Finite Group Databasebeta

← #22599105

Subgroups — #22599105

Order 9088.

472 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986510C211yesmaximal, characteristic, complemented
2#17986334C211yesmaximal, characteristic, complemented
3#17986320C212yesmaximal, complemented
4#17986317C211yesmaximal, characteristic, complemented
5#17986233C212yesmaximal, complemented
6#17986226C212yesmaximal, complemented
7#17986150C211yesmaximal, characteristic, complemented
8#17986142C212yesmaximal, complemented
9#17986133C211yesmaximal, characteristic, complemented
10C2×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
11C2×#121×C71C211yesmaximal, characteristic, complemented
12C2×#11765865C2211yescharacteristic, complemented
13C2×#11765853C2212yescomplemented
14C2×#11765850C2212yescomplemented
15C22×C71⋊C8C2212yescomplemented
16#14556111—21no—
17#14556107C2211yescharacteristic
18#14556107C2211yescharacteristic, complemented
19#14556102C2211yescharacteristic
20#14556102C2211yescharacteristic, complemented
21#14556097—21no—
22C4×C22⋊D71—21no—
23C4×C22⋊D71—21no—
24#14556087C2211yescharacteristic, complemented
25#14556061C2212yescomplemented
26#14556059C2212yescomplemented
27#14556051C2212yescomplemented
28#14556049C2212yescomplemented
29#14556049C2212yescomplemented
30#14556048C2212yescomplemented
31C22⋊(C71⋊C8)C2211yescharacteristic
32C22⋊(C71⋊C8)C2211yescharacteristic
33C22⋊(C71⋊C8)C2212yescomplemented
34C22×D4×C71—21no—
35C22×D4×C71C2211yescharacteristic
36C23×C4×C71C2211yescharacteristic
37#121×C71—21no—
38#121×C71—21no—
39#121×C71C2212yes—
40#121×C71C2212yes—
41C2×C4⋊C4×C71C2211yescharacteristic
42C2×C22⋊C4×C71—21no—
43#11765865—42no—
44C2×Dic142C2311yescharacteristic
45C2×D284C2311yescharacteristic
46C2×C22⋊D71—21no—
47C2×C22⋊D71—21no—
48C2×C22⋊D71—21no—
49C2×C4×D71—21no—
50#11765853—42no—
51#11765850—42no—
52C2×C71⋊C8—42no—
53C2×C71⋊C8C2312yes—
54C2×C71⋊C8C2312yes—
55C4⋊Dic71C2311yescharacteristic
56C4⋊Dic71C2311yescharacteristic
57Dic71⋊C4—21no—
58Dic71⋊C4—21no—
59C22⋊Dic71—21no—
60C22⋊Dic71—21no—
61C22⋊Dic71—21no—
62(C2×D71)⋊C4—21no—
63(C2×D71)⋊C4—21no—
64C4×Dic71—21no—
65C24×C71—21no—
66C24×C71D411yescharacteristic
67C2×D4×C71—22no—
68C2×D4×C71—22no—
69C2×D4×C71—21no—
70C2×D4×C71—42no—
71C2×D4×C71—44no—
72C2×D4×C71C2312yes—
73C22×C4×C71—21no—
74C22×C4×C71—42no—
75C22×C4×C71D411yescharacteristic
76C22×C4×C71D411yescharacteristic
77C22×C4×C71D411yescharacteristic
78C22×C4×C71C2311yescharacteristic
79C22×C4×C71C2311yescharacteristic
80C22×C4×C71C2311yescharacteristic
81C4⋊C4×C71—21no—
82C4⋊C4×C71C2312yes—
83C22⋊C4×C71—22no—
84C22⋊C4×C71—22no—
85C23×C71—22no—
86C23×C71—22no—
87C23×C71—22no—
88C23×C71—41no—
89C23×C71—42no—
90C23×C71—42no—
91C23×C71—42no—
92C23×C71C2×D411yescharacteristic
93C23×C71C2×D411yescharacteristic
94C23×C71C2×D411yescharacteristic
95C22×D71—21no—
96D4×C71—22no—
97D4×C71—22no—
98D4×C71—42no—
99D4×C71—44no—
100D4×C71—44no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71—41no—
108C2×C4×C71—42no—
109C2×C4×C71C2×D411yescharacteristic
110C2×C4×C71C2×D411yescharacteristic
111C2×C4×C71C2×D411yescharacteristic
112C2×C4×C71C2411yescharacteristic
113C22⋊D71—42no—
114C22⋊D71—42no—
115C22⋊D71—42no—
116C2×Dic71—21no—
117C2×Dic71—21no—
118C2×Dic71—21no—
119D284—22no—
120C4×D71—42no—
121Dic142—22no—
122C71⋊C8—22no—
123C71⋊C8—22no—
124C22×C71—21no—
125C22×C71—21no—
126C22×C71—21no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71—44no—
132C22×C71—42no—
133C22×C71—42no—
134C22×C71—42no—
135C22×C71—84no—
136C22×C71#13512yes—
137C22×C71AGL(1,ℤ/8)12yes—
138C22×C71C22×D411yescharacteristic
139C2×D71—41no—
140C2×D71—41no—
141C2×D71—41no—
142C4×C71—21no—
143C4×C71—21no—
144C4×C71—21no—
145C4×C71—42no—
146C4×C71H(5,𝔽2)12yes—
147Dic71—41no—
148Dic71—41no—
149Dic71—41no—
150C2×C71—22no—
151C2×C71—22no—
152C2×C71—41no—
153C2×C71—82no—
154C2×C71#53411yescharacteristic
155C2×C71C2×#13511yescharacteristic
156C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
157D71—81no—
158#3188—711nomaximal, Sylow
159C71#318811yescharacteristic, complemented, Sylow
160#525—711no—
161C2×#121D7111yescharacteristic, complemented
162#484—711no—
163#482—712no—
164#481—711no—
165#466—712no—
166#465—712no—
167#450—711no—
168#449—712no—
169#447—711no—
170C2×C22⋊C8—711no—
171C2×#41—711no—
172C22×D4—21no—
173C22×D4C2×D7111yescharacteristic
174C23×C4C2×D7111yescharacteristic
175C2×SD16—712no—
176C2×D8—712no—
177C22×C8—712no—
178#123—1421no—
179C22⋊Q8—711no—
180C22⋊Q8—711no—
181#121—21no—
182#121—21no—
183#121—711no—
184#121—711no—
185#121C2×D7112yes—
186#121C2×D7112yes—
187#120—1421no—
188C4×D4—1421no—
189C4×D4—1421no—
190#117—711no—
191C2×C4⋊C4C2×D7111yescharacteristic
192C2×C22⋊C4—21no—
193#107—712no—
194#106—712no—
195#103—712no—
196#102—712no—
197#102—712no—
198#102—712no—
199C22⋊C8—711no—
200C22⋊C8—711no—
201C22⋊C8—712no—
202C24—21no—
203C24C22⋊D7111yescharacteristic
204#41—2842no—
205C2×Q8—711no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—711no—
212C2×D4—1421no—
213C2×D4—1421no—
214C2×D4—1421no—
215C2×D4C22×D7112yes—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—1421no—
219C22×C4C22⋊D7111yescharacteristic
220C22×C4C22⋊D7111yescharacteristic
221C22×C4C22⋊D7111yescharacteristic
222C22×C4C22×D7111yescharacteristic
223C22×C4C22×D7111yescharacteristic
224C22×C4C22×D7111yescharacteristic
225SD16—2842no—
226D8—2842no—
227C2×C8—2842no—
228C2×C8—712no—
229C2×C8—712no—
230C4⋊C4—21no—
231C4⋊C4—711no—
232C4⋊C4—711no—
233C4⋊C4—1421no—
234C4⋊C4—1421no—
235C4⋊C4C22×D7112yes—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—1421no—
239C22⋊C4—1421no—
240C22⋊C4—1421no—
241C22⋊C4—1421no—
242C22⋊C4—1421no—
243C42—1421no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—1421no—
252C23C2×C22⋊D7111yescharacteristic
253C23C2×C22⋊D7111yescharacteristic
254C23C2×C22⋊D7111yescharacteristic
255Q8—1422no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—44no—
260D4—44no—
261D4—2842no—
262D4—2842no—
263D4—2842no—
264D4—1422no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—2842no—
274C2×C4—1421no—
275C2×C4—1421no—
276C2×C4—1421no—
277C2×C4C2×C22⋊D7111yescharacteristic
278C2×C4C2×C22⋊D7111yescharacteristic
279C2×C4C2×C22⋊D7111yescharacteristic
280C2×C4C23×D7111yescharacteristic
281C8—1422no—
282C8—1422no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—44no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—84no—
295C22—2841no—
296C22—2841no—
297C22—2841no—
298C22#1455614812yes—
299C22#1455615412yes—
300C22C22×C22⋊D7111yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—42no—
305C4—2841no—
306C4—2841no—
307C4—2841no—
308C4#1455617312yes—
309C2—22no—
310C2—22no—
311C2—41no—
312C2—82no—
313C2—5681no—
314C2#1798655811yescharacteristic
315C2C2×#1455614811yescharacteristic
316C2C2×#1455615411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#121×C71C2
socleC22×C71C22×D4
hypercenterC2×#121D71
hypocenterC71#3188
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #17986510C2semidirect
2. #17986334C2semidirect
3. #17986320C2semidirect
4. #17986317C2semidirect
5. #17986233C2semidirect
6. #17986226C2semidirect
7. #17986150C2semidirect
8. #17986142C2semidirect
9. #17986133C2semidirect
10. C2×C22⋊(C71⋊C8)C2semidirect
11. C2×#121×C71C2semidirect
12. C2×#11765865C22semidirect
13. C2×#11765853C22semidirect
14. C2×#11765850C22semidirect
15. C22×C71⋊C8C22semidirect
18. #14556107C22semidirect
20. #14556102C22semidirect
24. #14556087C22semidirect
25. #14556061C22semidirect
26. #14556059C22semidirect
27. #14556051C22semidirect
28. #14556049C22semidirect
29. #14556049C22semidirect
30. #14556048C22semidirect
33. C22⋊(C71⋊C8)C22semidirect
159. C71#3188semidirect
161. C2×#121D71semidirect