Finite Group Databasebeta

← #22599948

Subgroups — #22599948

Order 9088.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556156C211yesmaximal, characteristic, complemented
2C2×#14556155C211yesmaximal, characteristic, complemented
3#17986646C211yesmaximal, characteristic, complemented
4#17986640C211yesmaximal, characteristic, complemented
5#17986380C211yesmaximal, characteristic, complemented
6#17986246C211yesmaximal, characteristic, complemented
7#17986217C211yesmaximal, characteristic, complemented
8#17986213C211yesmaximal, characteristic, complemented
9#17986164C211yesmaximal, characteristic, complemented
10#17986159C211yesmaximal, characteristic, complemented
11#17986134C211yesmaximal, characteristic, complemented
12#17986132C211yesmaximal, characteristic, complemented
13#17986081C211yesmaximal, characteristic, complemented
14#17986075C211yesmaximal, characteristic, complemented
15#468×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×#11765865C2211yescharacteristic
18C22×D284C2211yescharacteristic, complemented
19C2×D4×D71C2211yescharacteristic, complemented
20#14556156—24no—
21#14556155—24no—
22C2×#11765855C2211yescharacteristic, complemented
23C2×#11765854C2211yescharacteristic, complemented
24C2×#11765854C2211yescharacteristic, complemented
25C2×#11765852C2211yescharacteristic, complemented
26C2×D568C2211yescharacteristic
27C2×D568C2211yescharacteristic
28C2×#11765850C2211yescharacteristic
29C2×#11765850C2211yescharacteristic
30C2×#11765848C2211yescharacteristic, complemented
31#14556117C2211yescharacteristic, complemented
32#14556115C2211yescharacteristic, complemented
33#14556113—21no—
34#14556110—21no—
35#14556108C2211yescharacteristic, complemented
36#14556103—21no—
37#14556103C2211yescharacteristic
38#14556099—21no—
39#14556098—21no—
40#14556096—21no—
41#14556090C2211yescharacteristic, complemented
42#14556090C2211yescharacteristic, complemented
43C4×C22⋊D71C2211yescharacteristic
44C22⋊C4×D71—21no—
45C22⋊C4×D71—21no—
46Dic71⋊C8C2211yescharacteristic, complemented
47#14556051C2211yescharacteristic, complemented
48#14556048C2211yescharacteristic, complemented
49#14556048C2211yescharacteristic, complemented
50#14556048C2211yescharacteristic, complemented
51C22⋊(C71⋊C8)C2211yescharacteristic, complemented
52(C2×D71)⋊C8C2211yescharacteristic, complemented
53C8×Dic71C2211yescharacteristic, complemented
54C2×SD16×C71C2211yescharacteristic, complemented
55C2×M16×C71C2211yescharacteristic, complemented
56C22⋊Q8×C71C2211yescharacteristic, complemented
57#121×C71C2211yescharacteristic
58#107×C71C2211yescharacteristic, complemented
59#103×C71C2211yescharacteristic, complemented
60#102×C71C2211yescharacteristic, complemented
61C23×D71—21no—
62C23×D71—21no—
63#11765866—22no—
64#11765866—41no—
65#11765865—22no—
66#11765865—41no—
67C2×Dic142C2311yescharacteristic
68C2×D284—22no—
69C2×D284—21no—
70C2×D284—21no—
71C2×D284—21no—
72C2×D284—41no—
73C2×D284C2311yescharacteristic
74C2×D284C2311yescharacteristic
75C2×D284C2311yescharacteristic
76C2×C22⋊D71—21no—
77C2×C22⋊D71—21no—
78C2×C22⋊D71D411yescharacteristic
79D4×D71—22no—
80D4×D71—41no—
81C2×C4×D71—21no—
82C2×C4×D71—21no—
83C2×C4×D71—21no—
84C2×C4×D71—21no—
85C2×C4×D71C2311yescharacteristic
86#11765855—41no—
87#11765854—22no—
88#11765854—22no—
89#11765852—22no—
90D568—22no—
91D568D414yescomplemented
92#11765850—22no—
93#11765850—41no—
94#11765848—22no—
95C2×C71⋊C8C2311yescharacteristic
96C2×C71⋊C8C2311yescharacteristic
97C4⋊Dic71—21no—
98Dic71⋊C4D411yescharacteristic
99C22⋊Dic71—21no—
100C22⋊Dic71—21no—
101C22⋊Dic71D411yescharacteristic
102(C2×D71)⋊C4—21no—
103(C2×D71)⋊C4—21no—
104(C2×D71)⋊C4—21no—
105(C2×D71)⋊C4—21no—
106(C2×D71)⋊C4—21no—
107(C2×D71)⋊C4—21no—
108(C2×D71)⋊C4D411yescharacteristic
109C4×Dic71C2311yescharacteristic
110C2×Q8×C71C2311yescharacteristic
111C2×D4×C71—21no—
112C2×D4×C71C2311yescharacteristic
113C22×C4×C71C2311yescharacteristic
114SD16×C71—22no—
115M16×C71—22no—
116C2×C8×C71C2311yescharacteristic
117C2×C8×C71C2311yescharacteristic
118C4⋊C4×C71—21no—
119C4⋊C4×C71C2311yescharacteristic
120C4⋊C4×C71C2311yescharacteristic
121C22⋊C4×C71—21no—
122C22⋊C4×C71—21no—
123C23×C71—21no—
124C23×C71C2×D411yescharacteristic
125C22×D71—21no—
126C22×D71—21no—
127C22×D71—21no—
128C22×D71—42no—
129C22×D71—42no—
130C22×D71—42no—
131C22×D71—42no—
132C22×D71—81no—
133C22×D71—81no—
134C22×D71C2×D411yescharacteristic
135Q8×C71—21no—
136Q8×C71—21no—
137D4×C71—21no—
138D4×C71—21no—
139D4×C71—81no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71C2×D411yescharacteristic
146C2×C4×C71C2411yescharacteristic
147C22⋊D71—41no—
148C22⋊D71—42no—
149C22⋊D71—42no—
150C2×Dic71—21no—
151C2×Dic71C2×D411yescharacteristic
152C2×Dic71C2×D411yescharacteristic
153C2×Dic71C2×D411yescharacteristic
154D284—21no—
155D284—21no—
156D284—21no—
157D284—21no—
158D284—42no—
159D284—42no—
160D284—81no—
161D284C2×D412yes—
162D284C2×D412yes—
163C4×D71—21no—
164C4×D71—21no—
165C4×D71—42no—
166C4×D71—42no—
167C4×D71—81no—
168C4×D71—81no—
169Dic142—21no—
170Dic142—21no—
171C8×C71—21no—
172C8×C71C2×D412yes—
173C71⋊C8—21no—
174C71⋊C8—21no—
175C22×C71—21no—
176C22×C71—21no—
177C22×C71—21no—
178C22×C71—41no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71C22×D411yescharacteristic
182C2×D71—21no—
183C2×D71—22no—
184C2×D71—22no—
185C2×D71—21no—
186C2×D71—21no—
187C2×D71—41no—
188C2×D71—41no—
189C2×D71—41no—
190C2×D71—41no—
191C2×D71—41no—
192C2×D71—41no—
193C2×D71—41no—
194C2×D71—84no—
195C2×D71—84no—
196C4×C71—21no—
197C4×C71—41no—
198C4×C71—41no—
199C4×C71—41no—
200C4×C71C22×D411yescharacteristic
201C4×C71H(5,𝔽2)11yescharacteristic
202Dic71—21no—
203Dic71—21no—
204Dic71—21no—
205Dic71—41no—
206C2×C71—41no—
207C2×C71—81no—
208C2×C71D4211yescharacteristic
209C2×C71#57511yescharacteristic
210C2×C71#57611yescharacteristic
211D71—41no—
212D71—42no—
213D71—42no—
214D71—81no—
215#3273—711nomaximal, Sylow
216C71#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—711no—
218C2×AGL(1,ℤ/8)—711no—
219#546—711no—
220#546—711no—
221#495—711no—
222#468—711no—
223#468D7111yescharacteristic, complemented
224#463—711no—
225#463—711no—
226#452—711no—
227#452—711no—
228#447—711no—
229#447—711no—
230C4×D8—711no—
231#436—711no—
232C2×#41—711no—
233C2×#41—711no—
234C22×D4—711no—
235C22×D4—711no—
236AGL(1,ℤ/8)—1424no—
237AGL(1,ℤ/8)—1424no—
238C2×Q16—711no—
239C2×SD16—711no—
240C2×SD16—711no—
241C2×SD16—711no—
242C2×SD16C2×D7111yescharacteristic, complemented
243C2×D8—711no—
244C2×D8—711no—
245C2×D8—711no—
246C2×D8—711no—
247C2×M16—711no—
248C2×M16C2×D7111yescharacteristic, complemented
249#124—711no—
250#124—711no—
251#123—1421no—
252#123—1421no—
253C22⋊Q8—711no—
254C22⋊Q8C2×D7111yescharacteristic, complemented
255#121—711no—
256#121—1421no—
257#121—1421no—
258#121C2×D7111yescharacteristic
259#120—1421no—
260#120—1421no—
261C4×D4—711no—
262C4×D4—711no—
263C4×D4—711no—
264C2×C22⋊C4—1421no—
265C2×C22⋊C4—1421no—
266#107C2×D7111yescharacteristic, complemented
267C4⋊C8—711no—
268#103—711no—
269#103C2×D7111yescharacteristic, complemented
270#102—711no—
271#102—711no—
272#102—711no—
273#102C2×D7111yescharacteristic, complemented
274C22⋊C8—711no—
275C22⋊C8—711no—
276C4×C8—711no—
277C24—1421no—
278C24—1421no—
279#41—2841no—
280#41—2841no—
281#41—1422no—
282#41—1422no—
283C2×Q8—711no—
284C2×Q8C22×D7111yescharacteristic
285C2×D4—21no—
286C2×D4—2841no—
287C2×D4—2841no—
288C2×D4—711no—
289C2×D4—711no—
290C2×D4—711no—
291C2×D4—711no—
292C2×D4—1421no—
293C2×D4—1422no—
294C2×D4—1421no—
295C2×D4—1421no—
296C2×D4—1422no—
297C2×D4—1421no—
298C2×D4—1421no—
299C2×D4C22×D7111yescharacteristic
300C22×C4—711no—
301C22×C4—1421no—
302C22×C4—1421no—
303C22×C4—1421no—
304C22×C4—1421no—
305C22×C4C22×D7111yescharacteristic
306Q16—2841no—
307SD16—22no—
308SD16—1422no—
309SD16—1422no—
310SD16—1422no—
311D8—2841no—
312D8—714no—
313D8—1422no—
314D8—1422no—
315M16—22no—
316M16—1422no—
317C2×C8—711no—
318C2×C8—711no—
319C2×C8C22×D7111yescharacteristic
320C2×C8C22×D7111yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—711no—
323C4⋊C4—1421no—
324C4⋊C4C22×D7111yescharacteristic
325C4⋊C4C22×D7111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—711no—
329C22⋊C4—711no—
330C22⋊C4—1421no—
331C22⋊C4—1421no—
332C22⋊C4—1421no—
333C22⋊C4—1421no—
334C22⋊C4—1421no—
335C22⋊C4—1421no—
336C22⋊C4—1421no—
337C22⋊C4—1421no—
338C42—711no—
339C23—21no—
340C23—2842no—
341C23—2842no—
342C23—2842no—
343C23—2842no—
344C23—5681no—
345C23—5681no—
346C23—711no—
347C23—1421no—
348C23—1421no—
349C23—1421no—
350C23D4×D7111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—1421no—
354Q8—1421no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—2842no—
359D4—2842no—
360D4—2842no—
361D4—2841no—
362D4—2842no—
363D4—5681no—
364D4—712no—
365D4—712no—
366D4—1421no—
367D4—1421no—
368D4—1421no—
369D4—1421no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—2842no—
376C2×C4—2842no—
377C2×C4—5681no—
378C2×C4—5681no—
379C2×C4—711no—
380C2×C4—711no—
381C2×C4—711no—
382C2×C4—1421no—
383C2×C4—1421no—
384C2×C4—1421no—
385C2×C4D4×D7111yescharacteristic
386C2×C4C23×D7111yescharacteristic
387C8—21no—
388C8—1421no—
389C8—1421no—
390C8D4×D7112yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—2841no—
398C22—2841no—
399C22—2841no—
400C22—2841no—
401C22—2841no—
402C22—2841no—
403C22—2841no—
404C22—5684no—
405C22—5684no—
406C22—1422no—
407C22—1422no—
408C22—1421no—
409C22—1421no—
410C22—1421no—
411C22C2×D4×D7111yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—2841no—
417C4—1421no—
418C4—1421no—
419C4—1421no—
420C4C2×D4×D7111yescharacteristic
421C4#1455617411yescharacteristic
422C2—41no—
423C2—81no—
424C2—2842no—
425C2—2842no—
426C2—2841no—
427C2—5681no—
428C2#1798663311yescharacteristic
429C2#1798679811yescharacteristic
430C2#1798680311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#468×C71C2
socleC22×C71C22×D4
hypercenter#468D71
hypocenterC71#3273
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#14556156C2semidirect
2. C2×#14556155C2semidirect
3. #17986646C2semidirect
4. #17986640C2semidirect
5. #17986380C2semidirect
6. #17986246C2semidirect
7. #17986217C2semidirect
8. #17986213C2semidirect
9. #17986164C2semidirect
10. #17986159C2semidirect
11. #17986134C2semidirect
12. #17986132C2semidirect
13. #17986081C2semidirect
14. #17986075C2semidirect
15. #468×C71C2semidirect
18. C22×D284C22semidirect
19. C2×D4×D71C22semidirect
22. C2×#11765855C22semidirect
23. C2×#11765854C22semidirect
24. C2×#11765854C22semidirect
25. C2×#11765852C22semidirect
30. C2×#11765848C22semidirect
31. #14556117C22semidirect
32. #14556115C22semidirect
35. #14556108C22semidirect
41. #14556090C22semidirect
42. #14556090C22semidirect
46. Dic71⋊C8C22semidirect
47. #14556051C22semidirect
48. #14556048C22semidirect
49. #14556048C22semidirect
50. #14556048C22semidirect
51. C22⋊(C71⋊C8)C22semidirect
52. (C2×D71)⋊C8C22semidirect
53. C8×Dic71C22semidirect
54. C2×SD16×C71C22semidirect
55. C2×M16×C71C22semidirect
56. C22⋊Q8×C71C22semidirect
58. #107×C71C22semidirect
59. #103×C71C22semidirect
60. #102×C71C22semidirect
91. D568D4semidirect
216. C71#3273semidirect
223. #468D71semidirect
242. C2×SD16C2×D71semidirect
248. C2×M16C2×D71semidirect
254. C22⋊Q8C2×D71semidirect
266. #107C2×D71semidirect
269. #103C2×D71semidirect
273. #102C2×D71semidirect