Finite Group Databasebeta

← C73×#59928 #32969332

Subgroups — C73×#59928 #32969332

Order 18688.

1680 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2811C211yesmaximal, characteristic, complemented
2C73×#2809C211yesmaximal, characteristic, complemented
3C73×#2732C212yesmaximal
4C73×#2612C212yesmaximal, complemented
5C73×#2574C214yesmaximal, complemented
6C73×#2565C212yesmaximal, complemented
7C73×#2564C212yesmaximal
8C73×#2556C214yesmaximal, complemented
9C73×#2423C214yesmaximal, complemented
10C73×#2372C212yesmaximal
11C73×#2368C214yesmaximal, complemented
12C4×#124×C73C211yesmaximal, characteristic
13C73×#2295C212yesmaximal, complemented
14C2×#125×C73C2212yes—
15C2×#124×C73C2211yescharacteristic
16C2×C22⋊Q8×C73C2214yes—
17C2×C22⋊Q8×C73C2212yes—
18C2×C4×Q8×C73C2212yes—
19C2×C4×Q8×C73C2211yescharacteristic
20C2×C4×D4×C73C2211yescharacteristic
21#400×C73C2212yes—
22#399×C73C2214yes—
23#399×C73C2212yes—
24#398×C73C2214yes—
25#398×C73C2214yes—
26#398×C73C2214yes—
27#396×C73C2214yes—
28#395×C73C2214yes—
29#395×C73C2212yes—
30#393×C73C2212yes—
31#391×C73C2212yes—
32#391×C73C2212yes—
33#391×C73C2211yescharacteristic
34#391×C73C2212yes—
35#391×C73C2212yes—
36#391×C73C2214yes—
37#391×C73C2214yes—
38#390×C73C2211yescharacteristic
39#389×C73C2212yes—
40#389×C73C2212yes—
41#389×C73C2214yes—
42#388×C73C2214yes—
43#388×C73C2214yes—
44#388×C73C2214yes—
45#387×C73C2214yes—
46#387×C73C2214yes—
47#387×C73C2214yes—
48#387×C73C2214yes—
49#387×C73C2214yes—
50#387×C73C2214yes—
51#386×C73C2214yes—
52#385×C73C2214yes—
53#385×C73C2214yes—
54#384×C73C2212yes—
55#382×C73C2211yescharacteristic
56#382×C73C2212yes—
57#380×C73C2214yes—
58C4×C4⋊C4×C73C2214yes—
59C4×C4⋊C4×C73C2212yes—
60C4×C4⋊C4×C73C2212yes—
61C4×C22⋊C4×C73C2214yes—
62C4×C22⋊C4×C73C2214yes—
63C4×C22⋊C4×C73C2214yes—
64#376×C73C2214yes—
65C2×#95×C73C2212yes—
66C43×C73C2211yescharacteristic
67C22×Q8×C73C2312yes—
68C22×Q8×C73C2311yescharacteristic
69C22×Q8×C73C2312yes—
70C22×D4×C73C2311yescharacteristic
71C23×C4×C73C2314yes—
72C23×C4×C73C2312yes—
73#125×C73—44no—
74#124×C73Q818yescomplemented
75C22⋊Q8×C73—216no—
76C22⋊Q8×C73—44no—
77C4×Q8×C73—28no—
78C4×Q8×C73—42no—
79C4×D4×C73—42no—
80C2×C4⋊C4×C73C2314yes—
81C2×C4⋊C4×C73C2314yes—
82C2×C4⋊C4×C73C2314yes—
83C2×C4⋊C4×C73C2314yes—
84C2×C4⋊C4×C73C2314yes—
85C2×C4⋊C4×C73C2312yes—
86C2×C4⋊C4×C73C2312yes—
87C2×C4⋊C4×C73C2312yes—
88C2×C4⋊C4×C73C2314yes—
89C2×C4⋊C4×C73C2312yes—
90C2×C4⋊C4×C73C2312yes—
91C2×C4⋊C4×C73C2311yescharacteristic
92C2×C4⋊C4×C73C2311yescharacteristic
93C2×C4⋊C4×C73C2312yes—
94C2×C4⋊C4×C73C2312yes—
95C2×C4⋊C4×C73C2314yes—
96C2×C22⋊C4×C73C2314yes—
97C2×C22⋊C4×C73C2314yes—
98C2×C22⋊C4×C73C2314yes—
99C2×C22⋊C4×C73C2312yes—
100C2×C22⋊C4×C73C2314yes—
101C2×C22⋊C4×C73C2314yes—
102C2×C42×C73C2312yes—
103C2×C42×C73C2314yes—
104C2×C42×C73C2312yes—
105C2×C42×C73C2311yescharacteristic
106C2×C42×C73C2312yes—
107C2×C42×C73C2312yes—
108C2×C42×C73C2312yes—
109C2×C42×C73C2314yes—
110C2×C42×C73C2311yescharacteristic
111C2×C42×C73C2311yescharacteristic
112#95×C73C2314yes—
113#95×C73C2314yes—
114#95×C73C2314yes—
115#95×C73C2314yes—
116#95×C73C2314yes—
117#95×C73C2314yes—
118#95×C73C2314yes—
119#95×C73C2314yes—
120#95×C73C2314yes—
121#95×C73C2314yes—
122#95×C73C2314yes—
123#95×C73C2314yes—
124#95×C73C2312yes—
125#95×C73C2312yes—
126#95×C73C2314yes—
127C24×C73C2412yes—
128C2×Q8×C73—41no—
129C2×Q8×C73—41no—
130C2×Q8×C73—42no—
131C2×Q8×C73—42no—
132C2×Q8×C73—42no—
133C2×Q8×C73—44no—
134C2×Q8×C73C2×Q814yes—
135C2×Q8×C73#4118yes—
136C2×D4×C73—41no—
137C2×D4×C73—41no—
138C2×D4×C73C2×Q814yes—
139C22×C4×C73—28no—
140C22×C4×C73—24no—
141C22×C4×C73—44no—
142C22×C4×C73—44no—
143C22×C4×C73—44no—
144C22×C4×C73C2414yes—
145C22×C4×C73C2412yes—
146C22×C4×C73C2414yes—
147C22×C4×C73C2411yescharacteristic
148C22×C4×C73C2414yes—
149C22×C4×C73C2412yes—
150C22×C4×C73C2411yescharacteristic
151C22×C4×C73C2412yes—
152C22×C4×C73C2412yes—
153C22×C4×C73C2414yes—
154C22×C4×C73C2412yes—
155C22×C4×C73C2411yescharacteristic
156C4⋊C4×C73—24no—
157C4⋊C4×C73—28no—
158C4⋊C4×C73—28no—
159C4⋊C4×C73—28no—
160C4⋊C4×C73—28no—
161C4⋊C4×C73—28no—
162C4⋊C4×C73—42no—
163C4⋊C4×C73—42no—
164C4⋊C4×C73—44no—
165C4⋊C4×C73—41no—
166C4⋊C4×C73—42no—
167C4⋊C4×C73—41no—
168C4⋊C4×C73—42no—
169C4⋊C4×C73—42no—
170C4⋊C4×C73—42no—
171C4⋊C4×C73—44no—
172C22⋊C4×C73—28no—
173C22⋊C4×C73—28no—
174C22⋊C4×C73—44no—
175C22⋊C4×C73—42no—
176C22⋊C4×C73—44no—
177C22⋊C4×C73C2×Q8116yes—
178C42×C73—24no—
179C42×C73—28no—
180C42×C73—22no—
181C42×C73—24no—
182C42×C73—42no—
183C42×C73—44no—
184C42×C73—42no—
185C42×C73—42no—
186C42×C73—41no—
187C42×C73C2×Q814yes—
188C23×C73—24no—
189C23×C73—22no—
190C23×C73—24no—
191C23×C73—22no—
192C23×C73C22×Q814yes—
193C23×C73C2511yescharacteristic
194Q8×C73—44no—
195Q8×C73—48no—
196Q8×C73—84no—
197D4×C73—44no—
198C2×C4×C73—22no—
199C2×C4×C73—22no—
200C2×C4×C73—21no—
201C2×C4×C73—22no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—24no—
207C2×C4×C73—21no—
208C2×C4×C73—24no—
209C2×C4×C73—22no—
210C2×C4×C73—24no—
211C2×C4×C73—24no—
212C2×C4×C73—24no—
213C2×C4×C73—24no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—24no—
217C2×C4×C73—24no—
218C2×C4×C73—22no—
219C2×C4×C73—24no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—88no—
225C2×C4×C73—84no—
226C2×C4×C73C22×Q814yes—
227C2×C4×C73C22×Q812yes—
228C2×C4×C73C22×Q814yes—
229C2×C4×C73C2×#4114yes—
230C2×C4×C73C2×#4118yes—
231C2×C4×C73H(5,𝔽2)12yes—
232C22×C73—28no—
233C22×C73—24no—
234C22×C73—42no—
235C22×C73—44no—
236C22×C73—42no—
237C22×C73C23×Q811yescharacteristic
238C22×C73C22×#4112yes—
239C22×C73C2×H(5,𝔽2)11yescharacteristic
240C22×C73#58512yes—
241C22×C73#58511yescharacteristic
242C4×C73—24no—
243C4×C73—28no—
244C4×C73—24no—
245C4×C73—22no—
246C4×C73—24no—
247C4×C73—22no—
248C4×C73—41no—
249C4×C73—44no—
250C4×C73—42no—
251C4×C73—44no—
252C4×C73—42no—
253C4×C73—44no—
254#59928C7311yesmaximal, characteristic, direct product factor, Sylow
255C2×C73—44no—
256C2×C73Q8×#4112yes—
257C2×C73#348311yescharacteristic
258C2×C73#349112yes—
259C2×C73#350111yescharacteristic
260C2×C73#354511yescharacteristic
261#2811C2×C7311yescharacteristic, complemented
262#2809C2×C7311yescharacteristic, complemented
263#2732C2×C7312yes—
264#2612C2×C7312yescomplemented
265#2574C2×C7314yescomplemented
266#2565C2×C7312yescomplemented
267#2564C2×C7312yes—
268#2556C2×C7314yescomplemented
269#2423C2×C7314yescomplemented
270#2372C2×C7312yes—
271#2368C2×C7314yescomplemented
272C4×#124C2×C7311yescharacteristic
273#2295C2×C7312yescomplemented
274C73#5992811yescharacteristic, direct product factor, Sylow
275C2×#125C22×C7312yes—
276C2×#124C22×C7311yescharacteristic
277C2×C22⋊Q8C22×C7314yes—
278C2×C22⋊Q8C22×C7312yes—
279C2×C4×Q8C22×C7312yes—
280C2×C4×Q8C22×C7311yescharacteristic
281C2×C4×D4C22×C7311yescharacteristic
282#400C22×C7312yes—
283#399C22×C7312yes—
284#399C22×C7314yes—
285#398C22×C7314yes—
286#398C22×C7314yes—
287#398C22×C7314yes—
288#396C22×C7314yes—
289#395C22×C7312yes—
290#395C22×C7314yes—
291#393C22×C7312yes—
292#391C22×C7312yes—
293#391C22×C7312yes—
294#391C22×C7314yes—
295#391C22×C7312yes—
296#391C22×C7314yes—
297#391C22×C7312yes—
298#391C22×C7311yescharacteristic
299#390C22×C7311yescharacteristic
300#389C22×C7314yes—
301#389C22×C7312yes—
302#389C22×C7312yes—
303#388C22×C7314yes—
304#388C22×C7314yes—
305#388C22×C7314yes—
306#387C22×C7314yes—
307#387C22×C7314yes—
308#387C22×C7314yes—
309#387C22×C7314yes—
310#387C22×C7314yes—
311#387C22×C7314yes—
312#386C22×C7314yes—
313#385C22×C7314yes—
314#385C22×C7314yes—
315#384C22×C7312yes—
316#382C22×C7312yes—
317#382C22×C7311yescharacteristic
318#380C22×C7314yes—
319C4×C4⋊C4C22×C7312yes—
320C4×C4⋊C4C22×C7312yes—
321C4×C4⋊C4C22×C7314yes—
322C4×C22⋊C4C22×C7314yes—
323C4×C22⋊C4C22×C7314yes—
324C4×C22⋊C4C22×C7314yes—
325#376C22×C7314yes—
326C2×#95C22×C7312yes—
327C43C22×C7311yescharacteristic
328C22×Q8C23×C7312yes—
329C22×Q8C23×C7311yescharacteristic
330C22×Q8C23×C7312yes—
331C22×D4C23×C7311yescharacteristic
332C23×C4C23×C7312yes—
333C23×C4C23×C7314yes—
334#125—44no—
335#124Q8×C7318yescomplemented
336C22⋊Q8—216no—
337C22⋊Q8—44no—
338C4×Q8—28no—
339C4×Q8—42no—
340C4×D4—42no—
341C2×C4⋊C4C23×C7314yes—
342C2×C4⋊C4C23×C7312yes—
343C2×C4⋊C4C23×C7314yes—
344C2×C4⋊C4C23×C7314yes—
345C2×C4⋊C4C23×C7314yes—
346C2×C4⋊C4C23×C7314yes—
347C2×C4⋊C4C23×C7311yescharacteristic
348C2×C4⋊C4C23×C7312yes—
349C2×C4⋊C4C23×C7311yescharacteristic
350C2×C4⋊C4C23×C7312yes—
351C2×C4⋊C4C23×C7312yes—
352C2×C4⋊C4C23×C7312yes—
353C2×C4⋊C4C23×C7314yes—
354C2×C4⋊C4C23×C7312yes—
355C2×C4⋊C4C23×C7312yes—
356C2×C4⋊C4C23×C7314yes—
357C2×C22⋊C4C23×C7314yes—
358C2×C22⋊C4C23×C7314yes—
359C2×C22⋊C4C23×C7312yes—
360C2×C22⋊C4C23×C7314yes—
361C2×C22⋊C4C23×C7314yes—
362C2×C22⋊C4C23×C7314yes—
363C2×C42C23×C7311yescharacteristic
364C2×C42C23×C7312yes—
365C2×C42C23×C7312yes—
366C2×C42C23×C7314yes—
367C2×C42C23×C7312yes—
368C2×C42C23×C7311yescharacteristic
369C2×C42C23×C7314yes—
370C2×C42C23×C7312yes—
371C2×C42C23×C7312yes—
372C2×C42C23×C7311yescharacteristic
373#95C23×C7314yes—
374#95C23×C7312yes—
375#95C23×C7312yes—
376#95C23×C7314yes—
377#95C23×C7314yes—
378#95C23×C7314yes—
379#95C23×C7314yes—
380#95C23×C7314yes—
381#95C23×C7314yes—
382#95C23×C7314yes—
383#95C23×C7314yes—
384#95C23×C7314yes—
385#95C23×C7314yes—
386#95C23×C7314yes—
387#95C23×C7314yes—
388C24C24×C7312yes—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8C2×Q8×C7314yes—
396C2×Q8#41×C7318yes—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4C2×Q8×C7314yes—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4C24×C7314yes—
406C22×C4C24×C7312yes—
407C22×C4C24×C7314yes—
408C22×C4C24×C7311yescharacteristic
409C22×C4C24×C7314yes—
410C22×C4C24×C7312yes—
411C22×C4C24×C7311yescharacteristic
412C22×C4C24×C7312yes—
413C22×C4C24×C7312yes—
414C22×C4C24×C7314yes—
415C22×C4C24×C7312yes—
416C22×C4C24×C7311yescharacteristic
417C4⋊C4—24no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4C2×Q8×C73116yes—
439C42—24no—
440C42—28no—
441C42—22no—
442C42—24no—
443C42—42no—
444C42—44no—
445C42—42no—
446C42—42no—
447C42—41no—
448C42C2×Q8×C7314yes—
449C23—24no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23C22×Q8×C7314yes—
454C23C25×C7311yescharacteristic
455Q8—44no—
456Q8—48no—
457Q8—84no—
458D4—44no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—88no—
486C2×C4—84no—
487C2×C4C22×Q8×C7314yes—
488C2×C4C22×Q8×C7312yes—
489C2×C4C22×Q8×C7314yes—
490C2×C4C2×#41×C7314yes—
491C2×C4C2×#41×C7318yes—
492C2×C4H(5,𝔽2)×C7312yes—
493C22—28no—
494C22—24no—
495C22—42no—
496C22—44no—
497C22—42no—
498C22C23×Q8×C7311yescharacteristic
499C22C22×#41×C7312yes—
500C22C2×H(5,𝔽2)×C7311yescharacteristic
501C22#585×C7312yes—
502C22#585×C7311yescharacteristic
503C4—24no—
504C4—28no—
505C4—24no—
506C4—22no—
507C4—24no—
508C4—22no—
509C4—41no—
510C4—44no—
511C4—42no—
512C4—44no—
513C4—42no—
514C4—44no—
515C2—44no—
516C2Q8×#41×C7312yes—
517C2C73×#348311yescharacteristic
518C2C73×#349112yes—
519C2C73×#350111yescharacteristic
520C2C73×#354511yescharacteristic
subgroupG/H
center, socleC23×C73C25
commutator, Frattini, cosocleC23C25×C73
prime radicalC22×D4×C73C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#2811C2semidirect
2. C73×#2809C2semidirect
4. C73×#2612C2semidirect
5. C73×#2574C2semidirect
6. C73×#2565C2semidirect
8. C73×#2556C2semidirect
9. C73×#2423C2semidirect
11. C73×#2368C2semidirect
13. C73×#2295C2semidirect
74. #124×C73Q8semidirect
254. #59928C73direct (N × H)
261. #2811C2×C73semidirect
262. #2809C2×C73semidirect
264. #2612C2×C73semidirect
265. #2574C2×C73semidirect
266. #2565C2×C73semidirect
268. #2556C2×C73semidirect
269. #2423C2×C73semidirect
271. #2368C2×C73semidirect
273. #2295C2×C73semidirect
274. C73#59928direct (N × H)
335. #124Q8×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#28112
C73×#28092
C73×#27322
C73×#26122
C73×#25742
C73×#25652
C73×#25642
C73×#25562
C73×#24232
C73×#23722
C73×#23682
C4×#124×C732
C73×#22952
#5992873