Finite Group Databasebeta

← C73×#25994 #32935398

Subgroups — C73×#25994 #32935398

Order 18688.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2440C211yesmaximal, characteristic
2C73×#2345C211yesmaximal, characteristic
3C73×#2100C211yesmaximal, characteristic, complemented
4C73×#2087C211yesmaximal, characteristic, complemented
5C73×#2087C211yesmaximal, characteristic, complemented
6C73×#2057C211yesmaximal, characteristic, complemented
7C73×#1987C211yesmaximal, characteristic, complemented
8C73×#1947C211yesmaximal, characteristic, complemented
9C73×#1921C211yesmaximal, characteristic, complemented
10C73×#1921C211yesmaximal, characteristic, complemented
11C73×#1852C211yesmaximal, characteristic
12C73×#1850C211yesmaximal, characteristic
13C73×#1824C211yesmaximal, characteristic
14C73×#1808C211yesmaximal, characteristic
15C2×#340×C73C211yesmaximal, characteristic
16C23×C8×C73C2211yescharacteristic
17C2×C4⋊Q8×C73C2211yescharacteristic
18C2×C22⋊Q8×C73—21no—
19C22×C4⋊C4×C73C2211yescharacteristic
20C22×C4⋊C4×C73C2211yescharacteristic
21C2×#107×C73C2211yescharacteristic
22C2×#107×C73C2211yescharacteristic
23C2×#107×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27C2×C22⋊C8×C73C2211yescharacteristic
28#398×C73—21no—
29#395×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#384×C73C2211yescharacteristic
38#380×C73—21no—
39#380×C73C2211yescharacteristic
40#380×C73C2211yescharacteristic
41C4×C4⋊C4×C73C2211yescharacteristic
42C4×C4⋊C4×C73C2211yescharacteristic
43C4×C22⋊C4×C73—21no—
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
59C22×Q8×C73—21no—
60C23×C4×C73—21no—
61C23×C4×C73—21no—
62C23×C4×C73C2311yescharacteristic
63C22×C8×C73—22no—
64C22×C8×C73—22no—
65C22×C8×C73—41no—
66C22×C8×C73C2311yescharacteristic
67C22×C8×C73C2311yescharacteristic
68C22×C8×C73C2311yescharacteristic
69C22×C8×C73C2311yescharacteristic
70C4⋊Q8×C73—22no—
71C4⋊Q8×C73—41no—
72C22⋊Q8×C73—44no—
73C2×C4⋊C4×C73—21no—
74C2×C4⋊C4×C73—22no—
75C2×C4⋊C4×C73—22no—
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—21no—
83C2×C4⋊C4×C73—21no—
84C2×C4⋊C4×C73—41no—
85C2×C4⋊C4×C73Q814yescomplemented
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×C4⋊C4×C73C2311yescharacteristic
93C2×C4⋊C4×C73C2311yescharacteristic
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×C73C2311yescharacteristic
101C2×C42×C73C2311yescharacteristic
102#107×C73—22no—
103#107×C73—22no—
104#107×C73—41no—
105#107×C73—41no—
106C4⋊C8×C73—22no—
107C4⋊C8×C73—41no—
108C22⋊C8×C73—22no—
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
120C2×Q8×C73—42no—
121C2×Q8×C73—42no—
122C2×Q8×C73—81no—
123C22×C4×C73—21no—
124C22×C4×C73—21no—
125C22×C4×C73—21no—
126C22×C4×C73—21no—
127C22×C4×C73—21no—
128C22×C4×C73—21no—
129C22×C4×C73—21no—
130C22×C4×C73—24no—
131C22×C4×C73—21no—
132C22×C4×C73—21no—
133C22×C4×C73—21no—
134C22×C4×C73—21no—
135C22×C4×C73—21no—
136C22×C4×C73—42no—
137C22×C4×C73—42no—
138C22×C4×C73—81no—
139C22×C4×C73—81no—
140C22×C4×C73—81no—
141C22×C4×C73C2×D411yescharacteristic
142C22×C4×C73C2×D411yescharacteristic
143C22×C4×C73C2×D411yescharacteristic
144C22×C4×C73C2×D411yescharacteristic
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×Q812yes—
147C22×C4×C73#4112yes—
148C22×C4×C73C2411yescharacteristic
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—21no—
158C2×C8×C73—21no—
159C2×C8×C73—21no—
160C2×C8×C73—44no—
161C2×C8×C73#4112yes—
162C4⋊C4×C73—24no—
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—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—42no—
177C4⋊C4×C73—42no—
178C4⋊C4×C73—42no—
179C4⋊C4×C73—42no—
180C4⋊C4×C73—42no—
181C4⋊C4×C73—81no—
182C4⋊C4×C73—81no—
183C4⋊C4×C73—81no—
184C4⋊C4×C73C2×Q812yes—
185C4⋊C4×C73C2×Q812yes—
186C4⋊C4×C73#4112yes—
187C22⋊C4×C73—22no—
188C22⋊C4×C73—42no—
189C22⋊C4×C73—42no—
190C22⋊C4×C73Q1614yes—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—81no—
196C42×C73—81no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73C22⋊Q812yes—
202C23×C73C2×Q1612yes—
203C23×C73AGL(1,ℤ/8)12yes—
204C23×C73C22×D411yescharacteristic
205Q8×C73—84no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—21no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
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—41no—
245C2×C4×C73—41no—
246C2×C4×C73—84no—
247C2×C4×C73—84no—
248C2×C4×C73C22⋊Q812yes—
249C2×C4×C73C2×Q1612yes—
250C2×C4×C73C2×Q1612yes—
251C2×C4×C73C22×Q811yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73H(5,𝔽2)11yescharacteristic
256C2×C4×C732−1+411yescharacteristic
257C8×C73—22no—
258C8×C73—41no—
259C8×C73—41no—
260C8×C73—41no—
261C22×C73—22no—
262C22×C73—22no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73#47714yes—
268C22×C73C2×C22⋊Q811yescharacteristic
269C22×C73#54611yescharacteristic
270C22×C73#54711yescharacteristic
271C22×C73C22×Q1611yescharacteristic
272C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C73#57611yescharacteristic
274C22×C73#57711yescharacteristic
275C4×C73—22no—
276C4×C73—22no—
277C4×C73—22no—
278C4×C73—42no—
279C4×C73—42no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—42no—
283C4×C73—81no—
284C4×C73—81no—
285C4×C73—81no—
286C4×C73—81no—
287C4×C73—81no—
288C4×C73#53311yescharacteristic
289C4×C73#54411yescharacteristic
290C4×C73#55111yescharacteristic
291C4×C73#55211yescharacteristic
292#25994C7311yesmaximal, characteristic, direct product factor, Sylow
293C2×C73—24no—
294C2×C73#243811yescharacteristic
295C2×C73C2×#47711yescharacteristic
296C2×C73#308111yescharacteristic
297C2×C73#329911yescharacteristic
298C2×C73#331811yescharacteristic
299C2×C73#333811yescharacteristic
300C2×C73#334511yescharacteristic
301#2440C2×C7311yescharacteristic
302#2345C2×C7311yescharacteristic
303#2100C2×C7311yescharacteristic, complemented
304#2087C2×C7311yescharacteristic, complemented
305#2087C2×C7311yescharacteristic, complemented
306#2057C2×C7311yescharacteristic, complemented
307#1987C2×C7311yescharacteristic, complemented
308#1947C2×C7311yescharacteristic, complemented
309#1921C2×C7311yescharacteristic, complemented
310#1921C2×C7311yescharacteristic, complemented
311#1852C2×C7311yescharacteristic
312#1850C2×C7311yescharacteristic
313#1824C2×C7311yescharacteristic
314#1808C2×C7311yescharacteristic
315C2×#340C2×C7311yescharacteristic
316C73#2599411yescharacteristic, direct product factor, Sylow
317C23×C8C22×C7311yescharacteristic
318C2×C4⋊Q8C22×C7311yescharacteristic
319C2×C22⋊Q8—21no—
320C22×C4⋊C4C22×C7311yescharacteristic
321C22×C4⋊C4C22×C7311yescharacteristic
322C2×#107C22×C7311yescharacteristic
323C2×#107C22×C7311yescharacteristic
324C2×#107C22×C7311yescharacteristic
325C2×#107C22×C7311yescharacteristic
326C2×C4⋊C8C22×C7311yescharacteristic
327C2×C4⋊C8C22×C7311yescharacteristic
328C2×C22⋊C8C22×C7311yescharacteristic
329#398—21no—
330#395—21no—
331#389C22×C7311yescharacteristic
332#389C22×C7311yescharacteristic
333#389C22×C7311yescharacteristic
334#389C22×C7311yescharacteristic
335#387—21no—
336#387—21no—
337#385—21no—
338#384C22×C7311yescharacteristic
339#380—21no—
340#380C22×C7311yescharacteristic
341#380C22×C7311yescharacteristic
342C4×C4⋊C4C22×C7311yescharacteristic
343C4×C4⋊C4C22×C7311yescharacteristic
344C4×C22⋊C4—21no—
345C4×C22⋊C4C22×C7311yescharacteristic
346#340C22×C7311yescharacteristic
347#340C22×C7311yescharacteristic
348#340C22×C7311yescharacteristic
349#340C22×C7311yescharacteristic
350#340C22×C7311yescharacteristic
351#340C22×C7311yescharacteristic
352#340C22×C7311yescharacteristic
353#340C22×C7311yescharacteristic
354#340C22×C7311yescharacteristic
355#340C22×C7311yescharacteristic
356#340C22×C7311yescharacteristic
357#340C22×C7311yescharacteristic
358#336C22×C7311yescharacteristic
359#336C22×C7311yescharacteristic
360C22×Q8—21no—
361C23×C4—21no—
362C23×C4—21no—
363C23×C4C23×C7311yescharacteristic
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—41no—
367C22×C8C23×C7311yescharacteristic
368C22×C8C23×C7311yescharacteristic
369C22×C8C23×C7311yescharacteristic
370C22×C8C23×C7311yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—41no—
373C22⋊Q8—44no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—41no—
386C2×C4⋊C4Q8×C7314yescomplemented
387C2×C4⋊C4C23×C7311yescharacteristic
388C2×C4⋊C4C23×C7311yescharacteristic
389C2×C4⋊C4C23×C7311yescharacteristic
390C2×C4⋊C4C23×C7311yescharacteristic
391C2×C4⋊C4C23×C7311yescharacteristic
392C2×C4⋊C4C23×C7311yescharacteristic
393C2×C4⋊C4C23×C7311yescharacteristic
394C2×C4⋊C4C23×C7311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4D4×C7311yescharacteristic
398C2×C22⋊C4D4×C7311yescharacteristic
399C2×C42—21no—
400C2×C42—21no—
401C2×C42C23×C7311yescharacteristic
402C2×C42C23×C7311yescharacteristic
403#107—22no—
404#107—22no—
405#107—41no—
406#107—41no—
407C4⋊C8—22no—
408C4⋊C8—41no—
409C22⋊C8—22no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95D4×C7311yescharacteristic
419#95D4×C7311yescharacteristic
420C24C2×D4×C7311yescharacteristic
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—81no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4C2×D4×C7311yescharacteristic
443C22×C4C2×D4×C7311yescharacteristic
444C22×C4C2×D4×C7311yescharacteristic
445C22×C4C2×D4×C7311yescharacteristic
446C22×C4C2×D4×C7311yescharacteristic
447C22×C4C2×Q8×C7312yes—
448C22×C4#41×C7312yes—
449C22×C4C24×C7311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—44no—
462C2×C8#41×C7312yes—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4C2×Q8×C7312yes—
486C4⋊C4C2×Q8×C7312yes—
487C4⋊C4#41×C7312yes—
488C22⋊C4—22no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4Q16×C7314yes—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—81no—
497C42—81no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C22⋊Q8×C7312yes—
503C23C2×Q16×C7312yes—
504C23AGL(1,ℤ/8)×C7312yes—
505C23C22×D4×C7311yescharacteristic
506Q8—84no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
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—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4C22⋊Q8×C7312yes—
550C2×C4C2×Q16×C7312yes—
551C2×C4C2×Q16×C7312yes—
552C2×C4C22×Q8×C7311yescharacteristic
553C2×C4C2×#41×C7311yescharacteristic
554C2×C4C2×#41×C7311yescharacteristic
555C2×C4C2×#41×C7311yescharacteristic
556C2×C4H(5,𝔽2)×C7311yescharacteristic
557C2×C42−1+4×C7311yescharacteristic
558C8—22no—
559C8—41no—
560C8—41no—
561C8—41no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22#477×C7314yes—
569C22C2×C22⋊Q8×C7311yescharacteristic
570C22#546×C7311yescharacteristic
571C22#547×C7311yescharacteristic
572C22C22×Q16×C7311yescharacteristic
573C22C2×AGL(1,ℤ/8)×C7311yescharacteristic
574C22#576×C7311yescharacteristic
575C22#577×C7311yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4#533×C7311yescharacteristic
590C4#544×C7311yescharacteristic
591C4#551×C7311yescharacteristic
592C4#552×C7311yescharacteristic
593C2—24no—
594C2C73×#243811yescharacteristic
595C2C2×#477×C7311yescharacteristic
596C2C73×#308111yescharacteristic
597C2C73×#329911yescharacteristic
598C2C73×#331811yescharacteristic
599C2C73×#333811yescharacteristic
600C2C73×#334511yescharacteristic
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×#2100C2semidirect
4. C73×#2087C2semidirect
5. C73×#2087C2semidirect
6. C73×#2057C2semidirect
7. C73×#1987C2semidirect
8. C73×#1947C2semidirect
9. C73×#1921C2semidirect
10. C73×#1921C2semidirect
85. C2×C4⋊C4×C73Q8semidirect
292. #25994C73direct (N × H)
303. #2100C2×C73semidirect
304. #2087C2×C73semidirect
305. #2087C2×C73semidirect
306. #2057C2×C73semidirect
307. #1987C2×C73semidirect
308. #1947C2×C73semidirect
309. #1921C2×C73semidirect
310. #1921C2×C73semidirect
316. C73#25994direct (N × H)
386. C2×C4⋊C4Q8×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#24402
C73×#23452
C73×#21002
C73×#20872
C73×#20572
C73×#19872
C73×#19472
C73×#19212
C73×#18522
C73×#18502
C73×#18242
C73×#18082
C2×#340×C732
#2599473