Finite Group Databasebeta

← #22598015

Subgroups — #22598015

Order 9088.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556155C211yesmaximal, characteristic, complemented
2C2×#14556136C211yesmaximal, characteristic, complemented
3#17986579C211yesmaximal, characteristic, complemented
4#17986565C211yesmaximal, characteristic, complemented
5#17986252C211yesmaximal, characteristic, complemented
6#17986251C211yesmaximal, characteristic, complemented
7#17986248C211yesmaximal, characteristic, complemented
8#17986236C211yesmaximal, characteristic, complemented
9#17986233C211yesmaximal, characteristic, complemented
10#17986230C211yesmaximal, characteristic, complemented
11#17986225C211yesmaximal, characteristic, complemented
12#17986040C211yesmaximal, characteristic, complemented
13#17985994C211yesmaximal, characteristic, complemented
14#102×D71C211yesmaximal, characteristic, complemented
15#468×C71C211yesmaximal, characteristic, complemented
16C2×#11765866C2211yescharacteristic
17C2×#11765865C2211yescharacteristic
18C2×D4×D71C2211yescharacteristic
19#14556155—24no—
20C2×#11765852C2211yescharacteristic
21C2×D568C2211yescharacteristic
22C2×#11765850C2211yescharacteristic
23#14556136—24no—
24C2×#11765848C2211yescharacteristic
25C2×#11765848C2211yescharacteristic, complemented
26C22×C71⋊C8C2211yescharacteristic, complemented
27C2×C8×D71C2211yescharacteristic
28C2×C8×D71C2211yescharacteristic, complemented
29#14556124—21no—
30#14556114—21no—
31#14556106—21no—
32#14556104C2211yescharacteristic, complemented
33#14556103—21no—
34#14556103C2211yescharacteristic
35#14556103C2211yescharacteristic, complemented
36#14556102C2211yescharacteristic
37#14556102C2211yescharacteristic, complemented
38#14556101—21no—
39#14556101—21no—
40#14556099C2211yescharacteristic
41#14556098—21no—
42#14556096—21no—
43#14556086C2211yescharacteristic
44C4⋊C4×D71C2211yescharacteristic
45#14556061C2211yescharacteristic, complemented
46#14556060C2211yescharacteristic, complemented
47#14556059C2211yescharacteristic, complemented
48#14556052C2211yescharacteristic, complemented
49#14556050C2211yescharacteristic
50#14556050C2211yescharacteristic, complemented
51#14556049C2211yescharacteristic
52#14556048C2211yescharacteristic
53#14556048C2211yescharacteristic, complemented
54C2×SD16×C71C2211yescharacteristic
55C2×M16×C71C2211yescharacteristic, complemented
56C22⋊Q8×C71C2211yescharacteristic, complemented
57#121×C71C2211yescharacteristic
58#107×C71C2211yescharacteristic, complemented
59#103×C71C2211yescharacteristic, complemented
60#102×C71C2211yescharacteristic
61C23×D71—21no—
62#11765866—22no—
63#11765866—22no—
64#11765865—22no—
65#11765865—22no—
66C2×Dic142D411yescharacteristic
67C2×D284—21no—
68C2×D284—21no—
69C2×D284—21no—
70C2×D284D411yescharacteristic
71C2×D284C2311yescharacteristic
72C2×D284C2311yescharacteristic
73C2×C22⋊D71—21no—
74C2×C22⋊D71—21no—
75C2×C22⋊D71—21no—
76C2×C22⋊D71—21no—
77C2×C22⋊D71D411yescharacteristic
78C2×C22⋊D71D411yescharacteristic
79D4×D71—22no—
80D4×D71—22no—
81C2×C4×D71—21no—
82C2×C4×D71—21no—
83C2×C4×D71C2311yescharacteristic
84C2×C4×D71C2311yescharacteristic
85#11765852—22no—
86D568—22no—
87#11765850—22no—
88#11765848—22no—
89#11765848D414yescomplemented
90C2×C71⋊C8—22no—
91C2×C71⋊C8C2311yescharacteristic
92C2×C71⋊C8C2311yescharacteristic
93C8×D71—22no—
94C8×D71—22no—
95C4⋊Dic71C2311yescharacteristic
96C4⋊Dic71C2311yescharacteristic
97Dic71⋊C4—21no—
98Dic71⋊C4—21no—
99C22⋊Dic71—21no—
100(C2×D71)⋊C4—21no—
101(C2×D71)⋊C4—21no—
102(C2×D71)⋊C4—21no—
103(C2×D71)⋊C4—21no—
104(C2×D71)⋊C4—21no—
105(C2×D71)⋊C4—21no—
106C4×Dic71—21no—
107C2×Q8×C71C2311yescharacteristic
108C2×D4×C71—21no—
109C2×D4×C71C2311yescharacteristic
110C22×C4×C71C2311yescharacteristic
111SD16×C71—22no—
112M16×C71—22no—
113C2×C8×C71C2311yescharacteristic
114C2×C8×C71C2311yescharacteristic
115C4⋊C4×C71—21no—
116C4⋊C4×C71C2311yescharacteristic
117C4⋊C4×C71C2311yescharacteristic
118C22⋊C4×C71—21no—
119C22⋊C4×C71—21no—
120C23×C71—21no—
121C23×C71C2×D411yescharacteristic
122C22×D71—21no—
123C22×D71—21no—
124C22×D71—42no—
125C22×D71—81no—
126C22×D71—81no—
127C22×D71C2×D411yescharacteristic
128C22×D71C2×D411yescharacteristic
129Q8×C71—21no—
130Q8×C71—21no—
131D4×C71—21no—
132D4×C71—21no—
133D4×C71—81no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71C2×D411yescharacteristic
140C2×C4×C71C2411yescharacteristic
141C22⋊D71—22no—
142C22⋊D71—22no—
143C22⋊D71—42no—
144C22⋊D71—81no—
145C22⋊D71—81no—
146C22⋊D71—81no—
147C2×Dic71—21no—
148C2×Dic71—21no—
149C2×Dic71C2×D411yescharacteristic
150C2×Dic71C2×D411yescharacteristic
151D284—22no—
152D284—21no—
153D284—21no—
154D284—21no—
155D284—21no—
156D284—42no—
157D284—81no—
158D284—81no—
159C4×D71—21no—
160C4×D71—21no—
161C4×D71—42no—
162C4×D71—81no—
163C4×D71C2×D412yes—
164C4×D71#4112yes—
165Dic142—22no—
166C8×C71—21no—
167C8×C71C2×D412yes—
168C71⋊C8—21no—
169C71⋊C8C2×D412yes—
170C22×C71—21no—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71—41no—
174C22×C71—41no—
175C22×C71—41no—
176C22×C71C22×D411yescharacteristic
177C2×D71—21no—
178C2×D71—21no—
179C2×D71—21no—
180C2×D71—21no—
181C2×D71—21no—
182C2×D71—41no—
183C2×D71—41no—
184C2×D71—41no—
185C2×D71—41no—
186C2×D71—41no—
187C2×D71—41no—
188C2×D71—84no—
189C2×D71#12112yes—
190C4×C71—21no—
191C4×C71—41no—
192C4×C71—41no—
193C4×C71—41no—
194C4×C71C22×D411yescharacteristic
195C4×C71C2×#4111yescharacteristic
196Dic71—21no—
197Dic71—41no—
198Dic71—41no—
199Dic71#12112yes—
200C2×C71—41no—
201C2×C71—81no—
202C2×C71C2×#12111yescharacteristic
203C2×C71#57611yescharacteristic
204C2×C71#57711yescharacteristic
205D71—22no—
206D71—41no—
207D71—81no—
208D71—81no—
209#3062—711nomaximal, Sylow
210C71#306211yescharacteristic, complemented, Sylow
211C2×AGL(1,ℤ/8)—711no—
212C2×#131—711no—
213#537—711no—
214#535—711no—
215#469—711no—
216#469—711no—
217#468—711no—
218#468D7111yescharacteristic, complemented
219#466—711no—
220#466—711no—
221#465—711no—
222#465—711no—
223#428—711no—
224#416—711no—
225C2×#102—711no—
226C2×#41—711no—
227C2×#41—711no—
228C22×D4—711no—
229AGL(1,ℤ/8)—1424no—
230C2×SD16—711no—
231C2×SD16C2×D7111yescharacteristic
232C2×D8—711no—
233C2×D8—711no—
234#131—1424no—
235C2×M16—711no—
236C2×M16—711no—
237C2×M16C2×D7111yescharacteristic, complemented
238C22×C8—711no—
239C22×C8—711no—
240C22×C8—711no—
241#127—1421no—
242#124—1421no—
243C22⋊Q8—711no—
244C22⋊Q8—1421no—
245C22⋊Q8C2×D7111yescharacteristic, complemented
246#121—711no—
247#121—711no—
248#121—711no—
249#121—711no—
250#121—711no—
251#121—1421no—
252#121—1421no—
253#121—1421no—
254#121C2×D7111yescharacteristic
255#120—1421no—
256#120—1421no—
257#117—711no—
258C2×C4⋊C4—711no—
259#107—711no—
260#107C2×D7111yescharacteristic, complemented
261#106—711no—
262#106—711no—
263#103—711no—
264#103C2×D7111yescharacteristic, complemented
265#102—711no—
266#102—711no—
267#102—711no—
268#102—711no—
269#102—711no—
270#102C2×D7111yescharacteristic
271C24—1421no—
272#41—1422no—
273#41—1422no—
274#41—1422no—
275#41—1422no—
276C2×Q8—711no—
277C2×Q8C22×D7111yescharacteristic
278C2×D4—21no—
279C2×D4—711no—
280C2×D4—711no—
281C2×D4—711no—
282C2×D4—711no—
283C2×D4—711no—
284C2×D4—1421no—
285C2×D4—1421no—
286C2×D4—1421no—
287C2×D4—1422no—
288C2×D4—1422no—
289C2×D4—1421no—
290C2×D4—1421no—
291C2×D4—1421no—
292C2×D4—1421no—
293C2×D4C22×D7111yescharacteristic
294C22×C4—711no—
295C22×C4—711no—
296C22×C4—1421no—
297C22×C4—1421no—
298C22×C4C22×D7111yescharacteristic
299SD16—22no—
300SD16—1422no—
301D8—1422no—
302D8—1422no—
303M16—22no—
304M16—714no—
305M16—1422no—
306C2×C8—711no—
307C2×C8—711no—
308C2×C8—1422no—
309C2×C8—1422no—
310C2×C8—1422no—
311C2×C8C22×D7111yescharacteristic
312C2×C8C22×D7111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—711no—
315C4⋊C4—711no—
316C4⋊C4—1421no—
317C4⋊C4—1421no—
318C4⋊C4C22×D7111yescharacteristic
319C4⋊C4C22×D7111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—1421no—
323C22⋊C4—1421no—
324C22⋊C4—1421no—
325C22⋊C4—1421no—
326C22⋊C4—1421no—
327C22⋊C4—1421no—
328C22⋊C4—1421no—
329C42—1421no—
330C23—21no—
331C23—2842no—
332C23—5681no—
333C23—5681no—
334C23—711no—
335C23—711no—
336C23—1421no—
337C23—1421no—
338C23D4×D7111yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—1422no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—2842no—
346D4—2842no—
347D4—5681no—
348D4—5681no—
349D4—5681no—
350D4—5681no—
351D4—5681no—
352D4—1422no—
353D4—1422no—
354D4—1422no—
355D4—1421no—
356D4—1421no—
357D4—1421no—
358D4—1421no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—2842no—
365C2×C4—5681no—
366C2×C4—711no—
367C2×C4—712no—
368C2×C4—712no—
369C2×C4—711no—
370C2×C4—1421no—
371C2×C4—1421no—
372C2×C4—1421no—
373C2×C4—1421no—
374C2×C4D4×D7111yescharacteristic
375C2×C4C23×D7111yescharacteristic
376C8—21no—
377C8—712no—
378C8—1421no—
379C8D4×D7112yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—2841no—
387C22—2841no—
388C22—2841no—
389C22—2841no—
390C22—2841no—
391C22—2841no—
392C22—5684no—
393C22—712no—
394C22—1421no—
395C22—1421no—
396C22—1421no—
397C22—1421no—
398C22—1421no—
399C22C2×D4×D7111yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—2841no—
405C4—2841no—
406C4—712no—
407C4—1421no—
408C4C2×D4×D7111yescharacteristic
409C4#41×D7111yescharacteristic
410C2—41no—
411C2—81no—
412C2—2841no—
413C2—5681no—
414C2—5681no—
415C2—1422no—
416C2#121×D7111yescharacteristic
417C2#1798680311yescharacteristic
418C2#1798680811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#468×C71C2
socleC22×C71C22×D4
hypercenter#468D71
hypocenterC71#3062
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×#14556155C2semidirect
2. C2×#14556136C2semidirect
3. #17986579C2semidirect
4. #17986565C2semidirect
5. #17986252C2semidirect
6. #17986251C2semidirect
7. #17986248C2semidirect
8. #17986236C2semidirect
9. #17986233C2semidirect
10. #17986230C2semidirect
11. #17986225C2semidirect
12. #17986040C2semidirect
13. #17985994C2semidirect
14. #102×D71C2semidirect
15. #468×C71C2semidirect
25. C2×#11765848C22semidirect
26. C22×C71⋊C8C22semidirect
28. C2×C8×D71C22semidirect
32. #14556104C22semidirect
35. #14556103C22semidirect
37. #14556102C22semidirect
45. #14556061C22semidirect
46. #14556060C22semidirect
47. #14556059C22semidirect
48. #14556052C22semidirect
50. #14556050C22semidirect
53. #14556048C22semidirect
55. C2×M16×C71C22semidirect
56. C22⋊Q8×C71C22semidirect
58. #107×C71C22semidirect
59. #103×C71C22semidirect
89. #11765848D4semidirect
210. C71#3062semidirect
218. #468D71semidirect
237. C2×M16C2×D71semidirect
245. C22⋊Q8C2×D71semidirect
260. #107C2×D71semidirect
264. #103C2×D71semidirect