Finite Group Databasebeta

← #22599865

Subgroups — #22599865

Order 9088.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556152C211yesmaximal, characteristic, complemented
2#17986648C211yesmaximal, characteristic, complemented
3#17986637C211yesmaximal, characteristic, complemented
4#17986392C211yesmaximal, characteristic, complemented
5#17986254C211yesmaximal, characteristic, complemented
6#17986228C211yesmaximal, characteristic, complemented
7#17986216C211yesmaximal, characteristic, complemented
8#17986202C211yesmaximal, characteristic, complemented
9#17986161C211yesmaximal, characteristic, complemented
10#17986158C211yesmaximal, characteristic, complemented
11#17986148C211yesmaximal, characteristic, complemented
12#17986144C211yesmaximal, characteristic, complemented
13#17986111C211yesmaximal, characteristic, complemented
14#17986070C211yesmaximal, characteristic, complemented
15C2×#135×C71C211yesmaximal, characteristic, complemented
16C2×#11765864C2211yescharacteristic
17C2×D4×D71C2211yescharacteristic
18#14556152—24no—
19C2×#11765855C2211yescharacteristic, complemented
20C2×#11765854C2211yescharacteristic
21C2×#11765853C2211yescharacteristic
22C2×#11765852C2211yescharacteristic
23C2×#11765850C2211yescharacteristic
24C2×#11765848C2211yescharacteristic
25#14556126C2211yescharacteristic, complemented
26#14556117C2211yescharacteristic
27#14556111—21no—
28#14556111—21no—
29#14556109—21no—
30#14556107C2211yescharacteristic, complemented
31#14556104C2211yescharacteristic
32#14556104C2211yescharacteristic
33#14556102C2211yescharacteristic
34#14556100—21no—
35#14556100—21no—
36#14556096—21no—
37D4×Dic71C2211yescharacteristic
38D4×Dic71C2211yescharacteristic, complemented
39C4×C22⋊D71C2211yescharacteristic
40C2×Dic71⋊C4—21no—
41C2×(C2×D71)⋊C4—21no—
42#14556060C2211yescharacteristic, complemented
43Dic71⋊C8C2211yescharacteristic, complemented
44#14556053C2211yescharacteristic, complemented
45#14556052C2211yescharacteristic, complemented
46#14556052C2211yescharacteristic, complemented
47#14556050C2211yescharacteristic
48#14556049C2211yescharacteristic
49#14556049C2211yescharacteristic, complemented
50C22⋊(C71⋊C8)C2211yescharacteristic, complemented
51(C2×D71)⋊C8C2211yescharacteristic
52#14556038C2211yescharacteristic, complemented
53C2×#41×C71C2211yescharacteristic
54C2×#41×C71C2211yescharacteristic, complemented
55#135×C71—24no—
56C2×Q16×C71C2211yescharacteristic, complemented
57C2×SD16×C71C2211yescharacteristic
58C2×SD16×C71C2211yescharacteristic, complemented
59C2×D8×C71C2211yescharacteristic
60C22×C8×C71C2211yescharacteristic, complemented
61C23×D71—21no—
62#11765864—22no—
63#11765864—41no—
64C2×Dic142C2311yescharacteristic
65C2×D284C2311yescharacteristic
66C2×C22⋊D71—21no—
67C2×C22⋊D71—21no—
68C2×C22⋊D71—21no—
69C2×C22⋊D71D411yescharacteristic
70D4×D71—22no—
71D4×D71—41no—
72C22×Dic71—21no—
73C22×Dic71—21no—
74C22×Dic71—21no—
75C2×C4×D71C2311yescharacteristic
76#11765855—41no—
77#11765854—22no—
78#11765853—22no—
79#11765852—41no—
80#11765850—22no—
81#11765848—22no—
82C2×C71⋊C8C2311yescharacteristic
83C2×C71⋊C8C2311yescharacteristic
84C4⋊Dic71C2311yescharacteristic
85C4⋊Dic71C2311yescharacteristic
86Dic71⋊C4—21no—
87Dic71⋊C4—21no—
88Dic71⋊C4—21no—
89Dic71⋊C4—21no—
90Dic71⋊C4—21no—
91Dic71⋊C4D411yescharacteristic
92C22⋊Dic71—21no—
93C22⋊Dic71—21no—
94C22⋊Dic71—21no—
95C22⋊Dic71D411yescharacteristic
96(C2×D71)⋊C4—21no—
97(C2×D71)⋊C4—21no—
98(C2×D71)⋊C4—21no—
99(C2×D71)⋊C4—21no—
100(C2×D71)⋊C4—21no—
101(C2×D71)⋊C4D411yescharacteristic
102C4×Dic71C2311yescharacteristic
103#41×C71—22no—
104#41×C71—22no—
105#41×C71—41no—
106#41×C71—41no—
107C2×Q8×C71C2311yescharacteristic
108C2×Q8×C71C2311yescharacteristic
109C2×D4×C71—21no—
110C2×D4×C71—21no—
111C2×D4×C71C2311yescharacteristic
112C2×D4×C71C2311yescharacteristic
113C22×C4×C71—21no—
114C22×C4×C71—21no—
115C22×C4×C71C2311yescharacteristic
116Q16×C71—22no—
117SD16×C71—22no—
118SD16×C71—22no—
119D8×C71D414yescomplemented
120C2×C8×C71—22no—
121C2×C8×C71C2311yescharacteristic
122C2×C8×C71C2311yescharacteristic
123C23×C71—21no—
124C23×C71—21no—
125C23×C71C2×D411yescharacteristic
126C22×D71—21no—
127C22×D71—42no—
128C22×D71—42no—
129C22×D71—81no—
130C22×D71C2×D411yescharacteristic
131Q8×C71—21no—
132Q8×C71—21no—
133Q8×C71—21no—
134Q8×C71—21no—
135D4×C71—21no—
136D4×C71—21no—
137D4×C71—42no—
138D4×C71—42no—
139D4×C71C2×D412yes—
140D4×C71C2×D412yes—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—42no—
146C2×C4×C71—42no—
147C2×C4×C71C2×D411yescharacteristic
148C2×C4×C71C2411yescharacteristic
149C22⋊D71—41no—
150C22⋊D71—42no—
151C22⋊D71—42no—
152C22⋊D71—81no—
153C2×Dic71—21no—
154C2×Dic71—21no—
155C2×Dic71—21no—
156C2×Dic71—42no—
157C2×Dic71—42no—
158C2×Dic71—81no—
159C2×Dic71C2×D411yescharacteristic
160C2×Dic71C2×D411yescharacteristic
161C2×Dic71C2×D411yescharacteristic
162D284—21no—
163D284—21no—
164C4×D71—21no—
165C4×D71—21no—
166Dic142—21no—
167Dic142—21no—
168C8×C71—21no—
169C8×C71C2×D412yes—
170C71⋊C8—21no—
171C71⋊C8—21no—
172C22×C71—22no—
173C22×C71—22no—
174C22×C71—21no—
175C22×C71—21no—
176C22×C71—21no—
177C22×C71—41no—
178C22×C71—41no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71C22×D411yescharacteristic
182C2×D71—21no—
183C2×D71—21no—
184C2×D71—21no—
185C2×D71—41no—
186C2×D71—41no—
187C2×D71—41no—
188C2×D71—84no—
189C4×C71—21no—
190C4×C71—41no—
191C4×C71—41no—
192C4×C71C22×D411yescharacteristic
193C4×C71H(5,𝔽2)11yescharacteristic
194Dic71—21no—
195Dic71—21no—
196Dic71—21no—
197Dic71—41no—
198Dic71—41no—
199Dic71—41no—
200C2×C71—42no—
201C2×C71—41no—
202C2×C71—42no—
203C2×C71D4211yescharacteristic
204C2×C71#57511yescharacteristic
205C2×C71#57711yescharacteristic
206D71—41no—
207D71—81no—
208#3267—711nomaximal, Sylow
209C71#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—711no—
211C2×#135D7111yescharacteristic, complemented
212#547—711no—
213#546—711no—
214#497—711no—
215#470—711no—
216#465—711no—
217#463—711no—
218#461—711no—
219#452—711no—
220#452—711no—
221#450—711no—
222#449—711no—
223#442—711no—
224#435—711no—
225C2×#41—711no—
226C2×#41C2×D7111yescharacteristic
227C2×#41C2×D7111yescharacteristic, complemented
228C22×D4—711no—
229AGL(1,ℤ/8)—1424no—
230#135—24no—
231C2×Q16—711no—
232C2×Q16C2×D7111yescharacteristic, complemented
233C2×SD16—711no—
234C2×SD16—711no—
235C2×SD16—711no—
236C2×SD16C2×D7111yescharacteristic
237C2×SD16C2×D7111yescharacteristic, complemented
238C2×D8—711no—
239C2×D8C2×D7111yescharacteristic
240C2×M16—711no—
241C22×C8C2×D7111yescharacteristic, complemented
242C4⋊Q8—711no—
243#124—711no—
244#123—1421no—
245#123—1421no—
246#123—1421no—
247C22⋊Q8—711no—
248C22⋊Q8—711no—
249C22⋊Q8—711no—
250#121—711no—
251#121—1421no—
252#121—1421no—
253#120—1421no—
254C4×D4—711no—
255C4×D4—711no—
256C4×D4—711no—
257C2×C4⋊C4—1421no—
258C2×C22⋊C4—1421no—
259#106—711no—
260C4⋊C8—711no—
261#103—711no—
262#103—711no—
263#103—711no—
264#102—711no—
265#102—711no—
266#102—711no—
267C22⋊C8—711no—
268C22⋊C8—711no—
269#97—711no—
270C24—1421no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—2841no—
276#41—1422no—
277C2×Q8—711no—
278C2×Q8C22×D7111yescharacteristic
279C2×Q8C22×D7111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2841no—
283C2×D4—711no—
284C2×D4—711no—
285C2×D4—1421no—
286C2×D4—1421no—
287C2×D4—1422no—
288C2×D4—1421no—
289C2×D4C22×D7111yescharacteristic
290C2×D4C22×D7111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—711no—
294C22×C4—1421no—
295C22×C4—1421no—
296C22×C4—1421no—
297C22×C4C22×D7111yescharacteristic
298Q16—22no—
299Q16—2841no—
300SD16—22no—
301SD16—22no—
302SD16—2841no—
303SD16—1422no—
304SD16—1422no—
305D8—1422no—
306D8C22⋊D7114yescomplemented
307M16—1422no—
308C2×C8—22no—
309C2×C8—711no—
310C2×C8—711no—
311C2×C8C22×D7111yescharacteristic
312C2×C8C22×D7111yescharacteristic
313C4⋊C4—711no—
314C4⋊C4—711no—
315C4⋊C4—711no—
316C4⋊C4—1421no—
317C4⋊C4—1421no—
318C4⋊C4—1421no—
319C4⋊C4—1421no—
320C4⋊C4—1421no—
321C22⋊C4—711no—
322C22⋊C4—711no—
323C22⋊C4—1421no—
324C22⋊C4—1421no—
325C22⋊C4—1421no—
326C22⋊C4—1421no—
327C22⋊C4—1421no—
328C22⋊C4—1421no—
329C22⋊C4—1421no—
330C22⋊C4—1421no—
331C42—711no—
332C23—21no—
333C23—21no—
334C23—2842no—
335C23—2842no—
336C23—5681no—
337C23—711no—
338C23—1421no—
339C23D4×D7111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—1421no—
345Q8—1421no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—2842no—
351D4—2842no—
352D4—2841no—
353D4—5681no—
354D4—1421no—
355D4—1421no—
356D4C2×C22⋊D7112yes—
357D4C2×C22⋊D7112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—2842no—
365C2×C4—2842no—
366C2×C4—5681no—
367C2×C4—711no—
368C2×C4—711no—
369C2×C4—711no—
370C2×C4—1421no—
371C2×C4—1421no—
372C2×C4—1421no—
373C2×C4—1421no—
374C2×C4—1421no—
375C2×C4D4×D7111yescharacteristic
376C2×C4C23×D7111yescharacteristic
377C8—21no—
378C8—1421no—
379C8—1421no—
380C8C2×C22⋊D7112yes—
381C22—21no—
382C22—22no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—2841no—
391C22—2841no—
392C22—2841no—
393C22—5684no—
394C22—1421no—
395C22—1421no—
396C22—1421no—
397C22C2×D4×D7111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—2841no—
402C4—2841no—
403C4—2841no—
404C4—1421no—
405C4—1421no—
406C4—1421no—
407C4C22×C22⋊D7111yescharacteristic
408C4#1455617311yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—2841no—
413C2—5681no—
414C2D4×C22⋊D7111yescharacteristic
415C2#1798679611yescharacteristic
416C2#1798680611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC2×#135×C71C2
socleC22×C71C22×D4
hypercenterC2×#135D71
hypocenterC71#3267
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#14556152C2semidirect
2. #17986648C2semidirect
3. #17986637C2semidirect
4. #17986392C2semidirect
5. #17986254C2semidirect
6. #17986228C2semidirect
7. #17986216C2semidirect
8. #17986202C2semidirect
9. #17986161C2semidirect
10. #17986158C2semidirect
11. #17986148C2semidirect
12. #17986144C2semidirect
13. #17986111C2semidirect
14. #17986070C2semidirect
15. C2×#135×C71C2semidirect
19. C2×#11765855C22semidirect
25. #14556126C22semidirect
30. #14556107C22semidirect
38. D4×Dic71C22semidirect
42. #14556060C22semidirect
43. Dic71⋊C8C22semidirect
44. #14556053C22semidirect
45. #14556052C22semidirect
46. #14556052C22semidirect
49. #14556049C22semidirect
50. C22⋊(C71⋊C8)C22semidirect
52. #14556038C22semidirect
54. C2×#41×C71C22semidirect
56. C2×Q16×C71C22semidirect
58. C2×SD16×C71C22semidirect
60. C22×C8×C71C22semidirect
119. D8×C71D4semidirect
209. C71#3267semidirect
211. C2×#135D71semidirect
227. C2×#41C2×D71semidirect
232. C2×Q16C2×D71semidirect
237. C2×SD16C2×D71semidirect
241. C22×C8C2×D71semidirect
306. D8C22⋊D71semidirect