Finite Group Databasebeta

← #13059901

Subgroups — #13059901

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C22⋊D7C211yesmaximal, characteristic, complemented
2C2×#66683C211yesmaximal, characteristic, complemented
3C4×#4268C211yesmaximal, characteristic, complemented
4C2×C7×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5C3×#6493C211yesmaximal, characteristic, complemented
6#10621130C211yesmaximal, characteristic, complemented
7#10621098C211yesmaximal, characteristic, complemented
8#10621036C211yesmaximal, characteristic, complemented
9#10621035C211yesmaximal, characteristic, complemented
10#10621026C211yesmaximal, characteristic, complemented
11#10621023C211yesmaximal, characteristic, complemented
12#10621004C211yesmaximal, characteristic, complemented
13#10621000C211yesmaximal, characteristic, complemented
14#10620998C211yesmaximal, characteristic, complemented
15#10620970C211yesmaximal, characteristic, complemented
16#90181—31nomaximal, Hall
17C23×D3×C7C2211yescharacteristic
18C22×D3×D7C2211yescharacteristic
19C22×C3×C4×C7C2211yescharacteristic, complemented
20C2×#4268C2211yescharacteristic
21C2×C4×D21C2211yescharacteristic, complemented
22C22×C7×Dic3C2211yescharacteristic, complemented
23C2×C3×C22⋊D7C2211yescharacteristic
24C2×C3×C22⋊D7C2211yescharacteristic, complemented
25C2×C3×D28C2211yescharacteristic, complemented
26D3×C22⋊D7—24no—
27C2×#4248C2211yescharacteristic, complemented
28C2×#4247C2211yescharacteristic
29C2×#4246C2211yescharacteristic, complemented
30C2×#4245C2211yescharacteristic
31C2×#4245C2211yescharacteristic
32C2×#4245C2211yescharacteristic, complemented
33C2×D21⋊C4C2211yescharacteristic, complemented
34C2×D3×Dic7C2211yescharacteristic
35#66683—24no—
36C2×Dic3×D7C2211yescharacteristic, complemented
37#66636C2211yescharacteristic, complemented
38#66631C2211yescharacteristic, complemented
39#66629C2211yescharacteristic, complemented
40C4×Dic21C2211yescharacteristic, complemented
41C7×(C2×D3)⋊C4C2211yescharacteristic
42C7×(C2×D3)⋊C4C2211yescharacteristic
43C7×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C7×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C3×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C3×(C2×D7)⋊C4C2211yescharacteristic, complemented
47C3×C4⋊Dic7C2211yescharacteristic, complemented
48#66577C2211yescharacteristic, complemented
49#66577C2211yescharacteristic, complemented
50(C2×D3)⋊Dic7C2211yescharacteristic
51(C2×D3)⋊Dic7C2211yescharacteristic, complemented
52(C2×D7)⋊Dic3C2211yescharacteristic, complemented
53(C2×D7)⋊Dic3C2211yescharacteristic, complemented
54C22×C22⋊D7—31no—
55C2×#1157—31no—
56C2×C7×C22⋊C4—31no—
57#6516—31no—
58#6493D311yescharacteristic, complemented
59#6492—31no—
60C4×C22⋊D7—31no—
61#6448—31no—
62#6447—31no—
63#6447—31no—
64#6446—31no—
65#6445—31no—
66(C22⋊D7)⋊C4—31no—
67D7×C22⋊C4—31no—
68#6441—31no—
69#5610—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C22×D21C2311yescharacteristic
72C22×D3×C7—22no—
73C22×D3×C7—22no—
74C22×D3×C7—41no—
75C22×D3×C7C2311yescharacteristic
76C22×D3×C7C2311yescharacteristic
77C22×C3×D7C2311yescharacteristic
78C22×C3×D7C2311yescharacteristic
79C2×D3×D7—22no—
80C2×D3×D7—22no—
81C2×D3×D7—41no—
82C2×C3×C4×C7—41no—
83C2×C3×C4×C7C2311yescharacteristic
84C2×C3×C4×C7C2311yescharacteristic
85#4268D414yescomplemented
86C2×Dic21C2311yescharacteristic
87C2×Dic21C2311yescharacteristic
88C2×Dic21C2311yescharacteristic
89C4×D21—41no—
90C2×C7×Dic3—22no—
91C2×C7×Dic3C2311yescharacteristic
92C2×C7×Dic3C2311yescharacteristic
93C3×C22⋊D7—22no—
94C3×C22⋊D7—22no—
95C2×C3×Dic7C2311yescharacteristic
96C2×C3×Dic7C2311yescharacteristic
97C3×D28—41no—
98#4248—22no—
99#4247—22no—
100#4246—22no—
101#4245—22no—
102#4245—22no—
103#4245—41no—
104D21⋊C4—22no—
105D3×Dic7—22no—
106Dic3×D7—22no—
107C24×C7—31no—
108C23×D7—31no—
109C22×C4×C7—31no—
110C22×C4×C7C2×D311yescharacteristic, complemented
111C2×C22⋊D7—31no—
112C2×C22⋊D7—31no—
113C2×C22⋊D7—31no—
114C2×C22⋊D7—31no—
115C2×C22⋊D7—31no—
116C2×C22⋊D7—64no—
117C2×C22⋊D7C2×D311yescharacteristic
118C2×C22⋊D7C2×D311yescharacteristic, complemented
119C22×Dic7—31no—
120#1157—64no—
121C2×D28—31no—
122C2×D28C2×D311yescharacteristic, complemented
123C2×C4×D7—31no—
124C2×C4×D7—31no—
125C2×C4×D7—31no—
126C2×Dic14—31no—
127C7×C22⋊C4—31no—
128C7×C22⋊C4—31no—
129C7×C22⋊C4—31no—
130C7×C22⋊C4—31no—
131C22⋊Dic7—31no—
132C22⋊Dic7—31no—
133C22⋊Dic7—31no—
134(C2×D7)⋊C4—31no—
135(C2×D7)⋊C4—31no—
136(C2×D7)⋊C4—31no—
137(C2×D7)⋊C4C2×D311yescharacteristic, complemented
138(C2×D7)⋊C4C2×D311yescharacteristic, complemented
139C4⋊Dic7C2×D311yescharacteristic, complemented
140Dic7⋊C4—31no—
141Dic7⋊C4—31no—
142Dic7⋊C4—31no—
143C4×Dic7—31no—
144C2×#237—71no—
145C2×D3×D4—71no—
146C3×#121—71no—
147#939—71no—
148#937—71no—
149#937—71no—
150#935—71no—
151C4×#67—71no—
152C2×(C2×D3)⋊C4D711yescharacteristic, complemented
153#896—71no—
154#894—71no—
155#885—71no—
156#884—71no—
157#67⋊C4—71no—
158D3×C22⋊C4—71no—
159C22×C3×C7—21no—
160C22×C3×C7—21no—
161C22×C3×C7C2×D412yes—
162C22×C3×C7C2411yescharacteristic
163C2×D21—21no—
164C2×D21—21no—
165C2×D21C2×D412yes—
166C2×D3×C7—21no—
167C2×D3×C7—21no—
168C2×D3×C7—21no—
169C2×D3×C7—21no—
170C2×D3×C7—21no—
171C2×D3×C7—44no—
172C2×D3×C7#4112yescomplemented
173C2×C3×D7—21no—
174C2×C3×D7—21no—
175C2×C3×D7—21no—
176C2×C3×D7—21no—
177C2×C3×D7—21no—
178C2×C3×D7—21no—
179D3×D7—44no—
180C3×C4×C7—21no—
181C3×C4×C7—21no—
182Dic21—21no—
183Dic21—21no—
184Dic21C2×D412yes—
185C3×Dic7—21no—
186C3×Dic7—21no—
187C7×Dic3—21no—
188C7×Dic3—21no—
189#546—211noSylow
190C23×C7—31no—
191C23×C7—31no—
192C23×C7—62no—
193C23×C7—62no—
194C23×C7—121no—
195C23×C7C22×D311yescharacteristic
196C22×D7—31no—
197C22×D7—62no—
198C22×D7—62no—
199C22×D7—121no—
200C22×D7C22×D311yescharacteristic
201C22×D7C22×D311yescharacteristic
202C2×C4×C7—31no—
203C2×C4×C7—31no—
204C2×C4×C7—41no—
205C2×C4×C7—62no—
206C2×C4×C7C22×D311yescharacteristic
207C2×C4×C7C22×D311yescharacteristic
208C22⋊D7—22no—
209C22⋊D7—22no—
210C22⋊D7—34no—
211C22⋊D7—62no—
212C22⋊D7—62no—
213C22⋊D7—62no—
214C22⋊D7—121no—
215C2×Dic7—31no—
216C2×Dic7—31no—
217C2×Dic7—31no—
218C2×Dic7—62no—
219C2×Dic7C22×D311yescharacteristic
220C2×Dic7C22×D311yescharacteristic
221D28—41no—
222D28—62no—
223C4×D7—62no—
224C4×D7—62no—
225C4×D7—121no—
226Dic14—62no—
227C23×D3—71no—
228C23×D3C2×D711yescharacteristic
229C2×C3×D4—71no—
230C2×C3×D4—71no—
231C2×C3×D4—71no—
232C22×C3×C4C2×D711yescharacteristic, complemented
233C2×#67—71no—
234C2×#67—71no—
235C2×#67—71no—
236C2×#67—71no—
237C2×#67—71no—
238C22×Dic3—71no—
239C22×Dic3C2×D711yescharacteristic, complemented
240#237—144no—
241D3×D4—144no—
242C2×D12—71no—
243C2×C4×D3—71no—
244C2×C4×D3—71no—
245C2×C4×D3—71no—
246C2×Dic6—71no—
247C3×C4⋊C4—71no—
248C3×C22⋊C4—71no—
249C3×C22⋊C4—71no—
250#217—71no—
251#217—71no—
252#217—71no—
253(C2×D3)⋊C4—71no—
254(C2×D3)⋊C4—71no—
255(C2×D3)⋊C4—71no—
256(C2×D3)⋊C4C2×D711yescharacteristic
257(C2×D3)⋊C4C2×D711yescharacteristic
258(C2×D3)⋊C4C2×D711yescharacteristic, complemented
259(C2×D3)⋊C4C2×D711yescharacteristic, complemented
260Dic3⋊C4—71no—
261Dic3⋊C4—71no—
262Dic3⋊C4—71no—
263C4×Dic3—71no—
264C2×C3×C7—22no—
265C2×C3×C7C22×D411yescharacteristic
266C2×C3×C7C2×#4111yescharacteristic
267C2×C3×C7H(5,𝔽2)11yescharacteristic
268D21—22no—
269C3×D7—41no—
270C3×D7—41no—
271D3×C7—22no—
272D3×C7—41no—
273C2×#41—211no—
274C22×D4—211no—
275#124—211no—
276#123—211no—
277#123—211no—
278C22⋊Q8—211no—
279#121—71no—
280#121—211no—
281#121—211no—
282#120—211no—
283#120—211no—
284C4×D4—211no—
285C4×D4—211no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—211no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—32no—
291C22×C7—61no—
292C22×C7—61no—
293C22×C7—61no—
294C22×C7—61no—
295C22×C7—61no—
296C22×C7—124no—
297C22×C7D3×D412yes—
298C22×C7C23×D311yescharacteristic
299C2×D7—21no—
300C2×D7—21no—
301C2×D7—21no—
302C2×D7—21no—
303C2×D7—21no—
304C2×D7—21no—
305C2×D7—32no—
306C2×D7—61no—
307C2×D7—61no—
308C2×D7—124no—
309C4×C7—21no—
310C4×C7—21no—
311C4×C7—61no—
312C4×C7—61no—
313Dic7—21no—
314Dic7—21no—
315Dic7—32no—
316Dic7—61no—
317Dic7—61no—
318C23×C3—71no—
319C23×C3—71no—
320C23×C3C22×D711yescharacteristic
321C22×D3—22no—
322C22×D3—22no—
323C22×D3—41no—
324C22×D3—71no—
325C22×D3—142no—
326C22×D3—142no—
327C22×D3—281no—
328C22×D3C22×D711yescharacteristic
329C22×D3C22×D711yescharacteristic
330C3×D4—142no—
331C3×D4—142no—
332C3×D4—281no—
333C2×C3×C4—41no—
334C2×C3×C4—71no—
335C2×C3×C4—71no—
336C2×C3×C4C22×D711yescharacteristic
337C2×C3×C4C22×D711yescharacteristic
338#67—74no—
339#67—142no—
340#67—142no—
341#67—142no—
342#67—281no—
343C2×Dic3—22no—
344C2×Dic3—71no—
345C2×Dic3—71no—
346C2×Dic3—71no—
347C2×Dic3—142no—
348C2×Dic3C22×D711yescharacteristic
349C2×Dic3C22×D711yescharacteristic
350D12—142no—
351C4×D3—142no—
352C4×D3—142no—
353C4×D3—281no—
354Dic6—142no—
355C3×C7#54611yescharacteristic, complemented, Hall
356C24—31no—
357C24—211no—
358#41—424no—
359C2×Q8—211no—
360C2×D4—71no—
361C2×D4—71no—
362C2×D4—71no—
363C2×D4—211no—
364C2×D4—211no—
365C2×D4—211no—
366C2×D4—211no—
367C2×D4—211no—
368C2×D4—211no—
369C2×D4—424no—
370C22×C4—31no—
371C22×C4—211no—
372C22×C4—211no—
373C22×C4—211no—
374C22×C4—211no—
375C22×C4D3×D711yescharacteristic, complemented
376C4⋊C4—71no—
377C4⋊C4—211no—
378C4⋊C4—211no—
379C4⋊C4—211no—
380C22⋊C4—31no—
381C22⋊C4—31no—
382C22⋊C4—31no—
383C22⋊C4—31no—
384C22⋊C4—71no—
385C22⋊C4—71no—
386C22⋊C4—211no—
387C22⋊C4—211no—
388C22⋊C4—211no—
389C22⋊C4—211no—
390C22⋊C4—211no—
391C22⋊C4—211no—
392C42—211no—
393C2×C7—22no—
394C2×C7—62no—
395C2×C7—121no—
396C2×C7C2×D3×D411yescharacteristic
397C2×C7#100411yescharacteristic
398C2×C7D3×#4111yescharacteristic
399D7—41no—
400D7—41no—
401D7—62no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—141no—
405C22×C3—141no—
406C22×C3—141no—
407C22×C3—141no—
408C22×C3—141no—
409C22×C3—141no—
410C22×C3D4×D712yes—
411C22×C3C23×D711yescharacteristic
412C2×D3—21no—
413C2×D3—21no—
414C2×D3—21no—
415C2×D3—21no—
416C2×D3—21no—
417C2×D3—44no—
418C2×D3—72no—
419C2×D3—141no—
420C2×D3—141no—
421C2×D3—284no—
422C2×D3#115712yescomplemented
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—141no—
426C3×C4—141no—
427Dic3—21no—
428Dic3—21no—
429Dic3—72no—
430Dic3—141no—
431Dic3—141no—
432C23—31no—
433C23—31no—
434C23—62no—
435C23—62no—
436C23—71no—
437C23—71no—
438C23—121no—
439C23—211no—
440C23—422no—
441C23—422no—
442C23—841no—
443C23C2×D3×D711yescharacteristic
444Q8—422no—
445D4—142no—
446D4—142no—
447D4—214no—
448D4—281no—
449D4—422no—
450D4—422no—
451D4—422no—
452D4—422no—
453D4—841no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—41no—
457C2×C4—62no—
458C2×C4—71no—
459C2×C4—71no—
460C2×C4—211no—
461C2×C4—211no—
462C2×C4—211no—
463C2×C4—422no—
464C2×C4—422no—
465C2×C4—422no—
466C2×C4—841no—
467C2×C4C2×D3×D711yescharacteristic
468C2×C4C2×D3×D711yescharacteristic
469C7#561011yescharacteristic, complemented, Sylow
470C2×C3—22no—
471C2×C3—281no—
472C2×C3—281no—
473C2×C3C2×#115711yescharacteristic
474C2×C3C2×D4×D711yescharacteristic
475C2×C3#654811yescharacteristic
476D3—22no—
477D3—41no—
478D3—142no—
479C22—21no—
480C22—21no—
481C22—32no—
482C22—61no—
483C22—61no—
484C22—61no—
485C22—61no—
486C22—61no—
487C22—124no—
488C22—141no—
489C22—141no—
490C22—141no—
491C22—141no—
492C22—141no—
493C22—141no—
494C22—212no—
495C22—421no—
496C22—421no—
497C22—844no—
498C22#6668112yes—
499C22C22×D3×D711yescharacteristic
500C4—21no—
501C4—21no—
502C4—61no—
503C4—61no—
504C4—141no—
505C4—141no—
506C4—212no—
507C4—421no—
508C4—421no—
509C3#9018111yescharacteristic, complemented, Sylow
510C2—22no—
511C2—62no—
512C2—121no—
513C2—281no—
514C2—281no—
515C2—422no—
516C2C2×#6668111yescharacteristic
517C2D3×#115711yescharacteristic
518C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#546
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×C22⋊D7C2semidirect
2. C2×#66683C2semidirect
3. C4×#4268C2semidirect
4. C2×C7×(C2×D3)⋊C4C2semidirect
5. C3×#6493C2semidirect
6. #10621130C2semidirect
7. #10621098C2semidirect
8. #10621036C2semidirect
9. #10621035C2semidirect
10. #10621026C2semidirect
11. #10621023C2semidirect
12. #10621004C2semidirect
13. #10621000C2semidirect
14. #10620998C2semidirect
15. #10620970C2semidirect
19. C22×C3×C4×C7C22semidirect
21. C2×C4×D21C22semidirect
22. C22×C7×Dic3C22semidirect
24. C2×C3×C22⋊D7C22semidirect
25. C2×C3×D28C22semidirect
27. C2×#4248C22semidirect
29. C2×#4246C22semidirect
32. C2×#4245C22semidirect
33. C2×D21⋊C4C22semidirect
36. C2×Dic3×D7C22semidirect
37. #66636C22semidirect
38. #66631C22semidirect
39. #66629C22semidirect
40. C4×Dic21C22semidirect
43. C7×(C2×D3)⋊C4C22semidirect
44. C7×(C2×D3)⋊C4C22semidirect
45. C3×(C2×D7)⋊C4C22semidirect
46. C3×(C2×D7)⋊C4C22semidirect
47. C3×C4⋊Dic7C22semidirect
48. #66577C22semidirect
49. #66577C22semidirect
51. (C2×D3)⋊Dic7C22semidirect
52. (C2×D7)⋊Dic3C22semidirect
53. (C2×D7)⋊Dic3C22semidirect
58. #6493D3semidirect
85. #4268D4semidirect
110. C22×C4×C7C2×D3semidirect
118. C2×C22⋊D7C2×D3semidirect
122. C2×D28C2×D3semidirect
137. (C2×D7)⋊C4C2×D3semidirect
138. (C2×D7)⋊C4C2×D3semidirect
139. C4⋊Dic7C2×D3semidirect
152. C2×(C2×D3)⋊C4D7semidirect
172. C2×D3×C7#41semidirect
232. C22×C3×C4C2×D7semidirect
239. C22×Dic3C2×D7semidirect
258. (C2×D3)⋊C4C2×D7semidirect
259. (C2×D3)⋊C4C2×D7semidirect
355. C3×C7#546semidirect
375. C22×C4D3×D7semidirect
422. C2×D3#1157semidirect
469. C7#5610semidirect
509. C3#90181semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C22⋊D72
C2×#666832
C4×#42682
C2×C7×(C2×D3)⋊C42
C3×#64932
#106211302
#106210982
#106210362
#106210352
#106210262
#106210232
#106210042
#106210002
#106209982
#106209702
#901813
#56107