Finite Group Databasebeta

← #11743783

Subgroups — #11743783

Order 896.

1540 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6553C211yesmaximal, characteristic, complemented
2C2×#6551C211yesmaximal, characteristic, complemented
3C7×#548C211yesmaximal, characteristic, complemented
4#90522C211yesmaximal, characteristic, complemented
5#90508C211yesmaximal, characteristic, complemented
6#90367C213yesmaximal, complemented
7D7×#124C213yesmaximal, complemented
8#90321C213yesmaximal, complemented
9#90319C213yesmaximal, complemented
10#90303C213yesmaximal, complemented
11#90298C213yesmaximal, complemented
12#90267C211yesmaximal, characteristic, complemented
13C4×#1161C211yesmaximal, characteristic, complemented
14#90241C213yesmaximal, complemented
15#90239C213yesmaximal, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic, complemented
18#6553—28no—
19#6551—28no—
20C2×#1161C2211yescharacteristic
21C2×#1161C2211yescharacteristic
22C2×#1161C2213yescomplemented
23C2×Q8×D7C2211yescharacteristic, complemented
24C2×Q8×D7C2213yescomplemented
25C2×D4×D7C2213yes—
26C2×D4×D7C2213yes—
27C2×#1157C2213yes—
28C2×#1157C2213yescomplemented
29C2×#1157C2213yescomplemented
30C22×D28C2213yes—
31C7×#124C2213yescomplemented
32C7×C22⋊Q8C2213yescomplemented
33C7×#121C2213yescomplemented
34C4×C7×Q8C2211yescharacteristic, complemented
35C4×C7×D4C2213yescomplemented
36#6510C2213yescomplemented
37#6509C2213yescomplemented
38Q8×Dic7C2211yescharacteristic, complemented
39#6503C2213yescomplemented
40#6501C2213yescomplemented
41#6501C2213yescomplemented
42#6500C2216yescomplemented
43#6499C2213yescomplemented
44#6493C2213yescomplemented
45C4×C22⋊D7C2213yescomplemented
46C4×C22⋊D7C2213yescomplemented
47#6487C2213yescomplemented
48#6460C2213yescomplemented
49#6458C2213yescomplemented
50#6458C2213yescomplemented
51#6457C2216yescomplemented
52#6456C2213yescomplemented
53#6455C2213yescomplemented
54#6451C2213yescomplemented
55#6448C2211yescharacteristic, complemented
56#6448C2213yescomplemented
57#6448C2216yescomplemented
58#6447C2213yescomplemented
59#6447C2213yescomplemented
60#6446C2216yescomplemented
61#6445C2216yescomplemented
62D7×C22⋊C4C2216yescomplemented
63D7×C22⋊C4C2216yescomplemented
64#6438C2213yescomplemented
65C4×D28C2213yescomplemented
66C4×D28C2213yescomplemented
67C42×D7C2213yescomplemented
68#3500—71nomaximal, Sylow
69C23×D7C2316yes—
70C7×#41—24no—
71C2×C7×Q8—24no—
72C2×C7×Q8C2313yes—
73C2×C7×Q8C2311yescharacteristic
74C2×C7×D4C2313yes—
75C2×C7×D4C2313yes—
76C22×C4×C7C2313yes—
77C22×C4×C7C2313yes—
78C2×C22⋊D7C2313yes—
79C2×C22⋊D7C2313yes—
80C2×C22⋊D7C2313yes—
81C2×C22⋊D7C2311yescharacteristic
82C2×C22⋊D7C2313yes—
83C2×C22⋊D7C2313yes—
84#1161—24no—
85#1161—212no—
86#1161D418yescomplemented
87Q8×D7—24no—
88Q8×D7—212no—
89D4×D7—212no—
90D4×D7—212no—
91#1157—212no—
92#1157—212no—
93#1157—212no—
94C2×D28—212no—
95C2×D28C2313yes—
96C2×D28C2313yes—
97C2×D28C2313yes—
98C2×D28C2313yes—
99C2×D28C2313yes—
100C2×C4×D7C2313yes—
101C2×C4×D7C2313yes—
102C2×C4×D7C2316yes—
103C2×C4×D7C2313yes—
104C2×C4×D7C2313yes—
105C2×C4×D7C2313yes—
106C2×Dic14C2313yes—
107C2×Dic14C2313yes—
108C2×Dic14C2311yescharacteristic
109C7×C4⋊C4C2313yes—
110C7×C4⋊C4C2313yes—
111C7×C22⋊C4C2313yes—
112C7×C22⋊C4C2316yes—
113C7×C22⋊C4C2311yescharacteristic
114C42×C7C2313yes—
115C22⋊Dic7C2313yes—
116C22⋊Dic7C2316yes—
117C22⋊Dic7C2311yescharacteristic
118(C2×D7)⋊C4C2311yescharacteristic
119(C2×D7)⋊C4C2313yes—
120(C2×D7)⋊C4C2316yes—
121(C2×D7)⋊C4C2316yes—
122(C2×D7)⋊C4C2313yes—
123(C2×D7)⋊C4C2316yes—
124(C2×D7)⋊C4C2311yescharacteristic
125C4⋊Dic7C2313yes—
126C4⋊Dic7C2313yes—
127Dic7⋊C4C2313yes—
128Dic7⋊C4C2313yes—
129C4×Dic7C2311yescharacteristic
130C4×Dic7C2313yes—
131C4×Dic7C2313yes—
132C2×2−1+4—71no—
133C2×H(5,𝔽2)—71no—
134#548—71no—
135#548—71no—
136#548—71no—
137#548D711yescharacteristic, complemented
138#546—73no—
139#546—73no—
140#546—73no—
141#546—73no—
142#536—73no—
143#535—73no—
144#532—73no—
145C2×#124—73no—
146C4×#41—71no—
147C23×C7C2413yes—
148C23×C7C2411yescharacteristic
149C22×D7—212no—
150C22×D7—46no—
151C22×D7—46no—
152C22×D7C2413yes—
153C22×D7C2411yescharacteristic
154C22×D7C2413yes—
155C7×Q8—26no—
156C7×Q8C2×D414yes—
157C7×D4—26no—
158C7×D4—43no—
159C2×C4×C7—26no—
160C2×C4×C7—26no—
161C2×C4×C7C2413yes—
162C2×C4×C7C2411yescharacteristic
163C2×C4×C7C2411yescharacteristic
164C2×C4×C7C2413yes—
165C2×C4×C7C2413yes—
166C22⋊D7—26no—
167C22⋊D7—26no—
168C22⋊D7—22no—
169C22⋊D7—26no—
170C22⋊D7—43no—
171C22⋊D7—43no—
172C2×Dic7C2411yescharacteristic
173C2×Dic7C2413yes—
174C2×Dic7C2411yescharacteristic
175C2×Dic7C2413yes—
176C2×Dic7C2411yescharacteristic
177D28—26no—
178D28—26no—
179D28—26no—
180D28—43no—
181D28C2×D4112yes—
182C4×D7—212no—
183C4×D7—26no—
184C4×D7—26no—
185C4×D7—43no—
186C4×D7—43no—
187C4×D7C2×D4112yes—
188Dic14—26no—
189Dic14—22no—
190Dic14—26no—
1912−1+4—148no—
192H(5,𝔽2)—148no—
193C2×#41—71no—
194C2×#41—73no—
195C2×#41—71no—
196C2×#41—73no—
197C2×#41—73no—
198C2×#41—73no—
199C2×#41C2×D711yescharacteristic
200C22×Q8—73no—
201C22×Q8—71no—
202C22×Q8C2×D711yescharacteristic, complemented
203C22×D4—73no—
204C22×D4—73no—
205C22×D4—73no—
206C4⋊Q8—73no—
207#127—73no—
208#124—73no—
209#124—73no—
210#124—73no—
211#124—76no—
212#124—71no—
213#124—73no—
214#124C2×D713yescomplemented
215#123—76no—
216#123—76no—
217C22⋊Q8—73no—
218C22⋊Q8—73no—
219C22⋊Q8—73no—
220C22⋊Q8C2×D713yescomplemented
221#121—73no—
222#121—73no—
223#121—73no—
224#121—73no—
225#121—73no—
226#121—73no—
227#121—73no—
228#121C2×D713yescomplemented
229#120—76no—
230#120—76no—
231C4×Q8—71no—
232C4×Q8C2×D711yescharacteristic, complemented
233C4×D4—73no—
234C4×D4—73no—
235C4×D4—73no—
236C4×D4—73no—
237C4×D4—73no—
238C4×D4C2×D713yescomplemented
239#117—73no—
240C2×C22⋊C4—76no—
241C2×C22⋊C4—76no—
242C2×C42—73no—
243C22×C7—21no—
244C22×C7—23no—
245C22×C7—21no—
246C22×C7—23no—
247C22×C7—21no—
248C22×C7—23no—
249C22×C7C2511yescharacteristic
250C2×D7—23no—
251C2×D7—23no—
252C2×D7—21no—
253C2×D7—23no—
254C2×D7—21no—
255C2×D7—23no—
256C2×D7—21no—
257C2×D7—23no—
258C2×D7—812no—
259C2×D7C22×D416yes—
260C4×C7—23no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—23no—
264C4×C7C22×D416yes—
265Dic7—23no—
266Dic7—21no—
267Dic7—21no—
268Dic7—23no—
269Dic7C22×D412yes—
270C24—76no—
271#41—24no—
272#41—78no—
273#41—1412no—
274#41—1412no—
275#41—1412no—
276#41—1412no—
277#41—144no—
278C2×Q8—24no—
279C2×Q8—73no—
280C2×Q8—73no—
281C2×Q8—71no—
282C2×Q8—1412no—
283C2×Q8—144no—
284C2×Q8C22×D713yes—
285C2×Q8C22×D711yescharacteristic
286C2×D4—73no—
287C2×D4—73no—
288C2×D4—73no—
289C2×D4—71no—
290C2×D4—73no—
291C2×D4—73no—
292C2×D4—73no—
293C2×D4—73no—
294C2×D4—73no—
295C2×D4—73no—
296C2×D4—73no—
297C2×D4—1412no—
298C2×D4—1412no—
299C2×D4—1412no—
300C2×D4C22×D713yes—
301C2×D4C22×D713yes—
302C22×C4—73no—
303C22×C4—76no—
304C22×C4—73no—
305C22×C4—73no—
306C22×C4—73no—
307C22×C4—73no—
308C22×C4C22×D713yes—
309C22×C4C22×D713yes—
310C4⋊C4—73no—
311C4⋊C4—73no—
312C4⋊C4—73no—
313C4⋊C4—73no—
314C4⋊C4C22×D713yes—
315C4⋊C4C22×D713yes—
316C22⋊C4—73no—
317C22⋊C4—76no—
318C22⋊C4—71no—
319C22⋊C4—73no—
320C22⋊C4—73no—
321C22⋊C4—76no—
322C22⋊C4—76no—
323C22⋊C4—76no—
324C22⋊C4—71no—
325C22⋊C4—71no—
326C22⋊C4C22×D713yes—
327C22⋊C4C22×D716yes—
328C22⋊C4C22×D711yescharacteristic
329C42—71no—
330C42—73no—
331C42—73no—
332C42C22×D713yes—
333C2×C7—43no—
334C2×C7—41no—
335C2×C7C23×D411yescharacteristic
336C2×C7#58511yescharacteristic
337C2×C7#58511yescharacteristic
338D7—26no—
339D7—41no—
340D7—43no—
341C23—73no—
342C23—73no—
343C23—71no—
344C23—1412no—
345C23—286no—
346C23—286no—
347C23C23×D713yes—
348C23C23×D711yescharacteristic
349Q8—26no—
350Q8—146no—
351Q8—142no—
352Q8—146no—
353Q8D4×D714yes—
354D4—26no—
355D4—43no—
356D4—712no—
357D4—146no—
358D4—146no—
359D4—146no—
360D4—142no—
361D4—146no—
362D4—146no—
363D4—146no—
364D4—283no—
365D4—283no—
366D4—283no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—71no—
370C2×C4—712no—
371C2×C4—71no—
372C2×C4—73no—
373C2×C4—73no—
374C2×C4—71no—
375C2×C4—1412no—
376C2×C4—146no—
377C2×C4—146no—
378C2×C4—283no—
379C2×C4—283no—
380C2×C4C23×D711yescharacteristic
381C2×C4C23×D713yes—
382C2×C4C23×D713yes—
383C2×C4C23×D711yescharacteristic
384C2×C4C23×D713yes—
385C7#350011yescharacteristic, complemented, Sylow
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—21no—
391C22—23no—
392C22—76no—
393C22—143no—
394C22—143no—
395C22—141no—
396C22—143no—
397C22—141no—
398C22—143no—
399C22—141no—
400C22—143no—
401C22—5612no—
402C22C24×D711yescharacteristic
403C4—21no—
404C4—23no—
405C4—23no—
406C4—21no—
407C4—72no—
408C4—141no—
409C4—143no—
410C4—141no—
411C4—143no—
412C4C2×D4×D716yes—
413C2—43no—
414C2—41no—
415C2—146no—
416C2—283no—
417C2—281no—
418C2C22×D4×D711yescharacteristic
419C2#9061611yescharacteristic
420C2#9062011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D7
commutator, socle, cosocleC22×C7C25
FittingC7×#548C2
hypercenter#548D7
hypocenterC7#3500
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#6553C2semidirect
2. C2×#6551C2semidirect
3. C7×#548C2semidirect
4. #90522C2semidirect
5. #90508C2semidirect
6. #90367C2semidirect
7. D7×#124C2semidirect
8. #90321C2semidirect
9. #90319C2semidirect
10. #90303C2semidirect
11. #90298C2semidirect
12. #90267C2semidirect
13. C4×#1161C2semidirect
14. #90241C2semidirect
15. #90239C2semidirect
17. C22×C7×Q8C22semidirect
22. C2×#1161C22semidirect
23. C2×Q8×D7C22semidirect
24. C2×Q8×D7C22semidirect
28. C2×#1157C22semidirect
29. C2×#1157C22semidirect
31. C7×#124C22semidirect
32. C7×C22⋊Q8C22semidirect
33. C7×#121C22semidirect
34. C4×C7×Q8C22semidirect
35. C4×C7×D4C22semidirect
36. #6510C22semidirect
37. #6509C22semidirect
38. Q8×Dic7C22semidirect
39. #6503C22semidirect
40. #6501C22semidirect
41. #6501C22semidirect
42. #6500C22semidirect
43. #6499C22semidirect
44. #6493C22semidirect
45. C4×C22⋊D7C22semidirect
46. C4×C22⋊D7C22semidirect
47. #6487C22semidirect
48. #6460C22semidirect
49. #6458C22semidirect
50. #6458C22semidirect
51. #6457C22semidirect
52. #6456C22semidirect
53. #6455C22semidirect
54. #6451C22semidirect
55. #6448C22semidirect
56. #6448C22semidirect
57. #6448C22semidirect
58. #6447C22semidirect
59. #6447C22semidirect
60. #6446C22semidirect
61. #6445C22semidirect
62. D7×C22⋊C4C22semidirect
63. D7×C22⋊C4C22semidirect
64. #6438C22semidirect
65. C4×D28C22semidirect
66. C4×D28C22semidirect
67. C42×D7C22semidirect
86. #1161D4semidirect
137. #548D7semidirect
202. C22×Q8C2×D7semidirect
214. #124C2×D7semidirect
220. C22⋊Q8C2×D7semidirect
228. #121C2×D7semidirect
232. C4×Q8C2×D7semidirect
238. C4×D4C2×D7semidirect
385. C7#3500semidirect

Maximal subgroups

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

subgroupindex
C2×#65532
C2×#65512
C7×#5482
#905222
#905082
#903672
D7×#1242
#903212
#903192
#903032
#902982
#902672
C4×#11612
#902412
#902392
#35007