Finite Group Databasebeta

← #31892960

Subgroups — #31892960

Order 18176.

546 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 135 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#22597421C211yesmaximal, characteristic, complemented
2#22591047C211yesmaximal, characteristic
3#22586730C212yesmaximal
4#22586168C212yesmaximal, complemented
5C71×#1461C211yesmaximal, characteristic, complemented
6C2×#14556117C411yescharacteristic
7C2×C4×D284C2211yescharacteristic
8Q8×C71⋊C8—24no—
9#17986052—22no—
10#17986052C2212yes—
11#17985906C411yescharacteristic
12#17985900C411yescharacteristic
13#17985890—22no—
14#17985870—22no—
15#17985868C411yescharacteristic
16#17985845C2211yescharacteristic
17#17985726—24no—
18C2×C4×Q8×C71C2211yescharacteristic
19#431×C71C2212yes—
20#324×C71—22no—
21C22×D284C2×C411yescharacteristic
22C22×C4×D71—21no—
23#14556117—22no—
24#14556117C814yescomplemented
25C4×D284—22no—
26C4×D284—22no—
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×Dic71⋊C4—22no—
29C2×(C2×D71)⋊C4—22no—
30C2×(C2×D71)⋊C4—22no—
31C2×(C2×D71)⋊C4—21no—
32C2×C4×Dic71C2×C411yescharacteristic
33C2×C4×Dic71C2×C411yescharacteristic
34C4⋊(C71⋊C8)—24no—
35C4⋊(C71⋊C8)—22no—
36C22⋊(C71⋊C8)—22no—
37C4×C71⋊C8—24no—
38C4×C71⋊C8D414yes—
39C71⋊#95—22no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C22×Q8×C71C2×C411yescharacteristic
43C4×Q8×C71—22no—
44C4×Q8×C71—22no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71D412yes—
47C2×C42×C71D412yes—
48C2×C42×C71C2311yescharacteristic
49C4⋊C8×C71—22no—
50C22⋊C8×C71—22no—
51C4×C8×C71—22no—
52C23×D71—21no—
53C2×D284—22no—
54C2×D284—22no—
55C2×D284—21no—
56C2×D284C2×C812yes—
57C22×Dic71—21no—
58C22×Dic71—21no—
59C22×Dic71—21no—
60C2×C4×D71—42no—
61C2×C4×D71—82no—
62C2×C71⋊C8—24no—
63C4⋊Dic71—22no—
64Dic71⋊C4—82no—
65(C2×D71)⋊C4—28no—
66(C2×D71)⋊C4—41no—
67(C2×D71)⋊C4—41no—
68(C2×D71)⋊C4—82no—
69C4×Dic71—21no—
70C4×Dic71—22no—
71C4×Dic71C2×C812yes—
72C2×Q8×C71—22no—
73C2×Q8×C71—22no—
74C2×Q8×C71C2×C812yes—
75C2×Q8×C71M1612yes—
76C22×C4×C71C22⋊C412yes—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22×C411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C2×C8×C71—42no—
82C4⋊C4×C71—22no—
83C4⋊C4×C71—24no—
84C42×C71—22no—
85C42×C71—22no—
86C42×C71C2×D412yes—
87C42×C71#4112yes—
88C23×C71C2×C22⋊C411yescharacteristic
89C22×D71—22no—
90C22×D71—42no—
91C22×D71—42no—
92C22×D71—41no—
93C22×D71—41no—
94Q8×C71—24no—
95Q8×C71—24no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—41no—
100C2×C4×C71—41no—
101C2×C4×C71—42no—
102C2×C4×C71C22⋊C814yes—
103C2×C4×C71C4×D412yes—
104C2×C4×C71#12011yescharacteristic
105C2×C4×C71#12112yes—
106C2×C4×C71#12311yescharacteristic
107C2×C4×C71C22×C811yescharacteristic
108C2×C4×C71C2×M1611yescharacteristic
109C2×C4×C71#13112yes—
110C2×Dic71—22no—
111C2×Dic71—41no—
112C2×Dic71—41no—
113C2×Dic71—41no—
114C2×Dic71—42no—
115C2×Dic71—41no—
116C2×Dic71—41no—
117C2×Dic71—41no—
118D284—24no—
119D284—42no—
120C4×D71—84no—
121C8×C71—82no—
122C71⋊C8—44no—
123C22×C71—21no—
124C22×C71—21no—
125C22×C71#38611yescharacteristic
126C22×C71C2×C22⋊C811yescharacteristic
127C22×C71#40811yescharacteristic
128C2×D71—44no—
129C2×D71—42no—
130C2×D71—82no—
131C2×D71—81no—
132C2×D71—81no—
133C4×C71—24no—
134C4×C71—22no—
135C4×C71—21no—
136C4×C71—21no—
137C4×C71—42no—
138C4×C71C8×D412yes—
139C4×C71#43512yes—
140Dic71—42no—
141Dic71—81no—
142Dic71—81no—
143#9102—711nomaximal, Sylow
144C2×C71—21no—
145C2×C71—21no—
146C2×C71#186911yescharacteristic
147C2×C71#191011yescharacteristic
148C2×C71#191211yescharacteristic
149D71—82no—
150#2982—711no—
151#2326—711no—
152#1478—712no—
153#1461D7111yescharacteristic, complemented
154#1274—712no—
155C71#910211yescharacteristic, complemented, Sylow
156C2×#124—711no—
157C2×C4×Q8C2×D7111yescharacteristic
158C2×C4×D4—711no—
159C8×Q8—1424no—
160#431—712no—
161#431—1422no—
162#431C2×D7112yes—
163#391—711no—
164#390—711no—
165#388—1422no—
166#385—1422no—
167#384—711no—
168#376—711no—
169D4⋊C8—1424no—
170#324—22no—
171C22×Q8C4×D7111yescharacteristic
172C22×D4—711no—
173C23×C4—1421no—
174#124—714no—
175#124—1422no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1422no—
179C4×D4—1422no—
180C2×C4⋊C4—711no—
181C2×C4⋊C4—1422no—
182C2×C4⋊C4C4×D7111yescharacteristic
183C2×C4⋊C4C22⋊D7112yes—
184C2×C22⋊C4—1421no—
185C2×C22⋊C4—1422no—
186C2×C22⋊C4—1422no—
187C2×C42—711no—
188C2×C42—711no—
189C2×C42D28412yes—
190C2×C42C22×D7111yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—1424no—
193C4⋊C8—1422no—
194C22⋊C8—22no—
195C22⋊C8—1422no—
196C4×C8—22no—
197C4×C8—714no—
198C4×C8—1424no—
199#95—1422no—
200#95—1421no—
201#95—1421no—
202C24—1421no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8C8×D7112yes—
206C2×Q8#1176584812yes—
207C2×D4—712no—
208C2×D4—1422no—
209C2×D4—1421no—
210C2×D4—1422no—
211C22×C4—2842no—
212C22×C4—5682no—
213C22×C4—1421no—
214C22×C4—1421no—
215C22×C4—1421no—
216C22×C4(C2×D71)⋊C412yes—
217C22×C4(C2×D71)⋊C412yes—
218C22×C4C2×C4×D7111yescharacteristic
219C22×C4C2×C22⋊D7111yescharacteristic
220C22×C4C2×D28411yescharacteristic
221C2×C8—42no—
222C2×C8—1424no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—5682no—
226C4⋊C4—1422no—
227C22⋊C4—2841no—
228C22⋊C4—2841no—
229C22⋊C4—5682no—
230C22⋊C4—1428no—
231C42—22no—
232C42—22no—
233C42—712no—
234C42—1422no—
235C42—1421no—
236C42D4×D7112yes—
237C42#1176586412yes—
238C23—2842no—
239C23—2842no—
240C23—2841no—
241C23—2841no—
242C23—1422no—
243C23C2×(C2×D71)⋊C411yescharacteristic
244Q8—24no—
245Q8—24no—
246D4—2842no—
247D4—1424no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—2841no—
255C2×C4—2841no—
256C2×C4—2841no—
257C2×C4—2842no—
258C2×C4—2841no—
259C2×C4—2841no—
260C2×C4—2841no—
261C2×C4—5684no—
262C2×C4—1422no—
263C2×C4(C2×D71)⋊C814yes—
264C2×C4(C22⋊D71)⋊C412yes—
265C2×C4#1455609811yescharacteristic
266C2×C4#1455610012yes—
267C2×C4#1455611211yescharacteristic
268C2×C4C2×C8×D7111yescharacteristic
269C2×C4C2×#1176584811yescharacteristic
270C2×C4#1455613612yes—
271C8—82no—
272C8—2844no—
273C22—21no—
274C22—21no—
275C22—2844no—
276C22—2842no—
277C22—5682no—
278C22—5681no—
279C22—5681no—
280C22#1798587611yescharacteristic
281C22C2×(C2×D71)⋊C811yescharacteristic
282C22#1798596711yescharacteristic
283C4—22no—
284C4—21no—
285C4—21no—
286C4—24no—
287C4—42no—
288C4—2842no—
289C4—5681no—
290C4—5681no—
291C4(C22⋊D71)⋊C812yes—
292C4#1798606712yes—
293C2—21no—
294C2—21no—
295C2—5682no—
296C2#2258863011yescharacteristic
297C2#2258886511yescharacteristic
298C2#2258887511yescharacteristic
subgroupG/H
centerC22#17985876
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC71×#1461C2
socleC22×C71#386
prime radicalC22×D284C2×C4
cosocleC2×C42×C71C23
hypercenter#1461D71
hypocenterC71#9102
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #22597421C2semidirect
4. #22586168C2semidirect
5. C71×#1461C2semidirect
24. #14556117C8semidirect
153. #1461D71semidirect
155. C71#9102semidirect

Maximal subgroups

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

subgroupindex
#225974212
#225910472
#225867302
#225861682
C71×#14612
#910271