Finite Group Databasebeta

← #22600102

Subgroups — #22600102

Order 9088.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986807C211yesmaximal, characteristic, complemented
2#17986802C211yesmaximal, characteristic, complemented
3#17986801C211yesmaximal, characteristic, complemented
4#17986385C211yesmaximal, characteristic, complemented
5#17986269C211yesmaximal, characteristic, complemented
6#17986267C211yesmaximal, characteristic, complemented
7#17986264C211yesmaximal, characteristic, complemented
8#17986259C211yesmaximal, characteristic, complemented
9#17986175C211yesmaximal, characteristic, complemented
10#17986172C211yesmaximal, characteristic, complemented
11#17986167C211yesmaximal, characteristic, complemented
12#17986165C211yesmaximal, characteristic, complemented
13#17986121C211yesmaximal, characteristic, complemented
14#17986119C211yesmaximal, characteristic, complemented
15#577×C71C211yesmaximal, characteristic, complemented
16#14556174C2211yescharacteristic
17#14556174C2211yescharacteristic, complemented
18#14556159—21no—
19#14556159C2211yescharacteristic, complemented
20#14556155—21no—
21#14556155—21no—
22#14556154—21no—
23#14556154C2211yescharacteristic
24#14556153—21no—
25#14556153—21no—
26#14556153C2211yescharacteristic
27#14556153C2211yescharacteristic, complemented
28#14556153C2211yescharacteristic, complemented
29#14556153C2211yescharacteristic, complemented
30#14556151—21no—
31#14556148—21no—
32#14556148—21no—
33#14556148C2211yescharacteristic
34C2×#11765852—21no—
35C2×#11765852C2211yescharacteristic, complemented
36C2×D568C2211yescharacteristic, complemented
37C2×#11765850—21no—
38C2×#11765850C2211yescharacteristic, complemented
39D8×D71—21no—
40#14556135C2211yescharacteristic, complemented
41#14556135C2211yescharacteristic, complemented
42#14556134C2211yescharacteristic, complemented
43#14556124C2211yescharacteristic, complemented
44#14556117C2211yescharacteristic, complemented
45#14556064C2211yescharacteristic, complemented
46#14556063C2211yescharacteristic, complemented
47#14556055C2211yescharacteristic, complemented
48#14556055C2211yescharacteristic, complemented
49#14556055C2211yescharacteristic, complemented
50#14556054C2211yescharacteristic, complemented
51#14556047C2211yescharacteristic, complemented
52#14556045C2211yescharacteristic, complemented
53#14556044C2211yescharacteristic, complemented
54#14556044C2211yescharacteristic, complemented
55#14556038C2211yescharacteristic, complemented
562−1+4×C71C2211yescharacteristic, complemented
57H(5,𝔽2)×C71C2211yescharacteristic, complemented
58#137×C71—21no—
59#137×C71C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C71—21no—
61AGL(1,ℤ/8)×C71C2211yescharacteristic, complemented
62#135×C71—21no—
63#135×C71C2211yescharacteristic
64C2×SD16×C71—21no—
65C2×SD16×C71C2211yescharacteristic, complemented
66#131×C71C2211yescharacteristic, complemented
67#11765866—21no—
68#11765866—21no—
69#11765865—21no—
70#11765865—21no—
71#11765865C2311yescharacteristic
72C2×D284—21no—
73C2×D284—21no—
74C2×D284C2311yescharacteristic
75C2×D284C2311yescharacteristic
76C2×C22⋊D71—41no—
77D4×D71—21no—
78D4×D71—21no—
79D4×D71—41no—
80D4×D71—41no—
81#11765855—21no—
82#11765855D411yescharacteristic, complemented
83#11765853—21no—
84#11765853D411yescharacteristic, complemented
85#11765852—21no—
86#11765852—21no—
87#11765852—21no—
88#11765852—21no—
89#11765852—21no—
90#11765852—21no—
91#11765852—21no—
92#11765852D411yescharacteristic, complemented
93D568—21no—
94D568—21no—
95#11765850—21no—
96#11765850—21no—
97#11765850—21no—
98#11765850—21no—
99#11765850—21no—
100#11765850—21no—
101#11765850—21no—
102#11765850D411yescharacteristic, complemented
103#11765849—21no—
104#11765849—21no—
105#11765849C2311yescharacteristic
106#11765849C2311yescharacteristic
107#11765849C2311yescharacteristic
108#11765848—21no—
109C2×C71⋊C8—21no—
110C2×C71⋊C8—21no—
111C2×C71⋊C8C2311yescharacteristic
112C8×D71—21no—
113(C2×D71)⋊C4—41no—
114C4×Dic71C2311yescharacteristic
115#41×C71—21no—
116#41×C71—21no—
117#41×C71—21no—
118#41×C71—21no—
119#41×C71—41no—
120#41×C71C2311yescharacteristic
121#41×C71C2311yescharacteristic
122#41×C71C2311yescharacteristic
123C2×Q8×C71—21no—
124C2×Q8×C71—21no—
125C2×Q8×C71C2311yescharacteristic
126C2×D4×C71—21no—
127C2×D4×C71—21no—
128C2×D4×C71—41no—
129C2×D4×C71C2311yescharacteristic
130Q16×C71—21no—
131Q16×C71D411yescharacteristic, complemented
132SD16×C71—21no—
133SD16×C71—21no—
134SD16×C71—21no—
135SD16×C71—21no—
136SD16×C71D411yescharacteristic, complemented
137SD16×C71D411yescharacteristic, complemented
138D8×C71—21no—
139D8×C71D411yescharacteristic, complemented
140M16×C71—21no—
141M16×C71C2311yescharacteristic
142C2×C8×C71—21no—
143C2×C8×C71C2311yescharacteristic
144C23×C71—21no—
145C23×C71—41no—
146C22×D71—21no—
147C22×D71—21no—
148C22×D71—41no—
149C22×D71—41no—
150Q8×C71—21no—
151Q8×C71—21no—
152Q8×C71—41no—
153Q8×C71C2×D411yescharacteristic
154Q8×C71C2×D411yescharacteristic
155Q8×C71C2×D411yescharacteristic
156D4×C71—21no—
157D4×C71—21no—
158D4×C71—21no—
159D4×C71—21no—
160D4×C71—21no—
161D4×C71—41no—
162D4×C71—41no—
163D4×C71—41no—
164D4×C71C2×D411yescharacteristic
165D4×C71C2×D411yescharacteristic
166D4×C71C2×D411yescharacteristic
167C2×C4×C71—21no—
168C2×C4×C71—21no—
169C2×C4×C71—21no—
170C2×C4×C71—21no—
171C2×C4×C71—41no—
172C2×C4×C71—41no—
173C2×C4×C71—41no—
174C2×C4×C71C2411yescharacteristic
175C22⋊D71—21no—
176C22⋊D71—41no—
177C22⋊D71—41no—
178C22⋊D71—82no—
179C2×Dic71—21no—
180D284—21no—
181D284—21no—
182D284—21no—
183D284—21no—
184D284—41no—
185D284—41no—
186D284C2×D411yescharacteristic
187C4×D71—21no—
188C4×D71—41no—
189C4×D71—41no—
190Dic142C2×D411yescharacteristic
191C8×C71—21no—
192C8×C71C2×D411yescharacteristic
193C8×C71C2×D411yescharacteristic
194C71⋊C8—21no—
195C71⋊C8—21no—
196C71⋊C8—21no—
197C71⋊C8C2×D411yescharacteristic
198C71⋊C8C2×D411yescharacteristic
199C22×C71—21no—
200C22×C71—21no—
201C22×C71—21no—
202C22×C71—41no—
203C22×C71—81no—
204C22×C71—161no—
205C22×C71H(5,𝔽2)11yescharacteristic
206C2×D71—21no—
207C2×D71—41no—
208C2×D71—41no—
209C2×D71—81no—
210C2×D71—81no—
211C2×D71—161no—
212C2×D71—161no—
213C4×C71—21no—
214C4×C71—21no—
215C4×C71—21no—
216C4×C71—41no—
217C4×C71C22×D411yescharacteristic
218C4×C71C22×D411yescharacteristic
219Dic71—21no—
220Dic71—41no—
221C2×C71—21no—
222C2×C71—41no—
223C2×C71—41no—
224C2×C71—41no—
225C2×C71—81no—
226C2×C71D4211yescharacteristic
227D71—41no—
228D71—81no—
229D71—81no—
230#3288—711nomaximal, Sylow
231C71#328811yescharacteristic, complemented, Sylow
232#577—711no—
233#577D7111yescharacteristic, complemented
234#576—711no—
235#576—711no—
236#496—711no—
237#472—711no—
238#472—711no—
239#471—711no—
240#471—711no—
241#455—711no—
242#454—711no—
243#453—711no—
244#453—711no—
245#444—711no—
246#444—711no—
2472−1+4C2×D7111yescharacteristic, complemented
248H(5,𝔽2)—711no—
249H(5,𝔽2)—711no—
250H(5,𝔽2)C2×D7111yescharacteristic, complemented
251#137—21no—
252#137—711no—
253#137—1421no—
254#137C2×D7111yescharacteristic, complemented
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—711no—
257AGL(1,ℤ/8)—711no—
258AGL(1,ℤ/8)—711no—
259AGL(1,ℤ/8)—711no—
260AGL(1,ℤ/8)—711no—
261AGL(1,ℤ/8)—1421no—
262AGL(1,ℤ/8)—1421no—
263AGL(1,ℤ/8)—1421no—
264AGL(1,ℤ/8)—1421no—
265AGL(1,ℤ/8)—1421no—
266AGL(1,ℤ/8)C2×D7111yescharacteristic, complemented
267#135—21no—
268#135—711no—
269#135—1421no—
270#135—1421no—
271#135—1421no—
272#135C2×D7111yescharacteristic
273C2×SD16—21no—
274C2×SD16—711no—
275C2×SD16—1421no—
276C2×SD16C2×D7111yescharacteristic, complemented
277C2×D8—711no—
278C2×D8—711no—
279C2×D8—1421no—
280C2×D8—1421no—
281#131—711no—
282#131—711no—
283#131—711no—
284#131C2×D7111yescharacteristic, complemented
285#127—711no—
286#124—711no—
287#108—711no—
288#108—711no—
289#104—711no—
290#104—711no—
291#104—711no—
292#104—711no—
293#101—711no—
294#100—711no—
295#100—711no—
296#100—711no—
297#97—711no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—41no—
303#41—711no—
304#41—1421no—
305#41—1421no—
306#41—1421no—
307#41—1421no—
308#41C22×D7111yescharacteristic
309#41C22×D7111yescharacteristic
310#41C22×D7111yescharacteristic
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8C22×D7111yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—2841no—
318C2×D4—2841no—
319C2×D4—2841no—
320C2×D4—711no—
321C2×D4—711no—
322C2×D4—1421no—
323C2×D4—1421no—
324C2×D4—1421no—
325C2×D4—1421no—
326C2×D4C22×D7111yescharacteristic
327Q16—21no—
328Q16—711no—
329Q16—1421no—
330Q16C22⋊D7111yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—711no—
336SD16—711no—
337SD16—1421no—
338SD16—1421no—
339SD16—1421no—
340SD16—1421no—
341SD16—1421no—
342SD16—1421no—
343SD16—1421no—
344SD16—1421no—
345SD16C22⋊D7111yescharacteristic, complemented
346SD16C22⋊D7111yescharacteristic, complemented
347D8—21no—
348D8—711no—
349D8—1421no—
350D8—1421no—
351D8—1421no—
352D8—1421no—
353D8—1421no—
354D8—1421no—
355D8—1421no—
356D8—1421no—
357D8—1421no—
358D8C22⋊D7111yescharacteristic, complemented
359M16—21no—
360M16—711no—
361M16—711no—
362M16—711no—
363M16—1421no—
364M16—1421no—
365M16—1421no—
366M16C22×D7111yescharacteristic
367C2×C8—21no—
368C2×C8—711no—
369C2×C8—1421no—
370C2×C8—1421no—
371C2×C8—1421no—
372C2×C8C22×D7111yescharacteristic
373C22⋊C4—2841no—
374C42—711no—
375C23—21no—
376C23—41no—
377C23—2841no—
378C23—2841no—
379C23—1421no—
380C23—1421no—
381Q8—21no—
382Q8—21no—
383Q8—41no—
384Q8—711no—
385Q8D4×D7111yescharacteristic
386Q8C2×C22⋊D7111yescharacteristic
387Q8C2×C22⋊D7111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—2841no—
397D4—2841no—
398D4—2841no—
399D4—2841no—
400D4—5682no—
401D4—711no—
402D4—1421no—
403D4—1421no—
404D4—1421no—
405D4—1421no—
406D4—1421no—
407D4D4×D7111yescharacteristic
408D4C2×C22⋊D7111yescharacteristic
409D4C2×C22⋊D7111yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—2841no—
418C2×C4—2841no—
419C2×C4—1421no—
420C2×C4—1421no—
421C2×C4C23×D7111yescharacteristic
422C8—21no—
423C8—711no—
424C8—711no—
425C8—1421no—
426C8—1421no—
427C8—1421no—
428C8C2×C22⋊D7111yescharacteristic
429C8C2×C22⋊D7111yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—81no—
435C22—161no—
436C22—2841no—
437C22—2841no—
438C22—5681no—
439C22—5681no—
440C22—11361no—
441C22—11361no—
442C22—1421no—
443C22#1455617311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—41no—
448C4—2841no—
449C4—1421no—
450C4C2×D4×D7111yescharacteristic
451C4C22×C22⋊D7111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—2841no—
458C2—5681no—
459C2—5681no—
460C2D4×C22⋊D7111yescharacteristic
subgroupG/H
centerC2D4×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#577×C71C2
socleC2×C71D42
hypercenter#577D71
hypocenterC71#3288
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #17986807C2semidirect
2. #17986802C2semidirect
3. #17986801C2semidirect
4. #17986385C2semidirect
5. #17986269C2semidirect
6. #17986267C2semidirect
7. #17986264C2semidirect
8. #17986259C2semidirect
9. #17986175C2semidirect
10. #17986172C2semidirect
11. #17986167C2semidirect
12. #17986165C2semidirect
13. #17986121C2semidirect
14. #17986119C2semidirect
15. #577×C71C2semidirect
17. #14556174C22semidirect
19. #14556159C22semidirect
27. #14556153C22semidirect
28. #14556153C22semidirect
29. #14556153C22semidirect
35. C2×#11765852C22semidirect
36. C2×D568C22semidirect
38. C2×#11765850C22semidirect
40. #14556135C22semidirect
41. #14556135C22semidirect
42. #14556134C22semidirect
43. #14556124C22semidirect
44. #14556117C22semidirect
45. #14556064C22semidirect
46. #14556063C22semidirect
47. #14556055C22semidirect
48. #14556055C22semidirect
49. #14556055C22semidirect
50. #14556054C22semidirect
51. #14556047C22semidirect
52. #14556045C22semidirect
53. #14556044C22semidirect
54. #14556044C22semidirect
55. #14556038C22semidirect
56. 2−1+4×C71C22semidirect
57. H(5,𝔽2)×C71C22semidirect
59. #137×C71C22semidirect
61. AGL(1,ℤ/8)×C71C22semidirect
65. C2×SD16×C71C22semidirect
66. #131×C71C22semidirect
82. #11765855D4semidirect
84. #11765853D4semidirect
92. #11765852D4semidirect
102. #11765850D4semidirect
131. Q16×C71D4semidirect
136. SD16×C71D4semidirect
137. SD16×C71D4semidirect
139. D8×C71D4semidirect
231. C71#3288semidirect
233. #577D71semidirect
247. 2−1+4C2×D71semidirect
250. H(5,𝔽2)C2×D71semidirect
254. #137C2×D71semidirect
266. AGL(1,ℤ/8)C2×D71semidirect
276. C2×SD16C2×D71semidirect
284. #131C2×D71semidirect
330. Q16C22⋊D71semidirect
345. SD16C22⋊D71semidirect
346. SD16C22⋊D71semidirect
358. D8C22⋊D71semidirect