Finite Group Databasebeta

← C73×#36728 #32946132

Subgroups — C73×#36728 #32946132

Order 18688.

1854 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×#2848C212yesmaximal
2C73×#2747C214yesmaximal, complemented
3C73×#2722C214yesmaximal, complemented
4C73×#2704C214yesmaximal, complemented
5C73×#2690C214yesmaximal, complemented
6C73×#2659C211yesmaximal, characteristic, complemented
7C73×#2646C211yesmaximal, characteristic, complemented
8C73×#2480C214yesmaximal
9C73×#2473C214yesmaximal, complemented
10C73×#2329C212yesmaximal, complemented
11C2×#380×C73C211yesmaximal, characteristic
12C24×C4×C73C2211yescharacteristic
13C2×#126×C73C2214yes—
14C2×#125×C73C2214yescomplemented
15C2×C22⋊Q8×C73C2212yes—
16C2×C22⋊Q8×C73C2212yes—
17C2×C22⋊Q8×C73C2212yes—
18C22×C4⋊C4×C73C2212yes—
19C22×C22⋊C4×C73C2212yes—
20#400×C73C2214yescomplemented
21#400×C73C2214yescomplemented
22#398×C73C2214yescomplemented
23#398×C73C2214yescomplemented
24#398×C73C2214yescomplemented
25#398×C73C2214yescomplemented
26#398×C73C2214yescomplemented
27#397×C73C2214yes—
28#397×C73C2214yes—
29#397×C73C2214yes—
30#396×C73C2214yes—
31#395×C73C2214yescomplemented
32#391×C73C2214yescomplemented
33#389×C73C2214yescomplemented
34#389×C73C2214yescomplemented
35#388×C73C2214yes—
36#388×C73C2214yes—
37#388×C73C2214yes—
38#387×C73C2214yescomplemented
39#387×C73C2214yescomplemented
40#387×C73C2214yescomplemented
41#387×C73C2214yescomplemented
42#385×C73C2214yes—
43#385×C73C2214yes—
44#385×C73C2214yes—
45#385×C73C2214yes—
46#385×C73C2214yes—
47#380×C73C2214yes—
48#380×C73C2212yes—
49#380×C73C2214yes—
50#380×C73C2214yes—
51#380×C73C2212yes—
52#380×C73C2212yes—
53#379×C73C2214yes—
54C4×C22⋊C4×C73C2214yes—
55C2×#95×C73C2211yescharacteristic
56C2×#95×C73C2211yescharacteristic
57C25×C73C2311yescharacteristic
58C22×Q8×C73C2312yes—
59C23×C4×C73—24no—
60C23×C4×C73—44no—
61C23×C4×C73C2311yescharacteristic
62C23×C4×C73C2312yes—
63C23×C4×C73C2312yes—
64C23×C4×C73C2312yes—
65C23×C4×C73C2312yes—
66C23×C4×C73C2311yescharacteristic
67#126×C73—48no—
68#125×C73—48no—
69C22⋊Q8×C73—28no—
70C22⋊Q8×C73—44no—
71C22⋊Q8×C73—44no—
72C2×C4⋊C4×C73—44no—
73C2×C4⋊C4×C73C2312yes—
74C2×C4⋊C4×C73C2314yes—
75C2×C4⋊C4×C73C2314yes—
76C2×C4⋊C4×C73C2314yes—
77C2×C4⋊C4×C73C2314yes—
78C2×C4⋊C4×C73C2312yes—
79C2×C4⋊C4×C73C2314yes—
80C2×C4⋊C4×C73C2312yes—
81C2×C4⋊C4×C73C2314yes—
82C2×C4⋊C4×C73C2314yes—
83C2×C4⋊C4×C73C2314yes—
84C2×C4⋊C4×C73C2314yes—
85C2×C4⋊C4×C73C2314yes—
86C2×C4⋊C4×C73C2314yes—
87C2×C22⋊C4×C73—44no—
88C2×C22⋊C4×C73C2314yes—
89C2×C22⋊C4×C73C2314yes—
90C2×C22⋊C4×C73C2314yes—
91C2×C22⋊C4×C73C2314yes—
92C2×C22⋊C4×C73C2314yes—
93C2×C22⋊C4×C73C2314yes—
94C2×C22⋊C4×C73C2314yes—
95C2×C22⋊C4×C73C2314yes—
96C2×C42×C73C2314yes—
97C2×C42×C73C2314yes—
98#95×C73C2314yes—
99#95×C73C2314yes—
100#95×C73C2314yes—
101#95×C73C2314yes—
102#95×C73C2314yes—
103#95×C73C2314yes—
104#95×C73C2314yes—
105#95×C73C2314yes—
106#95×C73C2314yes—
107#95×C73C2314yes—
108#95×C73C2314yes—
109#95×C73C2314yes—
110#95×C73C2314yes—
111C24×C73—22no—
112C24×C73—22no—
113C24×C73—22no—
114C24×C73—24no—
115C24×C73—42no—
116C24×C73C2411yescharacteristic
117C24×C73C2411yescharacteristic
118C24×C73C2411yescharacteristic
119C2×Q8×C73—24no—
120C2×Q8×C73—42no—
121C2×Q8×C73—42no—
122C22×C4×C73—22no—
123C22×C4×C73—24no—
124C22×C4×C73—24no—
125C22×C4×C73—24no—
126C22×C4×C73—24no—
127C22×C4×C73—24no—
128C22×C4×C73—24no—
129C22×C4×C73—24no—
130C22×C4×C73—42no—
131C22×C4×C73—42no—
132C22×C4×C73—42no—
133C22×C4×C73—42no—
134C22×C4×C73—42no—
135C22×C4×C73—42no—
136C22×C4×C73—42no—
137C22×C4×C73—416no—
138C22×C4×C73#4114yes—
139C22×C4×C73C2414yes—
140C22×C4×C73C2414yes—
141C22×C4×C73C2414yes—
142C22×C4×C73C2414yes—
143C22×C4×C73C2414yes—
144C22×C4×C73C2414yes—
145C22×C4×C73C2412yes—
146C22×C4×C73C2412yes—
147C4⋊C4×C73—24no—
148C4⋊C4×C73—28no—
149C4⋊C4×C73—28no—
150C4⋊C4×C73—28no—
151C4⋊C4×C73—28no—
152C4⋊C4×C73—44no—
153C4⋊C4×C73—42no—
154C4⋊C4×C73—42no—
155C4⋊C4×C73—44no—
156C4⋊C4×C73—44no—
157C4⋊C4×C73—44no—
158C4⋊C4×C73—44no—
159C4⋊C4×C73—44no—
160C4⋊C4×C73—44no—
161C22⋊C4×C73—28no—
162C22⋊C4×C73—28no—
163C22⋊C4×C73—28no—
164C22⋊C4×C73—28no—
165C22⋊C4×C73—28no—
166C22⋊C4×C73—44no—
167C22⋊C4×C73—44no—
168C22⋊C4×C73—44no—
169C42×C73—44no—
170C42×C73—44no—
171C23×C73—22no—
172C23×C73—21no—
173C23×C73—21no—
174C23×C73—28no—
175C23×C73—21no—
176C23×C73—21no—
177C23×C73—21no—
178C23×C73—22no—
179C23×C73—22no—
180C23×C73—21no—
181C23×C73—22no—
182C23×C73—22no—
183C23×C73—48no—
184C23×C73—48no—
185C23×C73—44no—
186C23×C73—44no—
187C23×C73C2×#4112yes—
188C23×C73H(5,𝔽2)12yes—
189C23×C732−1+412yes—
190C23×C732−1+414yes—
191C23×C73C2511yescharacteristic
192Q8×C73—84no—
193C2×C4×C73—28no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—22no—
198C2×C4×C73—24no—
199C2×C4×C73—24no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—24no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—24no—
207C2×C4×C73—24no—
208C2×C4×C73—24no—
209C2×C4×C73—24no—
210C2×C4×C73—24no—
211C2×C4×C73—24no—
212C2×C4×C73—24no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—24no—
216C2×C4×C73—44no—
217C2×C4×C73—48no—
218C2×C4×C73—48no—
219C2×C4×C73—84no—
220C2×C4×C73—84no—
221C2×C4×C73C2×#4114yes—
222C2×C4×C73H(5,𝔽2)14yes—
223C22×C73—22no—
224C22×C73—22no—
225C22×C73—24no—
226C22×C73—24no—
227C22×C73—22no—
228C22×C73—22no—
229C22×C73—24no—
230C22×C73—24no—
231C22×C73—416no—
232C22×C73—41no—
233C22×C73—41no—
234C22×C73—41no—
235C22×C73—41no—
236C22×C73—42no—
237C22×C73—42no—
238C22×C73#54114yes—
239C22×C73C22×#4111yescharacteristic
240C22×C73C2×H(5,𝔽2)12yes—
241C22×C73C2×H(5,𝔽2)11yescharacteristic
242C22×C73C2×2−1+412yes—
243C22×C73C2×2−1+411yescharacteristic
244C4×C73—24no—
245C4×C73—24no—
246C4×C73—44no—
247C4×C73—44no—
248C4×C73—44no—
249C4×C73—44no—
250C4×C73—44no—
251C4×C73—44no—
252#36728C7311yesmaximal, characteristic, direct product factor, Sylow
253C2×C73—24no—
254C2×C73—42no—
255C2×C73—42no—
256C2×C73C2×#54111yescharacteristic
257C2×C73#349011yescharacteristic
258C2×C73#349211yescharacteristic
259C2×C73#353612yes—
260C2×C73#353812yes—
261#2848C2×C7312yes—
262#2747C2×C7314yescomplemented
263#2722C2×C7314yescomplemented
264#2704C2×C7314yescomplemented
265#2690C2×C7314yescomplemented
266#2659C2×C7311yescharacteristic, complemented
267#2646C2×C7311yescharacteristic, complemented
268#2480C2×C7314yes—
269#2473C2×C7314yescomplemented
270#2329C2×C7312yescomplemented
271C2×#380C2×C7311yescharacteristic
272C73#3672811yescharacteristic, direct product factor, Sylow
273C24×C4C22×C7311yescharacteristic
274C2×#126C22×C7314yes—
275C2×#125C22×C7314yescomplemented
276C2×C22⋊Q8C22×C7312yes—
277C2×C22⋊Q8C22×C7312yes—
278C2×C22⋊Q8C22×C7312yes—
279C22×C4⋊C4C22×C7312yes—
280C22×C22⋊C4C22×C7312yes—
281#400C22×C7314yescomplemented
282#400C22×C7314yescomplemented
283#398C22×C7314yescomplemented
284#398C22×C7314yescomplemented
285#398C22×C7314yescomplemented
286#398C22×C7314yescomplemented
287#398C22×C7314yescomplemented
288#397C22×C7314yes—
289#397C22×C7314yes—
290#397C22×C7314yes—
291#396C22×C7314yes—
292#395C22×C7314yescomplemented
293#391C22×C7314yescomplemented
294#389C22×C7314yescomplemented
295#389C22×C7314yescomplemented
296#388C22×C7314yes—
297#388C22×C7314yes—
298#388C22×C7314yes—
299#387C22×C7314yescomplemented
300#387C22×C7314yescomplemented
301#387C22×C7314yescomplemented
302#387C22×C7314yescomplemented
303#385C22×C7314yes—
304#385C22×C7314yes—
305#385C22×C7314yes—
306#385C22×C7314yes—
307#385C22×C7314yes—
308#380C22×C7312yes—
309#380C22×C7312yes—
310#380C22×C7312yes—
311#380C22×C7314yes—
312#380C22×C7314yes—
313#380C22×C7314yes—
314#379C22×C7314yes—
315C4×C22⋊C4C22×C7314yes—
316C2×#95C22×C7311yescharacteristic
317C2×#95C22×C7311yescharacteristic
318C25C23×C7311yescharacteristic
319C22×Q8C23×C7312yes—
320C23×C4—24no—
321C23×C4—44no—
322C23×C4C23×C7311yescharacteristic
323C23×C4C23×C7312yes—
324C23×C4C23×C7312yes—
325C23×C4C23×C7312yes—
326C23×C4C23×C7312yes—
327C23×C4C23×C7311yescharacteristic
328#126—48no—
329#125—48no—
330C22⋊Q8—28no—
331C22⋊Q8—44no—
332C22⋊Q8—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C23×C7314yes—
335C2×C4⋊C4C23×C7314yes—
336C2×C4⋊C4C23×C7314yes—
337C2×C4⋊C4C23×C7312yes—
338C2×C4⋊C4C23×C7314yes—
339C2×C4⋊C4C23×C7314yes—
340C2×C4⋊C4C23×C7312yes—
341C2×C4⋊C4C23×C7312yes—
342C2×C4⋊C4C23×C7314yes—
343C2×C4⋊C4C23×C7314yes—
344C2×C4⋊C4C23×C7314yes—
345C2×C4⋊C4C23×C7314yes—
346C2×C4⋊C4C23×C7314yes—
347C2×C4⋊C4C23×C7314yes—
348C2×C22⋊C4—44no—
349C2×C22⋊C4C23×C7314yes—
350C2×C22⋊C4C23×C7314yes—
351C2×C22⋊C4C23×C7314yes—
352C2×C22⋊C4C23×C7314yes—
353C2×C22⋊C4C23×C7314yes—
354C2×C22⋊C4C23×C7314yes—
355C2×C22⋊C4C23×C7314yes—
356C2×C22⋊C4C23×C7314yes—
357C2×C42C23×C7314yes—
358C2×C42C23×C7314yes—
359#95C23×C7314yes—
360#95C23×C7314yes—
361#95C23×C7314yes—
362#95C23×C7314yes—
363#95C23×C7314yes—
364#95C23×C7314yes—
365#95C23×C7314yes—
366#95C23×C7314yes—
367#95C23×C7314yes—
368#95C23×C7314yes—
369#95C23×C7314yes—
370#95C23×C7314yes—
371#95C23×C7314yes—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—42no—
377C24C24×C7311yescharacteristic
378C24C24×C7311yescharacteristic
379C24C24×C7311yescharacteristic
380C2×Q8—24no—
381C2×Q8—42no—
382C2×Q8—42no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4#41×C7314yes—
400C22×C4C24×C7314yes—
401C22×C4C24×C7314yes—
402C22×C4C24×C7314yes—
403C22×C4C24×C7314yes—
404C22×C4C24×C7314yes—
405C22×C4C24×C7314yes—
406C22×C4C24×C7312yes—
407C22×C4C24×C7312yes—
408C4⋊C4—24no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C42—44no—
431C42—44no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—28no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—21no—
442C23—22no—
443C23—22no—
444C23—48no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23C2×#41×C7312yes—
449C23H(5,𝔽2)×C7312yes—
450C232−1+4×C7312yes—
451C232−1+4×C7314yes—
452C23C25×C7311yescharacteristic
453Q8—84no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4C2×#41×C7314yes—
483C2×C4H(5,𝔽2)×C7314yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—416no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22#541×C7314yes—
500C22C22×#41×C7311yescharacteristic
501C22C2×H(5,𝔽2)×C7312yes—
502C22C2×H(5,𝔽2)×C7311yescharacteristic
503C22C2×2−1+4×C7311yescharacteristic
504C22C2×2−1+4×C7312yes—
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—44no—
510C4—44no—
511C4—44no—
512C4—44no—
513C2—24no—
514C2—42no—
515C2—42no—
516C2C2×#541×C7311yescharacteristic
517C2C73×#349011yescharacteristic
518C2C73×#349211yescharacteristic
519C2C73×#353612yes—
520C2C73×#353812yes—
subgroupG/H
center, socleC23×C73C25
commutator, Frattini, cosocleC23C25×C73
prime radicalC25×C73C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2747C2semidirect
3. C73×#2722C2semidirect
4. C73×#2704C2semidirect
5. C73×#2690C2semidirect
6. C73×#2659C2semidirect
7. C73×#2646C2semidirect
9. C73×#2473C2semidirect
10. C73×#2329C2semidirect
14. C2×#125×C73C22semidirect
20. #400×C73C22semidirect
21. #400×C73C22semidirect
22. #398×C73C22semidirect
23. #398×C73C22semidirect
24. #398×C73C22semidirect
25. #398×C73C22semidirect
26. #398×C73C22semidirect
31. #395×C73C22semidirect
32. #391×C73C22semidirect
33. #389×C73C22semidirect
34. #389×C73C22semidirect
38. #387×C73C22semidirect
39. #387×C73C22semidirect
40. #387×C73C22semidirect
41. #387×C73C22semidirect
252. #36728C73direct (N × H)
262. #2747C2×C73semidirect
263. #2722C2×C73semidirect
264. #2704C2×C73semidirect
265. #2690C2×C73semidirect
266. #2659C2×C73semidirect
267. #2646C2×C73semidirect
269. #2473C2×C73semidirect
270. #2329C2×C73semidirect
272. C73#36728direct (N × H)
275. C2×#125C22×C73semidirect
281. #400C22×C73semidirect
282. #400C22×C73semidirect
283. #398C22×C73semidirect
284. #398C22×C73semidirect
285. #398C22×C73semidirect
286. #398C22×C73semidirect
287. #398C22×C73semidirect
292. #395C22×C73semidirect
293. #391C22×C73semidirect
294. #389C22×C73semidirect
295. #389C22×C73semidirect
299. #387C22×C73semidirect
300. #387C22×C73semidirect
301. #387C22×C73semidirect
302. #387C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#28482
C73×#27472
C73×#27222
C73×#27042
C73×#26902
C73×#26592
C73×#26462
C73×#24802
C73×#24732
C73×#23292
C2×#380×C732
#3672873