Finite Group Databasebeta

← C73×#25998 #32935402

Subgroups — C73×#25998 #32935402

Order 18688.

758 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2442C211yesmaximal, characteristic
2C73×#2344C211yesmaximal, characteristic
3C73×#2101C211yesmaximal, characteristic, complemented
4C73×#2088C211yesmaximal, characteristic, complemented
5C73×#2087C211yesmaximal, characteristic, complemented
6C73×#2059C211yesmaximal, characteristic, complemented
7C73×#1989C211yesmaximal, characteristic, complemented
8C73×#1949C211yesmaximal, characteristic, complemented
9C73×#1921C211yesmaximal, characteristic, complemented
10C73×#1919C211yesmaximal, characteristic, complemented
11C73×#1852C211yesmaximal, characteristic
12C73×#1850C211yesmaximal, characteristic
13C73×#1823C211yesmaximal, characteristic
14C73×#1807C211yesmaximal, characteristic
15C2×#340×C73C211yesmaximal, characteristic
16C23×C8×C73C2211yescharacteristic
17C2×#125×C73C2211yescharacteristic
18C2×#117×C73—21no—
19C22×C4⋊C4×C73C2211yescharacteristic
20C22×C4⋊C4×C73C2211yescharacteristic
21C2×#107×C73C2211yescharacteristic
22C2×#107×C73C2211yescharacteristic
23C2×#106×C73C2211yescharacteristic
24C2×#106×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27C2×C22⋊C8×C73C2211yescharacteristic
28#400×C73—21no—
29#398×C73—21no—
30#389×C73C2211yescharacteristic
31#389×C73C2211yescharacteristic
32#389×C73C2211yescharacteristic
33#389×C73C2211yescharacteristic
34#387×C73—21no—
35#387×C73—21no—
36#385×C73—21no—
37#382×C73C2211yescharacteristic
38#381×C73—21no—
39#380×C73—21no—
40#380×C73C2211yescharacteristic
41#380×C73C2211yescharacteristic
42C4×C4⋊C4×C73C2211yescharacteristic
43C4×C4⋊C4×C73C2211yescharacteristic
44C4×C22⋊C4×C73C2211yescharacteristic
45#340×C73C2211yescharacteristic
46#340×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
59C23×C4×C73—21no—
60C23×C4×C73—21no—
61C23×C4×C73C2311yescharacteristic
62C22×C8×C73—22no—
63C22×C8×C73—22no—
64C22×C8×C73—41no—
65C22×C8×C73C2311yescharacteristic
66C22×C8×C73C2311yescharacteristic
67C22×C8×C73C2311yescharacteristic
68C22×C8×C73C2311yescharacteristic
69#125×C73—22no—
70#125×C73—41no—
71#117×C73—44no—
72C2×C4⋊C4×C73—21no—
73C2×C4⋊C4×C73—22no—
74C2×C4⋊C4×C73—22no—
75C2×C4⋊C4×C73—21no—
76C2×C4⋊C4×C73—21no—
77C2×C4⋊C4×C73—21no—
78C2×C4⋊C4×C73—21no—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73—41no—
83C2×C4⋊C4×C73Q814yes—
84C2×C4⋊C4×C73C2311yescharacteristic
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×C73D411yescharacteristic
95C2×C22⋊C4×C73D411yescharacteristic
96C2×C42×C73—21no—
97C2×C42×C73—21no—
98C2×C42×C73C2311yescharacteristic
99C2×C42×C73C2311yescharacteristic
100#107×C73—22no—
101#107×C73—41no—
102#106×C73—22no—
103#106×C73—41no—
104C4⋊C8×C73—22no—
105C4⋊C8×C73—41no—
106C22⋊C8×C73—22no—
107#95×C73—21no—
108#95×C73—21no—
109#95×C73—21no—
110#95×C73—21no—
111#95×C73—21no—
112#95×C73—21no—
113#95×C73—21no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119C24×C73C2×D411yescharacteristic
120C22×C4×C73—21no—
121C22×C4×C73—21no—
122C22×C4×C73—21no—
123C22×C4×C73—21no—
124C22×C4×C73—24no—
125C22×C4×C73—21no—
126C22×C4×C73—21no—
127C22×C4×C73—21no—
128C22×C4×C73—21no—
129C22×C4×C73—21no—
130C22×C4×C73—21no—
131C22×C4×C73—21no—
132C22×C4×C73—21no—
133C22×C4×C73—42no—
134C22×C4×C73—42no—
135C22×C4×C73—42no—
136C22×C4×C73—42no—
137C22×C4×C73—81no—
138C22×C4×C73C2×D411yescharacteristic
139C22×C4×C73C2×D411yescharacteristic
140C22×C4×C73C2×D411yescharacteristic
141C22×C4×C73C2×D411yescharacteristic
142C22×C4×C73C2×D411yescharacteristic
143C22×C4×C73C2×Q812yes—
144C22×C4×C73#4112yes—
145C22×C4×C73C2411yescharacteristic
146C2×C8×C73—21no—
147C2×C8×C73—21no—
148C2×C8×C73—21no—
149C2×C8×C73—21no—
150C2×C8×C73—21no—
151C2×C8×C73—21no—
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73—21no—
155C2×C8×C73—21no—
156C2×C8×C73—21no—
157C2×C8×C73—44no—
158C2×C8×C73#4112yes—
159C4⋊C4×C73—24no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—21no—
165C4⋊C4×C73—21no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—21no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—21no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—42no—
174C4⋊C4×C73—42no—
175C4⋊C4×C73—42no—
176C4⋊C4×C73—42no—
177C4⋊C4×C73—42no—
178C4⋊C4×C73—81no—
179C4⋊C4×C73—81no—
180C4⋊C4×C73C2×Q812yes—
181C4⋊C4×C73C2×Q812yes—
182C4⋊C4×C73#4112yes—
183C22⋊C4×C73—22no—
184C22⋊C4×C73—22no—
185C22⋊C4×C73—42no—
186C22⋊C4×C73—42no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—42no—
192C42×C73—42no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73C22⋊Q812yes—
198C23×C73#13512yes—
199C23×C73AGL(1,ℤ/8)12yes—
200C23×C73C22×D411yescharacteristic
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—22no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—48no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
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—84no—
245C2×C4×C73C22⋊Q812yes—
246C2×C4×C73C22×Q811yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73H(5,𝔽2)11yescharacteristic
251C2×C4×C732−1+411yescharacteristic
252C8×C73—22no—
253C8×C73—41no—
254C8×C73—41no—
255C8×C73—41no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73—22no—
261C22×C73—22no—
262C22×C73#47814yes—
263C22×C73C2×C22⋊Q811yescharacteristic
264C22×C73#54611yescharacteristic
265C22×C73#54711yescharacteristic
266C22×C73C2×#13511yescharacteristic
267C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C73#57711yescharacteristic
269C22×C73#57811yescharacteristic
270C4×C73—22no—
271C4×C73—42no—
272C4×C73—42no—
273C4×C73—41no—
274C4×C73—41no—
275C4×C73—41no—
276C4×C73—41no—
277C4×C73—42no—
278C4×C73—81no—
279C4×C73—81no—
280C4×C73—81no—
281C4×C73—81no—
282C4×C73—81no—
283C4×C73#53311yescharacteristic
284C4×C73#54411yescharacteristic
285C4×C73#55111yescharacteristic
286C4×C73#55211yescharacteristic
287#25998C7311yesmaximal, characteristic, direct product factor, Sylow
288C2×C73—24no—
289C2×C73#243811yescharacteristic
290C2×C73C2×#47811yescharacteristic
291C2×C73#308211yescharacteristic
292C2×C73#330011yescharacteristic
293C2×C73#330611yescharacteristic
294C2×C73#333711yescharacteristic
295C2×C73#334311yescharacteristic
296#2442C2×C7311yescharacteristic
297#2344C2×C7311yescharacteristic
298#2101C2×C7311yescharacteristic, complemented
299#2088C2×C7311yescharacteristic, complemented
300#2087C2×C7311yescharacteristic, complemented
301#2059C2×C7311yescharacteristic, complemented
302#1989C2×C7311yescharacteristic, complemented
303#1949C2×C7311yescharacteristic, complemented
304#1921C2×C7311yescharacteristic, complemented
305#1919C2×C7311yescharacteristic, complemented
306#1852C2×C7311yescharacteristic
307#1850C2×C7311yescharacteristic
308#1823C2×C7311yescharacteristic
309#1807C2×C7311yescharacteristic
310C2×#340C2×C7311yescharacteristic
311C73#2599811yescharacteristic, direct product factor, Sylow
312C23×C8C22×C7311yescharacteristic
313C2×#125C22×C7311yescharacteristic
314C2×#117—21no—
315C22×C4⋊C4C22×C7311yescharacteristic
316C22×C4⋊C4C22×C7311yescharacteristic
317C2×#107C22×C7311yescharacteristic
318C2×#107C22×C7311yescharacteristic
319C2×#106C22×C7311yescharacteristic
320C2×#106C22×C7311yescharacteristic
321C2×C4⋊C8C22×C7311yescharacteristic
322C2×C4⋊C8C22×C7311yescharacteristic
323C2×C22⋊C8C22×C7311yescharacteristic
324#400—21no—
325#398—21no—
326#389C22×C7311yescharacteristic
327#389C22×C7311yescharacteristic
328#389C22×C7311yescharacteristic
329#389C22×C7311yescharacteristic
330#387—21no—
331#387—21no—
332#385—21no—
333#382C22×C7311yescharacteristic
334#381—21no—
335#380—21no—
336#380C22×C7311yescharacteristic
337#380C22×C7311yescharacteristic
338C4×C4⋊C4C22×C7311yescharacteristic
339C4×C4⋊C4C22×C7311yescharacteristic
340C4×C22⋊C4C22×C7311yescharacteristic
341#340C22×C7311yescharacteristic
342#340C22×C7311yescharacteristic
343#340C22×C7311yescharacteristic
344#340C22×C7311yescharacteristic
345#340C22×C7311yescharacteristic
346#340C22×C7311yescharacteristic
347#340C22×C7311yescharacteristic
348#340C22×C7311yescharacteristic
349#340C22×C7311yescharacteristic
350#340C22×C7311yescharacteristic
351#340C22×C7311yescharacteristic
352#340C22×C7311yescharacteristic
353#336C22×C7311yescharacteristic
354#336C22×C7311yescharacteristic
355C23×C4—21no—
356C23×C4—21no—
357C23×C4C23×C7311yescharacteristic
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—41no—
361C22×C8C23×C7311yescharacteristic
362C22×C8C23×C7311yescharacteristic
363C22×C8C23×C7311yescharacteristic
364C22×C8C23×C7311yescharacteristic
365#125—22no—
366#125—41no—
367#117—44no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—41no—
379C2×C4⋊C4Q8×C7314yes—
380C2×C4⋊C4C23×C7311yescharacteristic
381C2×C4⋊C4C23×C7311yescharacteristic
382C2×C4⋊C4C23×C7311yescharacteristic
383C2×C4⋊C4C23×C7311yescharacteristic
384C2×C4⋊C4C23×C7311yescharacteristic
385C2×C4⋊C4C23×C7311yescharacteristic
386C2×C4⋊C4C23×C7311yescharacteristic
387C2×C4⋊C4C23×C7311yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4D4×C7311yescharacteristic
391C2×C22⋊C4D4×C7311yescharacteristic
392C2×C42—21no—
393C2×C42—21no—
394C2×C42C23×C7311yescharacteristic
395C2×C42C23×C7311yescharacteristic
396#107—22no—
397#107—41no—
398#106—22no—
399#106—41no—
400C4⋊C8—22no—
401C4⋊C8—41no—
402C22⋊C8—22no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95D4×C7311yescharacteristic
414#95D4×C7311yescharacteristic
415C24C2×D4×C7311yescharacteristic
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—81no—
434C22×C4C2×D4×C7311yescharacteristic
435C22×C4C2×D4×C7311yescharacteristic
436C22×C4C2×D4×C7311yescharacteristic
437C22×C4C2×D4×C7311yescharacteristic
438C22×C4C2×D4×C7311yescharacteristic
439C22×C4C2×Q8×C7312yes—
440C22×C4#41×C7312yes—
441C22×C4C24×C7311yescharacteristic
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—44no—
454C2×C8#41×C7312yes—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
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—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4C2×Q8×C7312yes—
477C4⋊C4C2×Q8×C7312yes—
478C4⋊C4#41×C7312yes—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—42no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23C22⋊Q8×C7312yes—
494C23#135×C7312yes—
495C23AGL(1,ℤ/8)×C7312yes—
496C23C22×D4×C7311yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4C22⋊Q8×C7312yes—
542C2×C4C22×Q8×C7311yescharacteristic
543C2×C4C2×#41×C7311yescharacteristic
544C2×C4C2×#41×C7311yescharacteristic
545C2×C4C2×#41×C7311yescharacteristic
546C2×C4H(5,𝔽2)×C7311yescharacteristic
547C2×C42−1+4×C7311yescharacteristic
548C8—22no—
549C8—41no—
550C8—41no—
551C8—41no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22#478×C7314yes—
559C22C2×C22⋊Q8×C7311yescharacteristic
560C22#546×C7311yescharacteristic
561C22#547×C7311yescharacteristic
562C22C2×#135×C7311yescharacteristic
563C22C2×AGL(1,ℤ/8)×C7311yescharacteristic
564C22#577×C7311yescharacteristic
565C22#578×C7311yescharacteristic
566C4—22no—
567C4—41no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4#533×C7311yescharacteristic
580C4#544×C7311yescharacteristic
581C4#551×C7311yescharacteristic
582C4#552×C7311yescharacteristic
583C2—24no—
584C2C73×#243811yescharacteristic
585C2C2×#478×C7311yescharacteristic
586C2C73×#308211yescharacteristic
587C2C73×#330011yescharacteristic
588C2C73×#330611yescharacteristic
589C2C73×#333711yescharacteristic
590C2C73×#334311yescharacteristic
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×#2087C2semidirect
6. C73×#2059C2semidirect
7. C73×#1989C2semidirect
8. C73×#1949C2semidirect
9. C73×#1921C2semidirect
10. C73×#1919C2semidirect
287. #25998C73direct (N × H)
298. #2101C2×C73semidirect
299. #2088C2×C73semidirect
300. #2087C2×C73semidirect
301. #2059C2×C73semidirect
302. #1989C2×C73semidirect
303. #1949C2×C73semidirect
304. #1921C2×C73semidirect
305. #1919C2×C73semidirect
311. C73#25998direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#24422
C73×#23442
C73×#21012
C73×#20882
C73×#20872
C73×#20592
C73×#19892
C73×#19492
C73×#19212
C73×#19192
C73×#18522
C73×#18502
C73×#18232
C73×#18072
C2×#340×C732
#2599873