Finite Group Databasebeta

← #31895177

Subgroups — #31895177

Order 18176.

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

normal subgroups only

This group has 512 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#22588420C211yesmaximal, characteristic
3#22586873C212yesmaximal
4#22586193C212yesmaximal
5C71×#2503C211yesmaximal, characteristic
6#17985963C2212yes—
7#17985885—24no—
8#17985862C2212yes—
9#17985859—21no—
10C4×C22⋊Dic71—21no—
11#17985768—22no—
12#17985736—24no—
13C2×C22⋊Q8×C71C2211yescharacteristic
14C22×C4⋊C4×C71C2211yescharacteristic
15#400×C71—22no—
16#391×C71C412yescomplemented
17#382×C71C412yescomplemented
18#381×C71—21no—
19#381×C71—21no—
20C4×C22⋊C4×C71C2211yescharacteristic
21C2×#11765849—22no—
22C2×C4⋊Dic71—22no—
23C2×C22⋊Dic71—22no—
24C2×C4×Dic71—22no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)—22no—
27C71⋊#95—22no—
28C71⋊#95—22no—
29C71⋊#95—22no—
30C71⋊#95D414yes—
31C22×Q8×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C23×C4×C71C2311yescharacteristic
35C22⋊Q8×C71—22no—
36C22⋊Q8×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C412yes—
42C2×C4⋊C4×C71D412yes—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C42×C71C2×C412yes—
46#95×C71—22no—
47#95×C71—22no—
48#95×C71—22no—
49#95×C71—22no—
50#95×C71—21no—
51C22×Dic71—24no—
52#11765849—44no—
53C2×C71⋊C8—42no—
54C4⋊Dic71—82no—
55C22⋊Dic71—44no—
56C4×Dic71—82no—
57C24×C71C2×D411yescharacteristic
58C2×Q8×C71—22no—
59C2×Q8×C71—42no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—22no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—42no—
68C22×C4×C71—42no—
69C22×C4×C71—42no—
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22⋊C412yes—
72C22×C4×C71C22×C411yescharacteristic
73C22×C4×C71C2×D412yes—
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71#4112yes—
76C4⋊C4×C71—24no—
77C4⋊C4×C71—24no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—42no—
80C22⋊C4×C71—42no—
81C22⋊C4×C71—41no—
82C22⋊C4×C71—41no—
83C42×C71—42no—
84C23×C71—22no—
85C23×C71—22no—
86C23×C71—41no—
87C23×C71C2×C22⋊C411yescharacteristic
88C23×C71#12011yescharacteristic
89C23×C71#12311yescharacteristic
90Q8×C71—82no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
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—42no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—81no—
109C2×C4×C71—81no—
110C2×C4×C71C4×D412yes—
111C2×C4×C71#12112yes—
112C2×Dic71—44no—
113C2×Dic71—44no—
114C2×Dic71—44no—
115C71⋊C8—82no—
116C22×C71—21no—
117C22×C71—21no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71—44no—
124C22×C71#38611yescharacteristic
125C22×C71#41012yes—
126C22×C71#45512yes—
127C4×C71—22no—
128C4×C71—22no—
129C4×C71—42no—
130C4×C71—42no—
131C4×C71—82no—
132C4×C71—81no—
133C4×C71—81no—
134Dic71—84no—
135#9460—711nomaximal, Sylow
136C2×C71—22no—
137C2×C71—22no—
138C2×C71—41no—
139C2×C71#188511yescharacteristic
140C2×C71#189911yescharacteristic
141C2×C71#191211yescharacteristic
142#2503D7111yescharacteristic
143#2313—711no—
144#1821—711no—
145#1511—712no—
146#1284—712no—
147C71#946011yescharacteristic, complemented, Sylow
148C2×C22⋊Q8C2×D7111yescharacteristic
149C22×C4⋊C4C2×D7111yescharacteristic
150#407—712no—
151#400—22no—
152#391Dic7112yescomplemented
153#387—1424no—
154#382Dic7112yescomplemented
155#381—21no—
156#381—21no—
157#381—712no—
158#380—1421no—
159C4×C22⋊C4—1421no—
160C4×C22⋊C4C2×D7111yescharacteristic
161#343—1422no—
162#328—1424no—
163C22×Q8C2×Dic7111yescharacteristic
164C23×C4C22⋊D7111yescharacteristic
165C23×C4C22⋊D7111yescharacteristic
166C23×C4C22×D7111yescharacteristic
167C2×M16—1422no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—1422no—
174C2×C4⋊C4C2×Dic7111yescharacteristic
175C2×C4⋊C4C2×Dic7112yes—
176C2×C4⋊C4C22⋊D7112yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—1422no—
180C2×C42—1422no—
181C2×C42C2×Dic7112yes—
182C22⋊C8—1422no—
183C22⋊C8—1422no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—714no—
190#95—1422no—
191#95—1422no—
192#95—1422no—
193C24C2×C22⋊D7111yescharacteristic
194C2×Q8—22no—
195C2×Q8—42no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—1424no—
207C22×C4C22⋊Dic7112yes—
208C22×C4C22⋊Dic7112yes—
209C22×C4C22×Dic7111yescharacteristic
210C22×C4D4×D7112yes—
211C22×C4C2×C22⋊D7111yescharacteristic
212C22×C4#1176586412yes—
213M16—2844no—
214C2×C8—2842no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—5682no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—2844no—
224C42—42no—
225C42—5682no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23C2×C22⋊Dic7111yescharacteristic
230C23#1455609611yescharacteristic
231C23#1455610911yescharacteristic
232Q8—82no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—2844no—
253C2×C4—2844no—
254C2×C4—2844no—
255C2×C4D4×Dic7112yes—
256C2×C4#1455610012yes—
257C8—5682no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22#1798587911yescharacteristic
267C22#1798597312yes—
268C22#1798617612yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—81no—
275C4—81no—
276C4—5684no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2#2258872911yescharacteristic
281C2#2258879911yescharacteristic
282C2#2258887811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2503C2
socleC22×C71#386
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2503D71
hypocenterC71#9460
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
16. #391×C71C4semidirect
17. #382×C71C4semidirect
147. C71#9460semidirect
152. #391Dic71semidirect
154. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909432
#225884202
#225868732
#225861932
C71×#25032
#946071