Finite Group Databasebeta

← C73×#19324 #32928728

Subgroups — C73×#19324 #32928728

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3138C211yesmaximal, characteristic, complemented
2C73×#3116C211yesmaximal, characteristic, complemented
3C73×#3038C211yesmaximal, characteristic
4C73×#2971C211yesmaximal, characteristic, complemented
5C73×#1701C212yesmaximal, complemented
6C73×#1697C212yesmaximal, complemented
7C73×#1682C212yesmaximal, complemented
8C73×#1678C212yesmaximal, complemented
9C73×#1569C211yesmaximal, characteristic, complemented
10C73×#1559C211yesmaximal, characteristic, complemented
11C73×#1555C211yesmaximal, characteristic, complemented
12C2×AGL(1,ℤ/8)×C73—21no—
13#533×C73C2211yescharacteristic
14C2×#127×C73C2211yescharacteristic
15C4×#41×C73C2211yescharacteristic
16#490×C73—21no—
17#490×C73—21no—
18#486×C73C2212yes—
19#486×C73C2212yes—
20#485×C73—21no—
21#477×C73C2211yescharacteristic, complemented
22#476×C73C2211yescharacteristic, complemented
23#462×C73C2211yescharacteristic, complemented
24#461×C73C2211yescharacteristic, complemented
25#460×C73—21no—
26#460×C73C2212yes—
27#459×C73—21no—
28#459×C73C2212yes—
29#452×C73—21no—
30C8×Q8×C73C2212yescomplemented
31#441×C73—21no—
32#440×C73—21no—
33#436×C73—22no—
34#436×C73C2212yescomplemented
35#432×C73—21no—
36#432×C73C2211yescharacteristic
37#432×C73C2211yescharacteristic
38#423×C73C2211yescharacteristic
39#418×C73—22no—
40#418×C73—21no—
41C4×M16×C73—21no—
42C4×M16×C73C2211yescharacteristic
43#335×C73C2212yescomplemented
44#334×C73C2212yescomplemented
45#333×C73C2212yescomplemented
46#332×C73C2212yescomplemented
47#331×C73C2212yes—
48#331×C73C2212yes—
49C2×#41×C73D411yescharacteristic
50C22×D4×C73—21no—
51C22×D4×C73—21no—
52C22×D4×C73—21no—
53AGL(1,ℤ/8)×C73—44no—
54C2×Q16×C73—21no—
55C2×SD16×C73—22no—
56C2×SD16×C73—21no—
57C2×D8×C73—22no—
58C2×M16×C73—21no—
59C2×M16×C73—22no—
60C2×M16×C73D412yes—
61C4⋊Q8×C73C2311yescharacteristic
62C4⋊Q8×C73C2311yescharacteristic
63#127×C73—21no—
64#127×C73—21no—
65#127×C73—21no—
66#127×C73C2312yes—
67#125×C73—21no—
68C22⋊Q8×C73—21no—
69C22⋊Q8×C73—21no—
70C4×Q8×C73—21no—
71C4×Q8×C73—21no—
72C4×Q8×C73D411yescharacteristic
73C4×Q8×C73C2311yescharacteristic
74C4×D4×C73—21no—
75C4×D4×C73—21no—
76C4×D4×C73D411yescharacteristic
77C4×D4×C73C2311yescharacteristic
78#117×C73—21no—
79#117×C73—21no—
80#117×C73D411yescharacteristic
81C2×C42×C73—21no—
82C2×C42×C73C2311yescharacteristic
83#107×C73—21no—
84#106×C73—21no—
85C4⋊C8×C73—22no—
86C4⋊C8×C73—21no—
87C4⋊C8×C73C2312yes—
88C4⋊C8×C73C2312yes—
89C4⋊C8×C73C2311yescharacteristic
90C4⋊C8×C73C2311yescharacteristic
91#103×C73—21no—
92#103×C73—21no—
93#103×C73—21no—
94#102×C73—21no—
95#102×C73—22no—
96#102×C73—22no—
97#102×C73—22no—
98#102×C73—22no—
99#102×C73—22no—
100#102×C73—22no—
101C22⋊C8×C73—21no—
102C22⋊C8×C73—21no—
103C22⋊C8×C73—22no—
104#97×C73—21no—
105#97×C73—21no—
106#97×C73D411yescharacteristic
107#97×C73D411yescharacteristic
108C4×C8×C73—22no—
109C4×C8×C73C2312yes—
110C24×C73—41no—
111#41×C73—22no—
112#41×C73—41no—
113C2×Q8×C73—21no—
114C2×Q8×C73—21no—
115C2×Q8×C73C2×D411yescharacteristic
116C2×D4×C73—21no—
117C2×D4×C73—22no—
118C2×D4×C73—22no—
119C2×D4×C73—22no—
120C2×D4×C73—42no—
121C2×D4×C73—44no—
122C2×D4×C73—42no—
123C2×D4×C73—41no—
124C2×D4×C73—41no—
125C2×D4×C73—41no—
126C2×D4×C73—82no—
127C2×D4×C73C2×D411yescharacteristic
128C22×C4×C73—21no—
129C22×C4×C73—21no—
130C22×C4×C73—21no—
131C22×C4×C73C2×D411yescharacteristic
132C22×C4×C73C2×D411yescharacteristic
133C22×C4×C73#4111yescharacteristic
134Q16×C73—42no—
135SD16×C73—42no—
136SD16×C73—44no—
137D8×C73—44no—
138M16×C73—42no—
139M16×C73—44no—
140M16×C73—42no—
141C2×C8×C73—22no—
142C2×C8×C73—21no—
143C2×C8×C73—21no—
144C2×C8×C73—22no—
145C2×C8×C73C2×D412yes—
146C2×C8×C73C2×D412yes—
147C4⋊C4×C73—21no—
148C4⋊C4×C73—21no—
149C4⋊C4×C73—21no—
150C4⋊C4×C73—21no—
151C4⋊C4×C73—21no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—41no—
154C4⋊C4×C73—41no—
155C4⋊C4×C73C2×D411yescharacteristic
156C4⋊C4×C73C2×D411yescharacteristic
157C22⋊C4×C73—21no—
158C22⋊C4×C73—21no—
159C22⋊C4×C73—21no—
160C22⋊C4×C73—41no—
161C42×C73—21no—
162C42×C73—21no—
163C42×C73—21no—
164C42×C73—41no—
165C42×C73C2×D411yescharacteristic
166C42×C73C2×D411yescharacteristic
167C42×C73#4111yescharacteristic
168C42×C73C2411yescharacteristic
169C23×C73—21no—
170C23×C73—42no—
171C23×C73—82no—
172C23×C73—82no—
173C23×C73—82no—
174C23×C73#12111yescharacteristic
175Q8×C73—21no—
176Q8×C73—41no—
177Q8×C73—41no—
178Q8×C73—81no—
179Q8×C73AGL(1,ℤ/8)12yes—
180D4×C73—21no—
181D4×C73—21no—
182D4×C73—42no—
183D4×C73—42no—
184D4×C73—44no—
185D4×C73—42no—
186D4×C73—82no—
187D4×C73—82no—
188D4×C73—82no—
189D4×C73—82no—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—22no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73—41no—
197C2×C4×C73—42no—
198C2×C4×C73—42no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—81no—
202C2×C4×C73#12111yescharacteristic
203C2×C4×C73#12111yescharacteristic
204C2×C4×C73#12111yescharacteristic
205C2×C4×C73C22×D411yescharacteristic
206C2×C4×C73C22×D411yescharacteristic
207C2×C4×C73C2×#4111yescharacteristic
208C8×C73—22no—
209C8×C73—22no—
210C8×C73—42no—
211C8×C73—41no—
212C8×C73—42no—
213C8×C73—41no—
214C22×C73—21no—
215C22×C73—21no—
216C22×C73—21no—
217C22×C73—41no—
218C22×C73—41no—
219C22×C73—41no—
220C22×C73—82no—
221C22×C73—82no—
222C22×C73—82no—
223C22×C73—164no—
224C22×C73C2×#12111yescharacteristic
225C4×C73—21no—
226C4×C73—21no—
227C4×C73—22no—
228C4×C73—21no—
229C4×C73—21no—
230C4×C73—41no—
231C4×C73—41no—
232C4×C73—81no—
233C4×C73—81no—
234C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
235C4×C73#57512yes—
236C4×C73#57511yescharacteristic
237#19324C7311yesmaximal, characteristic, direct product factor, Sylow
238C2×C73—41no—
239C2×C73—81no—
240C2×C73—162no—
241C2×C73#304011yescharacteristic
242C2×C73#305511yescharacteristic
243C2×C73#306811yescharacteristic
244#3138C2×C7311yescharacteristic, complemented
245#3116C2×C7311yescharacteristic, complemented
246#3038C2×C7311yescharacteristic
247#2971C2×C7311yescharacteristic, complemented
248#1701C2×C7312yescomplemented
249#1697C2×C7312yescomplemented
250#1682C2×C7312yescomplemented
251#1678C2×C7312yescomplemented
252#1569C2×C7311yescharacteristic, complemented
253#1559C2×C7311yescharacteristic, complemented
254#1555C2×C7311yescharacteristic, complemented
255C73#1932411yescharacteristic, direct product factor, Sylow
256C2×AGL(1,ℤ/8)—21no—
257#533C22×C7311yescharacteristic
258C2×#127C22×C7311yescharacteristic
259C4×#41C22×C7311yescharacteristic
260#490—21no—
261#490—21no—
262#486C22×C7312yes—
263#486C22×C7312yes—
264#485—21no—
265#477C22×C7311yescharacteristic, complemented
266#476C22×C7311yescharacteristic, complemented
267#462C22×C7311yescharacteristic, complemented
268#461C22×C7311yescharacteristic, complemented
269#460—21no—
270#460C22×C7312yes—
271#459—21no—
272#459C22×C7312yes—
273#452—21no—
274C8×Q8C22×C7312yescomplemented
275#441—21no—
276#440—21no—
277#436—22no—
278#436C22×C7312yescomplemented
279#432—21no—
280#432C22×C7311yescharacteristic
281#432C22×C7311yescharacteristic
282#423C22×C7311yescharacteristic
283#418—22no—
284#418—21no—
285C4×M16—21no—
286C4×M16C22×C7311yescharacteristic
287#335C22×C7312yescomplemented
288#334C22×C7312yescomplemented
289#333C22×C7312yescomplemented
290#332C22×C7312yescomplemented
291#331C22×C7312yes—
292#331C22×C7312yes—
293C2×#41D4×C7311yescharacteristic
294C22×D4—21no—
295C22×D4—21no—
296C22×D4—21no—
297AGL(1,ℤ/8)—44no—
298C2×Q16—21no—
299C2×SD16—22no—
300C2×SD16—21no—
301C2×D8—22no—
302C2×M16—21no—
303C2×M16—22no—
304C2×M16D4×C7312yes—
305C4⋊Q8C23×C7311yescharacteristic
306C4⋊Q8C23×C7311yescharacteristic
307#127—21no—
308#127—21no—
309#127—21no—
310#127C23×C7312yes—
311#125—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8D4×C7311yescharacteristic
317C4×Q8C23×C7311yescharacteristic
318C4×D4—21no—
319C4×D4—21no—
320C4×D4D4×C7311yescharacteristic
321C4×D4C23×C7311yescharacteristic
322#117—21no—
323#117—21no—
324#117D4×C7311yescharacteristic
325C2×C42—21no—
326C2×C42C23×C7311yescharacteristic
327#107—21no—
328#106—21no—
329C4⋊C8—22no—
330C4⋊C8—21no—
331C4⋊C8C23×C7312yes—
332C4⋊C8C23×C7311yescharacteristic
333C4⋊C8C23×C7311yescharacteristic
334C4⋊C8C23×C7312yes—
335#103—21no—
336#103—21no—
337#103—21no—
338#102—21no—
339#102—22no—
340#102—22no—
341#102—22no—
342#102—22no—
343#102—22no—
344#102—22no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8—22no—
348#97—21no—
349#97—21no—
350#97D4×C7311yescharacteristic
351#97D4×C7311yescharacteristic
352C4×C8—22no—
353C4×C8C23×C7312yes—
354C24—41no—
355#41—22no—
356#41—41no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8C2×D4×C7311yescharacteristic
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—42no—
365C2×D4—44no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—82no—
371C2×D4C2×D4×C7311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4C2×D4×C7311yescharacteristic
376C22×C4C2×D4×C7311yescharacteristic
377C22×C4#41×C7311yescharacteristic
378Q16—42no—
379SD16—42no—
380SD16—44no—
381D8—44no—
382M16—42no—
383M16—44no—
384M16—42no—
385C2×C8—22no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8C2×D4×C7312yes—
390C2×C8C2×D4×C7312yes—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4C2×D4×C7311yescharacteristic
400C4⋊C4C2×D4×C7311yescharacteristic
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—41no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—41no—
409C42C2×D4×C7311yescharacteristic
410C42C2×D4×C7311yescharacteristic
411C42#41×C7311yescharacteristic
412C42C24×C7311yescharacteristic
413C23—21no—
414C23—42no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23#121×C7311yescharacteristic
419Q8—21no—
420Q8—41no—
421Q8—41no—
422Q8—81no—
423Q8AGL(1,ℤ/8)×C7312yes—
424D4—21no—
425D4—21no—
426D4—42no—
427D4—42no—
428D4—44no—
429D4—42no—
430D4—82no—
431D4—82no—
432D4—82no—
433D4—82no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4#121×C7311yescharacteristic
447C2×C4#121×C7311yescharacteristic
448C2×C4#121×C7311yescharacteristic
449C2×C4C22×D4×C7311yescharacteristic
450C2×C4C22×D4×C7311yescharacteristic
451C2×C4C2×#41×C7311yescharacteristic
452C8—22no—
453C8—22no—
454C8—42no—
455C8—41no—
456C8—42no—
457C8—41no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22C2×#121×C7311yescharacteristic
469C4—21no—
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—81no—
478C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
479C4#575×C7312yes—
480C4#575×C7311yescharacteristic
481C2—41no—
482C2—81no—
483C2—162no—
484C2C73×#304011yescharacteristic
485C2C73×#305511yescharacteristic
486C2C73×#306811yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3038C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3138C2semidirect
2. C73×#3116C2semidirect
4. C73×#2971C2semidirect
5. C73×#1701C2semidirect
6. C73×#1697C2semidirect
7. C73×#1682C2semidirect
8. C73×#1678C2semidirect
9. C73×#1569C2semidirect
10. C73×#1559C2semidirect
11. C73×#1555C2semidirect
21. #477×C73C22semidirect
22. #476×C73C22semidirect
23. #462×C73C22semidirect
24. #461×C73C22semidirect
30. C8×Q8×C73C22semidirect
34. #436×C73C22semidirect
43. #335×C73C22semidirect
44. #334×C73C22semidirect
45. #333×C73C22semidirect
46. #332×C73C22semidirect
237. #19324C73direct (N × H)
244. #3138C2×C73semidirect
245. #3116C2×C73semidirect
247. #2971C2×C73semidirect
248. #1701C2×C73semidirect
249. #1697C2×C73semidirect
250. #1682C2×C73semidirect
251. #1678C2×C73semidirect
252. #1569C2×C73semidirect
253. #1559C2×C73semidirect
254. #1555C2×C73semidirect
255. C73#19324direct (N × H)
265. #477C22×C73semidirect
266. #476C22×C73semidirect
267. #462C22×C73semidirect
268. #461C22×C73semidirect
274. C8×Q8C22×C73semidirect
278. #436C22×C73semidirect
287. #335C22×C73semidirect
288. #334C22×C73semidirect
289. #333C22×C73semidirect
290. #332C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#31382
C73×#31162
C73×#30382
C73×#29712
C73×#17012
C73×#16972
C73×#16822
C73×#16782
C73×#15692
C73×#15592
C73×#15552
#1932473