Finite Group Databasebeta

← #13058054

Subgroups — #13058054

Order 1344.

660 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66694C211yesmaximal, characteristic
2C2×D21⋊Q8C211yesmaximal, characteristic, complemented
3C22⋊C4×D21C211yesmaximal, characteristic, complemented
4C7×#885C211yesmaximal, characteristic, complemented
5C3×#6448C211yesmaximal, characteristic, complemented
6#10621123C211yesmaximal, characteristic, complemented
7#10621096C211yesmaximal, characteristic, complemented
8#10621087C211yesmaximal, characteristic, complemented
9#10621035C211yesmaximal, characteristic, complemented
10#10621035C211yesmaximal, characteristic, complemented
11#10621017C211yesmaximal, characteristic, complemented
12C4×D21⋊C4C211yesmaximal, characteristic, complemented
13#10620915C211yesmaximal, characteristic, complemented
14#10620912C211yesmaximal, characteristic, complemented
15#10620888C211yesmaximal, characteristic, complemented
16D7×#124—31nomaximal, Hall
17C23×D21C2211yescharacteristic
18C22×D3×D7C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C2×C4×D21C2211yescharacteristic
21C2×C7×#67C2211yescharacteristic
22C2×C7×Dic6C2211yescharacteristic, complemented
23C2×C3×C22⋊D7C2211yescharacteristic
24C2×C3×Dic14C2211yescharacteristic, complemented
25#66694—24no—
26C2×#4248C2211yescharacteristic, complemented
27C2×#4248C2211yescharacteristic, complemented
28C2×#4247C2211yescharacteristic
29C2×#4246C2211yescharacteristic
30C2×D21⋊C4C2211yescharacteristic
31C2×D21⋊C4C2211yescharacteristic
32C2×D21⋊C4C2211yescharacteristic
33D21⋊Q8—24no—
34C3×C7×C22⋊C4C2211yescharacteristic, complemented
35#66636C2211yescharacteristic, complemented
36#66631C2211yescharacteristic, complemented
37#66631C2211yescharacteristic, complemented
38C7×#217C2211yescharacteristic, complemented
39C7×(C2×D3)⋊C4C2211yescharacteristic, complemented
40C7×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C4×C7×Dic3C2211yescharacteristic, complemented
42C3×C22⋊Dic7C2211yescharacteristic, complemented
43C3×(C2×D7)⋊C4C2211yescharacteristic, complemented
44C3×(C2×D7)⋊C4C2211yescharacteristic, complemented
45C3×C4×Dic7C2211yescharacteristic, complemented
46#66575C2211yescharacteristic, complemented
47#66575C2211yescharacteristic, complemented
48(C2×D3)⋊Dic7C2211yescharacteristic, complemented
49(C2×D3)⋊Dic7C2211yescharacteristic, complemented
50(C2×D7)⋊Dic3C2211yescharacteristic, complemented
51(C2×D7)⋊Dic3C2211yescharacteristic, complemented
52Dic3×Dic7C2211yescharacteristic, complemented
53Dic3×Dic7C2211yescharacteristic, complemented
54C2×Q8×D7—31no—
55C2×D4×D7—31no—
56C7×#124—31no—
57#6510—31no—
58#6499—31no—
59#6448—31no—
60#6448—31no—
61#6448—31no—
62#6448D311yescharacteristic, complemented
63D7×C22⋊C4—31no—
64D7×C22⋊C4—31no—
65D7×C22⋊C4—31no—
66D7×C22⋊C4—31no—
67#6438—31no—
68C42×D7—31no—
69D3×#124—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C22×D21—22no—
72C22×D21—22no—
73C22×D21—22no—
74C22×D21C2311yescharacteristic
75C22×D21C2311yescharacteristic
76C22×D3×C7C2311yescharacteristic
77C22×C3×D7C2311yescharacteristic
78C2×D3×D7—22no—
79C2×D3×D7—22no—
80C2×D3×D7—22no—
81C2×C3×C4×C7C2311yescharacteristic
82C2×C3×C4×C7C2311yescharacteristic
83C2×Dic21C2311yescharacteristic
84C2×Dic21C2311yescharacteristic
85C4×D21—22no—
86C4×D21—22no—
87C7×#67—22no—
88C2×C7×Dic3C2311yescharacteristic
89C2×C7×Dic3C2311yescharacteristic
90C2×C7×Dic3C2311yescharacteristic
91C7×Dic6—22no—
92C3×C22⋊D7—22no—
93C2×C3×Dic7C2311yescharacteristic
94C2×C3×Dic7C2311yescharacteristic
95C2×C3×Dic7C2311yescharacteristic
96C3×Dic14—22no—
97#4248—22no—
98#4248—22no—
99#4247—22no—
100#4246—22no—
101D21⋊C4—22no—
102D21⋊C4—22no—
103D21⋊C4D414yes—
104C23×D7—31no—
105C23×D7—31no—
106C2×C7×Q8—31no—
107C2×C7×D4—31no—
108C2×C22⋊D7—31no—
109C2×C22⋊D7C2×D311yescharacteristic
110Q8×D7—64no—
111D4×D7—64no—
112C2×D28—31no—
113C2×C4×D7—31no—
114C2×C4×D7—31no—
115C2×C4×D7—31no—
116C2×C4×D7—31no—
117C2×C4×D7—31no—
118C2×Dic14—31no—
119C2×Dic14—31no—
120C2×Dic14C2×D311yescharacteristic, complemented
121C7×C22⋊C4—31no—
122C7×C22⋊C4—31no—
123C7×C22⋊C4—31no—
124C7×C22⋊C4C2×D311yescharacteristic, complemented
125C42×C7—31no—
126C22⋊Dic7—31no—
127C22⋊Dic7—31no—
128C22⋊Dic7—31no—
129C22⋊Dic7C2×D311yescharacteristic, complemented
130(C2×D7)⋊C4—31no—
131(C2×D7)⋊C4—31no—
132(C2×D7)⋊C4—31no—
133(C2×D7)⋊C4—31no—
134(C2×D7)⋊C4—31no—
135(C2×D7)⋊C4—31no—
136(C2×D7)⋊C4C2×D311yescharacteristic, complemented
137(C2×D7)⋊C4C2×D311yescharacteristic, complemented
138C4×Dic7—31no—
139C4×Dic7—31no—
140C4×Dic7C2×D311yescharacteristic, complemented
141C2×D3×Q8—71no—
142C2×D3×D4—71no—
143C3×#124—71no—
144#947—71no—
145#936—71no—
146#885—71no—
147#885—71no—
148#885—71no—
149#885D711yescharacteristic, complemented
150D3×C22⋊C4—71no—
151D3×C22⋊C4—71no—
152D3×C22⋊C4—71no—
153D3×C22⋊C4—71no—
154#875—71no—
155C42×D3—71no—
156C22×C3×C7—21no—
157C22×C3×C7—21no—
158C22×C3×C7—21no—
159C22×C3×C7C2411yescharacteristic
160C2×D21—21no—
161C2×D21—21no—
162C2×D21—21no—
163C2×D21—44no—
164C2×D21C2×D412yes—
165C2×D21#4112yes—
166C2×D21#4112yes—
167C2×D3×C7—21no—
168C2×D3×C7—21no—
169C2×D3×C7—21no—
170C2×C3×D7—21no—
171C2×C3×D7—21no—
172C2×C3×D7—21no—
173D3×D7—44no—
174C3×C4×C7—21no—
175C3×C4×C7—21no—
176Dic21—21no—
177Dic21—21no—
178C3×Dic7—21no—
179C3×Dic7—21no—
180C3×Dic7C2×D412yes—
181C7×Dic3—21no—
182C7×Dic3—21no—
183C7×Dic3C2×D412yes—
184C2×#124—211noSylow
185C23×C7—31no—
186C23×C7C22×D311yescharacteristic
187C22×D7—31no—
188C22×D7—31no—
189C22×D7—62no—
190C22×D7—62no—
191C22×D7—62no—
192C22×D7—62no—
193C22×D7—62no—
194C22×D7—62no—
195C22×D7C22×D311yescharacteristic
196C7×Q8—62no—
197C7×D4—62no—
198C2×C4×C7—31no—
199C2×C4×C7—31no—
200C2×C4×C7—31no—
201C2×C4×C7C22×D311yescharacteristic
202C2×C4×C7C22×D311yescharacteristic
203C22⋊D7—22no—
204C22⋊D7—62no—
205C2×Dic7—31no—
206C2×Dic7—31no—
207C2×Dic7C22×D311yescharacteristic
208C2×Dic7C22×D311yescharacteristic
209C2×Dic7C22×D311yescharacteristic
210D28—62no—
211C4×D7—34no—
212C4×D7—62no—
213C4×D7—62no—
214C4×D7—62no—
215C4×D7—62no—
216Dic14—22no—
217Dic14—62no—
218Dic14—62no—
219C23×D3—71no—
220C23×D3—71no—
221C2×C3×Q8—71no—
222C2×C3×D4—71no—
223C2×#67—71no—
224C2×#67C2×D711yescharacteristic
225D3×Q8—144no—
226D3×D4—144no—
227C2×D12—71no—
228C2×C4×D3—71no—
229C2×C4×D3—71no—
230C2×C4×D3—71no—
231C2×C4×D3—71no—
232C2×C4×D3—71no—
233C2×Dic6—71no—
234C2×Dic6—71no—
235C2×Dic6C2×D711yescharacteristic, complemented
236C3×C22⋊C4—71no—
237C3×C22⋊C4—71no—
238C3×C22⋊C4—71no—
239C3×C22⋊C4C2×D711yescharacteristic, complemented
240C3×C42—71no—
241#217—71no—
242#217—71no—
243#217—71no—
244#217C2×D711yescharacteristic, complemented
245(C2×D3)⋊C4—71no—
246(C2×D3)⋊C4—71no—
247(C2×D3)⋊C4—71no—
248(C2×D3)⋊C4—71no—
249(C2×D3)⋊C4—71no—
250(C2×D3)⋊C4—71no—
251(C2×D3)⋊C4C2×D711yescharacteristic, complemented
252(C2×D3)⋊C4C2×D711yescharacteristic, complemented
253C4×Dic3—71no—
254C4×Dic3—71no—
255C4×Dic3C2×D711yescharacteristic, complemented
256C2×C3×C7—41no—
257C2×C3×C7C22×D411yescharacteristic
258C2×C3×C7C2×#4111yescharacteristic
259C2×C3×C7C2×#4111yescharacteristic
260D21—41no—
261D21#12414yescomplemented
262C3×D7—41no—
263D3×C7—41no—
264C22×Q8—211no—
265C22×D4—211no—
266#124—31no—
267#124—71no—
268#124—211no—
269#124—211no—
270#124—211no—
271#124—211no—
272#124—211no—
273#124—211no—
274C2×C22⋊C4—211no—
275C2×C22⋊C4—211no—
276C2×C22⋊C4—211no—
277C2×C22⋊C4—211no—
278C2×C42—211no—
279C22×C7—21no—
280C22×C7—21no—
281C22×C7—21no—
282C22×C7—61no—
283C22×C7—61no—
284C22×C7—61no—
285C22×C7C23×D311yescharacteristic
286C2×D7—21no—
287C2×D7—21no—
288C2×D7—21no—
289C2×D7—32no—
290C2×D7—32no—
291C2×D7—32no—
292C2×D7—61no—
293C2×D7—61no—
294C2×D7—61no—
295C2×D7—124no—
296C2×D7—124no—
297C4×C7—21no—
298C4×C7—21no—
299C4×C7—32no—
300C4×C7—61no—
301C4×C7—61no—
302Dic7—21no—
303Dic7—21no—
304Dic7—61no—
305Dic7—61no—
306Dic7D3×D412yes—
307C23×C3—71no—
308C23×C3C22×D711yescharacteristic
309C22×D3—71no—
310C22×D3—71no—
311C22×D3—142no—
312C22×D3—142no—
313C22×D3—142no—
314C22×D3—142no—
315C22×D3—142no—
316C22×D3—142no—
317C22×D3C22×D711yescharacteristic
318C3×Q8—142no—
319C3×D4—142no—
320C2×C3×C4—71no—
321C2×C3×C4—71no—
322C2×C3×C4—71no—
323C2×C3×C4C22×D711yescharacteristic
324C2×C3×C4C22×D711yescharacteristic
325#67—22no—
326#67—142no—
327C2×Dic3—71no—
328C2×Dic3—71no—
329C2×Dic3C22×D711yescharacteristic
330C2×Dic3C22×D711yescharacteristic
331C2×Dic3C22×D711yescharacteristic
332D12—142no—
333C4×D3—74no—
334C4×D3—142no—
335C4×D3—142no—
336C4×D3—142no—
337C4×D3—142no—
338Dic6—22no—
339Dic6—142no—
340Dic6—142no—
341C3×C7C2×#12411yescharacteristic, complemented, Hall
342C24—211no—
343C24—211no—
344C2×Q8—31no—
345C2×Q8—71no—
346C2×Q8—211no—
347C2×Q8—211no—
348C2×Q8—424no—
349C2×D4—31no—
350C2×D4—71no—
351C2×D4—211no—
352C2×D4—211no—
353C2×D4—424no—
354C22×C4—211no—
355C22×C4—211no—
356C22×C4—211no—
357C22×C4—211no—
358C22×C4—211no—
359C22⋊C4—31no—
360C22⋊C4—31no—
361C22⋊C4—31no—
362C22⋊C4—71no—
363C22⋊C4—71no—
364C22⋊C4—71no—
365C22⋊C4—211no—
366C22⋊C4—211no—
367C22⋊C4—211no—
368C22⋊C4—211no—
369C22⋊C4—211no—
370C22⋊C4—211no—
371C22⋊C4—211no—
372C22⋊C4—211no—
373C22⋊C4—211no—
374C22⋊C4D3×D711yescharacteristic, complemented
375C42—31no—
376C42—71no—
377C42—211no—
378C42—211no—
379C2×C7—41no—
380C2×C7—121no—
381C2×C7C2×D3×D411yescharacteristic
382C2×C7D3×#4111yescharacteristic
383C2×C7D3×#4111yescharacteristic
384D7—34no—
385D7—41no—
386D7—121no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—141no—
391C22×C3—141no—
392C22×C3—141no—
393C22×C3C23×D711yescharacteristic
394C2×D3—21no—
395C2×D3—21no—
396C2×D3—21no—
397C2×D3—72no—
398C2×D3—72no—
399C2×D3—72no—
400C2×D3—141no—
401C2×D3—141no—
402C2×D3—141no—
403C2×D3—284no—
404C2×D3—284no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—72no—
408C3×C4—141no—
409C3×C4—141no—
410Dic3—21no—
411Dic3—21no—
412Dic3—141no—
413Dic3—141no—
414Dic3D4×D712yes—
415C23—31no—
416C23—71no—
417C23—211no—
418C23—211no—
419C23—422no—
420C23—422no—
421C23—422no—
422C23—422no—
423C23—422no—
424C23—422no—
425C23C2×D3×D711yescharacteristic
426Q8—62no—
427Q8—142no—
428Q8—422no—
429Q8—422no—
430D4—62no—
431D4—142no—
432D4—422no—
433D4—422no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—71no—
438C2×C4—71no—
439C2×C4—71no—
440C2×C4—214no—
441C2×C4—211no—
442C2×C4—211no—
443C2×C4—422no—
444C2×C4—422no—
445C2×C4—422no—
446C2×C4—422no—
447C2×C4C2×D3×D711yescharacteristic
448C2×C4C2×D3×D711yescharacteristic
449C7D3×#12411yescharacteristic, complemented, Sylow
450C2×C3—41no—
451C2×C3—281no—
452C2×C3C2×D4×D711yescharacteristic
453C2×C3D7×#4111yescharacteristic
454C2×C3D7×#4111yescharacteristic
455D3—41no—
456D3—74no—
457D3—281no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—61no—
462C22—61no—
463C22—61no—
464C22—141no—
465C22—141no—
466C22—141no—
467C22—212no—
468C22—212no—
469C22—212no—
470C22—421no—
471C22—421no—
472C22—421no—
473C22—844no—
474C22—844no—
475C22C22×D3×D711yescharacteristic
476C4—21no—
477C4—21no—
478C4—32no—
479C4—61no—
480C4—61no—
481C4—72no—
482C4—141no—
483C4—141no—
484C4—421no—
485C4—421no—
486C3D7×#12411yescharacteristic, complemented, Sylow
487C2—41no—
488C2—121no—
489C2—214no—
490C2—281no—
491C2—841no—
492C2#1062156311yescharacteristic
493C2D3×D4×D711yescharacteristic
494C2#1062157111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radicalC2×#66694C2
hypercenterC22⋊C4D3×D7
hypocenterC3×C7C2×#124
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×D21⋊Q8C2semidirect
3. C22⋊C4×D21C2semidirect
4. C7×#885C2semidirect
5. C3×#6448C2semidirect
6. #10621123C2semidirect
7. #10621096C2semidirect
8. #10621087C2semidirect
9. #10621035C2semidirect
10. #10621035C2semidirect
11. #10621017C2semidirect
12. C4×D21⋊C4C2semidirect
13. #10620915C2semidirect
14. #10620912C2semidirect
15. #10620888C2semidirect
22. C2×C7×Dic6C22semidirect
24. C2×C3×Dic14C22semidirect
26. C2×#4248C22semidirect
27. C2×#4248C22semidirect
34. C3×C7×C22⋊C4C22semidirect
35. #66636C22semidirect
36. #66631C22semidirect
37. #66631C22semidirect
38. C7×#217C22semidirect
39. C7×(C2×D3)⋊C4C22semidirect
40. C7×(C2×D3)⋊C4C22semidirect
41. C4×C7×Dic3C22semidirect
42. C3×C22⋊Dic7C22semidirect
43. C3×(C2×D7)⋊C4C22semidirect
44. C3×(C2×D7)⋊C4C22semidirect
45. C3×C4×Dic7C22semidirect
46. #66575C22semidirect
47. #66575C22semidirect
48. (C2×D3)⋊Dic7C22semidirect
49. (C2×D3)⋊Dic7C22semidirect
50. (C2×D7)⋊Dic3C22semidirect
51. (C2×D7)⋊Dic3C22semidirect
52. Dic3×Dic7C22semidirect
53. Dic3×Dic7C22semidirect
62. #6448D3semidirect
120. C2×Dic14C2×D3semidirect
124. C7×C22⋊C4C2×D3semidirect
129. C22⋊Dic7C2×D3semidirect
136. (C2×D7)⋊C4C2×D3semidirect
137. (C2×D7)⋊C4C2×D3semidirect
140. C4×Dic7C2×D3semidirect
149. #885D7semidirect
235. C2×Dic6C2×D7semidirect
239. C3×C22⋊C4C2×D7semidirect
244. #217C2×D7semidirect
251. (C2×D3)⋊C4C2×D7semidirect
252. (C2×D3)⋊C4C2×D7semidirect
255. C4×Dic3C2×D7semidirect
261. D21#124semidirect
341. C3×C7C2×#124semidirect
374. C22⋊C4D3×D7semidirect
449. C7D3×#124semidirect
486. C3D7×#124semidirect

Maximal subgroups

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

subgroupindex
C2×#666942
C2×D21⋊Q82
C22⋊C4×D212
C7×#8852
C3×#64482
#106211232
#106210962
#106210872
#106210352
#106210172
C4×D21⋊C42
#106209152
#106209122
#106208882
D7×#1243
D3×#1247