Finite Group Databasebeta

← #31890767

Subgroups — #31890767

Order 18176.

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

normal subgroups only

This group has 384 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#22597315C211yesmaximal, characteristic
2#22587103C211yesmaximal, characteristic, complemented
3#22586863C212yesmaximal
4#22586196C212yesmaximal, complemented
5C4×#106×C71C211yesmaximal, characteristic, complemented
6#17986533C2211yescharacteristic
7#17986129C411yescharacteristic, complemented
8C8×Dic142C411yescharacteristic, complemented
9#17986078C411yescharacteristic, complemented
10#17986070—21no—
11C8×D284C411yescharacteristic, complemented
12C8×C22⋊D71—21no—
13#17986058—21no—
14#17986056C2212yes—
15#17986056C2211yescharacteristic
16#17985953—21no—
17#17985752—21no—
18#17985750—21no—
19#17985735—22no—
20#17985732—22no—
21C2×#106×C71—21no—
22C2×C4×C8×C71C2211yescharacteristic
23C4×C4⋊C4×C71C2212yes—
24#340×C71—21no—
25C2×#11765848—21no—
26C2×C8×D71—21no—
27#14556122—21no—
28#14556120Q811yescharacteristic
29#14556116Q811yescharacteristic
30#14556111—21no—
31#14556107D411yescharacteristic
32#14556102D411yescharacteristic
33C4×Dic142C2×C411yescharacteristic
34C4×D284C2×C411yescharacteristic
35C4×C22⋊D71—21no—
36#14556085—21no—
37C4⋊(C71⋊C8)—22no—
38C4⋊(C71⋊C8)C2×C411yescharacteristic
39C4⋊(C71⋊C8)C2×C411yescharacteristic
40Dic71⋊C8—21no—
41Dic71⋊C8—21no—
42C22⋊(C71⋊C8)—21no—
43C22⋊(C71⋊C8)—22no—
44(C2×D71)⋊C8—21no—
45(C2×D71)⋊C8—21no—
46#14556039—21no—
47#14556039—22no—
48#14556038—21no—
49C8×Dic71—21no—
50C22×C8×C71D411yescharacteristic
51C22×C8×C71D411yescharacteristic
52C2×C4⋊C4×C71—22no—
53C2×C42×C71—22no—
54C2×C42×C71C2311yescharacteristic
55#106×C71—24no—
56C4×C8×C71C2×C411yescharacteristic
57C4×C8×C71C2×C411yescharacteristic
58C4×C8×C71D411yescharacteristic
59C4×C8×C71D411yescharacteristic
60#95×C71—22no—
61C2×Dic142C4⋊C411yescharacteristic
62C2×D284C4⋊C411yescharacteristic
63C2×C22⋊D71—21no—
64C2×C4×D71—21no—
65#11765848—42no—
66C2×C71⋊C8—21no—
67C2×C71⋊C8—21no—
68C2×C71⋊C8—42no—
69C8×D71—42no—
70C4⋊Dic71C4⋊C411yescharacteristic
71C4⋊Dic71C4⋊C411yescharacteristic
72Dic71⋊C4—21no—
73Dic71⋊C4—21no—
74Dic71⋊C4—21no—
75Dic71⋊C4—21no—
76C22⋊Dic71—21no—
77(C2×D71)⋊C4—21no—
78(C2×D71)⋊C4—21no—
79(C2×D71)⋊C4—21no—
80(C2×D71)⋊C4—21no—
81C4×Dic71—21no—
82C22×C4×C71—42no—
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71#4111yescharacteristic
86C2×C8×C71—22no—
87C2×C8×C71—41no—
88C2×C8×C71C22⋊C411yescharacteristic
89C2×C8×C71C22⋊C411yescharacteristic
90C2×C8×C71C22⋊C411yescharacteristic
91C2×C8×C71C22⋊C411yescharacteristic
92C4⋊C4×C71—24no—
93C4⋊C4×C71—42no—
94C42×C71—82no—
95C42×C71C22×C411yescharacteristic
96C42×C71C2×D411yescharacteristic
97C42×C71C2×Q811yescharacteristic
98C42×C71#4111yescharacteristic
99C23×C71#12111yescharacteristic
100C22×D71—21no—
101C2×C4×C71—22no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—41no—
105C2×C4×C71—44no—
106C2×C4×C71—82no—
107C2×C4×C71—82no—
108C2×C4×C71C2×C22⋊C411yescharacteristic
109C2×C4×C71C2×C4⋊C411yescharacteristic
110C2×C4×C71#11711yescharacteristic
111C2×C4×C71#12111yescharacteristic
112C2×C4×C71C22⋊Q811yescharacteristic
113C2×C4×C71C22⋊Q811yescharacteristic
114C22⋊D71—42no—
115C2×Dic71—21no—
116C2×Dic71—21no—
117C2×Dic71—21no—
118D284—41no—
119C4×D71—41no—
120C4×D71—41no—
121Dic142—41no—
122C8×C71—21no—
123C8×C71—21no—
124C8×C71—21no—
125C8×C71#10412yes—
126C71⋊C8—41no—
127C71⋊C8—41no—
128C71⋊C8—82no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71#38011yescharacteristic
132C22×C71#47112yes—
133C2×D71—41no—
134C2×D71—41no—
135C2×D71—41no—
136C4×C71—21no—
137C4×C71—21no—
138C4×C71—21no—
139C4×C71—21no—
140C4×C71—21no—
141C4×C71—84no—
142C4×C71C2×#10411yescharacteristic
143C4×C71#42111yescharacteristic
144Dic71—41no—
145Dic71—41no—
146Dic71—41no—
147#8696—711nomaximal, Sylow
148C2×C71—22no—
149C2×C71#180611yescharacteristic
150C2×C71#181211yescharacteristic
151C2×C71#181511yescharacteristic
152D71—81no—
153#2966—711no—
154C4×#106D7111yescharacteristic, complemented
155#1568—711no—
156#1509—712no—
157#1285—712no—
158C71#869611yescharacteristic, complemented, Sylow
159#529—711no—
160#446—711no—
161C8×Q8—711no—
162#436—711no—
163#435—1421no—
164C8×D4—711no—
165C8×D4—1421no—
166#433—1421no—
167#432—711no—
168#432—712no—
169C2×#106—21no—
170#405—1421no—
171C2×C4×C8C2×D7111yescharacteristic
172C4×C4⋊C4C2×D7112yes—
173#340—21no—
174#335—1421no—
175#334—1421no—
176#328—1422no—
177#327—1422no—
178C2×M16—1421no—
179C22×C8—1421no—
180C22×C8C22⋊D7111yescharacteristic
181C22×C8C22⋊D7111yescharacteristic
182#126—1421no—
183#125—711no—
184#124—711no—
185#123—1421no—
186C22⋊Q8—711no—
187#121—711no—
188C4×Q8—711no—
189C4×D4—711no—
190C4×D4—1421no—
191#117—1421no—
192C2×C4⋊C4—22no—
193C2×C42—22no—
194C2×C42C22×D7111yescharacteristic
195#106—24no—
196C4⋊C8—711no—
197C4⋊C8—711no—
198C4⋊C8—1421no—
199C4⋊C8—1421no—
200C4⋊C8—1422no—
201C22⋊C8—1422no—
202C22⋊C8—1421no—
203C22⋊C8—1421no—
204C22⋊C8—1421no—
205#97—1421no—
206#97—1421no—
207#97—1422no—
208C4×C8—1421no—
209C4×C8C4×D7111yescharacteristic
210C4×C8C4×D7111yescharacteristic
211C4×C8D28411yescharacteristic
212C4×C8D28411yescharacteristic
213#95—22no—
214C2×Q8—711no—
215C2×D4—711no—
216C2×D4—1421no—
217C22×C4—42no—
218C22×C4—1421no—
219C22×C4D4×D7111yescharacteristic
220C22×C4C2×C22⋊D7111yescharacteristic
221C22×C4#1176586411yescharacteristic
222M16—2842no—
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—2842no—
226C2×C8—2842no—
227C2×C8—1421no—
228C2×C8—1421no—
229C2×C8(C2×D71)⋊C411yescharacteristic
230C2×C8(C2×D71)⋊C411yescharacteristic
231C2×C8(C2×D71)⋊C411yescharacteristic
232C2×C8(C2×D71)⋊C411yescharacteristic
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—711no—
236C4⋊C4—711no—
237C4⋊C4—1421no—
238C4⋊C4—1421no—
239C4⋊C4—1421no—
240C4⋊C4—1421no—
241C22⋊C4—1421no—
242C22⋊C4—1421no—
243C22⋊C4—1421no—
244C22⋊C4—1421no—
245C22⋊C4—1421no—
246C42—82no—
247C42—1421no—
248C42C2×C4×D7111yescharacteristic
249C42C2×D28411yescharacteristic
250C42Q8×D7111yescharacteristic
251C42#1176586611yescharacteristic
252C23—1421no—
253C23#1455609911yescharacteristic
254Q8—2841no—
255D4—2841no—
256D4—2842no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—2841no—
265C2×C4—2841no—
266C2×C4—1421no—
267C2×C4—1421no—
268C2×C4—1421no—
269C2×C4C2×(C2×D71)⋊C411yescharacteristic
270C2×C4C4⋊C4×D7111yescharacteristic
271C2×C4#1455608611yescharacteristic
272C2×C4#1455610311yescharacteristic
273C2×C4#1455610411yescharacteristic
274C2×C4#1455610811yescharacteristic
275C8—21no—
276C8—21no—
277C8—21no—
278C8—2841no—
279C8—2841no—
280C8—5682no—
281C8#1455605612yes—
282C22—21no—
283C22—21no—
284C22—2841no—
285C22—2841no—
286C22—2841no—
287C22#1798585611yescharacteristic
288C22#1798625912yes—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—84no—
295C4—2841no—
296C4—2841no—
297C4—2841no—
298C4C2×#1455605611yescharacteristic
299C4#1798601711yescharacteristic
300C2—22no—
301C2—5681no—
302C2#2258836111yescharacteristic
303C2#2258838311yescharacteristic
304C2#2258840011yescharacteristic
subgroupG/H
centerC22#17985856
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC4×#106×C71C2
socleC22×C71#380
prime radical#14556102D4
cosocleC2×C42×C71C23
hypercenterC4×#106D71
hypocenterC71#8696
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #22587103C2semidirect
4. #22586196C2semidirect
5. C4×#106×C71C2semidirect
7. #17986129C4semidirect
8. C8×Dic142C4semidirect
9. #17986078C4semidirect
11. C8×D284C4semidirect
154. C4×#106D71semidirect
158. C71#8696semidirect

Maximal subgroups

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

subgroupindex
#225973152
#225871032
#225868632
#225861962
C4×#106×C712
#869671