Finite Group Databasebeta

← #31892110

Subgroups — #31892110

Order 18176.

566 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 139 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
2C4×#14556117C211yesmaximal, characteristic
3#22586732C212yesmaximal
4#22586172C212yesmaximal, complemented
5C71×#1461C211yesmaximal, characteristic, complemented
6C2×#14556117C411yescharacteristic
7C2×C4×D284C2211yescharacteristic
8Q8×C71⋊C8—24no—
9#17986052—22no—
10#17986052C2212yes—
11#17985906C411yescharacteristic
12#17985900C411yescharacteristic
13#17985889—22no—
14#17985864C411yescharacteristic
15C4×(C2×D71)⋊C4—22no—
16C42×Dic71C2211yescharacteristic
17#17985728—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×(C2×D71)⋊C4—22no—
29C2×(C2×D71)⋊C4—22no—
30C2×(C2×D71)⋊C4—21no—
31C2×C4×Dic71—21no—
32C2×C4×Dic71—21no—
33C2×C4×Dic71C2×C411yescharacteristic
34C2×C4×Dic71C2×C411yescharacteristic
35C4⋊(C71⋊C8)—24no—
36C4⋊(C71⋊C8)D414yes—
37C22⋊(C71⋊C8)—22no—
38C4×C71⋊C8—24no—
39C4×C71⋊C8—22no—
40C71⋊#95—22no—
41C71⋊#95—22no—
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—44no—
61C2×C4×D71—41no—
62C2×C4×D71—41no—
63C2×C71⋊C8—24no—
64C4⋊Dic71—22no—
65(C2×D71)⋊C4—28no—
66(C2×D71)⋊C4—44no—
67(C2×D71)⋊C4—42no—
68C4×Dic71—21no—
69C4×Dic71—22no—
70C4×Dic71—42no—
71C4×Dic71—42no—
72C4×Dic71C2×C812yes—
73C2×Q8×C71—22no—
74C2×Q8×C71—22no—
75C2×Q8×C71C2×C812yes—
76C2×Q8×C71M1612yes—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C22⋊C412yes—
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C2×C8×C71—42no—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—24no—
85C42×C71—22no—
86C42×C71—22no—
87C42×C71C2×D412yes—
88C42×C71#4112yes—
89C23×C71C2×C22⋊C411yescharacteristic
90C22×D71—22no—
91C22×D71—42no—
92C22×D71—42no—
93C22×D71—41no—
94C22×D71—41no—
95Q8×C71—24no—
96Q8×C71—24no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—41no—
102C2×C4×C71—42no—
103C2×C4×C71C22⋊C814yes—
104C2×C4×C71C4×D412yes—
105C2×C4×C71#12011yescharacteristic
106C2×C4×C71#12112yes—
107C2×C4×C71#12311yescharacteristic
108C2×C4×C71C22×C811yescharacteristic
109C2×C4×C71C2×M1611yescharacteristic
110C2×C4×C71#13112yes—
111C2×Dic71—22no—
112C2×Dic71—22no—
113C2×Dic71—41no—
114C2×Dic71—41no—
115C2×Dic71—41no—
116C2×Dic71—41no—
117C2×Dic71—42no—
118C2×Dic71—41no—
119D284—24no—
120D284—42no—
121C4×D71—82no—
122C4×D71—82no—
123C8×C71—82no—
124C71⋊C8—44no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71#38611yescharacteristic
128C22×C71C2×C22⋊C811yescharacteristic
129C22×C71#40811yescharacteristic
130C2×D71—44no—
131C2×D71—42no—
132C2×D71—82no—
133C2×D71—81no—
134C2×D71—81no—
135C4×C71—24no—
136C4×C71—22no—
137C4×C71—21no—
138C4×C71—42no—
139C4×C71C8×D412yes—
140C4×C71#43512yes—
141C4×C71#45412yes—
142Dic71—42no—
143Dic71—42no—
144Dic71—81no—
145#8960—711nomaximal, Sylow
146C2×C71—21no—
147C2×C71—21no—
148C2×C71#186911yescharacteristic
149C2×C71#188411yescharacteristic
150C2×C71#188811yescharacteristic
151D71—82no—
152#2982—711no—
153C4×#124—711no—
154#1478—712no—
155#1461D7111yescharacteristic, complemented
156#1276—712no—
157C71#896011yescharacteristic, complemented, Sylow
158C2×#124—711no—
159C2×C4×Q8C2×D7111yescharacteristic
160C2×C4×D4—711no—
161C8×Q8—1424no—
162#431—712no—
163#431—1422no—
164#431C2×D7112yes—
165#391—711no—
166#390—711no—
167#388—1422no—
168#382—711no—
169C4×C22⋊C4—1422no—
170C43—711no—
171D4⋊C8—1424no—
172#324—22no—
173C22×Q8C4×D7111yescharacteristic
174C22×D4—711no—
175C23×C4—1421no—
176#124—714no—
177#124—1422no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—1422no—
181C4×D4—1422no—
182C2×C4⋊C4—711no—
183C2×C4⋊C4C4×D7111yescharacteristic
184C2×C4⋊C4C22⋊D7112yes—
185C2×C22⋊C4—1422no—
186C2×C22⋊C4—1421no—
187C2×C22⋊C4—1422no—
188C2×C42—711no—
189C2×C42—711no—
190C2×C42—1421no—
191C2×C42—1421no—
192C2×C42D28412yes—
193C2×C42C22×D7111yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—714no—
196C4⋊C8—1424no—
197C22⋊C8—22no—
198C22⋊C8—1422no—
199C4×C8—22no—
200C4×C8—1424no—
201C4×C8—1422no—
202#95—1422no—
203#95—1422no—
204C24—1421no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8C8×D7112yes—
208C2×Q8#1176584812yes—
209C2×D4—712no—
210C2×D4—1422no—
211C2×D4—1421no—
212C2×D4—1422no—
213C22×C4—2844no—
214C22×C4—2841no—
215C22×C4—2841no—
216C22×C4—1421no—
217C22×C4—1421no—
218C22×C4—1421no—
219C22×C4(C2×D71)⋊C412yes—
220C22×C4(C2×D71)⋊C412yes—
221C22×C4C2×C4×D7111yescharacteristic
222C22×C4C2×C22⋊D7111yescharacteristic
223C22×C4C2×D28411yescharacteristic
224C2×C8—42no—
225C2×C8—1424no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—1422no—
229C22⋊C4—2844no—
230C22⋊C4—2842no—
231C22⋊C4—1428no—
232C42—22no—
233C42—22no—
234C42—2842no—
235C42—2842no—
236C42—712no—
237C42—1422no—
238C42—1421no—
239C42D4×D7112yes—
240C42#1176586412yes—
241C23—2842no—
242C23—2842no—
243C23—2841no—
244C23—2841no—
245C23—1422no—
246C23C2×(C2×D71)⋊C411yescharacteristic
247Q8—24no—
248Q8—24no—
249D4—2842no—
250D4—1424no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—2841no—
258C2×C4—2841no—
259C2×C4—2841no—
260C2×C4—2841no—
261C2×C4—2842no—
262C2×C4—2841no—
263C2×C4—5682no—
264C2×C4—5682no—
265C2×C4—1422no—
266C2×C4—1422no—
267C2×C4(C2×D71)⋊C814yes—
268C2×C4(C22⋊D71)⋊C412yes—
269C2×C4#1455609811yescharacteristic
270C2×C4#1455610012yes—
271C2×C4#1455611211yescharacteristic
272C2×C4C2×C8×D7111yescharacteristic
273C2×C4C2×#1176584811yescharacteristic
274C2×C4#1455613612yes—
275C8—82no—
276C8—2844no—
277C22—21no—
278C22—21no—
279C22—2844no—
280C22—2842no—
281C22—5682no—
282C22—5681no—
283C22—5681no—
284C22#1798587611yescharacteristic
285C22C2×(C2×D71)⋊C811yescharacteristic
286C22#1798596711yescharacteristic
287C4—22no—
288C4—21no—
289C4—24no—
290C4—42no—
291C4—2842no—
292C4—2842no—
293C4—5681no—
294C4(C22⋊D71)⋊C812yes—
295C4#1798606712yes—
296C4#1798617412yes—
297C2—21no—
298C2—21no—
299C2—5682no—
300C2#2258863011yescharacteristic
301C2#2258872111yescharacteristic
302C2#2258874111yescharacteristic
subgroupG/H
centerC2×C4#14556098
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC71×#1461C2
socleC22×C71#386
prime radicalC22×D284C2×C4
cosocleC2×C42×C71C23
hypercenter#1461D71
hypocenterC71#8960
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #22597421C2semidirect
4. #22586172C2semidirect
5. C71×#1461C2semidirect
24. #14556117C8semidirect
155. #1461D71semidirect
157. C71#8960semidirect

Maximal subgroups

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

subgroupindex
#225974212
C4×#145561172
#225867322
#225861722
C71×#14612
#896071