Finite Group Databasebeta

← C3×C7×#5077 #72246832

Subgroups — C3×C7×#5077 #72246832

Order 4032.

780 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×#833C211yesmaximal, characteristic, complemented
2C3×C7×#805C212yesmaximal, complemented
3C3×C7×#1004C211yesmaximal, characteristic, complemented
4C32×C7×#127C211yesmaximal, characteristic, complemented
5C3×C7×#932C212yesmaximal, complemented
6C3×C7×#453—31nomaximal
7C7×#5077C311yesmaximal, characteristic, direct product factor
8C2×C32×C7×D4—22no—
9C2×C32×C7×D4C2211yescharacteristic
10C2×C32×C7×D4C2211yescharacteristic, complemented
11C2×C3×C7×#67—22no—
12C3×C7×#237—22no—
13C3×D3×C7×D4—22no—
14C3×C7×#235C2212yes—
15C32×C42×C7C2211yescharacteristic, complemented
16C3×C7×#214—22no—
17C3×C7×#213—22no—
18C3×C7×#208C2212yescomplemented
19C3×C7×H(5,𝔽2)—31no—
20C7×#1004C2×C311yescharacteristic, complemented
21C3×C7×AGL(1,ℤ/8)—32no—
22C3×C7×#127—21no—
23C3×C7×#127D311yescharacteristic, complemented
24C3×C7×#127C2×C311yescharacteristic, complemented
25C7×#932C2×C312yescomplemented
26C3×C7×#104—32no—
27C3×C7×#100—31no—
28C7×#833C2×C311yescharacteristic, complemented
29C7×#805C2×C312yescomplemented
30C3×#5077C711yesmaximal, characteristic, direct product factor, Hall
31C23×C32×C7—21no—
32C23×C32×C7D412yes—
33C22×C3×D3×C7—22no—
34C32×C7×D4—22no—
35C32×C7×D4—22no—
36C32×C7×D4—44no—
37C2×C32×C4×C7—21no—
38C2×C32×C4×C7C2311yescharacteristic
39C3×C7×#67—24no—
40C3×C7×#67—22no—
41C2×C3×C7×Dic3—22no—
42C3×C7×D12D412yescomplemented
43C3×C4×D3×C7—22no—
44C3×C7×Dic6D412yescomplemented
45C3×C7×C3⋊C8—22no—
46C7×#453—31noHall
47C3×C7×#41—32no—
48C3×C7×#41—62no—
49C2×C3×C7×D4—22no—
50C2×C3×C7×D4—22no—
51C2×C3×C7×D4—22no—
52C2×C3×C7×D4—22no—
53C2×C3×C7×D4—21no—
54C2×C3×C7×D4—21no—
55C2×C3×C7×D4—62no—
56C2×C3×C7×D4—62no—
57C2×C3×C7×D4C2×D311yescharacteristic
58C2×C3×C7×D4C2×D311yescharacteristic, complemented
59C2×C3×C7×D4C22×C311yescharacteristic
60C2×C3×C7×D4C22×C311yescharacteristic, complemented
61C2×C7×#67—22no—
62C7×#237—22no—
63D3×C7×D4—22no—
64C7×#235C22×C312yes—
65C3×C7×SD16—62no—
66C3×C7×D8—62no—
67C3×C7×M16—32no—
68C3×C42×C7—21no—
69C3×C42×C7C2×D311yescharacteristic, complemented
70C3×C42×C7C22×C311yescharacteristic, complemented
71C7×#214—22no—
72C7×#213—22no—
73C7×#208C22×C312yescomplemented
74C3×#1004C2×C711yescharacteristic, complemented
75C32×#127C2×C711yescharacteristic, complemented
76C3×#932C2×C712yescomplemented
77C3×#833C2×C711yescharacteristic, complemented
78C3×#805C2×C712yescomplemented
79C22×C32×C7—22no—
80C22×C32×C7—42no—
81C22×C32×C7—42no—
82C22×C32×C7—41no—
83C22×C32×C7C2×D411yescharacteristic
84C2×C3×D3×C7—22no—
85C2×C3×D3×C7—82no—
86C32×C4×C7—22no—
87C32×C4×C7C2×D412yes—
88C3×C7×Dic3—22no—
89C7×H(5,𝔽2)—31no—
90C7×AGL(1,ℤ/8)—32no—
91C7×#127C3×D311yescharacteristic, complemented
92C7×#104—32no—
93C7×#100—31no—
94C3×#453—31no—
95#5077C3×C711yescharacteristic, direct product factor
96C23×C3×C7—21no—
97C23×C3×C7—21no—
98C23×C3×C7—21no—
99C23×C3×C7—21no—
100C23×C3×C7—22no—
101C23×C3×C7—62no—
102C23×C3×C7#6712yes—
103C23×C3×C7C3×D412yes—
104C22×D3×C7—22no—
105C3×C7×Q8—32no—
106C3×C7×D4—22no—
107C3×C7×D4—22no—
108C3×C7×D4—22no—
109C3×C7×D4—22no—
110C3×C7×D4—32no—
111C3×C7×D4—42no—
112C3×C7×D4—42no—
113C3×C7×D4—44no—
114C3×C7×D4—44no—
115C3×C7×D4—44no—
116C3×C7×D4—44no—
117C3×C7×D4—64no—
118C3×C7×D4—62no—
119C2×C3×C4×C7—21no—
120C2×C3×C4×C7—21no—
121C2×C3×C4×C7—21no—
122C2×C3×C4×C7—21no—
123C2×C3×C4×C7—21no—
124C2×C3×C4×C7—62no—
125C2×C3×C4×C7—62no—
126C2×C3×C4×C7C22×D311yescharacteristic
127C2×C3×C4×C7C23×C311yescharacteristic
128C7×#67—24no—
129C7×#67—22no—
130C2×C7×Dic3—22no—
131C7×D12C3×D412yescomplemented
132C4×D3×C7—22no—
133C7×Dic6C3×D412yescomplemented
134C3×C7×C8—62no—
135C7×C3⋊C8—22no—
136C2×C32×D4—22no—
137C2×C32×D4C22×C711yescharacteristic
138C2×C32×D4C22×C711yescharacteristic, complemented
139C2×C3×#67—22no—
140C3×#237—22no—
141C3×D3×D4—22no—
142C3×#235C22×C712yes—
143C32×C42C22×C711yescharacteristic, complemented
144C3×#214—22no—
145C3×#213—22no—
146C3×#208C22×C712yescomplemented
147C2×C32×C7—21no—
148C2×C32×C7—42no—
149C2×C32×C7—81no—
150C2×C32×C7#12011yescharacteristic
151C3×D3×C7—42no—
152C7×#41—32no—
153C7×#41—62no—
154C2×C7×D4—22no—
155C2×C7×D4—62no—
156C2×C7×D4—62no—
157C2×C7×D4C2×C3×D311yescharacteristic
158C2×C7×D4C2×C3×D311yescharacteristic, complemented
159C7×SD16—62no—
160C7×D8—62no—
161C7×M16—32no—
162C42×C7C2×C3×D311yescharacteristic, complemented
163C3×H(5,𝔽2)—31no—
164#1004C2×C3×C711yescharacteristic, complemented
165C3×AGL(1,ℤ/8)—32no—
166C3×#127—21no—
167C3×#127D3×C711yescharacteristic, complemented
168C3×#127C2×C3×C711yescharacteristic, complemented
169#932C2×C3×C712yescomplemented
170C3×#104—32no—
171C3×#100—31no—
172#833C2×C3×C711yescharacteristic, complemented
173#805C2×C3×C712yescomplemented
174C22×C3×C7—22no—
175C22×C3×C7—22no—
176C22×C3×C7—21no—
177C22×C3×C7—42no—
178C22×C3×C7—42no—
179C22×C3×C7—42no—
180C22×C3×C7—42no—
181C22×C3×C7—42no—
182C22×C3×C7—42no—
183C22×C3×C7—42no—
184C22×C3×C7—42no—
185C22×C3×C7—42no—
186C22×C3×C7—41no—
187C22×C3×C7—41no—
188C22×C3×C7—41no—
189C22×C3×C7—41no—
190C22×C3×C7—62no—
191C22×C3×C7—242no—
192C22×C3×C7C2×#6711yescharacteristic
193C22×C3×C7C2×C3×D411yescharacteristic
194C2×D3×C7—22no—
195C2×D3×C7—82no—
196C3×C4×C7—22no—
197C3×C4×C7—22no—
198C3×C4×C7—22no—
199C3×C4×C7—22no—
200C3×C4×C7—22no—
201C3×C4×C7—62no—
202C3×C4×C7D3×D412yes—
203C3×C4×C7C2×C3×D412yes—
204C7×Dic3—22no—
205C23×C32—21no—
206C23×C32C7×D412yes—
207C22×C3×D3—22no—
208C32×D4—22no—
209C32×D4—22no—
210C32×D4—44no—
211C2×C32×C4—21no—
212C2×C32×C4C23×C711yescharacteristic
213C3×#67—24no—
214C3×#67—22no—
215C2×C3×Dic3—22no—
216C3×D12C7×D412yescomplemented
217C3×C4×D3—22no—
218C3×Dic6C7×D412yescomplemented
219C3×C3⋊C8—22no—
220#453—31noSylow
221C32×C7#45311yescharacteristic, complemented, Hall
222C23×C7—21no—
223C23×C7—62no—
224C23×C7C3×#6712yes—
225C7×Q8—32no—
226C7×D4—22no—
227C7×D4—22no—
228C7×D4—32no—
229C7×D4—44no—
230C7×D4—64no—
231C7×D4—62no—
232C2×C4×C7—21no—
233C2×C4×C7—62no—
234C2×C4×C7—62no—
235C2×C4×C7C22×C3×D311yescharacteristic
236C7×C8—62no—
237C3×#41—32no—
238C3×#41—62no—
239C2×C3×D4—22no—
240C2×C3×D4—22no—
241C2×C3×D4—22no—
242C2×C3×D4—22no—
243C2×C3×D4—21no—
244C2×C3×D4—21no—
245C2×C3×D4—62no—
246C2×C3×D4—62no—
247C2×C3×D4C2×D3×C711yescharacteristic
248C2×C3×D4C2×D3×C711yescharacteristic, complemented
249C2×C3×D4C22×C3×C711yescharacteristic
250C2×C3×D4C22×C3×C711yescharacteristic, complemented
251C2×#67—22no—
252#237—22no—
253D3×D4—22no—
254#235C22×C3×C712yes—
255C3×SD16—62no—
256C3×D8—62no—
257C3×M16—32no—
258C3×C42—21no—
259C3×C42C2×D3×C711yescharacteristic, complemented
260C3×C42C22×C3×C711yescharacteristic, complemented
261#214—22no—
262#213—22no—
263#208C22×C3×C712yescomplemented
264C2×C3×C7—21no—
265C2×C3×C7—21no—
266C2×C3×C7—21no—
267C2×C3×C7—21no—
268C2×C3×C7—21no—
269C2×C3×C7—42no—
270C2×C3×C7—42no—
271C2×C3×C7—82no—
272C2×C3×C7—81no—
273C2×C3×C7—81no—
274C2×C3×C7—81no—
275C2×C3×C7—81no—
276C2×C3×C7—122no—
277C2×C3×C7#93711yescharacteristic
278C2×C3×C7C3×#12011yescharacteristic
279D3×C7—42no—
280C22×C32—22no—
281C22×C32—42no—
282C22×C32—42no—
283C22×C32—41no—
284C22×C32C2×C7×D411yescharacteristic
285C2×C3×D3—22no—
286C2×C3×D3—82no—
287C32×C4—22no—
288C32×C4C2×C7×D412yes—
289C3×Dic3—22no—
290H(5,𝔽2)—31no—
291AGL(1,ℤ/8)—32no—
292#127C3×D3×C711yescharacteristic, complemented
293#104—32no—
294#100—31no—
295C22×C7—22no—
296C22×C7—42no—
297C22×C7—42no—
298C22×C7—41no—
299C22×C7—62no—
300C22×C7—242no—
301C22×C7C2×C3×#6711yescharacteristic
302C4×C7—22no—
303C4×C7—62no—
304C4×C7C3×D3×D412yes—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—22no—
310C23×C3—62no—
311C23×C3C7×#6712yes—
312C23×C3C3×C7×D412yes—
313C22×D3—22no—
314C3×Q8—32no—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—32no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—44no—
323C3×D4—44no—
324C3×D4—44no—
325C3×D4—44no—
326C3×D4—64no—
327C3×D4—62no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4C22×D3×C711yescharacteristic
336C2×C3×C4C23×C3×C711yescharacteristic
337#67—24no—
338#67—22no—
339C2×Dic3—22no—
340D12C3×C7×D412yescomplemented
341C4×D3—22no—
342Dic6C3×C7×D412yescomplemented
343C3×C8—62no—
344C3⋊C8—22no—
345C3×C7—21no—
346C3×C7#507711yescharacteristic, direct product factor
347C3×C7C3×#45311yescharacteristic, complemented
348C2×C32—21no—
349C2×C32—42no—
350C2×C32—81no—
351C2×C32C7×#12011yescharacteristic
352C3×D3—42no—
353#41—32no—
354#41—62no—
355C2×D4—22no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4C2×C3×D3×C711yescharacteristic
359C2×D4C2×C3×D3×C711yescharacteristic, complemented
360SD16—62no—
361D8—62no—
362M16—32no—
363C42C2×C3×D3×C711yescharacteristic, complemented
364C2×C7—21no—
365C2×C7—42no—
366C2×C7—81no—
367C2×C7—122no—
368C2×C7C3×#93711yescharacteristic
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—41no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—62no—
386C22×C3—242no—
387C22×C3C2×C7×#6711yescharacteristic
388C22×C3C2×C3×C7×D411yescharacteristic
389C2×D3—22no—
390C2×D3—82no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—62no—
397C3×C4D3×C7×D412yes—
398C3×C4C2×C3×C7×D412yes—
399Dic3—22no—
400C32C7×#45311yescharacteristic, complemented, Sylow
401C23—21no—
402C23—62no—
403C23C3×C7×#6712yes—
404Q8—32no—
405D4—22no—
406D4—22no—
407D4—32no—
408D4—44no—
409D4—64no—
410D4—62no—
411C2×C4—21no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4C22×C3×D3×C711yescharacteristic
415C8—62no—
416C7C3×#507711yescharacteristic, direct product factor, Sylow
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—42no—
423C2×C3—42no—
424C2×C3—82no—
425C2×C3—81no—
426C2×C3—81no—
427C2×C3—81no—
428C2×C3—81no—
429C2×C3—122no—
430C2×C3C7×#93711yescharacteristic
431C2×C3C3×C7×#12011yescharacteristic
432D3—42no—
433C22—22no—
434C22—42no—
435C22—42no—
436C22—41no—
437C22—62no—
438C22—242no—
439C22C2×C3×C7×#6711yescharacteristic
440C4—22no—
441C4—62no—
442C4C3×D3×C7×D412yes—
443C3—21no—
444C3C7×#507711yescharacteristic, direct product factor
445C3C3×C7×#45311yescharacteristic, complemented
446C2—21no—
447C2—42no—
448C2—81no—
449C2—122no—
450C2C3×C7×#93711yescharacteristic
subgroupG/H
centerC2×C3×C7#937
commutator, cosocleC2×C3×C4C23×C3×C7
FrattiniC2×C4C22×C3×D3×C7
FittingC32×C7×#127C2
socleC2×C32×C7#120
hypercenterC3×C7×#127D3
hypocenterC3C3×C7×#453
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C7 ◁ C2×C3×C4×C7 ◁ C3×C7×#127

Complements

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

NH ≅ G/Ntype
1. C3×C7×#833C2semidirect
2. C3×C7×#805C2semidirect
3. C3×C7×#1004C2semidirect
4. C32×C7×#127C2semidirect
5. C3×C7×#932C2semidirect
7. C7×#5077C3direct (N × H)
10. C2×C32×C7×D4C22semidirect
15. C32×C42×C7C22semidirect
18. C3×C7×#208C22semidirect
20. C7×#1004C2×C3semidirect
23. C3×C7×#127D3semidirect
24. C3×C7×#127C2×C3semidirect
25. C7×#932C2×C3semidirect
28. C7×#833C2×C3semidirect
29. C7×#805C2×C3semidirect
30. C3×#5077C7direct (N × H)
42. C3×C7×D12D4semidirect
44. C3×C7×Dic6D4semidirect
58. C2×C3×C7×D4C2×D3semidirect
60. C2×C3×C7×D4C22×C3semidirect
69. C3×C42×C7C2×D3semidirect
70. C3×C42×C7C22×C3semidirect
73. C7×#208C22×C3semidirect
74. C3×#1004C2×C7semidirect
75. C32×#127C2×C7semidirect
76. C3×#932C2×C7semidirect
77. C3×#833C2×C7semidirect
78. C3×#805C2×C7semidirect
91. C7×#127C3×D3semidirect
95. #5077C3×C7direct (N × H)
131. C7×D12C3×D4semidirect
133. C7×Dic6C3×D4semidirect
138. C2×C32×D4C22×C7semidirect
143. C32×C42C22×C7semidirect
146. C3×#208C22×C7semidirect
158. C2×C7×D4C2×C3×D3semidirect
162. C42×C7C2×C3×D3semidirect
164. #1004C2×C3×C7semidirect
167. C3×#127D3×C7semidirect
168. C3×#127C2×C3×C7semidirect
169. #932C2×C3×C7semidirect
172. #833C2×C3×C7semidirect
173. #805C2×C3×C7semidirect
216. C3×D12C7×D4semidirect
218. C3×Dic6C7×D4semidirect
221. C32×C7#453semidirect
248. C2×C3×D4C2×D3×C7semidirect
250. C2×C3×D4C22×C3×C7semidirect
259. C3×C42C2×D3×C7semidirect
260. C3×C42C22×C3×C7semidirect
263. #208C22×C3×C7semidirect
292. #127C3×D3×C7semidirect
340. D12C3×C7×D4semidirect
342. Dic6C3×C7×D4semidirect
346. C3×C7#5077direct (N × H)
347. C3×C7C3×#453semidirect
359. C2×D4C2×C3×D3×C7semidirect
363. C42C2×C3×D3×C7semidirect
400. C32C7×#453semidirect
416. C7C3×#5077direct (N × H)
444. C3C7×#5077direct (N × H)
445. C3C3×C7×#453semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#8332
C3×C7×#8052
C3×C7×#10042
C32×C7×#1272
C3×C7×#9322
C3×C7×#4533
C7×#50773
C3×#50777