Finite Group Databasebeta

← C73×#9117 #32918521

Subgroups — C73×#9117 #32918521

Order 18688.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#407×C73C211yesmaximal, characteristic, complemented
2C73×#2847C211yesmaximal, characteristic
3C2×#327×C73C212yesmaximal, complemented
4C73×C22⋊#95C211yesmaximal, characteristic, complemented
5C73×#1294C212yesmaximal, complemented
6C24×C4×C73C2211yescharacteristic
7C22×M16×C73—21no—
8C2×#121×C73C411yescharacteristic, complemented
9C2×#121×C73C411yescharacteristic, complemented
10C2×#121×C73C2211yescharacteristic
11C2×#120×C73—21no—
12C22×C22⋊C4×C73—21no—
13#407×C73—24no—
14C2×C22⋊C8×C73—21no—
15C2×C22⋊C8×C73C2212yes—
16#386×C73—21no—
17#386×C73—21no—
18#386×C73—21no—
19#380×C73C411yescharacteristic, complemented
20#380×C73C411yescharacteristic, complemented
21#380×C73C2211yescharacteristic
22C2×#95×C73—21no—
23C2×#95×C73C2212yes—
24#327×C73—24no—
25C25×C73D411yescharacteristic
26C22×D4×C73—21no—
27C22×D4×C73—21no—
28C22×D4×C73—21no—
29C22×D4×C73C2×C411yescharacteristic
30C23×C4×C73—22no—
31C23×C4×C73—22no—
32C23×C4×C73—22no—
33C23×C4×C73—42no—
34C23×C4×C73—42no—
35C23×C4×C73C2×C411yescharacteristic
36C23×C4×C73C2×C411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C23×C4×C73C2311yescharacteristic
41C2×M16×C73—24no—
42C2×M16×C73—42no—
43C2×M16×C73—42no—
44C22×C8×C73—22no—
45#121×C73—22no—
46#121×C73—22no—
47#121×C73—42no—
48#121×C73—42no—
49#120×C73—44no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C411yescharacteristic
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C22⋊C4×C73—24no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73—21no—
57C2×C22⋊C4×C73—21no—
58C2×C22⋊C4×C73—21no—
59C2×C22⋊C4×C73—21no—
60C2×C22⋊C4×C73—42no—
61C2×C22⋊C4×C73—42no—
62C22⋊C8×C73—24no—
63C22⋊C8×C73—22no—
64C22⋊C8×C73D414yescomplemented
65#95×C73—21no—
66#95×C73—21no—
67#95×C73—21no—
68#95×C73—22no—
69#95×C73—22no—
70#95×C73—42no—
71C24×C73—22no—
72C24×C73—22no—
73C24×C73—21no—
74C24×C73—21no—
75C24×C73—22no—
76C24×C73—22no—
77C24×C73—21no—
78C24×C73—41no—
79C24×C73—41no—
80C24×C73C22⋊C411yescharacteristic
81C24×C73C22⋊C411yescharacteristic
82C24×C73C2×D411yescharacteristic
83C2×D4×C73—22no—
84C2×D4×C73—41no—
85C2×D4×C73—41no—
86C2×D4×C73—42no—
87C2×D4×C73—42no—
88C2×D4×C73—42no—
89C2×D4×C73—42no—
90C2×D4×C73—42no—
91C2×D4×C73—42no—
92C2×D4×C73—42no—
93C2×D4×C73—42no—
94C2×D4×C73—41no—
95C2×D4×C73—41no—
96C22×C4×C73—22no—
97C22×C4×C73—22no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—22no—
101C22×C4×C73—22no—
102C22×C4×C73—22no—
103C22×C4×C73—22no—
104C22×C4×C73—22no—
105C22×C4×C73—22no—
106C22×C4×C73—21no—
107C22×C4×C73—21no—
108C22×C4×C73—24no—
109C22×C4×C73—21no—
110C22×C4×C73—21no—
111C22×C4×C73—21no—
112C22×C4×C73—41no—
113C22×C4×C73—41no—
114C22×C4×C73—41no—
115C22×C4×C73—41no—
116C22×C4×C73—41no—
117C22×C4×C73—41no—
118C22×C4×C73—41no—
119C22×C4×C73—48no—
120C22×C4×C73—48no—
121C22×C4×C73—42no—
122C22×C4×C73—42no—
123C22×C4×C73—42no—
124C22×C4×C73—42no—
125C22×C4×C73—42no—
126C22×C4×C73—42no—
127C22×C4×C73C22⋊C411yescharacteristic
128C22×C4×C73C22⋊C411yescharacteristic
129C22×C4×C73C22×C411yescharacteristic
130C22×C4×C73C2×D411yescharacteristic
131C22×C4×C73C2×D412yes—
132C22×C4×C73#4112yes—
133M16×C73—48no—
134C2×C8×C73—24no—
135C2×C8×C73—42no—
136C2×C8×C73—42no—
137C4⋊C4×C73—22no—
138C4⋊C4×C73—41no—
139C4⋊C4×C73—41no—
140C22⋊C4×C73—42no—
141C22⋊C4×C73—48no—
142C22⋊C4×C73—42no—
143C22⋊C4×C73—42no—
144C22⋊C4×C73—42no—
145C22⋊C4×C73—42no—
146C22⋊C4×C73—41no—
147C22⋊C4×C73—41no—
148C23×C73—21no—
149C23×C73—21no—
150C23×C73—21no—
151C23×C73—21no—
152C23×C73—24no—
153C23×C73—24no—
154C23×C73—21no—
155C23×C73—21no—
156C23×C73—21no—
157C23×C73—21no—
158C23×C73—21no—
159C23×C73—21no—
160C23×C73—21no—
161C23×C73—21no—
162C23×C73—21no—
163C23×C73—21no—
164C23×C73—21no—
165C23×C73—21no—
166C23×C73—44no—
167C23×C73—44no—
168C23×C73—44no—
169C23×C73—44no—
170C23×C73—42no—
171C23×C73—42no—
172C23×C73—42no—
173C23×C73—42no—
174C23×C73—41no—
175C23×C73—41no—
176C23×C73—41no—
177C23×C73—41no—
178C23×C73—41no—
179C23×C73—41no—
180C23×C73—41no—
181C23×C73#9912yes—
182C23×C73#10412yes—
183C23×C73C2×C22⋊C411yescharacteristic
184C23×C73#12112yes—
185C23×C73AGL(1,ℤ/8)11yescharacteristic
186C23×C73AGL(1,ℤ/8)12yes—
187C23×C73#13711yescharacteristic
188D4×C73—82no—
189D4×C73—84no—
190D4×C73—84no—
191D4×C73—82no—
192D4×C73—82no—
193C2×C4×C73—24no—
194C2×C4×C73—24no—
195C2×C4×C73—21no—
196C2×C4×C73—21no—
197C2×C4×C73—21no—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—42no—
205C2×C4×C73—42no—
206C2×C4×C73—44no—
207C2×C4×C73—44no—
208C2×C4×C73—44no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—44no—
212C2×C4×C73—44no—
213C2×C4×C73—44no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—41no—
221C2×C4×C73—41no—
222C2×C4×C73—41no—
223C2×C4×C73—82no—
224C2×C4×C73—82no—
225C2×C4×C73—82no—
226C2×C4×C73—82no—
227C2×C4×C73#9912yes—
228C2×C4×C73C4×D412yes—
229C2×C4×C73#12011yescharacteristic
230C2×C4×C73#12311yescharacteristic
231C8×C73—44no—
232C22×C73—22no—
233C22×C73—22no—
234C22×C73—21no—
235C22×C73—21no—
236C22×C73—22no—
237C22×C73—22no—
238C22×C73—21no—
239C22×C73—21no—
240C22×C73—22no—
241C22×C73—22no—
242C22×C73—22no—
243C22×C73—22no—
244C22×C73—22no—
245C22×C73—22no—
246C22×C73—48no—
247C22×C73—48no—
248C22×C73—41no—
249C22×C73—41no—
250C22×C73—41no—
251C22×C73—41no—
252C22×C73—41no—
253C22×C73—41no—
254C22×C73—41no—
255C22×C73—41no—
256C22×C73—81no—
257C22×C73—81no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73#38611yescharacteristic
264C22×C73C2×#9911yescharacteristic
265C22×C73C2×#9911yescharacteristic
266C22×C73#41711yescharacteristic
267C22×C73#41811yescharacteristic
268C22×C73C2×#10411yescharacteristic
269C22×C73#42111yescharacteristic
270C22×C73#46812yes—
271C22×C73#46912yes—
272C4×C73—21no—
273C4×C73—22no—
274C4×C73—21no—
275C4×C73—22no—
276C4×C73—41no—
277C4×C73—84no—
278C4×C73—81no—
279C4×C73—81no—
280C4×C73—81no—
281#9117C7311yesmaximal, characteristic, direct product factor, Sylow
282C2×C73—22no—
283C2×C73—22no—
284C2×C73—42no—
285C2×C73—42no—
286C2×C73—161no—
287C2×C73#150411yescharacteristic
288C2×C73#150711yescharacteristic
289C2×C73#151011yescharacteristic
290C2×C73#151311yescharacteristic
291C2×C73#187111yescharacteristic
292C2×C73#189511yescharacteristic
293C2×C73#189811yescharacteristic
294C2×#407C2×C7311yescharacteristic, complemented
295#2847C2×C7311yescharacteristic
296C2×#327C2×C7312yescomplemented
297C22⋊#95C2×C7311yescharacteristic, complemented
298#1294C2×C7312yescomplemented
299C73#911711yescharacteristic, direct product factor, Sylow
300C24×C4C22×C7311yescharacteristic
301C22×M16—21no—
302C2×#121C4×C7311yescharacteristic, complemented
303C2×#121C4×C7311yescharacteristic, complemented
304C2×#121C22×C7311yescharacteristic
305C2×#120—21no—
306C22×C22⋊C4—21no—
307#407—24no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8C22×C7312yes—
310#386—21no—
311#386—21no—
312#386—21no—
313#380C4×C7311yescharacteristic, complemented
314#380C4×C7311yescharacteristic, complemented
315#380C22×C7311yescharacteristic
316C2×#95—21no—
317C2×#95C22×C7312yes—
318#327—24no—
319C25D4×C7311yescharacteristic
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4C2×C4×C7311yescharacteristic
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—42no—
328C23×C4—42no—
329C23×C4C2×C4×C7311yescharacteristic
330C23×C4C2×C4×C7311yescharacteristic
331C23×C4D4×C7311yescharacteristic
332C23×C4D4×C7311yescharacteristic
333C23×C4D4×C7311yescharacteristic
334C23×C4C23×C7311yescharacteristic
335C2×M16—24no—
336C2×M16—42no—
337C2×M16—42no—
338C22×C8—22no—
339#121—22no—
340#121—22no—
341#121—42no—
342#121—42no—
343#120—44no—
344C2×C4⋊C4C2×C4×C7311yescharacteristic
345C2×C4⋊C4C2×C4×C7311yescharacteristic
346C2×C4⋊C4C2×C4×C7311yescharacteristic
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C22⋊C8—24no—
357C22⋊C8—22no—
358C22⋊C8D4×C7314yescomplemented
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—42no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—41no—
373C24—41no—
374C24C22⋊C4×C7311yescharacteristic
375C24C22⋊C4×C7311yescharacteristic
376C24C2×D4×C7311yescharacteristic
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22⋊C4×C7311yescharacteristic
422C22×C4C22⋊C4×C7311yescharacteristic
423C22×C4C22×C4×C7311yescharacteristic
424C22×C4C2×D4×C7311yescharacteristic
425C22×C4C2×D4×C7312yes—
426C22×C4#41×C7312yes—
427M16—48no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—48no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23#99×C7312yes—
476C23#104×C7312yes—
477C23C2×C22⋊C4×C7311yescharacteristic
478C23#121×C7312yes—
479C23AGL(1,ℤ/8)×C7311yescharacteristic
480C23AGL(1,ℤ/8)×C7312yes—
481C23#137×C7311yescharacteristic
482D4—82no—
483D4—84no—
484D4—84no—
485D4—82no—
486D4—82no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#99×C7312yes—
522C2×C4C4×D4×C7312yes—
523C2×C4#120×C7311yescharacteristic
524C2×C4#123×C7311yescharacteristic
525C8—44no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#386×C7311yescharacteristic
558C22C2×#99×C7311yescharacteristic
559C22C2×#99×C7311yescharacteristic
560C22#417×C7311yescharacteristic
561C22#418×C7311yescharacteristic
562C22C2×#104×C7311yescharacteristic
563C22#421×C7311yescharacteristic
564C22#468×C7312yes—
565C22#469×C7312yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—84no—
572C4—81no—
573C4—81no—
574C4—81no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—161no—
580C2C73×#150411yescharacteristic
581C2C73×#150711yescharacteristic
582C2C73×#151011yescharacteristic
583C2C73×#151311yescharacteristic
584C2C73×#187111yescharacteristic
585C2C73×#189511yescharacteristic
586C2C73×#189811yescharacteristic
subgroupG/H
center, socleC23×C73C2×C22⋊C4
commutatorC22×C4C22×C4×C73
Frattini, cosocleC23×C4C23×C73
prime radicalC73×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C73 ◁ C24×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#407×C73C2semidirect
3. C2×#327×C73C2semidirect
4. C73×C22⋊#95C2semidirect
5. C73×#1294C2semidirect
8. C2×#121×C73C4semidirect
9. C2×#121×C73C4semidirect
19. #380×C73C4semidirect
20. #380×C73C4semidirect
64. C22⋊C8×C73D4semidirect
281. #9117C73direct (N × H)
294. C2×#407C2×C73semidirect
296. C2×#327C2×C73semidirect
297. C22⋊#95C2×C73semidirect
298. #1294C2×C73semidirect
299. C73#9117direct (N × H)
302. C2×#121C4×C73semidirect
303. C2×#121C4×C73semidirect
313. #380C4×C73semidirect
314. #380C4×C73semidirect
358. C22⋊C8D4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#407×C732
C73×#28472
C2×#327×C732
C73×C22⋊#952
C73×#12942
#911773