Finite Group Databasebeta

← #11737569

Subgroups — #11737569

Order 896.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90515C211yesmaximal, characteristic, complemented
2C2×#6512C211yesmaximal, characteristic, complemented
3C2×#6474C211yesmaximal, characteristic, complemented
4#90300C211yesmaximal, characteristic, complemented
5C7×#449C211yesmaximal, characteristic, complemented
6#89947C211yesmaximal, characteristic, complemented
7#89941C211yesmaximal, characteristic, complemented
8#89922C211yesmaximal, characteristic, complemented
9#89814C211yesmaximal, characteristic, complemented
10#89590C211yesmaximal, characteristic, complemented
11#89584C211yesmaximal, characteristic, complemented
12#89553C211yesmaximal, characteristic, complemented
13#89528C211yesmaximal, characteristic, complemented
14#89503C211yesmaximal, characteristic, complemented
15#89501C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic, complemented
17C2×#1159C2211yescharacteristic, complemented
18C2×D4×D7—21no—
19C2×D4×D7C2211yescharacteristic
20C22×D28C2211yescharacteristic
21C2×C7×SD16C2211yescharacteristic, complemented
22C2×C7×D8C2211yescharacteristic, complemented
23C7×#121C2211yescharacteristic
24#6512—24no—
25#6509C2211yescharacteristic, complemented
26#6507C2211yescharacteristic, complemented
27C2×#1143C2211yescharacteristic
28C2×#1143C2211yescharacteristic, complemented
29#6503C2211yescharacteristic
30#6502—21no—
31#6500—21no—
32#6498—21no—
33D4×Dic7C2211yescharacteristic, complemented
34C2×#1142C2211yescharacteristic, complemented
35C2×#1141C2211yescharacteristic
36C2×#1141C2211yescharacteristic, complemented
37#6492—21no—
38C4×C22⋊D7C2211yescharacteristic
39#6487C2211yescharacteristic, complemented
40C2×Dic7⋊C4—21no—
41C2×#1136C2211yescharacteristic, complemented
42#6474—24no—
43C2×#1132C2211yescharacteristic
44C2×#1132C2211yescharacteristic, complemented
45C2×#1131C2211yescharacteristic, complemented
46#6458C2211yescharacteristic
47#6456C2211yescharacteristic
48#6447—21no—
49#6445—21no—
50C7×#103C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic, complemented
52C7×C22⋊C8C2211yescharacteristic, complemented
53#6406C2211yescharacteristic, complemented
54#6395C2211yescharacteristic
55#6395C2211yescharacteristic, complemented
56(C2×D7)⋊C8C2211yescharacteristic, complemented
57Dic7⋊C8C2211yescharacteristic, complemented
58C8×Dic7C2211yescharacteristic, complemented
59#6384C2211yescharacteristic, complemented
60#6381C2211yescharacteristic, complemented
61#3272—71nomaximal, Sylow
62C23×D7—21no—
63C23×D7—21no—
64C7×#41—22no—
65C7×#41—41no—
66C2×C7×Q8C2311yescharacteristic
67C2×C7×D4—21no—
68C2×C7×D4—21no—
69C2×C7×D4C2311yescharacteristic
70C2×C7×D4C2311yescharacteristic
71C22×C4×C7—21no—
72C22×C4×C7C2311yescharacteristic
73C2×C22⋊D7—21no—
74C2×C22⋊D7—21no—
75C2×C22⋊D7—21no—
76C2×C22⋊D7—21no—
77C2×C22⋊D7D411yescharacteristic
78C22×Dic7—21no—
79C22×Dic7—21no—
80#1159—22no—
81#1159—41no—
82D4×D7—22no—
83D4×D7—41no—
84D4×D7—44no—
85C2×D28—22no—
86C2×D28—21no—
87C2×D28—21no—
88C2×D28—41no—
89C2×D28C2311yescharacteristic
90C2×D28C2311yescharacteristic
91C2×C4×D7—21no—
92C2×C4×D7C2311yescharacteristic
93C2×Dic14C2311yescharacteristic
94C7×SD16—41no—
95C7×D8—22no—
96C2×C7×C8C2311yescharacteristic
97C2×C7×C8C2311yescharacteristic
98C7×C4⋊C4C2311yescharacteristic
99C7×C22⋊C4—21no—
100C22⋊Dic7—21no—
101C22⋊Dic7—21no—
102C22⋊Dic7D411yescharacteristic
103#1143—22no—
104#1143—22no—
105#1142—22no—
106#1141—22no—
107#1141D414yescomplemented
108(C2×D7)⋊C4—21no—
109(C2×D7)⋊C4—21no—
110(C2×D7)⋊C4—21no—
111(C2×D7)⋊C4D411yescharacteristic
112C4⋊Dic7C2311yescharacteristic
113Dic7⋊C4—21no—
114Dic7⋊C4—21no—
115Dic7⋊C4—21no—
116Dic7⋊C4—21no—
117Dic7⋊C4D411yescharacteristic
118C4×Dic7C2311yescharacteristic
119#1136—22no—
120C2×C7⋊C8C2311yescharacteristic
121C2×C7⋊C8C2311yescharacteristic
122#1132—22no—
123#1132—41no—
124#1131—22no—
125C2×AGL(1,ℤ/8)—71no—
126C2×AGL(1,ℤ/8)—71no—
127#547—71no—
128D42—71no—
129#492—71no—
130#468—71no—
131#468—71no—
132#461—71no—
133#459—71no—
134#450—71no—
135#450—71no—
136#449—71no—
137#449D711yescharacteristic, complemented
138C4×D8—71no—
139#436—71no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7C2×D411yescharacteristic
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—42no—
146C22×D7—42no—
147C22×D7—42no—
148C22×D7—42no—
149C22×D7—81no—
150C22×D7—81no—
151C22×D7C2×D411yescharacteristic
152C7×Q8—21no—
153C7×Q8—21no—
154C7×D4—21no—
155C7×D4—21no—
156C7×D4—21no—
157C7×D4—42no—
158C7×D4—42no—
159C7×D4C2×D412yescomplemented
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—42no—
165C2×C4×C7C2×D411yescharacteristic
166C2×C4×C7C2411yescharacteristic
167C22⋊D7—22no—
168C22⋊D7—42no—
169C22⋊D7—42no—
170C22⋊D7—42no—
171C22⋊D7—81no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—42no—
175C2×Dic7—42no—
176C2×Dic7C2×D411yescharacteristic
177C2×Dic7C2×D411yescharacteristic
178C2×Dic7C2×D411yescharacteristic
179D28—21no—
180D28—21no—
181D28—21no—
182D28—42no—
183D28—42no—
184D28C2×D412yes—
185C4×D7—21no—
186C4×D7—21no—
187C4×D7—42no—
188Dic14—21no—
189Dic14—21no—
190C7×C8—21no—
191C7×C8—21no—
192C7⋊C8—21no—
193C7⋊C8C2×D412yescomplemented
194C2×#41—71no—
195C2×#41C2×D711yescharacteristic, complemented
196C22×D4—71no—
197C22×D4—71no—
198C22×D4—141no—
199AGL(1,ℤ/8)—144no—
200AGL(1,ℤ/8)—144no—
201C2×SD16—71no—
202C2×SD16—71no—
203C2×SD16—71no—
204C2×SD16—71no—
205C2×SD16—71no—
206C2×SD16C2×D711yescharacteristic, complemented
207C2×D8—71no—
208C2×D8—71no—
209C2×D8C2×D711yescharacteristic, complemented
210C2×M16—71no—
211C2×M16—71no—
212C4⋊Q8—71no—
213#127—71no—
214#123—141no—
215#123—141no—
216C22⋊Q8—71no—
217C22⋊Q8—71no—
218#121—71no—
219#121—141no—
220#121—141no—
221#121C2×D711yescharacteristic
222#120—141no—
223#120—141no—
224C4×D4—71no—
225C4×D4—71no—
226C4×D4—71no—
227C2×C4⋊C4—141no—
228#107—71no—
229C4⋊C8—71no—
230#103—71no—
231#103C2×D711yescharacteristic, complemented
232#102—71no—
233#102—71no—
234#102—71no—
235#102C2×D711yescharacteristic, complemented
236C22⋊C8—71no—
237C22⋊C8C2×D711yescharacteristic, complemented
238C4×C8—71no—
239C22×C7—22no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7C22×D411yescharacteristic
249C2×D7—21no—
250C2×D7—22no—
251C2×D7—21no—
252C2×D7—21no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—41no—
256C2×D7—41no—
257C2×D7—41no—
258C2×D7—84no—
259C2×D7—84no—
260C4×C7—21no—
261C4×C7—41no—
262C4×C7—41no—
263C4×C7C22×D411yescharacteristic
264C4×C7H(5,𝔽2)11yescharacteristic
265Dic7—21no—
266Dic7—21no—
267Dic7—41no—
268Dic7—41no—
269Dic7AGL(1,ℤ/8)12yescomplemented
270C24—141no—
271C24—141no—
272#41—22no—
273#41—41no—
274#41—142no—
275#41—281no—
276C2×Q8—71no—
277C2×Q8C22×D711yescharacteristic
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—71no—
281C2×D4—71no—
282C2×D4—71no—
283C2×D4—142no—
284C2×D4—141no—
285C2×D4—141no—
286C2×D4—141no—
287C2×D4—141no—
288C2×D4—142no—
289C2×D4—141no—
290C2×D4—141no—
291C2×D4—281no—
292C2×D4—281no—
293C2×D4—284no—
294C2×D4C22×D711yescharacteristic
295C2×D4C22×D711yescharacteristic
296C22×C4—21no—
297C22×C4—71no—
298C22×C4—141no—
299C22×C4—141no—
300C22×C4—141no—
301C22×C4C22×D711yescharacteristic
302SD16—41no—
303SD16—142no—
304SD16—142no—
305SD16—142no—
306SD16—142no—
307SD16—281no—
308D8—22no—
309D8—74no—
310D8—142no—
311M16—142no—
312M16—142no—
313C2×C8—71no—
314C2×C8—71no—
315C2×C8C22×D711yescharacteristic
316C2×C8C22×D711yescharacteristic
317C4⋊C4—71no—
318C4⋊C4—71no—
319C4⋊C4—141no—
320C4⋊C4—141no—
321C4⋊C4—141no—
322C4⋊C4—141no—
323C4⋊C4C22×D711yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—71no—
326C22⋊C4—71no—
327C22⋊C4—141no—
328C22⋊C4—141no—
329C22⋊C4—141no—
330C22⋊C4—141no—
331C22⋊C4—141no—
332C42—71no—
333C2×C7—41no—
334C2×C7—42no—
335C2×C7—81no—
336C2×C7D4211yescharacteristic
337C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
338C2×C7#57711yescharacteristic
339D7—41no—
340D7—42no—
341D7—81no—
342C23—21no—
343C23—21no—
344C23—71no—
345C23—141no—
346C23—141no—
347C23—282no—
348C23—282no—
349C23—282no—
350C23—282no—
351C23—561no—
352C23—561no—
353C23D4×D711yescharacteristic
354Q8—21no—
355Q8—21no—
356Q8—141no—
357Q8—141no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—42no—
363D4—72no—
364D4—142no—
365D4—141no—
366D4—141no—
367D4—141no—
368D4—282no—
369D4—282no—
370D4—282no—
371D4—282no—
372D4—282no—
373D4—561no—
374D4D4×D712yescomplemented
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—71no—
381C2×C4—71no—
382C2×C4—71no—
383C2×C4—141no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4—282no—
388C2×C4—282no—
389C2×C4—282no—
390C2×C4D4×D711yescharacteristic
391C2×C4C23×D711yescharacteristic
392C8—21no—
393C8—21no—
394C8—72no—
395C8—141no—
396C7#327211yescharacteristic, complemented, Sylow
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—141no—
407C22—141no—
408C22—141no—
409C22—142no—
410C22—281no—
411C22—281no—
412C22—281no—
413C22—281no—
414C22—281no—
415C22—564no—
416C22—564no—
417C22C2×D4×D711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—72no—
422C4—141no—
423C4—141no—
424C4—281no—
425C4—281no—
426C4C2×D4×D711yescharacteristic
427C4#654811yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—281no—
432C2—282no—
433C2—561no—
434C2#9028111yescharacteristic
435C2#9046111yescharacteristic
436C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#449C2
socleC22×C7C22×D4
hypercenter#449D7
hypocenterC7#3272
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #90515C2semidirect
2. C2×#6512C2semidirect
3. C2×#6474C2semidirect
4. #90300C2semidirect
5. C7×#449C2semidirect
6. #89947C2semidirect
7. #89941C2semidirect
8. #89922C2semidirect
9. #89814C2semidirect
10. #89590C2semidirect
11. #89584C2semidirect
12. #89553C2semidirect
13. #89528C2semidirect
14. #89503C2semidirect
15. #89501C2semidirect
16. C2×C7×#41C22semidirect
17. C2×#1159C22semidirect
21. C2×C7×SD16C22semidirect
22. C2×C7×D8C22semidirect
25. #6509C22semidirect
26. #6507C22semidirect
28. C2×#1143C22semidirect
33. D4×Dic7C22semidirect
34. C2×#1142C22semidirect
36. C2×#1141C22semidirect
39. #6487C22semidirect
41. C2×#1136C22semidirect
44. C2×#1132C22semidirect
45. C2×#1131C22semidirect
50. C7×#103C22semidirect
51. C7×#102C22semidirect
52. C7×C22⋊C8C22semidirect
53. #6406C22semidirect
55. #6395C22semidirect
56. (C2×D7)⋊C8C22semidirect
57. Dic7⋊C8C22semidirect
58. C8×Dic7C22semidirect
59. #6384C22semidirect
60. #6381C22semidirect
107. #1141D4semidirect
137. #449D7semidirect
159. C7×D4C2×D4semidirect
193. C7⋊C8C2×D4semidirect
195. C2×#41C2×D7semidirect
206. C2×SD16C2×D7semidirect
209. C2×D8C2×D7semidirect
231. #103C2×D7semidirect
235. #102C2×D7semidirect
237. C22⋊C8C2×D7semidirect
269. Dic7AGL(1,ℤ/8)semidirect
374. D4D4×D7semidirect
396. C7#3272semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#905152
C2×#65122
C2×#64742
#903002
C7×#4492
#899472
#899412
#899222
#898142
#895902
#895842
#895532
#895282
#895032
#895012
#32727