Finite Group Databasebeta

← #22597884

Subgroups — #22597884

Order 9088.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#14556154C211yesmaximal, characteristic, complemented
2C2×#14556150C211yesmaximal, characteristic, complemented
3#17986566C211yesmaximal, characteristic, complemented
4#17986480C211yesmaximal, characteristic, complemented
5#17986220C212yesmaximal, complemented
6#17986213C212yesmaximal, complemented
7#17986198C212yesmaximal, complemented
8#17986193C212yesmaximal, complemented
9#17986025C211yesmaximal, characteristic, complemented
10#17985998C211yesmaximal, characteristic, complemented
11C2×#102×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic
13C2×#11765865C2211yescharacteristic
14#14556154—24no—
15#14556150—24no—
16C2×Dic284C2211yescharacteristic, complemented
17C2×#11765854C2211yescharacteristic, complemented
18C2×#11765854C2211yescharacteristic, complemented
19C2×#11765853C2212yes—
20C2×D568C2211yescharacteristic, complemented
21C2×#11765850C2212yes—
22C2×#11765849C2211yescharacteristic
23#14556124C2212yes—
24#14556115C2212yes—
25#14556099—21no—
26#14556099—21no—
27#14556097—22no—
28#14556093C2212yescomplemented
29#14556090C2212yescomplemented
30C4×C22⋊D71—22no—
31#14556087C2211yescharacteristic
32#14556086—21no—
33C4⋊C4×D71—21no—
34Dic71⋊C8C2212yescomplemented
35Dic71⋊C8C2212yescomplemented
36#14556051C2212yescomplemented
37#14556048C2212yescomplemented
38C22×D4×C71C2211yescharacteristic
39C22×C8×C71C2211yescharacteristic
40C2×C4⋊C4×C71C2211yescharacteristic
41#102×C71C2212yes—
42#102×C71C2212yes—
43#11765865—22no—
44#11765865—22no—
45#11765865—22no—
46#11765865D414yescomplemented
47C2×Dic142C2311yescharacteristic
48C2×Dic142C2311yescharacteristic
49C2×D284C2311yescharacteristic
50C2×D284C2311yescharacteristic
51C2×C22⋊D71—21no—
52C2×C22⋊D71—21no—
53C2×C22⋊D71—22no—
54C2×C22⋊D71—22no—
55C2×C4×D71—21no—
56C2×C4×D71—21no—
57C2×C4×D71—22no—
58Dic284—22no—
59#11765854—22no—
60#11765854—22no—
61#11765853—24no—
62D568—22no—
63#11765850—24no—
64#11765849—22no—
65C2×C71⋊C8C2312yes—
66C4⋊Dic71D411yescharacteristic
67C4⋊Dic71D411yescharacteristic
68Dic71⋊C4—22no—
69C22⋊Dic71—22no—
70C22⋊Dic71—22no—
71C22⋊Dic71D412yes—
72(C2×D71)⋊C4—22no—
73C4×Dic71—22no—
74C4×Dic71C2312yes—
75C24×C71—21no—
76C2×D4×C71—22no—
77C2×D4×C71—21no—
78C2×D4×C71—22no—
79C2×D4×C71C2312yes—
80C22×C4×C71—21no—
81C22×C4×C71C2311yescharacteristic
82C2×C8×C71—22no—
83C2×C8×C71C2311yescharacteristic
84C2×C8×C71C2311yescharacteristic
85C4⋊C4×C71—21no—
86C4⋊C4×C71C2312yes—
87C23×C71—22no—
88C23×C71—42no—
89C23×C71—81no—
90C23×C71—81no—
91C23×C71C2×D411yescharacteristic
92C22×D71—21no—
93C22×D71—21no—
94D4×C71—22no—
95D4×C71—22no—
96D4×C71—42no—
97C2×C4×C71—22no—
98C2×C4×C71—81no—
99C2×C4×C71C2×D411yescharacteristic
100C2×C4×C71C2×D412yes—
101C2×C4×C71#4112yes—
102C2×C4×C71C2411yescharacteristic
103C22⋊D71—24no—
104C22⋊D71—42no—
105C22⋊D71—82no—
106C22⋊D71—82no—
107C2×Dic71—21no—
108C2×Dic71—21no—
109C2×Dic71C2×D412yes—
110C2×Dic71C2×D412yes—
111D284—21no—
112D284—21no—
113D284—21no—
114D284C2×D412yes—
115C4×D71—24no—
116C4×D71—42no—
117C4×D71—82no—
118Dic142—21no—
119Dic142—21no—
120Dic142—21no—
121Dic142C2×D412yes—
122C8×C71—21no—
123C8×C71—21no—
124C71⋊C8—22no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—42no—
128C22×C71—42no—
129C22×C71—42no—
130C22×C71—84no—
131C22×C71#12112yes—
132C22×C71C22×D411yescharacteristic
133C2×D71—22no—
134C2×D71—41no—
135C2×D71—41no—
136C2×D71—41no—
137C2×D71—41no—
138C2×D71—41no—
139C4×C71—42no—
140C4×C71#12112yes—
141C4×C71C22×D411yescharacteristic
142C4×C71C2×#4111yescharacteristic
143Dic71—22no—
144Dic71—22no—
145Dic71—22no—
146Dic71—41no—
147C2×C71—22no—
148C2×C71—82no—
149C2×C71C2×#12111yescharacteristic
150C2×C71#57611yescharacteristic
151C2×C71#57711yescharacteristic
152D71—42no—
153D71—81no—
154#3045—711nomaximal, Sylow
155C71#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)—711no—
157C2×#135—711no—
158#535—711no—
159#520—711no—
160#464—712no—
161#463—712no—
162#460—712no—
163#459—712no—
164#424—711no—
165#417—711no—
166C2×#102D7111yescharacteristic, complemented
167C2×#41—711no—
168C2×#41—711no—
169C22×D4C2×D7111yescharacteristic
170AGL(1,ℤ/8)—1424no—
171#135—1424no—
172C2×Q16—711no—
173C2×SD16—712no—
174C2×SD16—711no—
175C2×SD16—711no—
176C2×D8—712no—
177C2×D8—711no—
178C2×M16—711no—
179C22×C8C2×D7111yescharacteristic
180#127—712no—
181#124—712no—
182#121—1421no—
183#121—1421no—
184#120—1422no—
185C4×Q8—712no—
186C4×D4—712no—
187C4×D4—1422no—
188#117—711no—
189#117—1421no—
190C2×C4⋊C4—1421no—
191C2×C4⋊C4C2×D7111yescharacteristic
192C4⋊C8—712no—
193C4⋊C8—712no—
194#103—712no—
195#102—712no—
196#102C2×D7112yes—
197#102C2×D7112yes—
198C24—21no—
199#41—714no—
200#41—1422no—
201#41—1422no—
202#41—1422no—
203C2×Q8—711no—
204C2×Q8—711no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—711no—
209C2×D4—711no—
210C2×D4—1422no—
211C2×D4—1422no—
212C2×D4—1421no—
213C2×D4—1421no—
214C2×D4C22×D7112yes—
215C22×C4—21no—
216C22×C4—1422no—
217C22×C4—1421no—
218C22×C4—1421no—
219C22×C4C22×D7111yescharacteristic
220Q16—1422no—
221SD16—1422no—
222SD16—1422no—
223SD16—1424no—
224D8—1422no—
225D8—1424no—
226M16—1422no—
227C2×C8—22no—
228C2×C8—712no—
229C2×C8C22×D7111yescharacteristic
230C2×C8C22×D7111yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—711no—
233C4⋊C4—711no—
234C4⋊C4—1422no—
235C4⋊C4C22×D7112yes—
236C22⋊C4—712no—
237C22⋊C4—1422no—
238C22⋊C4—1422no—
239C22⋊C4—1422no—
240C42—712no—
241C42—1422no—
242C23—22no—
243C23—42no—
244C23—81no—
245C23—81no—
246C23—1421no—
247C23—1421no—
248C23D4×D7111yescharacteristic
249Q8—712no—
250Q8—1421no—
251Q8—1421no—
252Q8—1421no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2842no—
257D4—5682no—
258D4—5682no—
259D4—712no—
260D4—1424no—
261D4—1421no—
262D4—1421no—
263D4—1421no—
264C2×C4—22no—
265C2×C4—81no—
266C2×C4—2842no—
267C2×C4—5682no—
268C2×C4—712no—
269C2×C4—712no—
270C2×C4—1421no—
271C2×C4—1421no—
272C2×C4—1424no—
273C2×C4D4×D7112yes—
274C2×C4D4×D7111yescharacteristic
275C2×C4#1176586512yes—
276C2×C4C23×D7111yescharacteristic
277C8—21no—
278C8—21no—
279C8—1422no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—2841no—
287C22—2841no—
288C22—2841no—
289C22—2841no—
290C22—2841no—
291C22—1422no—
292C22#1455610112yes—
293C22C2×D4×D7111yescharacteristic
294C4—42no—
295C4—2841no—
296C4—1422no—
297C4—1422no—
298C4—1422no—
299C4#1455610112yes—
300C4C2×D4×D7111yescharacteristic
301C4C2×#1176586511yescharacteristic
302C2—22no—
303C2—82no—
304C2—2842no—
305C2—5681no—
306C2C2×#1455610111yescharacteristic
307C2#1798680011yescharacteristic
308C2#1798680511yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#102×C71C2
socleC22×C71C22×D4
hypercenterC2×#102D71
hypocenterC71#3045
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#14556154C2semidirect
2. C2×#14556150C2semidirect
3. #17986566C2semidirect
4. #17986480C2semidirect
5. #17986220C2semidirect
6. #17986213C2semidirect
7. #17986198C2semidirect
8. #17986193C2semidirect
9. #17986025C2semidirect
10. #17985998C2semidirect
11. C2×#102×C71C2semidirect
16. C2×Dic284C22semidirect
17. C2×#11765854C22semidirect
18. C2×#11765854C22semidirect
20. C2×D568C22semidirect
28. #14556093C22semidirect
29. #14556090C22semidirect
34. Dic71⋊C8C22semidirect
35. Dic71⋊C8C22semidirect
36. #14556051C22semidirect
37. #14556048C22semidirect
46. #11765865D4semidirect
155. C71#3045semidirect
166. C2×#102D71semidirect