Finite Group Databasebeta

← C73×#17947 #32927351

Subgroups — C73×#17947 #32927351

Order 18688.

568 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3108C211yesmaximal, characteristic
2C73×#2993C211yesmaximal, characteristic
3C73×#2983C211yesmaximal, characteristic, complemented
4C73×#1593C211yesmaximal, characteristic, complemented
5C73×#1592C211yesmaximal, characteristic, complemented
6C73×#1591C211yesmaximal, characteristic, complemented
7C73×#1590C211yesmaximal, characteristic, complemented
8C73×#1587C211yesmaximal, characteristic, complemented
9C73×#1585C211yesmaximal, characteristic, complemented
10C73×#1584C211yesmaximal, characteristic, complemented
11C73×#1582C211yesmaximal, characteristic, complemented
12C73×#1574C211yesmaximal, characteristic, complemented
13C73×#1487C211yesmaximal, characteristic, complemented
14C73×#1482C211yesmaximal, characteristic
15C73×#1454C211yesmaximal, characteristic, complemented
16#529×C73C2211yescharacteristic
17C2×C4×Q8×C73C2211yescharacteristic
18#483×C73C411yescharacteristic
19#480×C73C411yescharacteristic
20#479×C73C411yescharacteristic
21#479×C73C411yescharacteristic
22#464×C73C411yescharacteristic
23#464×C73C411yescharacteristic
24#451×C73C411yescharacteristic
25#448×C73C411yescharacteristic
26#446×C73—21no—
27#446×C73—21no—
28#446×C73C2211yescharacteristic
29#446×C73C2211yescharacteristic
30#446×C73C2211yescharacteristic
31#446×C73C2211yescharacteristic
32#446×C73C2211yescharacteristic
33C8×Q8×C73C2211yescharacteristic
34C4×Q16×C73C2211yescharacteristic
35C4×Q16×C73C2211yescharacteristic
36C4×SD16×C73C2211yescharacteristic
37C4×SD16×C73C2211yescharacteristic
38#436×C73—21no—
39#436×C73C2211yescharacteristic
40#435×C73—21no—
41C8×D4×C73C2211yescharacteristic
42#433×C73—21no—
43#432×C73—21no—
44#432×C73—21no—
45#431×C73C2211yescharacteristic
46#431×C73C2211yescharacteristic
47#431×C73C2211yescharacteristic
48#431×C73C2211yescharacteristic
49#424×C73—21no—
50#423×C73C2211yescharacteristic
51#335×C73C2211yescharacteristic
52#335×C73C2211yescharacteristic
53#334×C73C2211yescharacteristic
54#334×C73C2211yescharacteristic
55Q8⋊C8×C73C2211yescharacteristic
56Q8⋊C8×C73C2211yescharacteristic
57Q8⋊C8×C73C2211yescharacteristic
58Q8⋊C8×C73C2211yescharacteristic
59Q8⋊C8×C73C2211yescharacteristic
60Q8⋊C8×C73C2211yescharacteristic
61D4⋊C8×C73C2211yescharacteristic
62D4⋊C8×C73C2211yescharacteristic
63#322×C73C2211yescharacteristic
64#322×C73C2211yescharacteristic
65#322×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic
67C22×Q8×C73C2×C411yescharacteristic
68C2×Q16×C73C2×C411yescharacteristic
69C2×Q16×C73C2×C411yescharacteristic
70C2×SD16×C73C2×C411yescharacteristic
71C2×SD16×C73C2×C411yescharacteristic
72C2×M16×C73—21no—
73C2×M16×C73—21no—
74C22×C8×C73—21no—
75#126×C73—21no—
76#125×C73C2×C411yescharacteristic
77#124×C73C2×C411yescharacteristic
78#123×C73—21no—
79C22⋊Q8×C73C2×C411yescharacteristic
80#121×C73C2×C411yescharacteristic
81C4×Q8×C73—21no—
82C4×Q8×C73—21no—
83C4×Q8×C73C2×C411yescharacteristic
84C4×Q8×C73C2×C411yescharacteristic
85C4×Q8×C73C2311yescharacteristic
86C4×Q8×C73C2311yescharacteristic
87C4×Q8×C73C2311yescharacteristic
88C4×D4×C73—21no—
89C4×D4×C73C2311yescharacteristic
90#117×C73—21no—
91C2×C4⋊C4×C73—21no—
92C2×C4⋊C4×C73C2×C411yescharacteristic
93C2×C42×C73—21no—
94C2×C42×C73C2311yescharacteristic
95#107×C73C2×C411yescharacteristic
96#107×C73C2×C411yescharacteristic
97#106×C73C2×C411yescharacteristic
98#106×C73C2×C411yescharacteristic
99C4⋊C8×C73—21no—
100C4⋊C8×C73—21no—
101C4⋊C8×C73—21no—
102C4⋊C8×C73—21no—
103C4⋊C8×C73—21no—
104C4⋊C8×C73—21no—
105C4⋊C8×C73—21no—
106C4⋊C8×C73—21no—
107C4⋊C8×C73C2×C411yescharacteristic
108C4⋊C8×C73C2×C411yescharacteristic
109C4⋊C8×C73D411yescharacteristic
110C4⋊C8×C73D411yescharacteristic
111C4⋊C8×C73C2311yescharacteristic
112C4⋊C8×C73C2311yescharacteristic
113C4⋊C8×C73C2311yescharacteristic
114C4⋊C8×C73C2311yescharacteristic
115#103×C73C2×C411yescharacteristic
116#103×C73C2×C411yescharacteristic
117#103×C73C2×C411yescharacteristic
118#103×C73C2×C411yescharacteristic
119#103×C73C2×C411yescharacteristic
120#103×C73C2×C411yescharacteristic
121#102×C73C2×C411yescharacteristic
122#102×C73C2×C411yescharacteristic
123C22⋊C8×C73—21no—
124C22⋊C8×C73—21no—
125C22⋊C8×C73—21no—
126C22⋊C8×C73—21no—
127C22⋊C8×C73C2×C411yescharacteristic
128C22⋊C8×C73C2×C411yescharacteristic
129C22⋊C8×C73D411yescharacteristic
130C22⋊C8×C73D411yescharacteristic
131#97×C73—21no—
132#97×C73—21no—
133#97×C73—21no—
134#97×C73—21no—
135#97×C73—21no—
136C4×C8×C73—21no—
137C4×C8×C73—21no—
138C4×C8×C73C2311yescharacteristic
139C4×C8×C73C2311yescharacteristic
140C4×C8×C73C2311yescharacteristic
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C2×Q8×C73—21no—
145C2×Q8×C73—41no—
146C2×Q8×C73—41no—
147C2×Q8×C73C22×C411yescharacteristic
148C2×Q8×C73C22×C411yescharacteristic
149C2×Q8×C73C22×C411yescharacteristic
150C2×D4×C73—21no—
151C2×D4×C73C22×C411yescharacteristic
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73C22×C411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73#4111yescharacteristic
158Q16×C73—41no—
159Q16×C73—41no—
160SD16×C73—41no—
161SD16×C73—41no—
162M16×C73—81no—
163M16×C73—81no—
164C2×C8×C73—21no—
165C2×C8×C73—21no—
166C2×C8×C73—21no—
167C2×C8×C73—21no—
168C2×C8×C73—21no—
169C2×C8×C73—21no—
170C2×C8×C73—81no—
171C2×C8×C73C22×C411yescharacteristic
172C2×C8×C73C22×C411yescharacteristic
173C2×C8×C73C22×C411yescharacteristic
174C2×C8×C73C22×C411yescharacteristic
175C2×C8×C73C2×D411yescharacteristic
176C2×C8×C73C2×D411yescharacteristic
177C2×C8×C73C2×D411yescharacteristic
178C2×C8×C73C2×D411yescharacteristic
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73C22×C411yescharacteristic
189C4⋊C4×C73C22×C411yescharacteristic
190C4⋊C4×C73C22×C411yescharacteristic
191C4⋊C4×C73C22×C411yescharacteristic
192C22⋊C4×C73—21no—
193C22⋊C4×C73—21no—
194C22⋊C4×C73—21no—
195C22⋊C4×C73—21no—
196C22⋊C4×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73C22×C411yescharacteristic
203C42×C73C2×D411yescharacteristic
204C42×C73#4111yescharacteristic
205C42×C73C2411yescharacteristic
206C23×C73—21no—
207C23×C73C4×D411yescharacteristic
208Q8×C73—21no—
209Q8×C73—21no—
210Q8×C73—21no—
211Q8×C73—21no—
212Q8×C73—21no—
213Q8×C73—21no—
214Q8×C73—42no—
215D4×C73—21no—
216D4×C73—21no—
217D4×C73—81no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—41no—
230C2×C4×C73—42no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—81no—
234C2×C4×C73C4×D411yescharacteristic
235C2×C4×C73C4×D411yescharacteristic
236C2×C4×C73C4×D411yescharacteristic
237C2×C4×C73C23×C411yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C8×C73—21no—
241C8×C73—21no—
242C8×C73—21no—
243C8×C73—21no—
244C8×C73—21no—
245C8×C73—21no—
246C8×C73—21no—
247C8×C73—21no—
248C8×C73—41no—
249C8×C73—41no—
250C8×C73—41no—
251C8×C73—41no—
252C8×C73—41no—
253C8×C73—41no—
254C22×C73—21no—
255C22×C73—21no—
256C22×C73—21no—
257C22×C73—41no—
258C22×C73—41no—
259C22×C73—41no—
260C22×C73C2×C4×D411yescharacteristic
261C4×C73—21no—
262C4×C73—21no—
263C4×C73—21no—
264C4×C73—21no—
265C4×C73—41no—
266C4×C73—41no—
267C4×C73—41no—
268C4×C73—41no—
269C4×C73—41no—
270C4×C73—41no—
271C4×C73—41no—
272C4×C73#56811yescharacteristic
273C4×C73#56811yescharacteristic
274C4×C73#57711yescharacteristic
275C4×C73#57811yescharacteristic
276#17947C7311yesmaximal, characteristic, direct product factor, Sylow
277C2×C73—41no—
278C2×C73—81no—
279C2×C73#293011yescharacteristic
280C2×C73#294711yescharacteristic
281C2×C73#295711yescharacteristic
282#3108C2×C7311yescharacteristic
283#2993C2×C7311yescharacteristic
284#2983C2×C7311yescharacteristic, complemented
285#1593C2×C7311yescharacteristic, complemented
286#1592C2×C7311yescharacteristic, complemented
287#1591C2×C7311yescharacteristic, complemented
288#1590C2×C7311yescharacteristic, complemented
289#1587C2×C7311yescharacteristic, complemented
290#1585C2×C7311yescharacteristic, complemented
291#1584C2×C7311yescharacteristic, complemented
292#1582C2×C7311yescharacteristic, complemented
293#1574C2×C7311yescharacteristic, complemented
294#1487C2×C7311yescharacteristic, complemented
295#1482C2×C7311yescharacteristic
296#1454C2×C7311yescharacteristic, complemented
297C73#1794711yescharacteristic, direct product factor, Sylow
298#529C22×C7311yescharacteristic
299C2×C4×Q8C22×C7311yescharacteristic
300#483C4×C7311yescharacteristic
301#480C4×C7311yescharacteristic
302#479C4×C7311yescharacteristic
303#479C4×C7311yescharacteristic
304#464C4×C7311yescharacteristic
305#464C4×C7311yescharacteristic
306#451C4×C7311yescharacteristic
307#448C4×C7311yescharacteristic
308#446—21no—
309#446—21no—
310#446C22×C7311yescharacteristic
311#446C22×C7311yescharacteristic
312#446C22×C7311yescharacteristic
313#446C22×C7311yescharacteristic
314#446C22×C7311yescharacteristic
315C8×Q8C22×C7311yescharacteristic
316C4×Q16C22×C7311yescharacteristic
317C4×Q16C22×C7311yescharacteristic
318C4×SD16C22×C7311yescharacteristic
319C4×SD16C22×C7311yescharacteristic
320#436—21no—
321#436C22×C7311yescharacteristic
322#435—21no—
323C8×D4C22×C7311yescharacteristic
324#433—21no—
325#432—21no—
326#432—21no—
327#431C22×C7311yescharacteristic
328#431C22×C7311yescharacteristic
329#431C22×C7311yescharacteristic
330#431C22×C7311yescharacteristic
331#424—21no—
332#423C22×C7311yescharacteristic
333#335C22×C7311yescharacteristic
334#335C22×C7311yescharacteristic
335#334C22×C7311yescharacteristic
336#334C22×C7311yescharacteristic
337Q8⋊C8C22×C7311yescharacteristic
338Q8⋊C8C22×C7311yescharacteristic
339Q8⋊C8C22×C7311yescharacteristic
340Q8⋊C8C22×C7311yescharacteristic
341Q8⋊C8C22×C7311yescharacteristic
342Q8⋊C8C22×C7311yescharacteristic
343D4⋊C8C22×C7311yescharacteristic
344D4⋊C8C22×C7311yescharacteristic
345#322C22×C7311yescharacteristic
346#322C22×C7311yescharacteristic
347#322C22×C7311yescharacteristic
348C82C22×C7311yescharacteristic
349C22×Q8C2×C4×C7311yescharacteristic
350C2×Q16C2×C4×C7311yescharacteristic
351C2×Q16C2×C4×C7311yescharacteristic
352C2×SD16C2×C4×C7311yescharacteristic
353C2×SD16C2×C4×C7311yescharacteristic
354C2×M16—21no—
355C2×M16—21no—
356C22×C8—21no—
357#126—21no—
358#125C2×C4×C7311yescharacteristic
359#124C2×C4×C7311yescharacteristic
360#123—21no—
361C22⋊Q8C2×C4×C7311yescharacteristic
362#121C2×C4×C7311yescharacteristic
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8C2×C4×C7311yescharacteristic
366C4×Q8C2×C4×C7311yescharacteristic
367C4×Q8C23×C7311yescharacteristic
368C4×Q8C23×C7311yescharacteristic
369C4×Q8C23×C7311yescharacteristic
370C4×D4—21no—
371C4×D4C23×C7311yescharacteristic
372#117—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C2×C4×C7311yescharacteristic
375C2×C42—21no—
376C2×C42C23×C7311yescharacteristic
377#107C2×C4×C7311yescharacteristic
378#107C2×C4×C7311yescharacteristic
379#106C2×C4×C7311yescharacteristic
380#106C2×C4×C7311yescharacteristic
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8C2×C4×C7311yescharacteristic
390C4⋊C8C2×C4×C7311yescharacteristic
391C4⋊C8D4×C7311yescharacteristic
392C4⋊C8D4×C7311yescharacteristic
393C4⋊C8C23×C7311yescharacteristic
394C4⋊C8C23×C7311yescharacteristic
395C4⋊C8C23×C7311yescharacteristic
396C4⋊C8C23×C7311yescharacteristic
397#103C2×C4×C7311yescharacteristic
398#103C2×C4×C7311yescharacteristic
399#103C2×C4×C7311yescharacteristic
400#103C2×C4×C7311yescharacteristic
401#103C2×C4×C7311yescharacteristic
402#103C2×C4×C7311yescharacteristic
403#102C2×C4×C7311yescharacteristic
404#102C2×C4×C7311yescharacteristic
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409C22⋊C8C2×C4×C7311yescharacteristic
410C22⋊C8C2×C4×C7311yescharacteristic
411C22⋊C8D4×C7311yescharacteristic
412C22⋊C8D4×C7311yescharacteristic
413#97—21no—
414#97—21no—
415#97—21no—
416#97—21no—
417#97—21no—
418C4×C8—21no—
419C4×C8—21no—
420C4×C8C23×C7311yescharacteristic
421C4×C8C23×C7311yescharacteristic
422C4×C8C23×C7311yescharacteristic
423C4×C8C23×C7311yescharacteristic
424C4×C8C23×C7311yescharacteristic
425C4×C8C23×C7311yescharacteristic
426C2×Q8—21no—
427C2×Q8—41no—
428C2×Q8—41no—
429C2×Q8C22×C4×C7311yescharacteristic
430C2×Q8C22×C4×C7311yescharacteristic
431C2×Q8C22×C4×C7311yescharacteristic
432C2×D4—21no—
433C2×D4C22×C4×C7311yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C22×C4×C7311yescharacteristic
438C22×C4C2×D4×C7311yescharacteristic
439C22×C4#41×C7311yescharacteristic
440Q16—41no—
441Q16—41no—
442SD16—41no—
443SD16—41no—
444M16—81no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—81no—
453C2×C8C22×C4×C7311yescharacteristic
454C2×C8C22×C4×C7311yescharacteristic
455C2×C8C22×C4×C7311yescharacteristic
456C2×C8C22×C4×C7311yescharacteristic
457C2×C8C2×D4×C7311yescharacteristic
458C2×C8C2×D4×C7311yescharacteristic
459C2×C8C2×D4×C7311yescharacteristic
460C2×C8C2×D4×C7311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C22×C4×C7311yescharacteristic
471C4⋊C4C22×C4×C7311yescharacteristic
472C4⋊C4C22×C4×C7311yescharacteristic
473C4⋊C4C22×C4×C7311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42C22×C4×C7311yescharacteristic
485C42C2×D4×C7311yescharacteristic
486C42#41×C7311yescharacteristic
487C42C24×C7311yescharacteristic
488C23—21no—
489C23C4×D4×C7311yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—42no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4C4×D4×C7311yescharacteristic
517C2×C4C4×D4×C7311yescharacteristic
518C2×C4C4×D4×C7311yescharacteristic
519C2×C4C23×C4×C7311yescharacteristic
520C2×C4C22×D4×C7311yescharacteristic
521C2×C4C2×#41×C7311yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—41no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—41no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×C4×D4×C7311yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4#568×C7311yescharacteristic
555C4#568×C7311yescharacteristic
556C4#577×C7311yescharacteristic
557C4#578×C7311yescharacteristic
558C2—41no—
559C2—81no—
560C2C73×#293011yescharacteristic
561C2C73×#294711yescharacteristic
562C2C73×#295711yescharacteristic
subgroupG/H
centerC2×C4×C73C22×D4
commutatorC2×C4C23×C4×C73
Frattini, cosocleC42C24×C73
socleC22×C73C2×C4×D4
prime radical#121×C73C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C73 ◁ #431×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#2983C2semidirect
4. C73×#1593C2semidirect
5. C73×#1592C2semidirect
6. C73×#1591C2semidirect
7. C73×#1590C2semidirect
8. C73×#1587C2semidirect
9. C73×#1585C2semidirect
10. C73×#1584C2semidirect
11. C73×#1582C2semidirect
12. C73×#1574C2semidirect
13. C73×#1487C2semidirect
15. C73×#1454C2semidirect
276. #17947C73direct (N × H)
284. #2983C2×C73semidirect
285. #1593C2×C73semidirect
286. #1592C2×C73semidirect
287. #1591C2×C73semidirect
288. #1590C2×C73semidirect
289. #1587C2×C73semidirect
290. #1585C2×C73semidirect
291. #1584C2×C73semidirect
292. #1582C2×C73semidirect
293. #1574C2×C73semidirect
294. #1487C2×C73semidirect
296. #1454C2×C73semidirect
297. C73#17947direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#31082
C73×#29932
C73×#29832
C73×#15932
C73×#15922
C73×#15912
C73×#15902
C73×#15872
C73×#15852
C73×#15842
C73×#15822
C73×#15742
C73×#14872
C73×#14822
C73×#14542
#1794773