Finite Group Databasebeta

← C73×#26415 #32935819

Subgroups — C73×#26415 #32935819

Order 18688.

746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2439C211yesmaximal, characteristic
2C73×#2362C211yesmaximal, characteristic
3C73×#2101C211yesmaximal, characteristic, complemented
4C73×#2088C211yesmaximal, characteristic, complemented
5C73×#2066C211yesmaximal, characteristic, complemented
6C73×#2024C211yesmaximal, characteristic, complemented
7C73×#1987C211yesmaximal, characteristic, complemented
8C73×#1918C211yesmaximal, characteristic, complemented
9C73×#1892C211yesmaximal, characteristic
10C73×#1873C211yesmaximal, characteristic
11C73×#1863C211yesmaximal, characteristic, complemented
12C73×#1854C211yesmaximal, characteristic
13C73×#1850C211yesmaximal, characteristic
14C73×#1805C211yesmaximal, characteristic, complemented
15C73×#1735C211yesmaximal, characteristic
16C23×C8×C73C2211yescharacteristic
17C2×C4⋊Q8×C73C2211yescharacteristic
18C2×C22⋊Q8×C73C2211yescharacteristic
19C2×C4×Q8×C73C2211yescharacteristic
20C2×#117×C73C2211yescharacteristic
21C2×#117×C73C2211yescharacteristic
22C2×#107×C73C2211yescharacteristic
23C2×#106×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×#103×C73C2211yescharacteristic
27C2×#103×C73C2211yescharacteristic
28C2×#103×C73C2211yescharacteristic
29C2×#103×C73C2211yescharacteristic
30C2×C22⋊C8×C73C2211yescharacteristic
31#396×C73—21no—
32#393×C73—21no—
33#391×C73—21no—
34#391×C73C2211yescharacteristic
35#389×C73C2211yescharacteristic
36#389×C73C2211yescharacteristic
37#389×C73C2211yescharacteristic
38#388×C73—21no—
39#388×C73—21no—
40#387×C73—21no—
41#387×C73—21no—
42#382×C73C2211yescharacteristic
43#380×C73C2211yescharacteristic
44C4×C4⋊C4×C73—21no—
45C4×C4⋊C4×C73C2211yescharacteristic
46C4×C22⋊C4×C73C2211yescharacteristic
47#340×C73C2211yescharacteristic
48#340×C73C2211yescharacteristic
49#340×C73C2211yescharacteristic
50#340×C73C2211yescharacteristic
51#340×C73C2211yescharacteristic
52#340×C73C2211yescharacteristic
53#340×C73C2211yescharacteristic
54#340×C73C2211yescharacteristic
55#340×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Q8×C73—21no—
60C22×Q8×C73C2311yescharacteristic
61C23×C4×C73C2311yescharacteristic
62C22×C8×C73—22no—
63C22×C8×C73—41no—
64C22×C8×C73—41no—
65C22×C8×C73C2311yescharacteristic
66C22×C8×C73C2311yescharacteristic
67C22×C8×C73C2311yescharacteristic
68C22×C8×C73C2311yescharacteristic
69C4⋊Q8×C73—22no—
70C4⋊Q8×C73—41no—
71C22⋊Q8×C73—22no—
72C22⋊Q8×C73Q814yescomplemented
73C4×Q8×C73—22no—
74C4×Q8×C73—41no—
75#117×C73—22no—
76#117×C73—22no—
77#117×C73—41no—
78#117×C73—41no—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73—21no—
83C2×C4⋊C4×C73—21no—
84C2×C4⋊C4×C73—21no—
85C2×C4⋊C4×C73C2311yescharacteristic
86C2×C4⋊C4×C73C2311yescharacteristic
87C2×C4⋊C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73C2311yescharacteristic
89C2×C4⋊C4×C73C2311yescharacteristic
90C2×C4⋊C4×C73C2311yescharacteristic
91C2×C4⋊C4×C73C2311yescharacteristic
92C2×C22⋊C4×C73—21no—
93C2×C22⋊C4×C73—21no—
94C2×C22⋊C4×C73—21no—
95C2×C22⋊C4×C73—21no—
96C2×C22⋊C4×C73D411yescharacteristic
97C2×C22⋊C4×C73D411yescharacteristic
98C2×C42×C73—21no—
99C2×C42×C73—21no—
100C2×C42×C73—21no—
101C2×C42×C73—21no—
102C2×C42×C73C2311yescharacteristic
103C2×C42×C73C2311yescharacteristic
104#107×C73—22no—
105#106×C73—41no—
106C4⋊C8×C73—41no—
107C4⋊C8×C73—41no—
108#103×C73—22no—
109#103×C73—41no—
110#103×C73—41no—
111#103×C73—41no—
112C22⋊C8×C73—22no—
113#95×C73—21no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73—21no—
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121#95×C73—21no—
122#95×C73D411yescharacteristic
123#95×C73D411yescharacteristic
124C24×C73C2×D411yescharacteristic
125C2×Q8×C73—21no—
126C2×Q8×C73—21no—
127C2×Q8×C73—21no—
128C2×Q8×C73—21no—
129C2×Q8×C73—21no—
130C2×Q8×C73—42no—
131C2×Q8×C73—42no—
132C2×Q8×C73—81no—
133C2×Q8×C73C2×Q812yes—
134C22×C4×C73—21no—
135C22×C4×C73—21no—
136C22×C4×C73—21no—
137C22×C4×C73—21no—
138C22×C4×C73—21no—
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73C2×D411yescharacteristic
148C22×C4×C73C2×D411yescharacteristic
149C22×C4×C73C2×D411yescharacteristic
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73C2×Q812yes—
152C22×C4×C73#4112yes—
153C22×C4×C73C2411yescharacteristic
154C2×C8×C73—21no—
155C2×C8×C73—21no—
156C2×C8×C73—21no—
157C2×C8×C73—21no—
158C2×C8×C73—21no—
159C2×C8×C73—21no—
160C2×C8×C73—21no—
161C2×C8×C73—21no—
162C2×C8×C73—21no—
163C2×C8×C73—21no—
164C2×C8×C73—21no—
165C2×C8×C73—21no—
166C2×C8×C73—44no—
167C4⋊C4×C73—24no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—21no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—42no—
180C4⋊C4×C73—42no—
181C4⋊C4×C73—42no—
182C4⋊C4×C73—42no—
183C4⋊C4×C73—81no—
184C4⋊C4×C73C2×Q812yes—
185C4⋊C4×C73#4112yes—
186C4⋊C4×C73#4112yes—
187C22⋊C4×C73—24no—
188C22⋊C4×C73—22no—
189C22⋊C4×C73—41no—
190C22⋊C4×C73—42no—
191C22⋊C4×C73—42no—
192C22⋊C4×C73—81no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—21no—
197C42×C73—42no—
198C42×C73—42no—
199C42×C73—42no—
200C42×C73—42no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73C22⋊Q812yes—
208C23×C73C22×D411yescharacteristic
209Q8×C73—22no—
210Q8×C73—22no—
211Q8×C73—41no—
212Q8×C73—41no—
213Q8×C73—84no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73C22⋊Q812yes—
256C2×C4×C73#13512yes—
257C2×C4×C73C22×Q811yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73C2×#4111yescharacteristic
261C2×C4×C73H(5,𝔽2)11yescharacteristic
262C2×C4×C73H(5,𝔽2)11yescharacteristic
263C8×C73—41no—
264C8×C73—41no—
265C8×C73—41no—
266C8×C73—41no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73C2×C22⋊Q811yescharacteristic
275C22×C73#54611yescharacteristic
276C22×C73#54611yescharacteristic
277C22×C73C2×#13511yescharacteristic
278C22×C73#57511yescharacteristic
279C22×C73#57711yescharacteristic
280C22×C73#57811yescharacteristic
281C4×C73—22no—
282C4×C73—22no—
283C4×C73—42no—
284C4×C73—42no—
285C4×C73—41no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—42no—
289C4×C73—41no—
290C4×C73—81no—
291C4×C73—81no—
292C4×C73—81no—
293C4×C73—81no—
294C4×C73#52911yescharacteristic
295C4×C73#53811yescharacteristic
296C4×C73#55411yescharacteristic
297C4×C73#55411yescharacteristic
298#26415C7311yesmaximal, characteristic, direct product factor, Sylow
299C2×C73—42no—
300C2×C73#245011yescharacteristic
301C2×C73#307611yescharacteristic
302C2×C73#308211yescharacteristic
303C2×C73#330011yescharacteristic
304C2×C73#330311yescharacteristic
305C2×C73#331111yescharacteristic
306C2×C73#331311yescharacteristic
307#2439C2×C7311yescharacteristic
308#2362C2×C7311yescharacteristic
309#2101C2×C7311yescharacteristic, complemented
310#2088C2×C7311yescharacteristic, complemented
311#2066C2×C7311yescharacteristic, complemented
312#2024C2×C7311yescharacteristic, complemented
313#1987C2×C7311yescharacteristic, complemented
314#1918C2×C7311yescharacteristic, complemented
315#1892C2×C7311yescharacteristic
316#1873C2×C7311yescharacteristic
317#1863C2×C7311yescharacteristic, complemented
318#1854C2×C7311yescharacteristic
319#1850C2×C7311yescharacteristic
320#1805C2×C7311yescharacteristic, complemented
321#1735C2×C7311yescharacteristic
322C73#2641511yescharacteristic, direct product factor, Sylow
323C23×C8C22×C7311yescharacteristic
324C2×C4⋊Q8C22×C7311yescharacteristic
325C2×C22⋊Q8C22×C7311yescharacteristic
326C2×C4×Q8C22×C7311yescharacteristic
327C2×#117C22×C7311yescharacteristic
328C2×#117C22×C7311yescharacteristic
329C2×#107C22×C7311yescharacteristic
330C2×#106C22×C7311yescharacteristic
331C2×C4⋊C8C22×C7311yescharacteristic
332C2×C4⋊C8C22×C7311yescharacteristic
333C2×#103C22×C7311yescharacteristic
334C2×#103C22×C7311yescharacteristic
335C2×#103C22×C7311yescharacteristic
336C2×#103C22×C7311yescharacteristic
337C2×C22⋊C8C22×C7311yescharacteristic
338#396—21no—
339#393—21no—
340#391—21no—
341#391C22×C7311yescharacteristic
342#389C22×C7311yescharacteristic
343#389C22×C7311yescharacteristic
344#389C22×C7311yescharacteristic
345#388—21no—
346#388—21no—
347#387—21no—
348#387—21no—
349#382C22×C7311yescharacteristic
350#380C22×C7311yescharacteristic
351C4×C4⋊C4—21no—
352C4×C4⋊C4C22×C7311yescharacteristic
353C4×C22⋊C4C22×C7311yescharacteristic
354#340C22×C7311yescharacteristic
355#340C22×C7311yescharacteristic
356#340C22×C7311yescharacteristic
357#340C22×C7311yescharacteristic
358#340C22×C7311yescharacteristic
359#340C22×C7311yescharacteristic
360#340C22×C7311yescharacteristic
361#340C22×C7311yescharacteristic
362#340C22×C7311yescharacteristic
363#340C22×C7311yescharacteristic
364#336C22×C7311yescharacteristic
365#336C22×C7311yescharacteristic
366C22×Q8—21no—
367C22×Q8C23×C7311yescharacteristic
368C23×C4C23×C7311yescharacteristic
369C22×C8—22no—
370C22×C8—41no—
371C22×C8—41no—
372C22×C8C23×C7311yescharacteristic
373C22×C8C23×C7311yescharacteristic
374C22×C8C23×C7311yescharacteristic
375C22×C8C23×C7311yescharacteristic
376C4⋊Q8—22no—
377C4⋊Q8—41no—
378C22⋊Q8—22no—
379C22⋊Q8Q8×C7314yescomplemented
380C4×Q8—22no—
381C4×Q8—41no—
382#117—22no—
383#117—22no—
384#117—41no—
385#117—41no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4C23×C7311yescharacteristic
393C2×C4⋊C4C23×C7311yescharacteristic
394C2×C4⋊C4C23×C7311yescharacteristic
395C2×C4⋊C4C23×C7311yescharacteristic
396C2×C4⋊C4C23×C7311yescharacteristic
397C2×C4⋊C4C23×C7311yescharacteristic
398C2×C4⋊C4C23×C7311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4D4×C7311yescharacteristic
404C2×C22⋊C4D4×C7311yescharacteristic
405C2×C42—21no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42C23×C7311yescharacteristic
410C2×C42C23×C7311yescharacteristic
411#107—22no—
412#106—41no—
413C4⋊C8—41no—
414C4⋊C8—41no—
415#103—22no—
416#103—41no—
417#103—41no—
418#103—41no—
419C22⋊C8—22no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95D4×C7311yescharacteristic
430#95D4×C7311yescharacteristic
431C24C2×D4×C7311yescharacteristic
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—81no—
440C2×Q8C2×Q8×C7312yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4C2×D4×C7311yescharacteristic
454C22×C4C2×D4×C7311yescharacteristic
455C22×C4C2×D4×C7311yescharacteristic
456C22×C4C2×D4×C7311yescharacteristic
457C22×C4C2×D4×C7311yescharacteristic
458C22×C4C2×Q8×C7312yes—
459C22×C4#41×C7312yes—
460C22×C4C24×C7311yescharacteristic
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—44no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4C2×Q8×C7312yes—
492C4⋊C4#41×C7312yes—
493C4⋊C4#41×C7312yes—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23C22⋊Q8×C7312yes—
515C23C22×D4×C7311yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520Q8—84no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4C22⋊Q8×C7312yes—
563C2×C4#135×C7312yes—
564C2×C4C22×Q8×C7311yescharacteristic
565C2×C4C2×#41×C7311yescharacteristic
566C2×C4C2×#41×C7311yescharacteristic
567C2×C4C2×#41×C7311yescharacteristic
568C2×C4H(5,𝔽2)×C7311yescharacteristic
569C2×C4H(5,𝔽2)×C7311yescharacteristic
570C8—41no—
571C8—41no—
572C8—41no—
573C8—41no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×C22⋊Q8×C7311yescharacteristic
582C22#546×C7311yescharacteristic
583C22#546×C7311yescharacteristic
584C22C2×#135×C7311yescharacteristic
585C22#575×C7311yescharacteristic
586C22#577×C7311yescharacteristic
587C22#578×C7311yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4#529×C7311yescharacteristic
602C4#538×C7311yescharacteristic
603C4#554×C7311yescharacteristic
604C4#554×C7311yescharacteristic
605C2—42no—
606C2C73×#245011yescharacteristic
607C2C73×#307611yescharacteristic
608C2C73×#308211yescharacteristic
609C2C73×#330011yescharacteristic
610C2C73×#330311yescharacteristic
611C2C73×#331111yescharacteristic
612C2C73×#331311yescharacteristic
subgroupG/H
center, socleC23×C73C22×D4
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC24×C73C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C4×C22⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#2101C2semidirect
4. C73×#2088C2semidirect
5. C73×#2066C2semidirect
6. C73×#2024C2semidirect
7. C73×#1987C2semidirect
8. C73×#1918C2semidirect
11. C73×#1863C2semidirect
14. C73×#1805C2semidirect
72. C22⋊Q8×C73Q8semidirect
298. #26415C73direct (N × H)
309. #2101C2×C73semidirect
310. #2088C2×C73semidirect
311. #2066C2×C73semidirect
312. #2024C2×C73semidirect
313. #1987C2×C73semidirect
314. #1918C2×C73semidirect
317. #1863C2×C73semidirect
320. #1805C2×C73semidirect
322. C73#26415direct (N × H)
379. C22⋊Q8Q8×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#24392
C73×#23622
C73×#21012
C73×#20882
C73×#20662
C73×#20242
C73×#19872
C73×#19182
C73×#18922
C73×#18732
C73×#18632
C73×#18542
C73×#18502
C73×#18052
C73×#17352
#2641573