Finite Group Databasebeta

← #41×C22⋊D7 #11744340

Subgroups — #41×C22⋊D7 #11744340

Order 896.

1648 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
1C22×C7×#41C211yesmaximal, characteristic, complemented
2C2×D7×#41C211yesmaximal, characteristic, complemented
3#90525C211yesmaximal, characteristic, complemented
4#90523C213yesmaximal, complemented
5#90522C211yesmaximal, characteristic, complemented
6#90520C213yesmaximal, complemented
7#90519C213yesmaximal, complemented
8#41×Dic7C211yesmaximal, characteristic, complemented
9#90515C213yesmaximal, complemented
10#90508C211yesmaximal, characteristic, complemented
11Q8×C22⋊D7C211yesmaximal, characteristic, complemented
12#90497C213yesmaximal, complemented
13D4×C22⋊D7C213yesmaximal, complemented
14#90482C213yesmaximal, complemented
15C2×C4×C22⋊D7C213yesmaximal, complemented
16C2×C7×#41—28no—
17C2×C7×#41C2211yescharacteristic
18C2×C7×#41C2213yescomplemented
19C2×C7×#41C2213yescomplemented
20C2×C7×#41C2211yescharacteristic, complemented
21C22×C7×Q8C2211yescharacteristic, complemented
22C22×C7×D4C2213yescomplemented
23C23×C4×C7C2213yescomplemented
24C22×C22⋊D7C2213yes—
25D7×#41—28no—
26C2×#1161C2211yescharacteristic, complemented
27C2×Q8×D7C2211yescharacteristic, complemented
28C2×#1159C2213yescomplemented
29C2×D4×D7C2213yescomplemented
30C2×#1157C2213yescomplemented
31C22×C4×D7C2213yescomplemented
32#6516C2216yescomplemented
33C2×C22⋊Dic7C2213yescomplemented
34#6510C2213yescomplemented
35#6509C2213yescomplemented
36#6509C2213yescomplemented
37Q8×Dic7C2211yescharacteristic, complemented
38Q8×Dic7C2211yescharacteristic, complemented
39#6507C2213yescomplemented
40#6503C2213yescomplemented
41#6502C2216yescomplemented
42#6502C2216yescomplemented
43#6501C2213yescomplemented
44#6501C2213yescomplemented
45#6500C2216yescomplemented
46#6499C2213yescomplemented
47#6498C2216yescomplemented
48D4×Dic7C2213yescomplemented
49D4×Dic7C2213yescomplemented
50#6493C2213yescomplemented
51#6493C2213yescomplemented
52#6492C2216yescomplemented
53C4×C22⋊D7C2211yescharacteristic
54C4×C22⋊D7C2213yes—
55C4×C22⋊D7C2213yescomplemented
56C4×C22⋊D7C2213yescomplemented
57C4×C22⋊D7C2213yescomplemented
58C4×C22⋊D7C2213yescomplemented
59C4×C22⋊D7C2213yescomplemented
60C4×C22⋊D7C2213yescomplemented
61C4×C22⋊D7C2216yescomplemented
62C2×(C2×D7)⋊C4C2213yescomplemented
63#6489C2213yescomplemented
64#6487C2213yescomplemented
65#6487C2213yescomplemented
66C2×Dic7⋊C4C2213yescomplemented
67C2×C4×Dic7C2213yescomplemented
68D4×#41—71nomaximal, Sylow
69C24×C7C2313yes—
70C23×D7C2313yes—
71C7×#41—24no—
72C7×#41—212no—
73C7×#41—212no—
74C7×#41D418yescomplemented
75C2×C7×Q8—24no—
76C2×C7×Q8C2313yes—
77C2×C7×Q8C2311yescharacteristic
78C2×C7×D4—212no—
79C2×C7×D4C2316yes—
80C2×C7×D4C2313yes—
81C2×C7×D4C2313yes—
82C22×C4×C7—212no—
83C22×C4×C7C2313yes—
84C22×C4×C7C2313yes—
85C22×C4×C7C2313yes—
86C22×C4×C7C2311yescharacteristic
87C22×C4×C7C2313yes—
88C22×C4×C7C2313yes—
89C2×C22⋊D7—212no—
90C2×C22⋊D7C2313yes—
91C2×C22⋊D7C2313yes—
92C2×C22⋊D7C2313yes—
93C2×C22⋊D7C2316yes—
94C2×C22⋊D7C2316yes—
95C2×C22⋊D7C2311yescharacteristic
96C22×Dic7C2313yes—
97C22×Dic7C2313yes—
98C22×Dic7C2313yes—
99#1161—24no—
100Q8×D7—24no—
101#1159—212no—
102D4×D7—212no—
103#1157—212no—
104C2×D28C2313yes—
105C2×C4×D7—212no—
106C2×C4×D7C2311yescharacteristic
107C2×C4×D7C2313yes—
108C2×C4×D7C2313yes—
109C2×C4×D7C2313yes—
110C2×Dic14C2313yes—
111C22⋊Dic7C2313yes—
112C22⋊Dic7C2316yes—
113C22⋊Dic7C2316yes—
114C22⋊Dic7C2311yescharacteristic
115C22⋊Dic7C2313yes—
116C22⋊Dic7C2313yes—
117(C2×D7)⋊C4C2311yescharacteristic
118(C2×D7)⋊C4C2313yes—
119(C2×D7)⋊C4C2313yes—
120(C2×D7)⋊C4C2313yes—
121(C2×D7)⋊C4C2316yes—
122C4⋊Dic7C2313yes—
123C4⋊Dic7C2313yes—
124Dic7⋊C4C2311yescharacteristic
125Dic7⋊C4C2313yes—
126Dic7⋊C4C2313yes—
127Dic7⋊C4C2313yes—
128Dic7⋊C4C2316yes—
129C4×Dic7C2311yescharacteristic
130C4×Dic7C2313yes—
131C4×Dic7C2313yes—
132C4×Dic7C2313yes—
133C22×#41—71no—
134C22×#41D711yescharacteristic, complemented
135#550—71no—
136D4×Q8—71no—
137#548—71no—
138#548—71no—
139#547—73no—
140#546—73no—
141#546—73no—
142D42—73no—
143#532—73no—
144#525—73no—
145#525—73no—
146C4×#41—71no—
147C2×C4×D4—73no—
148C23×C7—26no—
149C23×C7—26no—
150C23×C7—43no—
151C23×C7C2413yes—
152C23×C7C2413yes—
153C23×C7C2411yescharacteristic
154C22×D7—26no—
155C22×D7—26no—
156C22×D7—43no—
157C22×D7C2413yes—
158C22×D7C2411yescharacteristic
159C7×Q8—26no—
160C7×Q8C2×D414yescomplemented
161C7×D4—212no—
162C7×D4—26no—
163C7×D4C2×D4112yescomplemented
164C2×C4×C7—26no—
165C2×C4×C7—22no—
166C2×C4×C7—26no—
167C2×C4×C7—26no—
168C2×C4×C7—26no—
169C2×C4×C7C2×D4112yescomplemented
170C2×C4×C7C2411yescharacteristic
171C2×C4×C7C2413yes—
172C2×C4×C7C2413yes—
173C2×C4×C7C2411yescharacteristic
174C22⋊D7—212no—
175C22⋊D7—26no—
176C22⋊D7—26no—
177C22⋊D7—26no—
178C22⋊D7—46no—
179C22⋊D7#4114yesdirect product factor
180C2×Dic7—26no—
181C2×Dic7—26no—
182C2×Dic7—43no—
183C2×Dic7C2411yescharacteristic
184C2×Dic7C2413yes—
185C2×Dic7C2413yes—
186C2×Dic7C2413yes—
187C2×Dic7C2411yescharacteristic
188C2×Dic7C2411yescharacteristic
189D28—26no—
190C4×D7—22no—
191C4×D7—26no—
192C4×D7—26no—
193C4×D7—26no—
194Dic14—26no—
195C2×#41—28no—
196C2×#41—71no—
197C2×#41—73no—
198C2×#41—73no—
199C2×#41—148no—
200C2×#41C2×D711yescharacteristic
201C2×#41C2×D713yescomplemented
202C2×#41C2×D713yescomplemented
203C2×#41C2×D711yescharacteristic, complemented
204C22×Q8—71no—
205C22×Q8C2×D711yescharacteristic, complemented
206C22×D4—73no—
207C22×D4—73no—
208C22×D4C2×D713yescomplemented
209C23×C4—73no—
210C23×C4C2×D713yescomplemented
211C4⋊Q8—73no—
212#127—73no—
213#124—73no—
214#124—73no—
215#123—76no—
216#123—76no—
217C22⋊Q8—73no—
218C22⋊Q8—73no—
219C22⋊Q8—73no—
220C22⋊Q8—73no—
221#121—73no—
222#121—73no—
223#121—76no—
224#121—76no—
225#121—73no—
226#121—73no—
227#120—76no—
228#120—76no—
229C4×Q8—71no—
230C4×Q8—71no—
231C4×D4—73no—
232C4×D4—73no—
233C4×D4—71no—
234C4×D4—73no—
235C4×D4—73no—
236C4×D4—73no—
237C4×D4—73no—
238C4×D4—73no—
239C4×D4—73no—
240C4×D4—73no—
241C4×D4—76no—
242#117—73no—
243C2×C4⋊C4—73no—
244C2×C22⋊C4—73no—
245C2×C22⋊C4—73no—
246C2×C42—73no—
247C22×C7—21no—
248C22×C7—23no—
249C22×C7—23no—
250C22×C7—21no—
251C22×C7—23no—
252C22×C7—23no—
253C22×C7—23no—
254C22×C7—412no—
255C22×C7C22×D416yes—
256C22×C7C2×#4112yescomplemented
257C22×C7C2511yescharacteristic
258C2×D7—21no—
259C2×D7—21no—
260C2×D7—23no—
261C2×D7—23no—
262C2×D7—23no—
263C2×D7—412no—
264C2×D7C2×#4112yescomplemented
265C4×C7—21no—
266C4×C7—23no—
267C4×C7C22×D416yes—
268C4×C7C22×D412yes—
269Dic7—23no—
270Dic7—21no—
271Dic7—23no—
272Dic7—21no—
273Dic7—23no—
274Dic7C2×#4112yescomplemented
275C24—73no—
276C24C22×D713yes—
277#41—24no—
278#41—212no—
279#41—212no—
280#41—144no—
281#41—1412no—
282#41—1412no—
283#41C22⋊D718yesdirect product factor
284C2×Q8—24no—
285C2×Q8—73no—
286C2×Q8—144no—
287C2×Q8C22×D713yes—
288C2×Q8C22×D711yescharacteristic
289C2×D4—212no—
290C2×D4—71no—
291C2×D4—73no—
292C2×D4—73no—
293C2×D4—73no—
294C2×D4—76no—
295C2×D4—76no—
296C2×D4—73no—
297C2×D4—1412no—
298C2×D4—1412no—
299C2×D4C22×D716yes—
300C2×D4C22×D713yes—
301C2×D4C22×D713yes—
302C22×C4—212no—
303C22×C4—73no—
304C22×C4—73no—
305C22×C4—73no—
306C22×C4—73no—
307C22×C4—73no—
308C22×C4—71no—
309C22×C4—73no—
310C22×C4—1412no—
311C22×C4C22×D713yes—
312C22×C4C22×D713yes—
313C22×C4C22×D713yes—
314C22×C4C22×D711yescharacteristic
315C22×C4C22×D713yes—
316C22×C4C22×D713yes—
317C4⋊C4—71no—
318C4⋊C4—73no—
319C4⋊C4—73no—
320C4⋊C4—73no—
321C4⋊C4—76no—
322C4⋊C4—73no—
323C4⋊C4—73no—
324C22⋊C4—71no—
325C22⋊C4—73no—
326C22⋊C4—73no—
327C22⋊C4—73no—
328C22⋊C4—76no—
329C22⋊C4—76no—
330C22⋊C4—71no—
331C22⋊C4—73no—
332C22⋊C4—73no—
333C22⋊C4—73no—
334C22⋊C4—76no—
335C42—73no—
336C42—73no—
337C42—71no—
338C42—73no—
339C2×C7—26no—
340C2×C7—22no—
341C2×C7—43no—
342C2×C7C23×D411yescharacteristic
343C2×C7C22×#4111yescharacteristic
344C2×C7#58511yescharacteristic
345D7—22no—
346D7—43no—
347C23—26no—
348C23—26no—
349C23—43no—
350C23—71no—
351C23—73no—
352C23—146no—
353C23—146no—
354C23—283no—
355C23C23×D713yes—
356C23C23×D711yescharacteristic
357C23C23×D713yes—
358Q8—26no—
359Q8—146no—
360Q8C2×C22⋊D714yescomplemented
361D4—212no—
362D4—26no—
363D4—74no—
364D4—146no—
365D4—146no—
366D4—146no—
367D4—1412no—
368D4—146no—
369D4—286no—
370D4C2×C22⋊D7112yescomplemented
371C2×C4—26no—
372C2×C4—22no—
373C2×C4—26no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—71no—
377C2×C4—71no—
378C2×C4—73no—
379C2×C4—73no—
380C2×C4—71no—
381C2×C4—73no—
382C2×C4—146no—
383C2×C4—146no—
384C2×C4—142no—
385C2×C4—146no—
386C2×C4—146no—
387C2×C4—146no—
388C2×C4—283no—
389C2×C4C2×C22⋊D7112yescomplemented
390C2×C4C23×D711yescharacteristic
391C2×C4C23×D713yes—
392C2×C4C23×D713yes—
393C2×C4C23×D711yescharacteristic
394C7D4×#4111yescharacteristic, complemented, Sylow
395C22—23no—
396C22—23no—
397C22—21no—
398C22—21no—
399C22—23no—
400C22—23no—
401C22—23no—
402C22—412no—
403C22—72no—
404C22—141no—
405C22—141no—
406C22—143no—
407C22—143no—
408C22—143no—
409C22—2812no—
410C22D7×#4112yescomplemented
411C22C22×C22⋊D716yes—
412C22C24×D711yescharacteristic
413C4—21no—
414C4—23no—
415C4—72no—
416C4—143no—
417C4—141no—
418C4—143no—
419C4—141no—
420C4—143no—
421C4C22×C22⋊D712yes—
422C4C22×C22⋊D716yes—
423C2—22no—
424C2—26no—
425C2—43no—
426C2—142no—
427C2—283no—
428C2C2×D7×#4111yescharacteristic
429C2#9061611yescharacteristic
430C2C23×C22⋊D711yescharacteristic
subgroupG/H
centerC2×C4C23×D7
commutator, socle, cosocleC22×C7C25
FrattiniC22C24×D7
FittingC22×C7×#41C2
hypercenterC22×#41D7
hypocenterC7D4×#41
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C7×#41C2semidirect
2. C2×D7×#41C2semidirect
3. #90525C2semidirect
4. #90523C2semidirect
5. #90522C2semidirect
6. #90520C2semidirect
7. #90519C2semidirect
8. #41×Dic7C2semidirect
9. #90515C2semidirect
10. #90508C2semidirect
11. Q8×C22⋊D7C2semidirect
12. #90497C2semidirect
13. D4×C22⋊D7C2semidirect
14. #90482C2semidirect
15. C2×C4×C22⋊D7C2semidirect
18. C2×C7×#41C22semidirect
19. C2×C7×#41C22semidirect
20. C2×C7×#41C22semidirect
21. C22×C7×Q8C22semidirect
22. C22×C7×D4C22semidirect
23. C23×C4×C7C22semidirect
26. C2×#1161C22semidirect
27. C2×Q8×D7C22semidirect
28. C2×#1159C22semidirect
29. C2×D4×D7C22semidirect
30. C2×#1157C22semidirect
31. C22×C4×D7C22semidirect
32. #6516C22semidirect
33. C2×C22⋊Dic7C22semidirect
34. #6510C22semidirect
35. #6509C22semidirect
36. #6509C22semidirect
37. Q8×Dic7C22semidirect
38. Q8×Dic7C22semidirect
39. #6507C22semidirect
40. #6503C22semidirect
41. #6502C22semidirect
42. #6502C22semidirect
43. #6501C22semidirect
44. #6501C22semidirect
45. #6500C22semidirect
46. #6499C22semidirect
47. #6498C22semidirect
48. D4×Dic7C22semidirect
49. D4×Dic7C22semidirect
50. #6493C22semidirect
51. #6493C22semidirect
52. #6492C22semidirect
55. C4×C22⋊D7C22semidirect
56. C4×C22⋊D7C22semidirect
57. C4×C22⋊D7C22semidirect
58. C4×C22⋊D7C22semidirect
59. C4×C22⋊D7C22semidirect
60. C4×C22⋊D7C22semidirect
61. C4×C22⋊D7C22semidirect
62. C2×(C2×D7)⋊C4C22semidirect
63. #6489C22semidirect
64. #6487C22semidirect
65. #6487C22semidirect
66. C2×Dic7⋊C4C22semidirect
67. C2×C4×Dic7C22semidirect
74. C7×#41D4semidirect
134. C22×#41D7semidirect
160. C7×Q8C2×D4semidirect
163. C7×D4C2×D4semidirect
169. C2×C4×C7C2×D4semidirect
179. C22⋊D7#41direct (N × H)
201. C2×#41C2×D7semidirect
202. C2×#41C2×D7semidirect
203. C2×#41C2×D7semidirect
205. C22×Q8C2×D7semidirect
208. C22×D4C2×D7semidirect
210. C23×C4C2×D7semidirect
256. C22×C7C2×#41semidirect
264. C2×D7C2×#41semidirect
274. Dic7C2×#41semidirect
283. #41C22⋊D7direct (N × H)
360. Q8C2×C22⋊D7semidirect
370. D4C2×C22⋊D7semidirect
389. C2×C4C2×C22⋊D7semidirect
394. C7D4×#41semidirect
410. C22D7×#41semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#412
C2×D7×#412
#905252
#905232
#905222
#905202
#905192
#41×Dic72
#905152
#905082
Q8×C22⋊D72
#904972
D4×C22⋊D72
#904822
C2×C4×C22⋊D72
D4×#417