Finite Group Databasebeta

← #22600300

Subgroups — #22600300

Order 9088.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986652C211yesmaximal, characteristic, complemented
2#17986642C211yesmaximal, characteristic, complemented
3#17986328C211yesmaximal, characteristic, complemented
4#17986316C211yesmaximal, characteristic, complemented
5#17986286C211yesmaximal, characteristic, complemented
6#17986251C211yesmaximal, characteristic, complemented
7#17986231C211yesmaximal, characteristic, complemented
8#17986211C211yesmaximal, characteristic, complemented
9#17986147C211yesmaximal, characteristic, complemented
10#17986134C211yesmaximal, characteristic, complemented
11#17986112C211yesmaximal, characteristic, complemented
12#17986072C211yesmaximal, characteristic, complemented
13#17985999C211yesmaximal, characteristic, complemented
14#102×D71C211yesmaximal, characteristic, complemented
15#489×C71C211yesmaximal, characteristic, complemented
16C2×#11765866—21no—
17C2×#11765866C2211yescharacteristic
18C2×D4×D71C2211yescharacteristic
19C2×#11765852C2211yescharacteristic, complemented
20C2×D568C2211yescharacteristic
21C2×#11765850C2211yescharacteristic
22C2×#11765848C2211yescharacteristic, complemented
23C2×C8×D71C2211yescharacteristic
24#14556113—21no—
25#14556113—21no—
26#14556110—21no—
27#14556104—21no—
28#14556104C2211yescharacteristic
29#14556103C2211yescharacteristic
30#14556103C2211yescharacteristic, complemented
31#14556101—21no—
32#14556099C2211yescharacteristic
33#14556098—21no—
34C4×D284C2211yescharacteristic
35C4×D284C2211yescharacteristic
36C4×D284C2211yescharacteristic, complemented
37C4⋊C4×D71C2211yescharacteristic
38C4⋊C4×D71C2211yescharacteristic, complemented
39C22⋊C4×D71—21no—
40#14556060C2211yescharacteristic, complemented
41#14556059C2211yescharacteristic, complemented
42#14556059C2211yescharacteristic, complemented
43C4⋊(C71⋊C8)C2211yescharacteristic, complemented
44#14556052C2211yescharacteristic, complemented
45#14556050C2211yescharacteristic
46#14556050C2211yescharacteristic, complemented
47#14556049C2211yescharacteristic
48#14556048C2211yescharacteristic
49#14556048C2211yescharacteristic
50(C2×D71)⋊C8C2211yescharacteristic
51(C2×D71)⋊C8C2211yescharacteristic, complemented
52C4⋊Q8×C71C2211yescharacteristic, complemented
53#124×C71C2211yescharacteristic
54#103×C71C2211yescharacteristic, complemented
55#103×C71C2211yescharacteristic, complemented
56#102×C71C2211yescharacteristic
57#102×C71C2211yescharacteristic
58#97×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765866—22no—
61#11765866—41no—
62#11765866—44no—
63C2×D284—21no—
64C2×D284—21no—
65C2×D284—21no—
66C2×D284D411yescharacteristic
67C2×D284C2311yescharacteristic
68C2×D284C2311yescharacteristic
69C2×C22⋊D71—21no—
70C2×C22⋊D71—21no—
71D4×D71—22no—
72D4×D71—41no—
73C2×C4×D71—21no—
74C2×C4×D71—21no—
75C2×C4×D71—21no—
76C2×C4×D71—21no—
77C2×C4×D71—21no—
78C2×C4×D71C2311yescharacteristic
79C2×C4×D71C2311yescharacteristic
80#11765852—41no—
81D568—22no—
82#11765850—41no—
83#11765848—41no—
84C2×C71⋊C8C2311yescharacteristic
85C2×C71⋊C8C2311yescharacteristic
86C8×D71—41no—
87C4⋊Dic71D411yescharacteristic
88C4⋊Dic71C2311yescharacteristic
89C4⋊Dic71C2311yescharacteristic
90Dic71⋊C4—21no—
91Dic71⋊C4—21no—
92Dic71⋊C4—21no—
93C22⋊Dic71—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96(C2×D71)⋊C4—21no—
97(C2×D71)⋊C4—21no—
98(C2×D71)⋊C4—21no—
99(C2×D71)⋊C4D411yescharacteristic
100(C2×D71)⋊C4D411yescharacteristic
101C2×Q8×C71—21no—
102C2×Q8×C71C2311yescharacteristic
103C2×D4×C71C2311yescharacteristic
104C2×C8×C71C2311yescharacteristic
105C2×C8×C71C2311yescharacteristic
106C4⋊C4×C71—21no—
107C4⋊C4×C71C2311yescharacteristic
108C4⋊C4×C71C2311yescharacteristic
109C22⋊C4×C71—21no—
110C22⋊C4×C71—21no—
111C42×C71C2311yescharacteristic
112C23×C71—21no—
113C22×D71—21no—
114C22×D71—21no—
115C22×D71—42no—
116C22×D71—42no—
117C22×D71—81no—
118C22×D71C2×D411yescharacteristic
119C22×D71C2×D411yescharacteristic
120Q8×C71—21no—
121Q8×C71—21no—
122Q8×C71—42no—
123D4×C71—21no—
124D4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71C2×D411yescharacteristic
129C2×C4×C71C2×D411yescharacteristic
130C2×C4×C71C2411yescharacteristic
131C22⋊D71—42no—
132C22⋊D71—81no—
133C2×Dic71—21no—
134C2×Dic71—21no—
135C2×Dic71C2×D411yescharacteristic
136C2×Dic71C2×D411yescharacteristic
137D284—22no—
138D284—21no—
139D284—21no—
140D284—21no—
141D284—42no—
142D284—42no—
143D284—42no—
144D284#4112yescomplemented
145C4×D71—21no—
146C4×D71—21no—
147C4×D71—21no—
148C4×D71—21no—
149C4×D71—42no—
150C4×D71—42no—
151C4×D71—42no—
152C4×D71—42no—
153C4×D71—81no—
154C8×C71—21no—
155C8×C71—21no—
156C71⋊C8—21no—
157C71⋊C8—21no—
158C22×C71—41no—
159C22×C71—41no—
160C22×C71—41no—
161C22×C71C22×D411yescharacteristic
162C2×D71—21no—
163C2×D71—21no—
164C2×D71—21no—
165C2×D71—22no—
166C2×D71—21no—
167C2×D71—21no—
168C2×D71—41no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—41no—
172C2×D71—41no—
173C2×D71—84no—
174C2×D71AGL(1,ℤ/8)12yes—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—41no—
178C4×C71—41no—
179C4×C71—41no—
180C4×C71C2×#4111yescharacteristic
181C4×C71H(5,𝔽2)11yescharacteristic
182Dic71—21no—
183Dic71—21no—
184Dic71—41no—
185Dic71—41no—
186C2×C71—81no—
187C2×C71#54611yescharacteristic
188C2×C71C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C71#57711yescharacteristic
190D71—22no—
191D71—41no—
192D71—42no—
193D71—81no—
194#3306—711nomaximal, Sylow
195C71#330611yescharacteristic, complemented, Sylow
196#547—711no—
197#546—711no—
198#489D7111yescharacteristic, complemented
199#483—711no—
200#481—711no—
201#476—711no—
202#469—711no—
203#465—711no—
204#463—711no—
205#449—711no—
206#447—711no—
207#442—711no—
208#435—711no—
209#417—711no—
210C2×#102—711no—
211C2×#41—711no—
212C2×#41—1421no—
213C22×D4—711no—
214C2×SD16—711no—
215C2×D8—711no—
216C2×D8—711no—
217C2×M16—711no—
218C22×C8—711no—
219C4⋊Q8C2×D7111yescharacteristic, complemented
220#124C2×D7111yescharacteristic
221#123—1421no—
222#123—1421no—
223#123—1421no—
224C22⋊Q8—711no—
225C22⋊Q8—1421no—
226#121—711no—
227#121—711no—
228#121—711no—
229#121—1421no—
230#120—1421no—
231C4×D4—711no—
232C4×D4—711no—
233C4×D4—711no—
234C2×C4⋊C4—711no—
235C2×C4⋊C4—711no—
236C2×C22⋊C4—1421no—
237#106—711no—
238#106—711no—
239#106—711no—
240C4⋊C8—711no—
241#103—711no—
242#103C2×D7111yescharacteristic, complemented
243#103C2×D7111yescharacteristic, complemented
244#102—711no—
245#102—711no—
246#102—711no—
247#102—711no—
248#102—711no—
249#102C2×D7111yescharacteristic
250#102C2×D7111yescharacteristic
251C22⋊C8—711no—
252C22⋊C8—711no—
253#97C2×D7111yescharacteristic, complemented
254C24—1421no—
255#41—2844no—
256#41—2841no—
257#41—1422no—
258C2×Q8—21no—
259C2×Q8C22×D7111yescharacteristic
260C2×D4—2841no—
261C2×D4—711no—
262C2×D4—711no—
263C2×D4—711no—
264C2×D4—1421no—
265C2×D4—1421no—
266C2×D4—1421no—
267C2×D4—1421no—
268C2×D4—1422no—
269C2×D4—1421no—
270C2×D4C22×D7111yescharacteristic
271C22×C4—711no—
272C22×C4—711no—
273C22×C4—1421no—
274C22×C4—1421no—
275C22×C4—1421no—
276C22×C4—1421no—
277C22×C4—1421no—
278SD16—2841no—
279D8—2841no—
280D8—1422no—
281M16—2841no—
282C2×C8—2841no—
283C2×C8—711no—
284C2×C8—711no—
285C2×C8C22×D7111yescharacteristic
286C2×C8C22×D7111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—711no—
289C4⋊C4—711no—
290C4⋊C4—711no—
291C4⋊C4—1421no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C4⋊C4C22×D7111yescharacteristic
295C4⋊C4C22×D7111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—711no—
299C22⋊C4—711no—
300C22⋊C4—1421no—
301C22⋊C4—1421no—
302C22⋊C4—1421no—
303C22⋊C4—1421no—
304C22⋊C4—1421no—
305C22⋊C4—1421no—
306C42C22×D7111yescharacteristic
307C23—21no—
308C23—2842no—
309C23—2842no—
310C23—5681no—
311C23—711no—
312C23—711no—
313C23—1421no—
314C23—1421no—
315Q8—21no—
316Q8—21no—
317Q8—42no—
318D4—21no—
319D4—21no—
320D4—2842no—
321D4—2842no—
322D4—2842no—
323D4—2842no—
324D4—5681no—
325D4—712no—
326D4—1422no—
327D4—1421no—
328D4—1421no—
329D4—1421no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—2842no—
334C2×C4—2842no—
335C2×C4—2842no—
336C2×C4—2842no—
337C2×C4—5681no—
338C2×C4—711no—
339C2×C4—711no—
340C2×C4—1421no—
341C2×C4—1421no—
342C2×C4—1421no—
343C2×C4—1421no—
344C2×C4—1421no—
345C2×C4—1421no—
346C2×C4D4×D7111yescharacteristic
347C2×C4D4×D7111yescharacteristic
348C2×C4C23×D7111yescharacteristic
349C8—21no—
350C8—21no—
351C8—1421no—
352C8—1421no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2841no—
357C22—2841no—
358C22—2841no—
359C22—2841no—
360C22—2841no—
361C22—5684no—
362C22—712no—
363C22—1422no—
364C22—1421no—
365C22—1421no—
366C22—1421no—
367C22—1421no—
368C22—1421no—
369C22C2×D4×D7111yescharacteristic
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—2841no—
376C4—2841no—
377C4—1421no—
378C4—1421no—
379C4#41×D7111yescharacteristic
380C4#1455617411yescharacteristic
381C2—81no—
382C2—2842no—
383C2—2841no—
384C2—5681no—
385C2—1422no—
386C2#1798664211yescharacteristic
387C2AGL(1,ℤ/8)×D7111yescharacteristic
388C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#489×C71C2
socleC22×C71C22×D4
hypercenter#489D71
hypocenterC71#3306
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #17986652C2semidirect
2. #17986642C2semidirect
3. #17986328C2semidirect
4. #17986316C2semidirect
5. #17986286C2semidirect
6. #17986251C2semidirect
7. #17986231C2semidirect
8. #17986211C2semidirect
9. #17986147C2semidirect
10. #17986134C2semidirect
11. #17986112C2semidirect
12. #17986072C2semidirect
13. #17985999C2semidirect
14. #102×D71C2semidirect
15. #489×C71C2semidirect
19. C2×#11765852C22semidirect
22. C2×#11765848C22semidirect
30. #14556103C22semidirect
36. C4×D284C22semidirect
38. C4⋊C4×D71C22semidirect
40. #14556060C22semidirect
41. #14556059C22semidirect
42. #14556059C22semidirect
43. C4⋊(C71⋊C8)C22semidirect
44. #14556052C22semidirect
46. #14556050C22semidirect
51. (C2×D71)⋊C8C22semidirect
52. C4⋊Q8×C71C22semidirect
54. #103×C71C22semidirect
55. #103×C71C22semidirect
58. #97×C71C22semidirect
144. D284#41semidirect
195. C71#3306semidirect
198. #489D71semidirect
219. C4⋊Q8C2×D71semidirect
242. #103C2×D71semidirect
243. #103C2×D71semidirect
253. #97C2×D71semidirect