Finite Group Databasebeta

← C73×#19352 #32928756

Subgroups — C73×#19352 #32928756

Order 18688.

712 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3239C211yesmaximal, characteristic, complemented
2C73×#3217C211yesmaximal, characteristic
3C2×#492×C73C211yesmaximal, characteristic, complemented
4C73×#2966C211yesmaximal, characteristic, complemented
5C73×#1686C212yesmaximal, complemented
6C73×#1685C212yesmaximal, complemented
7C73×#1666C212yesmaximal, complemented
8C73×#1665C212yesmaximal, complemented
9C73×#1568C211yesmaximal, characteristic, complemented
10C73×#1556C211yesmaximal, characteristic, complemented
11C73×#1555C211yesmaximal, characteristic, complemented
12C22×SD16×C73—21no—
13C22×SD16×C73—21no—
14C2×C4⋊Q8×C73C2211yescharacteristic
15C2×#127×C73C2211yescharacteristic
16#529×C73C2211yescharacteristic
17#501×C73—21no—
18#496×C73—21no—
19#492×C73—21no—
20#492×C73—21no—
21#492×C73C2212yes—
22#492×C73C2212yes—
23#490×C73—21no—
24#489×C73—21no—
25#484×C73—21no—
26#481×C73—21no—
27#475×C73C2212yescomplemented
28#474×C73C2212yescomplemented
29#460×C73C2212yes—
30#459×C73C2212yes—
31#450×C73—21no—
32#447×C73—21no—
33#446×C73C2211yescharacteristic, complemented
34C8×Q8×C73C2211yescharacteristic, complemented
35#436×C73C2211yescharacteristic, complemented
36#435×C73—21no—
37C8×D4×C73—21no—
38C8×D4×C73C2211yescharacteristic, complemented
39#433×C73—21no—
40#432×C73C2211yescharacteristic
41#432×C73C2211yescharacteristic
42#432×C73C2211yescharacteristic
43#405×C73—21no—
44C2×C4×C8×C73C2211yescharacteristic
45#335×C73C2212yescomplemented
46#334×C73C2212yescomplemented
47#332×C73C2212yescomplemented
48#332×C73C2212yescomplemented
49#331×C73C2212yes—
50#331×C73C2212yes—
51C22×Q8×C73—21no—
52C22×D4×C73—21no—
53C22×D4×C73—21no—
54C22×D4×C73—21no—
55C2×SD16×C73—22no—
56C2×SD16×C73—22no—
57C2×SD16×C73—22no—
58C2×SD16×C73—22no—
59C2×SD16×C73—22no—
60C2×SD16×C73—44no—
61C2×SD16×C73—44no—
62C2×D8×C73—22no—
63C2×M16×C73—21no—
64C22×C8×C73—21no—
65C22×C8×C73D411yescharacteristic
66C22×C8×C73D411yescharacteristic
67C4⋊Q8×C73—21no—
68C4⋊Q8×C73—21no—
69C4⋊Q8×C73—21no—
70C4⋊Q8×C73C2312yes—
71#127×C73—21no—
72#127×C73—21no—
73#127×C73—21no—
74#127×C73C2312yes—
75#126×C73—21no—
76#125×C73D411yescharacteristic
77#124×C73D411yescharacteristic
78#123×C73—21no—
79C22⋊Q8×C73D411yescharacteristic
80#121×C73D411yescharacteristic
81C4×Q8×C73C2311yescharacteristic
82C4×D4×C73—21no—
83C4×D4×C73C2311yescharacteristic
84#117×C73—21no—
85C2×C4⋊C4×C73—21no—
86C2×C4⋊C4×C73—21no—
87C2×C42×C73C2311yescharacteristic
88#107×C73—22no—
89#106×C73—22no—
90C4⋊C8×C73—21no—
91C4⋊C8×C73—21no—
92C4⋊C8×C73C2311yescharacteristic
93C4⋊C8×C73C2311yescharacteristic
94C4⋊C8×C73C2312yes—
95C4⋊C8×C73C2312yes—
96#103×C73—22no—
97#102×C73—22no—
98#102×C73—22no—
99#102×C73—22no—
100C22⋊C8×C73—21no—
101C22⋊C8×C73—21no—
102C22⋊C8×C73—21no—
103C22⋊C8×C73—21no—
104C22⋊C8×C73—21no—
105#97×C73—21no—
106#97×C73—21no—
107#97×C73—21no—
108#97×C73—21no—
109C4×C8×C73—21no—
110C4×C8×C73D411yescharacteristic
111C4×C8×C73D411yescharacteristic
112C4×C8×C73C2311yescharacteristic
113C4×C8×C73C2311yescharacteristic
114C24×C73—41no—
115C2×Q8×C73—22no—
116C2×Q8×C73—42no—
117C2×Q8×C73—42no—
118C2×Q8×C73—41no—
119C2×Q8×C73C2×D411yescharacteristic
120C2×D4×C73—21no—
121C2×D4×C73—22no—
122C2×D4×C73—22no—
123C2×D4×C73—22no—
124C2×D4×C73—42no—
125C2×D4×C73—42no—
126C2×D4×C73—41no—
127C2×D4×C73—41no—
128C2×D4×C73—41no—
129C2×D4×C73—82no—
130C2×D4×C73—82no—
131C2×D4×C73C2×D411yescharacteristic
132C22×C4×C73—21no—
133C22×C4×C73—41no—
134C22×C4×C73C2×D411yescharacteristic
135C22×C4×C73C2×D411yescharacteristic
136C22×C4×C73#4111yescharacteristic
137SD16×C73—44no—
138SD16×C73—44no—
139SD16×C73—44no—
140SD16×C73—44no—
141SD16×C73—82no—
142D8×C73—82no—
143M16×C73—42no—
144C2×C8×C73—22no—
145C2×C8×C73—22no—
146C2×C8×C73—21no—
147C2×C8×C73—21no—
148C2×C8×C73—22no—
149C2×C8×C73—22no—
150C2×C8×C73—42no—
151C2×C8×C73C2×D411yescharacteristic
152C2×C8×C73C2×D411yescharacteristic
153C2×C8×C73C2×D411yescharacteristic
154C2×C8×C73C2×D411yescharacteristic
155C4⋊C4×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73—22no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73—41no—
162C4⋊C4×C73—41no—
163C4⋊C4×C73C2×D411yescharacteristic
164C4⋊C4×C73C2×D411yescharacteristic
165C22⋊C4×C73—21no—
166C22⋊C4×C73—21no—
167C22⋊C4×C73—21no—
168C22⋊C4×C73—21no—
169C22⋊C4×C73—21no—
170C42×C73—21no—
171C42×C73C2×D411yescharacteristic
172C42×C73C2×D411yescharacteristic
173C42×C73#4111yescharacteristic
174C42×C73C2411yescharacteristic
175C23×C73—21no—
176C23×C73—42no—
177C23×C73—82no—
178C23×C73—82no—
179C23×C73—82no—
180C23×C73#12111yescharacteristic
181Q8×C73—22no—
182Q8×C73—42no—
183Q8×C73—42no—
184Q8×C73—82no—
185D4×C73—22no—
186D4×C73—42no—
187D4×C73—42no—
188D4×C73—42no—
189D4×C73—42no—
190D4×C73—42no—
191D4×C73—82no—
192D4×C73—82no—
193D4×C73—82no—
194D4×C73—82no—
195C2×C4×C73—22no—
196C2×C4×C73—21no—
197C2×C4×C73—21no—
198C2×C4×C73—22no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—42no—
205C2×C4×C73—82no—
206C2×C4×C73#12111yescharacteristic
207C2×C4×C73#12111yescharacteristic
208C2×C4×C73#12111yescharacteristic
209C2×C4×C73C22×D411yescharacteristic
210C2×C4×C73C22×D411yescharacteristic
211C2×C4×C73C2×#4111yescharacteristic
212C8×C73—21no—
213C8×C73—21no—
214C8×C73—41no—
215C8×C73—41no—
216C8×C73—42no—
217C8×C73—42no—
218C8×C73AGL(1,ℤ/8)12yes—
219C8×C73AGL(1,ℤ/8)12yes—
220C22×C73—21no—
221C22×C73—21no—
222C22×C73—41no—
223C22×C73—41no—
224C22×C73—41no—
225C22×C73—82no—
226C22×C73—82no—
227C22×C73—82no—
228C22×C73—164no—
229C22×C73#47112yes—
230C22×C73C2×#12111yescharacteristic
231C4×C73—21no—
232C4×C73—21no—
233C4×C73—21no—
234C4×C73—21no—
235C4×C73—41no—
236C4×C73—41no—
237C4×C73—41no—
238C4×C73—82no—
239C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
241C4×C73#57712yes—
242#19352C7311yesmaximal, characteristic, direct product factor, Sylow
243C2×C73—22no—
244C2×C73—81no—
245C2×C73—162no—
246C2×C73#303811yescharacteristic
247C2×C73#305711yescharacteristic
248C2×C73C2×#47111yescharacteristic
249#3239C2×C7311yescharacteristic, complemented
250#3217C2×C7311yescharacteristic
251C2×#492C2×C7311yescharacteristic, complemented
252#2966C2×C7311yescharacteristic, complemented
253#1686C2×C7312yescomplemented
254#1685C2×C7312yescomplemented
255#1666C2×C7312yescomplemented
256#1665C2×C7312yescomplemented
257#1568C2×C7311yescharacteristic, complemented
258#1556C2×C7311yescharacteristic, complemented
259#1555C2×C7311yescharacteristic, complemented
260C73#1935211yescharacteristic, direct product factor, Sylow
261C22×SD16—21no—
262C22×SD16—21no—
263C2×C4⋊Q8C22×C7311yescharacteristic
264C2×#127C22×C7311yescharacteristic
265#529C22×C7311yescharacteristic
266#501—21no—
267#496—21no—
268#492—21no—
269#492—21no—
270#492C22×C7312yes—
271#492C22×C7312yes—
272#490—21no—
273#489—21no—
274#484—21no—
275#481—21no—
276#475C22×C7312yescomplemented
277#474C22×C7312yescomplemented
278#460C22×C7312yes—
279#459C22×C7312yes—
280#450—21no—
281#447—21no—
282#446C22×C7311yescharacteristic, complemented
283C8×Q8C22×C7311yescharacteristic, complemented
284#436C22×C7311yescharacteristic, complemented
285#435—21no—
286C8×D4—21no—
287C8×D4C22×C7311yescharacteristic, complemented
288#433—21no—
289#432C22×C7311yescharacteristic
290#432C22×C7311yescharacteristic
291#432C22×C7311yescharacteristic
292#405—21no—
293C2×C4×C8C22×C7311yescharacteristic
294#335C22×C7312yescomplemented
295#334C22×C7312yescomplemented
296#332C22×C7312yescomplemented
297#332C22×C7312yescomplemented
298#331C22×C7312yes—
299#331C22×C7312yes—
300C22×Q8—21no—
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—21no—
304C2×SD16—22no—
305C2×SD16—22no—
306C2×SD16—22no—
307C2×SD16—22no—
308C2×SD16—22no—
309C2×SD16—44no—
310C2×SD16—44no—
311C2×D8—22no—
312C2×M16—21no—
313C22×C8—21no—
314C22×C8D4×C7311yescharacteristic
315C22×C8D4×C7311yescharacteristic
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8C23×C7312yes—
320#127—21no—
321#127—21no—
322#127—21no—
323#127C23×C7312yes—
324#126—21no—
325#125D4×C7311yescharacteristic
326#124D4×C7311yescharacteristic
327#123—21no—
328C22⋊Q8D4×C7311yescharacteristic
329#121D4×C7311yescharacteristic
330C4×Q8C23×C7311yescharacteristic
331C4×D4—21no—
332C4×D4C23×C7311yescharacteristic
333#117—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C42C23×C7311yescharacteristic
337#107—22no—
338#106—22no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8C23×C7311yescharacteristic
342C4⋊C8C23×C7311yescharacteristic
343C4⋊C8C23×C7312yes—
344C4⋊C8C23×C7312yes—
345#103—22no—
346#102—22no—
347#102—22no—
348#102—22no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354#97—21no—
355#97—21no—
356#97—21no—
357#97—21no—
358C4×C8—21no—
359C4×C8D4×C7311yescharacteristic
360C4×C8D4×C7311yescharacteristic
361C4×C8C23×C7311yescharacteristic
362C4×C8C23×C7311yescharacteristic
363C24—41no—
364C2×Q8—22no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—41no—
368C2×Q8C2×D4×C7311yescharacteristic
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4C2×D4×C7311yescharacteristic
381C22×C4—21no—
382C22×C4—41no—
383C22×C4C2×D4×C7311yescharacteristic
384C22×C4C2×D4×C7311yescharacteristic
385C22×C4#41×C7311yescharacteristic
386SD16—44no—
387SD16—44no—
388SD16—44no—
389SD16—44no—
390SD16—82no—
391D8—82no—
392M16—42no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—42no—
400C2×C8C2×D4×C7311yescharacteristic
401C2×C8C2×D4×C7311yescharacteristic
402C2×C8C2×D4×C7311yescharacteristic
403C2×C8C2×D4×C7311yescharacteristic
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4C2×D4×C7311yescharacteristic
413C4⋊C4C2×D4×C7311yescharacteristic
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C42—21no—
420C42C2×D4×C7311yescharacteristic
421C42C2×D4×C7311yescharacteristic
422C42#41×C7311yescharacteristic
423C42C24×C7311yescharacteristic
424C23—21no—
425C23—42no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23#121×C7311yescharacteristic
430Q8—22no—
431Q8—42no—
432Q8—42no—
433Q8—82no—
434D4—22no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—82no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4#121×C7311yescharacteristic
456C2×C4#121×C7311yescharacteristic
457C2×C4#121×C7311yescharacteristic
458C2×C4C22×D4×C7311yescharacteristic
459C2×C4C22×D4×C7311yescharacteristic
460C2×C4C2×#41×C7311yescharacteristic
461C8—21no—
462C8—21no—
463C8—41no—
464C8—41no—
465C8—42no—
466C8—42no—
467C8AGL(1,ℤ/8)×C7312yes—
468C8AGL(1,ℤ/8)×C7312yes—
469C22—21no—
470C22—21no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—164no—
478C22#471×C7312yes—
479C22C2×#121×C7311yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
489C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
490C4#577×C7312yes—
491C2—22no—
492C2—81no—
493C2—162no—
494C2C73×#303811yescharacteristic
495C2C73×#305711yescharacteristic
496C2C2×#471×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3217C2
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×#3239C2semidirect
3. C2×#492×C73C2semidirect
4. C73×#2966C2semidirect
5. C73×#1686C2semidirect
6. C73×#1685C2semidirect
7. C73×#1666C2semidirect
8. C73×#1665C2semidirect
9. C73×#1568C2semidirect
10. C73×#1556C2semidirect
11. C73×#1555C2semidirect
27. #475×C73C22semidirect
28. #474×C73C22semidirect
33. #446×C73C22semidirect
34. C8×Q8×C73C22semidirect
35. #436×C73C22semidirect
38. C8×D4×C73C22semidirect
45. #335×C73C22semidirect
46. #334×C73C22semidirect
47. #332×C73C22semidirect
48. #332×C73C22semidirect
242. #19352C73direct (N × H)
249. #3239C2×C73semidirect
251. C2×#492C2×C73semidirect
252. #2966C2×C73semidirect
253. #1686C2×C73semidirect
254. #1685C2×C73semidirect
255. #1666C2×C73semidirect
256. #1665C2×C73semidirect
257. #1568C2×C73semidirect
258. #1556C2×C73semidirect
259. #1555C2×C73semidirect
260. C73#19352direct (N × H)
276. #475C22×C73semidirect
277. #474C22×C73semidirect
282. #446C22×C73semidirect
283. C8×Q8C22×C73semidirect
284. #436C22×C73semidirect
287. C8×D4C22×C73semidirect
294. #335C22×C73semidirect
295. #334C22×C73semidirect
296. #332C22×C73semidirect
297. #332C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#32392
C73×#32172
C2×#492×C732
C73×#29662
C73×#16862
C73×#16852
C73×#16662
C73×#16652
C73×#15682
C73×#15562
C73×#15552
#1935273