Finite Group Databasebeta

← C73×#16567 #32925971

Subgroups — C73×#16567 #32925971

Order 18688.

1194 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C73C211yesmaximal, characteristic
2C2×#389×C73C211yesmaximal, characteristic
3C2×#387×C73C211yesmaximal, characteristic
4C2×#382×C73C211yesmaximal, characteristic
5C73×#1445C211yesmaximal, characteristic
6C73×#1444C211yesmaximal, characteristic
7C73×#1442C212yesmaximal
8C73×#1441C211yesmaximal, characteristic
9C73×#1441C212yesmaximal
10C73×#1441C212yesmaximal
11C73×#1441C212yesmaximal
12C22×C4⋊C4×C73C2211yescharacteristic
13C22×C4⋊C4×C73C2211yescharacteristic
14C22×C4⋊C4×C73C2211yescharacteristic
15C22×C4⋊C4×C73C2212yes—
16C22×C4⋊C4×C73C2211yescharacteristic
17C22×C4⋊C4×C73C2212yes—
18C22×C4⋊C4×C73C2212yes—
19C22×C42×C73C2211yescharacteristic
20#400×C73—24no—
21#389×C73—24no—
22#387×C73—24no—
23#382×C73—24no—
24C2×#95×C73C2212yes—
25C2×#95×C73C2211yescharacteristic
26C2×#95×C73C2211yescharacteristic
27C2×#95×C73C2212yes—
28C2×#95×C73C2212yes—
29C2×#95×C73C2212yes—
30C2×#95×C73C2212yes—
31C2×#95×C73C2211yescharacteristic
32C2×#95×C73C2211yescharacteristic
33C2×#95×C73C2211yescharacteristic
34C2×#95×C73C2212yes—
35C2×#95×C73C2212yes—
36C2×#95×C73C2212yes—
37C2×#95×C73C2212yes—
38C2×#95×C73C2211yescharacteristic
39C23×C4×C73C2311yescharacteristic
40C23×C4×C73C2312yes—
41C23×C4×C73C2311yescharacteristic
42C23×C4×C73C2312yes—
43C23×C4×C73C2311yescharacteristic
44C23×C4×C73C2312yes—
45C23×C4×C73C2311yescharacteristic
46C23×C4×C73C2311yescharacteristic
47C23×C4×C73C2312yes—
48C23×C4×C73C2311yescharacteristic
49C23×C4×C73C2311yescharacteristic
50C2×C4⋊C4×C73—22no—
51C2×C4⋊C4×C73—22no—
52C2×C4⋊C4×C73—24no—
53C2×C4⋊C4×C73—22no—
54C2×C4⋊C4×C73—24no—
55C2×C4⋊C4×C73—22no—
56C2×C4⋊C4×C73—24no—
57C2×C4⋊C4×C73—22no—
58C2×C4⋊C4×C73—22no—
59C2×C4⋊C4×C73—22no—
60C2×C4⋊C4×C73—24no—
61C2×C4⋊C4×C73—24no—
62C2×C4⋊C4×C73—22no—
63C2×C4⋊C4×C73—22no—
64C2×C4⋊C4×C73—42no—
65C2×C4⋊C4×C73—42no—
66C2×C4⋊C4×C73—42no—
67C2×C4⋊C4×C73—42no—
68C2×C4⋊C4×C73—42no—
69C2×C4⋊C4×C73Q814yes—
70C2×C42×C73—22no—
71C2×C42×C73—22no—
72C2×C42×C73D414yes—
73#95×C73—24no—
74#95×C73—22no—
75#95×C73—22no—
76#95×C73—22no—
77#95×C73—22no—
78#95×C73—22no—
79#95×C73—24no—
80#95×C73—24no—
81#95×C73—24no—
82#95×C73—24no—
83#95×C73—42no—
84#95×C73—42no—
85#95×C73—42no—
86#95×C73—42no—
87#95×C73D414yes—
88C24×C73C2411yescharacteristic
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—22no—
92C22×C4×C73—21no—
93C22×C4×C73—22no—
94C22×C4×C73—21no—
95C22×C4×C73—21no—
96C22×C4×C73—21no—
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—22no—
100C22×C4×C73—21no—
101C22×C4×C73—21no—
102C22×C4×C73—21no—
103C22×C4×C73—22no—
104C22×C4×C73—22no—
105C22×C4×C73—22no—
106C22×C4×C73—21no—
107C22×C4×C73—22no—
108C22×C4×C73—21no—
109C22×C4×C73—22no—
110C22×C4×C73—22no—
111C22×C4×C73—22no—
112C22×C4×C73—22no—
113C22×C4×C73—22no—
114C22×C4×C73—22no—
115C22×C4×C73—22no—
116C22×C4×C73—22no—
117C22×C4×C73—22no—
118C22×C4×C73—22no—
119C22×C4×C73—22no—
120C22×C4×C73—21no—
121C22×C4×C73—21no—
122C22×C4×C73—21no—
123C22×C4×C73—22no—
124C22×C4×C73—21no—
125C22×C4×C73—22no—
126C22×C4×C73—22no—
127C22×C4×C73—22no—
128C22×C4×C73—22no—
129C22×C4×C73—22no—
130C22×C4×C73—21no—
131C22×C4×C73—22no—
132C22×C4×C73—22no—
133C22×C4×C73—22no—
134C22×C4×C73—22no—
135C22×C4×C73—22no—
136C22×C4×C73—22no—
137C22×C4×C73—22no—
138C22×C4×C73—21no—
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—21no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73C2×D414yes—
146C22×C4×C73C2×D412yes—
147C22×C4×C73C2×D412yes—
148C22×C4×C73C2×D412yes—
149C22×C4×C73C2×D412yes—
150C22×C4×C73C2×Q812yes—
151C22×C4×C73C2×Q812yes—
152C22×C4×C73C2×Q812yes—
153C22×C4×C73#4114yes—
154C22×C4×C73#4112yes—
155C22×C4×C73#4112yes—
156C4⋊C4×C73—28no—
157C4⋊C4×C73—28no—
158C4⋊C4×C73—48no—
159C4⋊C4×C73—48no—
160C4⋊C4×C73—48no—
161C4⋊C4×C73—44no—
162C4⋊C4×C73—44no—
163C42×C73—28no—
164C23×C73C22×D411yescharacteristic
165C23×C73C22×D411yescharacteristic
166C23×C73C22×Q811yescharacteristic
167C23×C73C2×#4111yescharacteristic
168C23×C73C2×#4111yescharacteristic
169C23×C73C2×#4112yes—
170C23×C73H(5,𝔽2)12yes—
171C23×C73H(5,𝔽2)11yescharacteristic
172C23×C732−1+412yes—
173C23×C732−1+412yes—
174C23×C732−1+411yescharacteristic
175C2×C4×C73—22no—
176C2×C4×C73—22no—
177C2×C4×C73—22no—
178C2×C4×C73—22no—
179C2×C4×C73—22no—
180C2×C4×C73—24no—
181C2×C4×C73—22no—
182C2×C4×C73—24no—
183C2×C4×C73—22no—
184C2×C4×C73—22no—
185C2×C4×C73—22no—
186C2×C4×C73—22no—
187C2×C4×C73—22no—
188C2×C4×C73—24no—
189C2×C4×C73—22no—
190C2×C4×C73—22no—
191C2×C4×C73—24no—
192C2×C4×C73—24no—
193C2×C4×C73—24no—
194C2×C4×C73—24no—
195C2×C4×C73—24no—
196C2×C4×C73—24no—
197C2×C4×C73—22no—
198C2×C4×C73—22no—
199C2×C4×C73—41no—
200C2×C4×C73—42no—
201C2×C4×C73—41no—
202C2×C4×C73—41no—
203C2×C4×C73—42no—
204C2×C4×C73—42no—
205C2×C4×C73—41no—
206C2×C4×C73—42no—
207C2×C4×C73—42no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—42no—
221C2×C4×C73—41no—
222C2×C4×C73—42no—
223C2×C4×C73—41no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73—41no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73#12114yes—
241C2×C4×C73C22⋊Q814yes—
242C22×C73C2×#12111yescharacteristic
243C22×C73C2×C22⋊Q811yescharacteristic
244C22×C73#52511yescharacteristic
245C22×C73#53511yescharacteristic
246C22×C73#53611yescharacteristic
247C22×C73#53611yescharacteristic
248C22×C73#53711yescharacteristic
249C22×C73#53712yes—
250C22×C73#53812yes—
251C22×C73#53912yes—
252C22×C73#54212yes—
253C22×C73#54212yes—
254C22×C73#54412yes—
255C22×C73#54611yescharacteristic
256C22×C73#54712yes—
257C22×C73D4×Q811yescharacteristic
258C22×C73#55212yes—
259C22×C73#55211yescharacteristic
260C22×C73#55411yescharacteristic
261C22×C73#55612yes—
262C22×C73#55712yes—
263C22×C73#56312yes—
264C22×C73#56412yes—
265C4×C73—24no—
266C4×C73—24no—
267C4×C73—42no—
268C4×C73—42no—
269C4×C73—44no—
270C4×C73—42no—
271C4×C73—44no—
272C4×C73—82no—
273C4×C73—81no—
274C4×C73—81no—
275C4×C73—82no—
276#16567C7311yesmaximal, characteristic, direct product factor, Sylow
277C2×C73#241311yescharacteristic
278C2×C73#241711yescharacteristic
279C2×C73#258111yescharacteristic
280C2×C73#262011yescharacteristic
281C2×C73#262811yescharacteristic
282C2×C73#263611yescharacteristic
283C2×C73#271512yes—
284C2×C73#272612yes—
285C2×C73#276712yes—
286C2×C73#277412yes—
287C2×C73#282211yescharacteristic
288C2×#400C2×C7311yescharacteristic
289C2×#389C2×C7311yescharacteristic
290C2×#387C2×C7311yescharacteristic
291C2×#382C2×C7311yescharacteristic
292#1445C2×C7311yescharacteristic
293#1444C2×C7311yescharacteristic
294#1442C2×C7312yes—
295#1441C2×C7311yescharacteristic
296#1441C2×C7312yes—
297#1441C2×C7312yes—
298#1441C2×C7312yes—
299C73#1656711yescharacteristic, direct product factor, Sylow
300C22×C4⋊C4C22×C7311yescharacteristic
301C22×C4⋊C4C22×C7311yescharacteristic
302C22×C4⋊C4C22×C7311yescharacteristic
303C22×C4⋊C4C22×C7312yes—
304C22×C4⋊C4C22×C7311yescharacteristic
305C22×C4⋊C4C22×C7312yes—
306C22×C4⋊C4C22×C7312yes—
307C22×C42C22×C7311yescharacteristic
308#400—24no—
309#389—24no—
310#387—24no—
311#382—24no—
312C2×#95C22×C7312yes—
313C2×#95C22×C7312yes—
314C2×#95C22×C7312yes—
315C2×#95C22×C7311yescharacteristic
316C2×#95C22×C7311yescharacteristic
317C2×#95C22×C7312yes—
318C2×#95C22×C7312yes—
319C2×#95C22×C7311yescharacteristic
320C2×#95C22×C7311yescharacteristic
321C2×#95C22×C7311yescharacteristic
322C2×#95C22×C7312yes—
323C2×#95C22×C7312yes—
324C2×#95C22×C7312yes—
325C2×#95C22×C7312yes—
326C2×#95C22×C7311yescharacteristic
327C23×C4C23×C7311yescharacteristic
328C23×C4C23×C7312yes—
329C23×C4C23×C7311yescharacteristic
330C23×C4C23×C7312yes—
331C23×C4C23×C7312yes—
332C23×C4C23×C7311yescharacteristic
333C23×C4C23×C7311yescharacteristic
334C23×C4C23×C7311yescharacteristic
335C23×C4C23×C7312yes—
336C23×C4C23×C7311yescharacteristic
337C23×C4C23×C7311yescharacteristic
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4Q8×C7314yes—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42D4×C7314yes—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—42no—
372#95—42no—
373#95—42no—
374#95—42no—
375#95D4×C7314yes—
376C24C24×C7311yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4C2×D4×C7314yes—
434C22×C4C2×D4×C7312yes—
435C22×C4C2×D4×C7312yes—
436C22×C4C2×D4×C7312yes—
437C22×C4C2×D4×C7312yes—
438C22×C4C2×Q8×C7312yes—
439C22×C4C2×Q8×C7312yes—
440C22×C4C2×Q8×C7312yes—
441C22×C4#41×C7314yes—
442C22×C4#41×C7312yes—
443C22×C4#41×C7312yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C42—28no—
452C23C22×D4×C7311yescharacteristic
453C23C22×D4×C7311yescharacteristic
454C23C22×Q8×C7311yescharacteristic
455C23C2×#41×C7311yescharacteristic
456C23C2×#41×C7311yescharacteristic
457C23C2×#41×C7312yes—
458C23H(5,𝔽2)×C7312yes—
459C23H(5,𝔽2)×C7311yescharacteristic
460C232−1+4×C7312yes—
461C232−1+4×C7312yes—
462C232−1+4×C7311yescharacteristic
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4#121×C7314yes—
529C2×C4C22⋊Q8×C7314yes—
530C22C2×#121×C7311yescharacteristic
531C22C2×C22⋊Q8×C7311yescharacteristic
532C22#525×C7311yescharacteristic
533C22#535×C7311yescharacteristic
534C22#536×C7311yescharacteristic
535C22#536×C7311yescharacteristic
536C22#537×C7312yes—
537C22#537×C7311yescharacteristic
538C22#538×C7312yes—
539C22#539×C7312yes—
540C22#542×C7312yes—
541C22#542×C7312yes—
542C22#544×C7312yes—
543C22#546×C7311yescharacteristic
544C22#547×C7312yes—
545C22D4×Q8×C7311yescharacteristic
546C22#552×C7311yescharacteristic
547C22#552×C7312yes—
548C22#554×C7311yescharacteristic
549C22#556×C7312yes—
550C22#557×C7312yes—
551C22#563×C7312yes—
552C22#564×C7312yes—
553C4—24no—
554C4—24no—
555C4—44no—
556C4—42no—
557C4—42no—
558C4—44no—
559C4—42no—
560C4—82no—
561C4—82no—
562C4—81no—
563C4—81no—
564C2C73×#241311yescharacteristic
565C2C73×#241711yescharacteristic
566C2C73×#258111yescharacteristic
567C2C73×#262011yescharacteristic
568C2C73×#262811yescharacteristic
569C2C73×#263611yescharacteristic
570C2C73×#271512yes—
571C2C73×#272612yes—
572C2C73×#276712yes—
573C2C73×#277412yes—
574C2C73×#282211yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C73C24
commutator, Frattini, cosocleC24C24×C73
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C73 ◁ G

Complements

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

NH ≅ G/Ntype
276. #16567C73direct (N × H)
299. C73#16567direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#400×C732
C2×#389×C732
C2×#387×C732
C2×#382×C732
C73×#14452
C73×#14442
C73×#14422
C73×#14412
#1656773