Finite Group Databasebeta

← #31894913

Subgroups — #31894913

Order 18176.

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

normal subgroups only

This group has 526 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#22586860C212yesmaximal
3#22586670C211yesmaximal, characteristic
4#22586226C212yesmaximal
5C71×#2583C211yesmaximal, characteristic
6#17985963C2212yes—
7#17985885—24no—
8#17985862—21no—
9#17985859—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—
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—24no—
29C71⋊#95D414yes—
30C22×Q8×C71C2×C411yescharacteristic
31C23×C4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71C2311yescharacteristic
34C22⋊Q8×C71—22no—
35C22⋊Q8×C71—22no—
36C2×C4⋊C4×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71C2×C412yes—
43C2×C4⋊C4×C71D412yes—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C42×C71—21no—
47#95×C71—22no—
48#95×C71—22no—
49#95×C71C2×C412yes—
50C22×Dic71—24no—
51#11765849—82no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—82no—
54C22⋊Dic71—44no—
55C4×Dic71—82no—
56C24×C71C2×D411yescharacteristic
57C2×Q8×C71—22no—
58C2×Q8×C71—81no—
59C22×C4×C71—22no—
60C22×C4×C71—21no—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—42no—
67C22×C4×C71—81no—
68C22×C4×C71—81no—
69C22×C4×C71C22⋊C412yes—
70C22×C4×C71C22⋊C412yes—
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D412yes—
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71#4112yes—
75C4⋊C4×C71—24no—
76C4⋊C4×C71—24no—
77C4⋊C4×C71—22no—
78C4⋊C4×C71—44no—
79C4⋊C4×C71—42no—
80C4⋊C4×C71—82no—
81C22⋊C4×C71—24no—
82C22⋊C4×C71—41no—
83C22⋊C4×C71—41no—
84C42×C71—81no—
85C23×C71—22no—
86C23×C71—22no—
87C23×C71—41no—
88C23×C71C2×C22⋊C411yescharacteristic
89C23×C71#12011yescharacteristic
90C23×C71#12311yescharacteristic
91Q8×C71—82no—
92C2×C4×C71—24no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—22no—
96C2×C4×C71—22no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71—42no—
101C2×C4×C71—42no—
102C2×C4×C71—42no—
103C2×C4×C71—42no—
104C2×C4×C71—42no—
105C2×C4×C71—42no—
106C2×C4×C71—42no—
107C2×C4×C71—42no—
108C2×C4×C71—41no—
109C2×C4×C71—41no—
110C2×C4×C71—42no—
111C2×C4×C71—81no—
112C2×C4×C71C4×D412yes—
113C2×C4×C71#12112yes—
114C2×Dic71—44no—
115C2×Dic71—44no—
116C2×Dic71—44no—
117C71⋊C8—82no—
118C22×C71—21no—
119C22×C71—21no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71—21no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71—44no—
126C22×C71#38611yescharacteristic
127C22×C71#42112yes—
128C22×C71#45612yes—
129C4×C71—22no—
130C4×C71—22no—
131C4×C71—44no—
132C4×C71—42no—
133C4×C71—42no—
134C4×C71—81no—
135C4×C71—81no—
136Dic71—84no—
137#9412—711nomaximal, Sylow
138C2×C71—22no—
139C2×C71—22no—
140C2×C71—41no—
141C2×C71#188611yescharacteristic
142C2×C71#189711yescharacteristic
143C2×C71#191311yescharacteristic
144#2583D7111yescharacteristic
145#2313—711no—
146#1508—712no—
147#1463—711no—
148#1297—712no—
149C71#941211yescharacteristic, complemented, Sylow
150C2×C22⋊Q8C2×D7111yescharacteristic
151C22×C4⋊C4C2×D7111yescharacteristic
152#407—712no—
153#398Dic7112yescomplemented
154#395Dic7112yescomplemented
155#389—22no—
156#387—1424no—
157#385—21no—
158#385—21no—
159#381—1421no—
160#380—1421no—
161C4×C22⋊C4—712no—
162C4×C22⋊C4C2×D7111yescharacteristic
163#328—1424no—
164#324—1422no—
165C22×Q8C2×Dic7111yescharacteristic
166C23×C4C22⋊D7111yescharacteristic
167C23×C4C22⋊D7111yescharacteristic
168C23×C4C22×D7111yescharacteristic
169C2×M16—1422no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—1422no—
178C2×C4⋊C4C2×Dic7111yescharacteristic
179C2×C4⋊C4C2×Dic7112yes—
180C2×C4⋊C4C22⋊D7112yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—1422no—
184C2×C42—21no—
185C2×C42—1422no—
186C22⋊C8—1422no—
187C22⋊C8—1422no—
188#95—22no—
189#95—22no—
190#95—714no—
191#95—1422no—
192#95—1424no—
193#95C2×Dic7112yes—
194C24C2×C22⋊D7111yescharacteristic
195C2×Q8—22no—
196C2×Q8—81no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—1424no—
208C22×C4C22⋊Dic7112yes—
209C22×C4C22⋊Dic7112yes—
210C22×C4C22×Dic7111yescharacteristic
211C22×C4D4×D7112yes—
212C22×C4C2×C22⋊D7111yescharacteristic
213C22×C4#1176586412yes—
214M16—5682no—
215C2×C8—2842no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—22no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—5682no—
223C22⋊C4—24no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—2844no—
227C42—81no—
228C42—5682no—
229C23—22no—
230C23—22no—
231C23—41no—
232C23C2×C22⋊Dic7111yescharacteristic
233C23#1455609611yescharacteristic
234C23#1455610911yescharacteristic
235Q8—82no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—2844no—
257C2×C4—2844no—
258C2×C4—2844no—
259C2×C4D4×Dic7112yes—
260C2×C4#1455610012yes—
261C8—5682no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1798587911yescharacteristic
271C22#1798601612yes—
272C22#1798618112yes—
273C4—22no—
274C4—22no—
275C4—44no—
276C4—42no—
277C4—42no—
278C4—81no—
279C4—81no—
280C4—5684no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#2258873411yescharacteristic
285C2#2258878911yescharacteristic
286C2#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#9412
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
149. C71#9412semidirect
153. #398Dic71semidirect
154. #395Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909432
#225868602
#225866702
#225862262
C71×#25832
#941271