Finite Group Databasebeta

← C73×#59596 #32969000

Subgroups — C73×#59596 #32969000

Order 18688.

1832 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2738C213yesmaximal, complemented
2C73×#2736C211yesmaximal, characteristic, complemented
3C73×#2589C213yesmaximal, complemented
4C73×#2577C213yesmaximal, complemented
5C73×#2572C213yesmaximal, complemented
6C73×#2470C213yesmaximal, complemented
7C73×#2468C213yesmaximal, complemented
8C73×#2418C213yesmaximal
9C73×#2347C213yesmaximal, complemented
10C73×#2342C213yesmaximal, complemented
11C4×#123×C73C213yesmaximal, complemented
12C2×#126×C73C2211yescharacteristic, complemented
13C2×#123×C73C2213yes—
14C2×#120×C73C2211yescharacteristic
15C2×C4×D4×C73C2213yes—
16C2×#117×C73C2213yes—
17C22×C22⋊C4×C73C2213yes—
18C22×C42×C73C2211yescharacteristic, complemented
19#400×C73C2213yescomplemented
20#400×C73C2213yescomplemented
21#399×C73C2213yes—
22#398×C73C2213yescomplemented
23#398×C73C2213yescomplemented
24#397×C73C2213yes—
25#397×C73C2213yes—
26#396×C73C2211yescharacteristic, complemented
27#394×C73C2213yes—
28#394×C73C2213yes—
29#393×C73C2213yes—
30#393×C73C2211yescharacteristic, complemented
31#391×C73C2213yescomplemented
32#389×C73C2213yescomplemented
33#389×C73C2213yescomplemented
34#388×C73C2213yes—
35#388×C73C2213yes—
36#388×C73C2213yes—
37#388×C73C2213yes—
38#388×C73C2213yes—
39#388×C73C2213yes—
40#388×C73C2213yes—
41#388×C73C2213yes—
42#388×C73C2213yescomplemented
43#387×C73C2213yescomplemented
44#387×C73C2213yescomplemented
45#387×C73C2213yescomplemented
46#387×C73C2213yescomplemented
47#386×C73C2213yes—
48#386×C73C2213yes—
49#385×C73C2213yes—
50#385×C73C2213yes—
51#385×C73C2213yes—
52#385×C73C2213yescomplemented
53#383×C73C2213yescomplemented
54#383×C73C2213yescomplemented
55#381×C73C2213yes—
56#381×C73C2213yescomplemented
57#380×C73C2213yes—
58#380×C73C2213yes—
59#379×C73C2213yes—
60C4×C4⋊C4×C73C2213yescomplemented
61C4×C4⋊C4×C73C2213yescomplemented
62C4×C4⋊C4×C73C2213yescomplemented
63C4×C22⋊C4×C73C2213yes—
64C4×C22⋊C4×C73C2213yes—
65C4×C22⋊C4×C73C2213yes—
66C4×C22⋊C4×C73C2213yescomplemented
67C25×C73C2311yescharacteristic
68C22×Q8×C73C2311yescharacteristic
69C22×D4×C73C2313yes—
70C23×C4×C73C2313yes—
71C23×C4×C73C2313yes—
72C23×C4×C73C2313yes—
73#126×C73—42no—
74#123×C73—212no—
75#120×C73—42no—
76C4×D4×C73—46no—
77#117×C73—212no—
78C2×C4⋊C4×C73C2313yes—
79C2×C4⋊C4×C73C2313yes—
80C2×C4⋊C4×C73C2313yes—
81C2×C4⋊C4×C73C2313yes—
82C2×C4⋊C4×C73C2313yes—
83C2×C4⋊C4×C73C2313yes—
84C2×C4⋊C4×C73C2313yes—
85C2×C4⋊C4×C73C2313yes—
86C2×C4⋊C4×C73C2313yes—
87C2×C4⋊C4×C73C2313yes—
88C2×C22⋊C4×C73—212no—
89C2×C22⋊C4×C73C2313yes—
90C2×C22⋊C4×C73C2313yes—
91C2×C22⋊C4×C73C2313yes—
92C2×C22⋊C4×C73C2313yes—
93C2×C22⋊C4×C73C2313yes—
94C2×C22⋊C4×C73C2313yes—
95C2×C22⋊C4×C73C2313yes—
96C2×C22⋊C4×C73C2313yes—
97C2×C22⋊C4×C73C2313yes—
98C2×C22⋊C4×C73C2313yes—
99C2×C22⋊C4×C73C2313yes—
100C2×C22⋊C4×C73C2313yes—
101C2×C22⋊C4×C73C2313yes—
102C2×C22⋊C4×C73C2313yes—
103C2×C42×C73—42no—
104C2×C42×C73C2313yes—
105C2×C42×C73C2313yes—
106C2×C42×C73C2313yes—
107C2×C42×C73C2313yes—
108C2×C42×C73C2313yes—
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111#95×C73C2313yes—
112#95×C73C2313yes—
113#95×C73C2313yes—
114#95×C73C2313yes—
115#95×C73C2313yes—
116#95×C73C2313yes—
117#95×C73C2313yes—
118#95×C73C2313yes—
119#95×C73C2313yes—
120#95×C73C2313yes—
121#95×C73C2313yes—
122#95×C73C2313yes—
123#95×C73C2313yes—
124#95×C73C2313yes—
125#95×C73C2313yes—
126#95×C73C2313yes—
127#95×C73C2313yes—
128#95×C73C2311yescharacteristic
129C24×C73—26no—
130C24×C73—43no—
131C24×C73—41no—
132C24×C73C2413yes—
133C24×C73C2411yescharacteristic
134C2×Q8×C73—43no—
135C2×D4×C73—26no—
136C2×D4×C73—26no—
137C2×D4×C73—43no—
138C22×C4×C73—26no—
139C22×C4×C73—26no—
140C22×C4×C73—26no—
141C22×C4×C73—26no—
142C22×C4×C73—26no—
143C22×C4×C73—43no—
144C22×C4×C73—43no—
145C22×C4×C73—43no—
146C22×C4×C73—43no—
147C22×C4×C73C2413yes—
148C22×C4×C73C2413yes—
149C22×C4×C73C2413yes—
150C22×C4×C73C2413yes—
151C22×C4×C73C2413yes—
152C22×C4×C73C2413yes—
153C22×C4×C73C2413yes—
154C22×C4×C73C2413yes—
155C22×C4×C73C2413yes—
156C4⋊C4×C73—26no—
157C4⋊C4×C73—26no—
158C4⋊C4×C73—26no—
159C4⋊C4×C73—26no—
160C4⋊C4×C73—43no—
161C4⋊C4×C73—43no—
162C4⋊C4×C73—43no—
163C4⋊C4×C73—43no—
164C4⋊C4×C73—43no—
165C4⋊C4×C73—43no—
166C22⋊C4×C73—26no—
167C22⋊C4×C73—26no—
168C22⋊C4×C73—26no—
169C22⋊C4×C73—26no—
170C22⋊C4×C73—26no—
171C22⋊C4×C73—26no—
172C22⋊C4×C73—26no—
173C22⋊C4×C73—43no—
174C22⋊C4×C73—43no—
175C22⋊C4×C73—43no—
176C22⋊C4×C73—43no—
177C22⋊C4×C73—43no—
178C22⋊C4×C73—43no—
179C22⋊C4×C73#41112yes—
180C42×C73—26no—
181C42×C73—26no—
182C42×C73—43no—
183C42×C73—41no—
184C42×C73—43no—
185C42×C73—43no—
186C42×C73—41no—
187C23×C73—23no—
188C23×C73—23no—
189C23×C73—23no—
190C23×C73—23no—
191C23×C73—23no—
192C23×C73—23no—
193C23×C73—23no—
194C23×C73—23no—
195C23×C73—412no—
196C23×C73—86no—
197C23×C73—82no—
198C23×C73C2×#4116yes—
199C23×C73H(5,𝔽2)12yes—
200C23×C73C2511yescharacteristic
201Q8×C73—82no—
202D4×C73—412no—
203C2×C4×C73—23no—
204C2×C4×C73—23no—
205C2×C4×C73—23no—
206C2×C4×C73—23no—
207C2×C4×C73—23no—
208C2×C4×C73—23no—
209C2×C4×C73—23no—
210C2×C4×C73—23no—
211C2×C4×C73—23no—
212C2×C4×C73—23no—
213C2×C4×C73—23no—
214C2×C4×C73—23no—
215C2×C4×C73—23no—
216C2×C4×C73—23no—
217C2×C4×C73—23no—
218C2×C4×C73—23no—
219C2×C4×C73—23no—
220C2×C4×C73—23no—
221C2×C4×C73—23no—
222C2×C4×C73—23no—
223C2×C4×C73—23no—
224C2×C4×C73—23no—
225C2×C4×C73—23no—
226C2×C4×C73—23no—
227C2×C4×C73—23no—
228C2×C4×C73—412no—
229C2×C4×C73—412no—
230C2×C4×C73—86no—
231C2×C4×C73C2×#4116yes—
232C2×C4×C73C2×#4116yes—
233C22×C73—26no—
234C22×C73—26no—
235C22×C73—26no—
236C22×C73—26no—
237C22×C73—43no—
238C22×C73—43no—
239C22×C73—43no—
240C22×C73—43no—
241C22×C73—41no—
242C22×C73—43no—
243C22×C73—88no—
244C22×C73C22×#4113yes—
245C22×C73C2×H(5,𝔽2)11yescharacteristic
246C22×C73#58513yes—
247C4×C73—26no—
248C4×C73—26no—
249C4×C73—43no—
250C4×C73—43no—
251C4×C73—43no—
252C4×C73—43no—
253C4×C73—43no—
254C4×C73—43no—
255C4×C73—43no—
256#59596C7311yesmaximal, characteristic, direct product factor, Sylow
257C2×C73—42no—
258C2×C73—46no—
259C2×C73#347513yes—
260C2×C73#351313yes—
261C2×C73#356811yescharacteristic
262#2738C2×C7313yescomplemented
263#2736C2×C7311yescharacteristic, complemented
264#2589C2×C7313yescomplemented
265#2577C2×C7313yescomplemented
266#2572C2×C7313yescomplemented
267#2470C2×C7313yescomplemented
268#2468C2×C7313yescomplemented
269#2418C2×C7313yes—
270#2347C2×C7313yescomplemented
271#2342C2×C7313yescomplemented
272C4×#123C2×C7313yescomplemented
273C73#5959611yescharacteristic, direct product factor, Sylow
274C2×#126C22×C7311yescharacteristic, complemented
275C2×#123C22×C7313yes—
276C2×#120C22×C7311yescharacteristic
277C2×C4×D4C22×C7313yes—
278C2×#117C22×C7313yes—
279C22×C22⋊C4C22×C7313yes—
280C22×C42C22×C7311yescharacteristic, complemented
281#400C22×C7313yescomplemented
282#400C22×C7313yescomplemented
283#399C22×C7313yes—
284#398C22×C7313yescomplemented
285#398C22×C7313yescomplemented
286#397C22×C7313yes—
287#397C22×C7313yes—
288#396C22×C7311yescharacteristic, complemented
289#394C22×C7313yes—
290#394C22×C7313yes—
291#393C22×C7313yes—
292#393C22×C7311yescharacteristic, complemented
293#391C22×C7313yescomplemented
294#389C22×C7313yescomplemented
295#389C22×C7313yescomplemented
296#388C22×C7313yes—
297#388C22×C7313yes—
298#388C22×C7313yes—
299#388C22×C7313yes—
300#388C22×C7313yes—
301#388C22×C7313yes—
302#388C22×C7313yes—
303#388C22×C7313yes—
304#388C22×C7313yescomplemented
305#387C22×C7313yescomplemented
306#387C22×C7313yescomplemented
307#387C22×C7313yescomplemented
308#387C22×C7313yescomplemented
309#386C22×C7313yes—
310#386C22×C7313yes—
311#385C22×C7313yes—
312#385C22×C7313yes—
313#385C22×C7313yes—
314#385C22×C7313yescomplemented
315#383C22×C7313yescomplemented
316#383C22×C7313yescomplemented
317#381C22×C7313yes—
318#381C22×C7313yescomplemented
319#380C22×C7313yes—
320#380C22×C7313yes—
321#379C22×C7313yes—
322C4×C4⋊C4C22×C7313yescomplemented
323C4×C4⋊C4C22×C7313yescomplemented
324C4×C4⋊C4C22×C7313yescomplemented
325C4×C22⋊C4C22×C7313yes—
326C4×C22⋊C4C22×C7313yes—
327C4×C22⋊C4C22×C7313yes—
328C4×C22⋊C4C22×C7313yescomplemented
329C25C23×C7311yescharacteristic
330C22×Q8C23×C7311yescharacteristic
331C22×D4C23×C7313yes—
332C23×C4C23×C7313yes—
333C23×C4C23×C7313yes—
334C23×C4C23×C7313yes—
335#126—42no—
336#123—212no—
337#120—42no—
338C4×D4—46no—
339#117—212no—
340C2×C4⋊C4C23×C7313yes—
341C2×C4⋊C4C23×C7313yes—
342C2×C4⋊C4C23×C7313yes—
343C2×C4⋊C4C23×C7313yes—
344C2×C4⋊C4C23×C7313yes—
345C2×C4⋊C4C23×C7313yes—
346C2×C4⋊C4C23×C7313yes—
347C2×C4⋊C4C23×C7313yes—
348C2×C4⋊C4C23×C7313yes—
349C2×C4⋊C4C23×C7313yes—
350C2×C22⋊C4—212no—
351C2×C22⋊C4C23×C7313yes—
352C2×C22⋊C4C23×C7313yes—
353C2×C22⋊C4C23×C7313yes—
354C2×C22⋊C4C23×C7313yes—
355C2×C22⋊C4C23×C7313yes—
356C2×C22⋊C4C23×C7313yes—
357C2×C22⋊C4C23×C7313yes—
358C2×C22⋊C4C23×C7313yes—
359C2×C22⋊C4C23×C7313yes—
360C2×C22⋊C4C23×C7313yes—
361C2×C22⋊C4C23×C7313yes—
362C2×C22⋊C4C23×C7313yes—
363C2×C22⋊C4C23×C7313yes—
364C2×C22⋊C4C23×C7313yes—
365C2×C42—42no—
366C2×C42C23×C7313yes—
367C2×C42C23×C7313yes—
368C2×C42C23×C7313yes—
369C2×C42C23×C7311yescharacteristic
370C2×C42C23×C7313yes—
371C2×C42C23×C7313yes—
372C2×C42C23×C7311yescharacteristic
373#95C23×C7313yes—
374#95C23×C7313yes—
375#95C23×C7313yes—
376#95C23×C7313yes—
377#95C23×C7313yes—
378#95C23×C7313yes—
379#95C23×C7313yes—
380#95C23×C7313yes—
381#95C23×C7313yes—
382#95C23×C7313yes—
383#95C23×C7313yes—
384#95C23×C7313yes—
385#95C23×C7313yes—
386#95C23×C7313yes—
387#95C23×C7313yes—
388#95C23×C7313yes—
389#95C23×C7313yes—
390#95C23×C7311yescharacteristic
391C24—26no—
392C24—43no—
393C24—41no—
394C24C24×C7313yes—
395C24C24×C7311yescharacteristic
396C2×Q8—43no—
397C2×D4—26no—
398C2×D4—26no—
399C2×D4—43no—
400C22×C4—26no—
401C22×C4—26no—
402C22×C4—26no—
403C22×C4—26no—
404C22×C4—26no—
405C22×C4—43no—
406C22×C4—43no—
407C22×C4—43no—
408C22×C4—43no—
409C22×C4C24×C7313yes—
410C22×C4C24×C7313yes—
411C22×C4C24×C7313yes—
412C22×C4C24×C7313yes—
413C22×C4C24×C7313yes—
414C22×C4C24×C7313yes—
415C22×C4C24×C7313yes—
416C22×C4C24×C7313yes—
417C22×C4C24×C7313yes—
418C4⋊C4—26no—
419C4⋊C4—26no—
420C4⋊C4—26no—
421C4⋊C4—26no—
422C4⋊C4—43no—
423C4⋊C4—43no—
424C4⋊C4—43no—
425C4⋊C4—43no—
426C4⋊C4—43no—
427C4⋊C4—43no—
428C22⋊C4—26no—
429C22⋊C4—26no—
430C22⋊C4—26no—
431C22⋊C4—26no—
432C22⋊C4—26no—
433C22⋊C4—26no—
434C22⋊C4—26no—
435C22⋊C4—43no—
436C22⋊C4—43no—
437C22⋊C4—43no—
438C22⋊C4—43no—
439C22⋊C4—43no—
440C22⋊C4—43no—
441C22⋊C4#41×C73112yes—
442C42—26no—
443C42—26no—
444C42—43no—
445C42—41no—
446C42—43no—
447C42—43no—
448C42—41no—
449C23—23no—
450C23—23no—
451C23—23no—
452C23—23no—
453C23—23no—
454C23—23no—
455C23—23no—
456C23—23no—
457C23—412no—
458C23—86no—
459C23—82no—
460C23C2×#41×C7316yes—
461C23H(5,𝔽2)×C7312yes—
462C23C25×C7311yescharacteristic
463Q8—82no—
464D4—412no—
465C2×C4—23no—
466C2×C4—23no—
467C2×C4—23no—
468C2×C4—23no—
469C2×C4—23no—
470C2×C4—23no—
471C2×C4—23no—
472C2×C4—23no—
473C2×C4—23no—
474C2×C4—23no—
475C2×C4—23no—
476C2×C4—23no—
477C2×C4—23no—
478C2×C4—23no—
479C2×C4—23no—
480C2×C4—23no—
481C2×C4—23no—
482C2×C4—23no—
483C2×C4—23no—
484C2×C4—23no—
485C2×C4—23no—
486C2×C4—23no—
487C2×C4—23no—
488C2×C4—23no—
489C2×C4—23no—
490C2×C4—412no—
491C2×C4—412no—
492C2×C4—86no—
493C2×C4C2×#41×C7316yes—
494C2×C4C2×#41×C7316yes—
495C22—26no—
496C22—26no—
497C22—26no—
498C22—26no—
499C22—43no—
500C22—43no—
501C22—43no—
502C22—43no—
503C22—43no—
504C22—41no—
505C22—88no—
506C22C22×#41×C7313yes—
507C22C2×H(5,𝔽2)×C7311yescharacteristic
508C22#585×C7313yes—
509C4—26no—
510C4—26no—
511C4—43no—
512C4—43no—
513C4—43no—
514C4—43no—
515C4—43no—
516C4—43no—
517C4—43no—
518C2—46no—
519C2—42no—
520C2C73×#347513yes—
521C2C73×#351313yes—
522C2C73×#356811yescharacteristic
subgroupG/H
center, socleC23×C73C25
commutator, Frattini, cosocleC23C25×C73
prime radicalC2×#120×C73C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#2738C2semidirect
2. C73×#2736C2semidirect
3. C73×#2589C2semidirect
4. C73×#2577C2semidirect
5. C73×#2572C2semidirect
6. C73×#2470C2semidirect
7. C73×#2468C2semidirect
9. C73×#2347C2semidirect
10. C73×#2342C2semidirect
11. C4×#123×C73C2semidirect
12. C2×#126×C73C22semidirect
18. C22×C42×C73C22semidirect
19. #400×C73C22semidirect
20. #400×C73C22semidirect
22. #398×C73C22semidirect
23. #398×C73C22semidirect
26. #396×C73C22semidirect
30. #393×C73C22semidirect
31. #391×C73C22semidirect
32. #389×C73C22semidirect
33. #389×C73C22semidirect
42. #388×C73C22semidirect
43. #387×C73C22semidirect
44. #387×C73C22semidirect
45. #387×C73C22semidirect
46. #387×C73C22semidirect
52. #385×C73C22semidirect
53. #383×C73C22semidirect
54. #383×C73C22semidirect
56. #381×C73C22semidirect
60. C4×C4⋊C4×C73C22semidirect
61. C4×C4⋊C4×C73C22semidirect
62. C4×C4⋊C4×C73C22semidirect
66. C4×C22⋊C4×C73C22semidirect
256. #59596C73direct (N × H)
262. #2738C2×C73semidirect
263. #2736C2×C73semidirect
264. #2589C2×C73semidirect
265. #2577C2×C73semidirect
266. #2572C2×C73semidirect
267. #2470C2×C73semidirect
268. #2468C2×C73semidirect
270. #2347C2×C73semidirect
271. #2342C2×C73semidirect
272. C4×#123C2×C73semidirect
273. C73#59596direct (N × H)
274. C2×#126C22×C73semidirect
280. C22×C42C22×C73semidirect
281. #400C22×C73semidirect
282. #400C22×C73semidirect
284. #398C22×C73semidirect
285. #398C22×C73semidirect
288. #396C22×C73semidirect
292. #393C22×C73semidirect
293. #391C22×C73semidirect
294. #389C22×C73semidirect
295. #389C22×C73semidirect
304. #388C22×C73semidirect
305. #387C22×C73semidirect
306. #387C22×C73semidirect
307. #387C22×C73semidirect
308. #387C22×C73semidirect
314. #385C22×C73semidirect
315. #383C22×C73semidirect
316. #383C22×C73semidirect
318. #381C22×C73semidirect
322. C4×C4⋊C4C22×C73semidirect
323. C4×C4⋊C4C22×C73semidirect
324. C4×C4⋊C4C22×C73semidirect
328. C4×C22⋊C4C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#27382
C73×#27362
C73×#25892
C73×#25772
C73×#25722
C73×#24702
C73×#24682
C73×#24182
C73×#23472
C73×#23422
C4×#123×C732
#5959673