Finite Group Databasebeta

← #13062250

Subgroups — #13062250

Order 1344.

1620 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66730C211yesmaximal, characteristic, complemented
2C7×#1004C211yesmaximal, characteristic, complemented
3C3×#6548C211yesmaximal, characteristic, complemented
4#10621596C214yesmaximal, complemented
5C2×#66684C214yesmaximal, complemented
6#10621572C214yesmaximal, complemented
7D3×#1159C214yesmaximal, complemented
8D7×#237C214yesmaximal, complemented
9#10621567C214yesmaximal, complemented
10#10621563C212yesmaximal, complemented
11#10621548C212yesmaximal, complemented
12#90618—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic, complemented
14C2×#4268C2212yescomplemented
15C22×Dic21C2212yescomplemented
16#66730C2214yescomplemented
17#66730C2214yescomplemented
18C2×C4×D21C2211yescharacteristic, complemented
19C2×Dic42C2211yescharacteristic, complemented
20C2×C7×#67C2214yescomplemented
21C7×#237C2214yescomplemented
22D3×C7×D4C2214yescomplemented
23C7×#235C2212yescomplemented
24C2×C3×C22⋊D7C2214yescomplemented
25C3×#1159C2214yescomplemented
26C3×D4×D7C2214yescomplemented
27C3×#1157C2212yescomplemented
28#66694C2214yescomplemented
29D3×C22⋊D7C2218yescomplemented
30D7×#67C2218yescomplemented
31C2×#4248C2214yescomplemented
32C2×#4245C2214yescomplemented
33#66686C2218yescomplemented
34C2×D3×Dic7C2214yescomplemented
35#66684C2218yescomplemented
36#66684C2214yescomplemented
37#66684C2218yescomplemented
38#66684C2218yescomplemented
39#66683C2218yescomplemented
40C2×Dic3×D7C2214yescomplemented
41#66681C2212yescomplemented
42C4×D3×D7C2214yescomplemented
43#66676C2214yescomplemented
44#66675C2214yescomplemented
45D21⋊Q8C2212yescomplemented
46#66672C2212yescomplemented
47D3×Dic14C2214yescomplemented
48#66670C2212yescomplemented
49#66669C2214yescomplemented
50D7×Dic6C2214yescomplemented
51C7×H(5,𝔽2)—31no—
52#6554—34no—
53#6554—32no—
54D7×#41—34no—
55D7×#41—32no—
56#6548—34no—
57#6548—34no—
58#6548D311yescharacteristic, complemented
59C2×#1159—31no—
60C2×#1159—34no—
61C2×#1159—34no—
62#5966—71nomaximal, Hall
63C23×C3×C7C2312yes—
64C22×D21C2311yescharacteristic
65C22×D3×C7C2314yes—
66C22×C3×D7C2314yes—
67C2×D3×D7C2314yes—
68C3×C7×D4C2314yes—
69C2×C3×C4×C7C2311yescharacteristic
70#4268C2318yes—
71C2×Dic21C2314yes—
72C2×Dic21C2312yes—
73C2×Dic21C2311yescharacteristic
74C2×Dic21C2314yes—
75C4×D21C2312yes—
76C4×D21C2312yes—
77Dic42C2314yes—
78C7×#67C2314yes—
79C7×#67C2318yes—
80C2×C7×Dic3C2314yes—
81C7×D12C2312yes—
82C4×D3×C7C2314yes—
83C7×Dic6C2312yes—
84C3×C22⋊D7C2318yes—
85C3×C22⋊D7C2314yes—
86C2×C3×Dic7C2314yes—
87C3×D28C2312yes—
88C3×C4×D7C2314yes—
89C3×Dic14C2312yes—
90#4248C2318yes—
91#4248C2314yes—
92#4247C2314yes—
93#4246C2314yes—
94#4245C2318yes—
95#4245C2314yes—
96D21⋊C4C2314yes—
97D3×Dic7C2318yes—
98D3×Dic7C2314yes—
99Dic3×D7C2318yes—
100Dic3×D7C2314yes—
101C7×#41—34no—
102C7×#41—32no—
103C2×C7×D4—34no—
104C2×C7×D4—34no—
105C2×C7×D4C2×D311yescharacteristic, complemented
106C2×C22⋊D7—32no—
107C2×C22⋊D7—34no—
108C2×C22⋊D7—38no—
109C2×C22⋊D7C2×D314yescomplemented
110C22×Dic7—32no—
111C22×Dic7—34no—
112#1161—32no—
113Q8×D7—34no—
114Q8×D7—32no—
115#1159—34no—
116#1159—34no—
117#1159—38no—
118#1159—34no—
119#1159—38no—
120#1159—38no—
121#1159—38no—
122#1159—34no—
123#1159—32no—
124#1159C2×D314yescomplemented
125D4×D7—38no—
126D4×D7—34no—
127D4×D7—32no—
128D4×D7C2×D314yescomplemented
129#1157—38no—
130#1157—34no—
131#1157—34no—
132#1157C2×D312yescomplemented
133C2×C4×D7—34no—
134C2×C4×D7—31no—
135C2×C4×D7—34no—
136C2×Dic14—31no—
137C2×Dic14—34no—
138C2×Dic14—34no—
139C3×H(5,𝔽2)—71no—
140#1010—74no—
141#1010—72no—
142D3×#41—74no—
143D3×#41—72no—
144#1004—74no—
145#1004—74no—
146#1004D711yescharacteristic, complemented
147C2×#237—71no—
148C2×#237—74no—
149C2×#237—74no—
150C22×C3×C7—42no—
151C22×C3×C7C2414yes—
152C22×C3×C7C2411yescharacteristic
153C2×D21—41no—
154C2×D21C2412yes—
155C2×D3×C7—44no—
156C2×D3×C7C2414yes—
157C2×C3×D7—44no—
158C2×C3×D7C2414yes—
159D3×D7—44no—
160C3×C4×C7C2412yes—
161Dic21C2411yescharacteristic
162Dic21C2414yes—
163Dic21C2411yescharacteristic
164C3×Dic7C2414yes—
165C7×Dic3C2414yes—
166#585—211noSylow
167C23×C7—34no—
168C23×C7C22×D312yes—
169C22×D7—31no—
170C22×D7—34no—
171C22×D7C22×D314yes—
172C7×Q8—32no—
173C7×D4—34no—
174C7×D4—38no—
175C7×D4—32no—
176C7×D4C22×D314yes—
177C2×C4×C7—34no—
178C2×C4×C7—34no—
179C2×C4×C7C22×D311yescharacteristic
180C22⋊D7—38no—
181C22⋊D7—38no—
182C22⋊D7—34no—
183C22⋊D7—34no—
184C22⋊D7C22×D318yes—
185C22⋊D7C22×D314yes—
186C2×Dic7—32no—
187C2×Dic7—34no—
188C2×Dic7—31no—
189C2×Dic7—34no—
190C2×Dic7—38no—
191C2×Dic7—34no—
192C2×Dic7C22×D314yes—
193D28—34no—
194D28C22×D312yes—
195C4×D7—34no—
196C4×D7—38no—
197C4×D7—32no—
198C4×D7—32no—
199C4×D7—34no—
200C4×D7C22×D314yes—
201Dic14—34no—
202Dic14—34no—
203Dic14—38no—
204Dic14C22×D312yes—
205C3×#41—74no—
206C3×#41—72no—
207C2×C3×D4—74no—
208C2×C3×D4—74no—
209C2×C3×D4C2×D711yescharacteristic, complemented
210C2×#67—72no—
211C2×#67—74no—
212C2×#67—78no—
213C2×#67C2×D714yescomplemented
214C22×Dic3—72no—
215C22×Dic3—74no—
216#239—72no—
217D3×Q8—74no—
218D3×Q8—72no—
219#237—74no—
220#237—74no—
221#237—78no—
222#237—74no—
223#237—78no—
224#237—78no—
225#237—78no—
226#237—74no—
227#237—72no—
228#237C2×D714yescomplemented
229D3×D4—78no—
230D3×D4—74no—
231D3×D4—72no—
232D3×D4C2×D714yescomplemented
233#235—78no—
234#235—74no—
235#235—74no—
236#235C2×D712yescomplemented
237C2×C4×D3—74no—
238C2×C4×D3—71no—
239C2×C4×D3—74no—
240C2×Dic6—71no—
241C2×Dic6—74no—
242C2×Dic6—74no—
243C2×C3×C7—24no—
244C2×C3×C7—21no—
245C2×C3×C7C2511yescharacteristic
246D21—22no—
247C3×D7—24no—
248D3×C7—24no—
2492−1+4—214no—
2502−1+4—212no—
251H(5,𝔽2)—31no—
252H(5,𝔽2)—71no—
253H(5,𝔽2)—214no—
254H(5,𝔽2)—214no—
255C2×#41—214no—
256C2×#41—214no—
257C2×#41—214no—
258C2×#41—212no—
259C2×#41—211no—
260C22×C7—34no—
261C22×C7—42no—
262C22×C7—124no—
263C22×C7C23×D311yescharacteristic
264C22×C7C23×D314yes—
265C2×D7—32no—
266C2×D7—44no—
267C2×D7—124no—
268C2×D7—121no—
269C2×D7C23×D314yes—
270C4×C7—34no—
271C4×C7C23×D312yes—
272Dic7—34no—
273Dic7—31no—
274Dic7—31no—
275Dic7C23×D314yes—
276C23×C3—74no—
277C23×C3C22×D712yes—
278C22×D3—71no—
279C22×D3—74no—
280C22×D3C22×D714yes—
281C3×Q8—72no—
282C3×D4—74no—
283C3×D4—78no—
284C3×D4—72no—
285C3×D4C22×D714yes—
286C2×C3×C4—74no—
287C2×C3×C4—74no—
288C2×C3×C4C22×D711yescharacteristic
289#67—78no—
290#67—78no—
291#67—74no—
292#67—74no—
293#67C22×D718yes—
294#67C22×D714yes—
295C2×Dic3—72no—
296C2×Dic3—74no—
297C2×Dic3—71no—
298C2×Dic3—74no—
299C2×Dic3—78no—
300C2×Dic3—74no—
301C2×Dic3C22×D714yes—
302D12—74no—
303D12C22×D712yes—
304C4×D3—74no—
305C4×D3—78no—
306C4×D3—72no—
307C4×D3—72no—
308C4×D3—74no—
309C4×D3C22×D714yes—
310Dic6—74no—
311Dic6—78no—
312Dic6—74no—
313Dic6C22×D712yes—
314C3×C7#58511yescharacteristic, complemented, Hall
315#41—34no—
316#41—32no—
317#41—74no—
318#41—72no—
319#41—214no—
320#41—218no—
321#41—214no—
322#41—218no—
323#41—214no—
324#41—214no—
325#41—218no—
326#41—218no—
327#41—218no—
328#41—212no—
329#41—212no—
330#41—214no—
331#41—214no—
332C2×Q8—214no—
333C2×Q8—214no—
334C2×Q8—214no—
335C2×Q8—212no—
336C2×Q8—211no—
337C2×D4—34no—
338C2×D4—34no—
339C2×D4—74no—
340C2×D4—74no—
341C2×D4—214no—
342C2×D4—212no—
343C2×D4—218no—
344C2×D4—218no—
345C2×D4—214no—
346C2×D4—212no—
347C2×D4D3×D711yescharacteristic, complemented
348C22×C4—214no—
349C22×C4—214no—
350C22×C4—214no—
351C22×C4—212no—
352C22×C4—211no—
353C2×C7—24no—
354C2×C7—21no—
355C2×C7—64no—
356C2×C7C24×D311yescharacteristic
357D7—24no—
358D7—62no—
359C22×C3—42no—
360C22×C3—74no—
361C22×C3—284no—
362C22×C3C23×D711yescharacteristic
363C22×C3C23×D714yes—
364C2×D3—44no—
365C2×D3—72no—
366C2×D3—284no—
367C2×D3—281no—
368C2×D3C23×D714yes—
369C3×C4—74no—
370C3×C4C23×D712yes—
371Dic3—74no—
372Dic3—71no—
373Dic3—71no—
374Dic3C23×D714yes—
375C23—34no—
376C23—74no—
377C23—214no—
378C23—211no—
379C23C2×D3×D712yes—
380Q8—32no—
381Q8—72no—
382Q8—214no—
383Q8—218no—
384Q8—214no—
385D4—38no—
386D4—34no—
387D4—32no—
388D4—78no—
389D4—74no—
390D4—72no—
391D4—214no—
392D4—214no—
393D4—214no—
394D4—218no—
395D4—218no—
396D4C2×D3×D714yes—
397C2×C4—34no—
398C2×C4—34no—
399C2×C4—74no—
400C2×C4—74no—
401C2×C4—214no—
402C2×C4—214no—
403C2×C4—218no—
404C2×C4—214no—
405C2×C4—218no—
406C2×C4—214no—
407C2×C4—211no—
408C2×C4—214no—
409C2×C4—212no—
410C2×C4—212no—
411C2×C4—212no—
412C2×C4C2×D3×D711yescharacteristic
413C7#596611yescharacteristic, complemented, Sylow
414C2×C3—24no—
415C2×C3—21no—
416C2×C3—144no—
417C2×C3C24×D711yescharacteristic
418D3—24no—
419D3—142no—
420C22—34no—
421C22—42no—
422C22—74no—
423C22—124no—
424C22—212no—
425C22—284no—
426C22—841no—
427C22—844no—
428C22C22×D3×D714yes—
429C22C22×D3×D711yescharacteristic
430C4—34no—
431C4—74no—
432C4—211no—
433C4—214no—
434C4—211no—
435C4C22×D3×D712yes—
436C3#9061811yescharacteristic, complemented, Sylow
437C2—21no—
438C2—24no—
439C2—64no—
440C2—144no—
441C2—422no—
442C2C23×D3×D711yescharacteristic
subgroupG/H
center, FrattiniC2C23×D3×D7
commutator, socle, cosocleC2×C3×C7C25
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#585
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66730C2semidirect
2. C7×#1004C2semidirect
3. C3×#6548C2semidirect
4. #10621596C2semidirect
5. C2×#66684C2semidirect
6. #10621572C2semidirect
7. D3×#1159C2semidirect
8. D7×#237C2semidirect
9. #10621567C2semidirect
10. #10621563C2semidirect
11. #10621548C2semidirect
13. C2×C3×C7×D4C22semidirect
14. C2×#4268C22semidirect
15. C22×Dic21C22semidirect
16. #66730C22semidirect
17. #66730C22semidirect
18. C2×C4×D21C22semidirect
19. C2×Dic42C22semidirect
20. C2×C7×#67C22semidirect
21. C7×#237C22semidirect
22. D3×C7×D4C22semidirect
23. C7×#235C22semidirect
24. C2×C3×C22⋊D7C22semidirect
25. C3×#1159C22semidirect
26. C3×D4×D7C22semidirect
27. C3×#1157C22semidirect
28. #66694C22semidirect
29. D3×C22⋊D7C22semidirect
30. D7×#67C22semidirect
31. C2×#4248C22semidirect
32. C2×#4245C22semidirect
33. #66686C22semidirect
34. C2×D3×Dic7C22semidirect
35. #66684C22semidirect
36. #66684C22semidirect
37. #66684C22semidirect
38. #66684C22semidirect
39. #66683C22semidirect
40. C2×Dic3×D7C22semidirect
41. #66681C22semidirect
42. C4×D3×D7C22semidirect
43. #66676C22semidirect
44. #66675C22semidirect
45. D21⋊Q8C22semidirect
46. #66672C22semidirect
47. D3×Dic14C22semidirect
48. #66670C22semidirect
49. #66669C22semidirect
50. D7×Dic6C22semidirect
58. #6548D3semidirect
105. C2×C7×D4C2×D3semidirect
109. C2×C22⋊D7C2×D3semidirect
124. #1159C2×D3semidirect
128. D4×D7C2×D3semidirect
132. #1157C2×D3semidirect
146. #1004D7semidirect
209. C2×C3×D4C2×D7semidirect
213. C2×#67C2×D7semidirect
228. #237C2×D7semidirect
232. D3×D4C2×D7semidirect
236. #235C2×D7semidirect
314. C3×C7#585semidirect
347. C2×D4D3×D7semidirect
413. C7#5966semidirect
436. C3#90618semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C7×#10042
C3×#65482
#106215962
C2×#666842
#106215722
D3×#11592
D7×#2372
#106215672
#106215632
#106215482
#906183
#59667