Finite Group Databasebeta

← #13062248

Subgroups — #13062248

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
1#10621657C211yesmaximal, characteristic, complemented
2C7×#1004C211yesmaximal, characteristic, complemented
3C2×C3×#1159C211yesmaximal, characteristic, complemented
4#10621596C214yesmaximal, complemented
5C2×#66686C214yesmaximal, complemented
6#10621574C214yesmaximal, complemented
7#10621571C214yesmaximal, complemented
8D3×#1159C214yesmaximal, complemented
9#10621567C214yesmaximal, complemented
10D7×#235C212yesmaximal, complemented
11#10621551C212yesmaximal, complemented
12#90618—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic, complemented
14C2×#4268C2214yescomplemented
15#66730C2214yescomplemented
16D4×D21C2214yescomplemented
17#66728C2212yescomplemented
18C2×C7×#67C2214yescomplemented
19C7×#237C2214yescomplemented
20D3×C7×D4C2214yescomplemented
21C7×#235C2212yescomplemented
22C2×C3×C22⋊D7C2212yescomplemented
23C22×C3×Dic7C2212yescomplemented
24C3×#1159C2214yescomplemented
25C3×#1159C2214yescomplemented
26C2×C3×C4×D7C2211yescharacteristic, complemented
27C2×C3×Dic14C2211yescharacteristic, complemented
28#66694C2218yescomplemented
29D3×C22⋊D7C2218yescomplemented
30D7×#67C2214yescomplemented
31C2×#4248C2214yescomplemented
32C2×#4247C2214yescomplemented
33C2×D21⋊C4C2214yescomplemented
34#66686C2218yescomplemented
35#66686C2214yescomplemented
36#66686C2218yescomplemented
37#66686C2218yescomplemented
38C2×D3×Dic7C2214yescomplemented
39#66684C2218yescomplemented
40#66683C2218yescomplemented
41D7×D12C2212yescomplemented
42C4×D3×D7C2214yescomplemented
43#66677C2212yescomplemented
44#66676C2212yescomplemented
45#66675C2214yescomplemented
46D21⋊Q8C2214yescomplemented
47#66673C2214yescomplemented
48#66672C2214yescomplemented
49D3×Dic14C2214yescomplemented
50D7×Dic6C2212yescomplemented
51C7×H(5,𝔽2)—31no—
52#6554—34no—
53#6554—32no—
54D7×#41—34no—
55D7×#41—32no—
56#6548—31no—
57#6548—34no—
58#6548—34no—
59C2×#1159—34no—
60C2×#1159—34no—
61C2×#1159D311yescharacteristic, complemented
62#5964—71nomaximal, Hall
63C23×C3×C7C2312yes—
64C22×D21C2314yes—
65C22×D3×C7C2314yes—
66C22×C3×D7C2311yescharacteristic
67C2×D3×D7C2314yes—
68C3×C7×D4C2314yes—
69C2×C3×C4×C7C2311yescharacteristic
70#4268C2318yes—
71#4268C2314yes—
72C2×Dic21C2314yes—
73D84C2312yes—
74C4×D21C2314yes—
75Dic42C2312yes—
76C7×#67C2318yes—
77C7×#67C2314yes—
78C2×C7×Dic3C2314yes—
79C7×D12C2312yes—
80C4×D3×C7C2314yes—
81C7×Dic6C2312yes—
82C3×C22⋊D7C2318yes—
83C2×C3×Dic7C2314yes—
84C2×C3×Dic7C2312yes—
85C2×C3×Dic7C2311yescharacteristic
86C2×C3×Dic7C2314yes—
87C3×C4×D7C2312yes—
88C3×C4×D7C2312yes—
89C3×Dic14C2314yes—
90#4248C2314yes—
91#4248C2318yes—
92#4247C2318yes—
93#4247C2314yes—
94#4246C2314yes—
95#4245C2314yes—
96D21⋊C4C2318yes—
97D21⋊C4C2314yes—
98D3×Dic7C2318yes—
99D3×Dic7C2314yes—
100Dic3×D7C2314yes—
101C7×#41—34no—
102C7×#41—32no—
103C2×C7×D4—34no—
104C2×C7×D4—34no—
105C2×C7×D4C2×D311yescharacteristic, complemented
106C2×C22⋊D7—34no—
107C2×C22⋊D7—38no—
108C2×C22⋊D7—34no—
109C2×C22⋊D7C2×D312yescomplemented
110C22×Dic7—34no—
111C22×Dic7C2×D312yescomplemented
112#1161—32no—
113Q8×D7—32no—
114Q8×D7—34no—
115#1159—34no—
116#1159—38no—
117#1159—38no—
118#1159—34no—
119#1159—38no—
120#1159—38no—
121#1159—32no—
122#1159—34no—
123#1159C2×D314yescomplemented
124#1159C2×D314yescomplemented
125D4×D7—34no—
126D4×D7—34no—
127D4×D7—38no—
128D4×D7—32no—
129#1157—32no—
130#1157—38no—
131#1157—34no—
132#1157—34no—
133C2×C4×D7—34no—
134C2×C4×D7—34no—
135C2×C4×D7C2×D311yescharacteristic, complemented
136C2×Dic14—34no—
137C2×Dic14—34no—
138C2×Dic14C2×D311yescharacteristic, complemented
139C2×C3×#41—71no—
140#1010—74no—
141#1009—74no—
142D3×#41—74no—
143D3×#41—74no—
144#1007—72no—
145#1004—71no—
146#1004—74no—
147#1004D711yescharacteristic, complemented
148C2×#235—74no—
149C2×#235—72no—
150C22×C3×C7—42no—
151C22×C3×C7C2411yescharacteristic
152C22×C3×C7C2414yes—
153C2×D21—44no—
154C2×D21C2414yes—
155C2×D3×C7—44no—
156C2×D3×C7C2414yes—
157C2×C3×D7—41no—
158C2×C3×D7C2412yes—
159D3×D7—44no—
160C3×C4×C7C2412yes—
161Dic21C2414yes—
162C3×Dic7C2411yescharacteristic
163C3×Dic7C2414yes—
164C3×Dic7C2411yescharacteristic
165C7×Dic3C2414yes—
166#585—211noSylow
167C23×C7—34no—
168C23×C7C22×D312yes—
169C22×D7—34no—
170C22×D7—34no—
171C22×D7C22×D311yescharacteristic
172C7×Q8—32no—
173C7×D4—38no—
174C7×D4—34no—
175C7×D4—32no—
176C7×D4C22×D314yes—
177C2×C4×C7—34no—
178C2×C4×C7—34no—
179C2×C4×C7C22×D311yescharacteristic
180C22⋊D7—38no—
181C22⋊D7—34no—
182C22⋊D7—34no—
183C22⋊D7—34no—
184C22⋊D7—38no—
185C22⋊D7C22×D318yes—
186C2×Dic7—34no—
187C2×Dic7—38no—
188C2×Dic7—34no—
189C2×Dic7C22×D314yes—
190C2×Dic7C22×D312yes—
191C2×Dic7C22×D314yes—
192C2×Dic7C22×D311yescharacteristic
193D28—34no—
194D28—32no—
195C4×D7—34no—
196C4×D7—34no—
197C4×D7—34no—
198C4×D7—38no—
199C4×D7C22×D312yes—
200C4×D7C22×D312yes—
201Dic14—32no—
202Dic14—34no—
203Dic14—38no—
204Dic14C22×D314yes—
205C3×#41—74no—
206C3×#41—74no—
207C2×C3×Q8—71no—
208C2×C3×D4—72no—
209C2×C3×D4C2×D711yescharacteristic, complemented
210C22×C3×C4—72no—
211C22×C3×C4—71no—
212C2×#67—74no—
213C2×#67—74no—
214C2×#67C2×D714yescomplemented
215#239—74no—
216#239—74no—
217D3×Q8—74no—
218D3×Q8—74no—
219#237—74no—
220#237—78no—
221#237—78no—
222#237C2×D714yescomplemented
223D3×D4—78no—
224D3×D4—74no—
225D3×D4—78no—
226D3×D4C2×D714yescomplemented
227#235—72no—
228#235—74no—
229#235—78no—
230#235—78no—
231#235—78no—
232#235—72no—
233#235—74no—
234#235—72no—
235#235C2×D712yescomplemented
236C2×D12—74no—
237C2×D12—72no—
238C2×C4×D3—74no—
239C2×C4×D3—74no—
240C2×C4×D3—74no—
241C2×Dic6—74no—
242C2×Dic6—72no—
243C2×C3×C7—21no—
244C2×C3×C7—24no—
245C2×C3×C7C2511yescharacteristic
246D21—24no—
247C3×D7—22no—
248D3×C7—24no—
2492−1+4—212no—
2502−1+4—214no—
251H(5,𝔽2)—31no—
252H(5,𝔽2)—214no—
253H(5,𝔽2)—214no—
254H(5,𝔽2)—211no—
255C2×#41—71no—
256C2×#41—214no—
257C2×#41—212no—
258C2×#41—214no—
259C2×#41—214no—
260C22×C7—34no—
261C22×C7—42no—
262C22×C7—124no—
263C22×C7C23×D314yes—
264C22×C7C23×D311yescharacteristic
265C2×D7—34no—
266C2×D7—41no—
267C2×D7—124no—
268C2×D7—124no—
269C2×D7C23×D312yes—
270C4×C7—34no—
271C4×C7C23×D312yes—
272Dic7—34no—
273Dic7C23×D311yescharacteristic
274Dic7C23×D314yes—
275Dic7C23×D311yescharacteristic
276C23×C3—71no—
277C23×C3C22×D712yes—
278C22×D3—74no—
279C22×D3—74no—
280C22×D3C22×D714yes—
281C3×Q8—74no—
282C3×D4—78no—
283C3×D4C22×D714yes—
284C2×C3×C4—74no—
285C2×C3×C4—74no—
286C2×C3×C4—71no—
287C2×C3×C4—72no—
288C2×C3×C4—72no—
289C2×C3×C4—72no—
290C2×C3×C4C22×D711yescharacteristic
291#67—74no—
292#67—78no—
293#67—74no—
294#67—74no—
295#67C22×D714yes—
296#67C22×D718yes—
297C2×Dic3—74no—
298C2×Dic3—74no—
299C2×Dic3C22×D714yes—
300D12—74no—
301D12—78no—
302D12—72no—
303D12C22×D712yes—
304C4×D3—74no—
305C4×D3—78no—
306C4×D3—74no—
307C4×D3—74no—
308C4×D3—78no—
309C4×D3C22×D714yes—
310Dic6—72no—
311Dic6—74no—
312Dic6—78no—
313Dic6C22×D712yes—
314C3×C7#58511yescharacteristic, complemented, Hall
315#41—34no—
316#41—32no—
317#41—74no—
318#41—74no—
319#41—214no—
320#41—218no—
321#41—214no—
322#41—214no—
323#41—218no—
324#41—212no—
325#41—218no—
326#41—218no—
327#41—212no—
328#41—218no—
329#41—214no—
330#41—214no—
331#41—212no—
332C2×Q8—71no—
333C2×Q8—214no—
334C2×Q8—214no—
335C2×Q8—212no—
336C2×Q8—214no—
337C2×D4—34no—
338C2×D4—34no—
339C2×D4—72no—
340C2×D4—212no—
341C2×D4—214no—
342C2×D4—218no—
343C2×D4—214no—
344C2×D4—218no—
345C2×D4—214no—
346C2×D4—214no—
347C2×D4D3×D711yescharacteristic, complemented
348C22×C4—72no—
349C22×C4—71no—
350C22×C4—214no—
351C22×C4—214no—
352C22×C4—214no—
353C2×C7—21no—
354C2×C7—24no—
355C2×C7—64no—
356C2×C7C24×D311yescharacteristic
357D7—22no—
358D7—64no—
359C22×C3—42no—
360C22×C3—72no—
361C22×C3—281no—
362C22×C3C23×D714yes—
363C22×C3C23×D711yescharacteristic
364C2×D3—44no—
365C2×D3—74no—
366C2×D3—284no—
367C2×D3—284no—
368C2×D3C23×D714yes—
369C3×C4—71no—
370C3×C4—74no—
371C3×C4—71no—
372C3×C4C23×D712yes—
373Dic3—74no—
374Dic3C23×D714yes—
375C23—34no—
376C23—71no—
377C23—214no—
378C23—214no—
379C23C2×D3×D712yes—
380Q8—32no—
381Q8—74no—
382Q8—214no—
383Q8—218no—
384Q8—212no—
385D4—38no—
386D4—34no—
387D4—32no—
388D4—78no—
389D4—214no—
390D4—214no—
391D4—218no—
392D4—214no—
393D4—218no—
394D4—214no—
395D4—212no—
396D4C2×D3×D714yes—
397C2×C4—34no—
398C2×C4—34no—
399C2×C4—71no—
400C2×C4—74no—
401C2×C4—72no—
402C2×C4—74no—
403C2×C4—72no—
404C2×C4—72no—
405C2×C4—214no—
406C2×C4—218no—
407C2×C4—214no—
408C2×C4—214no—
409C2×C4—218no—
410C2×C4—214no—
411C2×C4—214no—
412C2×C4C2×D3×D711yescharacteristic
413C7#596411yescharacteristic, complemented, Sylow
414C2×C3—24no—
415C2×C3—21no—
416C2×C3—142no—
417C2×C3C24×D711yescharacteristic
418D3—24no—
419D3—144no—
420C22—34no—
421C22—42no—
422C22—72no—
423C22—124no—
424C22—214no—
425C22—281no—
426C22—844no—
427C22—844no—
428C22C22×D3×D714yes—
429C22C22×D3×D711yescharacteristic
430C4—34no—
431C4—71no—
432C4—71no—
433C4—74no—
434C4—214no—
435C4C22×D3×D712yes—
436C3#9061811yescharacteristic, complemented, Sylow
437C2—21no—
438C2—24no—
439C2—64no—
440C2—142no—
441C2—424no—
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. #10621657C2semidirect
2. C7×#1004C2semidirect
3. C2×C3×#1159C2semidirect
4. #10621596C2semidirect
5. C2×#66686C2semidirect
6. #10621574C2semidirect
7. #10621571C2semidirect
8. D3×#1159C2semidirect
9. #10621567C2semidirect
10. D7×#235C2semidirect
11. #10621551C2semidirect
13. C2×C3×C7×D4C22semidirect
14. C2×#4268C22semidirect
15. #66730C22semidirect
16. D4×D21C22semidirect
17. #66728C22semidirect
18. C2×C7×#67C22semidirect
19. C7×#237C22semidirect
20. D3×C7×D4C22semidirect
21. C7×#235C22semidirect
22. C2×C3×C22⋊D7C22semidirect
23. C22×C3×Dic7C22semidirect
24. C3×#1159C22semidirect
25. C3×#1159C22semidirect
26. C2×C3×C4×D7C22semidirect
27. C2×C3×Dic14C22semidirect
28. #66694C22semidirect
29. D3×C22⋊D7C22semidirect
30. D7×#67C22semidirect
31. C2×#4248C22semidirect
32. C2×#4247C22semidirect
33. C2×D21⋊C4C22semidirect
34. #66686C22semidirect
35. #66686C22semidirect
36. #66686C22semidirect
37. #66686C22semidirect
38. C2×D3×Dic7C22semidirect
39. #66684C22semidirect
40. #66683C22semidirect
41. D7×D12C22semidirect
42. C4×D3×D7C22semidirect
43. #66677C22semidirect
44. #66676C22semidirect
45. #66675C22semidirect
46. D21⋊Q8C22semidirect
47. #66673C22semidirect
48. #66672C22semidirect
49. D3×Dic14C22semidirect
50. D7×Dic6C22semidirect
61. C2×#1159D3semidirect
105. C2×C7×D4C2×D3semidirect
109. C2×C22⋊D7C2×D3semidirect
111. C22×Dic7C2×D3semidirect
123. #1159C2×D3semidirect
124. #1159C2×D3semidirect
135. C2×C4×D7C2×D3semidirect
138. C2×Dic14C2×D3semidirect
147. #1004D7semidirect
209. C2×C3×D4C2×D7semidirect
214. C2×#67C2×D7semidirect
222. #237C2×D7semidirect
226. D3×D4C2×D7semidirect
235. #235C2×D7semidirect
314. C3×C7#585semidirect
347. C2×D4D3×D7semidirect
413. C7#5964semidirect
436. C3#90618semidirect

Maximal subgroups

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

subgroupindex
#106216572
C7×#10042
C2×C3×#11592
#106215962
C2×#666862
#106215742
#106215712
D3×#11592
#106215672
D7×#2352
#106215512
#906183
#59647