidirect

Maximal subgroups

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

subgroupindex
#225926272
#225884402
#225866702
#225861932
C71×#23132
#948271