Finite Group Databasebeta

← C73×#10231 #32919635

Subgroups — C73×#10231 #32919635

Order 18688.

728 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474×C73C211yesmaximal, characteristic, complemented
2C2×#461×C73C211yesmaximal, characteristic, complemented
3C73×#2537C211yesmaximal, characteristic
4C73×#1990C211yesmaximal, characteristic, complemented
5C2×D4⋊C8×C73C211yesmaximal, characteristic, complemented
6C2×D4⋊C8×C73C211yesmaximal, characteristic, complemented
7C73×#1296C211yesmaximal, characteristic, complemented
8C22×SD16×C73—21no—
9C2×C4⋊Q8×C73C2211yescharacteristic
10C2×#121×C73—21no—
11C2×#121×C73—21no—
12C2×C4×D4×C73C2211yescharacteristic
13C2×C4×D4×C73C2211yescharacteristic
14#474×C73—22no—
15#474×C73—22no—
16#461×C73—22no—
17#461×C73—22no—
18C2×#107×C73—21no—
19C2×C4⋊C8×C73C2211yescharacteristic
20C2×C4⋊C8×C73C2211yescharacteristic
21C2×#103×C73—21no—
22C2×#102×C73—21no—
23C2×C4×C8×C73C2211yescharacteristic
24#394×C73—22no—
25#384×C73C2211yescharacteristic
26#380×C73—21no—
27#380×C73—21no—
28#340×C73—22no—
29D4⋊C8×C73—22no—
30D4⋊C8×C73—22no—
31C22×Q8×C73—21no—
32C22×D4×C73—22no—
33C22×D4×C73D411yescharacteristic
34C22×D4×C73D411yescharacteristic
35C23×C4×C73—21no—
36C23×C4×C73—21no—
37C2×SD16×C73—24no—
38C2×SD16×C73—42no—
39C2×SD16×C73—42no—
40C22×C8×C73—21no—
41C22×C8×C73—21no—
42C22×C8×C73—21no—
43C4⋊Q8×C73—21no—
44C4⋊Q8×C73—21no—
45C4⋊Q8×C73—21no—
46C4⋊Q8×C73D412yescomplemented
47#121×C73—42no—
48#121×C73—42no—
49#121×C73—42no—
50#121×C73—42no—
51C4×D4×C73—21no—
52C4×D4×C73—21no—
53C4×D4×C73—21no—
54C4×D4×C73—21no—
55C4×D4×C73—21no—
56C4×D4×C73—21no—
57C4×D4×C73—21no—
58C4×D4×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73D411yescharacteristic
64C2×C4⋊C4×C73D411yescharacteristic
65C2×C4⋊C4×C73D412yes—
66C2×C22⋊C4×C73—22no—
67C2×C22⋊C4×C73—22no—
68C2×C22⋊C4×C73—21no—
69C2×C22⋊C4×C73—21no—
70C2×C42×C73C2311yescharacteristic
71#107×C73—24no—
72C4⋊C8×C73—21no—
73C4⋊C8×C73—21no—
74C4⋊C8×C73D412yescomplemented
75C4⋊C8×C73D412yescomplemented
76#103×C73—24no—
77#102×C73—24no—
78C4×C8×C73—21no—
79C4×C8×C73—21no—
80#95×C73—22no—
81C24×C73—21no—
82C24×C73—21no—
83C2×Q8×C73—22no—
84C2×Q8×C73—41no—
85C2×Q8×C73—41no—
86C2×Q8×C73—41no—
87C2×Q8×C73—41no—
88C2×Q8×C73—41no—
89C2×D4×C73—21no—
90C2×D4×C73—21no—
91C2×D4×C73—21no—
92C2×D4×C73—21no—
93C2×D4×C73—21no—
94C2×D4×C73—21no—
95C2×D4×C73—21no—
96C2×D4×C73—21no—
97C2×D4×C73—21no—
98C2×D4×C73—21no—
99C2×D4×C73—44no—
100C2×D4×C73—44no—
101C2×D4×C73—82no—
102C2×D4×C73D812yes—
103C2×D4×C73SD1612yescomplemented
104C22×C4×C73—21no—
105C22×C4×C73—21no—
106C22×C4×C73—22no—
107C22×C4×C73—41no—
108C22×C4×C73—41no—
109C22×C4×C73—41no—
110C22×C4×C73—41no—
111C22×C4×C73—41no—
112C22×C4×C73—41no—
113C22×C4×C73—41no—
114C22×C4×C73—41no—
115C22×C4×C73—41no—
116C22×C4×C73—41no—
117C22×C4×C73—41no—
118C22×C4×C73—41no—
119C22×C4×C73—41no—
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73C2×D411yescharacteristic
122C22×C4×C73C2×D411yescharacteristic
123SD16×C73—48no—
124C2×C8×C73—22no—
125C2×C8×C73—22no—
126C2×C8×C73—22no—
127C2×C8×C73—41no—
128C2×C8×C73—41no—
129C2×C8×C73—41no—
130C2×C8×C73—41no—
131C2×C8×C73—41no—
132C2×C8×C73—41no—
133C4⋊C4×C73—22no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—22no—
137C4⋊C4×C73—21no—
138C4⋊C4×C73—21no—
139C4⋊C4×C73—42no—
140C4⋊C4×C73—41no—
141C4⋊C4×C73—41no—
142C4⋊C4×C73—41no—
143C4⋊C4×C73—41no—
144C4⋊C4×C73—41no—
145C4⋊C4×C73—41no—
146C22⋊C4×C73—44no—
147C22⋊C4×C73—44no—
148C22⋊C4×C73—41no—
149C22⋊C4×C73—41no—
150C22⋊C4×C73—41no—
151C22⋊C4×C73—41no—
152C42×C73C2×D411yescharacteristic
153C42×C73C2×D411yescharacteristic
154C42×C73C2×D411yescharacteristic
155C42×C73#4111yescharacteristic
156C23×C73—22no—
157C23×C73—22no—
158C23×C73—41no—
159C23×C73—41no—
160C23×C73—41no—
161C23×C73—41no—
162C23×C73—41no—
163C23×C73—41no—
164C23×C73—41no—
165C23×C73—41no—
166C23×C73—41no—
167C23×C73—41no—
168C23×C73—41no—
169C23×C73—41no—
170C23×C73#12011yescharacteristic
171Q8×C73—42no—
172Q8×C73—42no—
173Q8×C73—81no—
174Q8×C73—81no—
175D4×C73—22no—
176D4×C73—22no—
177D4×C73—22no—
178D4×C73—22no—
179D4×C73—41no—
180D4×C73—41no—
181D4×C73—41no—
182D4×C73—41no—
183D4×C73—88no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—41no—
187C2×C4×C73—41no—
188C2×C4×C73—42no—
189C2×C4×C73—41no—
190C2×C4×C73—42no—
191C2×C4×C73—41no—
192C2×C4×C73—41no—
193C2×C4×C73—42no—
194C2×C4×C73—41no—
195C2×C4×C73—42no—
196C2×C4×C73—81no—
197C2×C4×C73—81no—
198C2×C4×C73—81no—
199C2×C4×C73—81no—
200C2×C4×C73—81no—
201C2×C4×C73—81no—
202C2×C4×C73—81no—
203C2×C4×C73—81no—
204C2×C4×C73—81no—
205C2×C4×C73—81no—
206C2×C4×C73#12011yescharacteristic
207C2×C4×C73#12011yescharacteristic
208C2×C4×C73#12111yescharacteristic
209C2×C4×C73#12111yescharacteristic
210C2×C4×C73#12111yescharacteristic
211C2×C4×C73#12711yescharacteristic
212C2×C4×C73C2×D811yescharacteristic
213C2×C4×C73C2×SD1611yescharacteristic
214C2×C4×C73#13511yescharacteristic
215C2×C4×C73#13511yescharacteristic
216C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
217C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
218C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
219C2×C4×C73#13711yescharacteristic
220C8×C73—42no—
221C8×C73—42no—
222C8×C73—42no—
223C22×C73—42no—
224C22×C73—42no—
225C22×C73—42no—
226C22×C73—42no—
227C22×C73—42no—
228C22×C73—42no—
229C22×C73—81no—
230C22×C73—81no—
231C22×C73—81no—
232C22×C73—81no—
233C22×C73—81no—
234C22×C73—81no—
235C22×C73—81no—
236C22×C73—81no—
237C22×C73#39211yescharacteristic
238C22×C73#44711yescharacteristic
239C22×C73#44911yescharacteristic
240C22×C73#45011yescharacteristic
241C22×C73#45211yescharacteristic
242C22×C73#45511yescharacteristic
243C22×C73#45611yescharacteristic
244C4×C73—21no—
245C4×C73—21no—
246C4×C73—82no—
247C4×C73—81no—
248C4×C73—82no—
249C4×C73—81no—
250C4×C73#44711yescharacteristic
251C4×C73#44811yescharacteristic
252C4×C73#44911yescharacteristic
253C4×C73#44911yescharacteristic
254C4×C73#46511yescharacteristic
255C4×C73#46611yescharacteristic
256C4×C73#46811yescharacteristic
257C4×C73#46911yescharacteristic
258#10231C7311yesmaximal, characteristic, direct product factor, Sylow
259C2×C73—82no—
260C2×C73—82no—
261C2×C73#163911yescharacteristic
262C2×C73#164611yescharacteristic
263C2×C73#164711yescharacteristic
264C2×C73#165411yescharacteristic
265C2×C73#199911yescharacteristic
266C2×C73#200011yescharacteristic
267C2×C73#200711yescharacteristic
268C2×#474C2×C7311yescharacteristic, complemented
269C2×#461C2×C7311yescharacteristic, complemented
270#2537C2×C7311yescharacteristic
271#1990C2×C7311yescharacteristic, complemented
272C2×D4⋊C8C2×C7311yescharacteristic, complemented
273C2×D4⋊C8C2×C7311yescharacteristic, complemented
274#1296C2×C7311yescharacteristic, complemented
275C73#1023111yescharacteristic, direct product factor, Sylow
276C22×SD16—21no—
277C2×C4⋊Q8C22×C7311yescharacteristic
278C2×#121—21no—
279C2×#121—21no—
280C2×C4×D4C22×C7311yescharacteristic
281C2×C4×D4C22×C7311yescharacteristic
282#474—22no—
283#474—22no—
284#461—22no—
285#461—22no—
286C2×#107—21no—
287C2×C4⋊C8C22×C7311yescharacteristic
288C2×C4⋊C8C22×C7311yescharacteristic
289C2×#103—21no—
290C2×#102—21no—
291C2×C4×C8C22×C7311yescharacteristic
292#394—22no—
293#384C22×C7311yescharacteristic
294#380—21no—
295#380—21no—
296#340—22no—
297D4⋊C8—22no—
298D4⋊C8—22no—
299C22×Q8—21no—
300C22×D4—22no—
301C22×D4D4×C7311yescharacteristic
302C22×D4D4×C7311yescharacteristic
303C23×C4—21no—
304C23×C4—21no—
305C2×SD16—24no—
306C2×SD16—42no—
307C2×SD16—42no—
308C22×C8—21no—
309C22×C8—21no—
310C22×C8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8D4×C7312yescomplemented
315#121—42no—
316#121—42no—
317#121—42no—
318#121—42no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4D4×C7311yescharacteristic
332C2×C4⋊C4D4×C7311yescharacteristic
333C2×C4⋊C4D4×C7312yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C42C23×C7311yescharacteristic
339#107—24no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8D4×C7312yescomplemented
343C4⋊C8D4×C7312yescomplemented
344#103—24no—
345#102—24no—
346C4×C8—21no—
347C4×C8—21no—
348#95—22no—
349C24—21no—
350C24—21no—
351C2×Q8—22no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—82no—
370C2×D4D8×C7312yes—
371C2×D4SD16×C7312yescomplemented
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4C2×D4×C7311yescharacteristic
389C22×C4C2×D4×C7311yescharacteristic
390C22×C4C2×D4×C7311yescharacteristic
391SD16—48no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C42C2×D4×C7311yescharacteristic
421C42C2×D4×C7311yescharacteristic
422C42C2×D4×C7311yescharacteristic
423C42#41×C7311yescharacteristic
424C23—22no—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23#120×C7311yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—81no—
442Q8—81no—
443D4—22no—
444D4—22no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—88no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4#120×C7311yescharacteristic
475C2×C4#120×C7311yescharacteristic
476C2×C4#121×C7311yescharacteristic
477C2×C4#121×C7311yescharacteristic
478C2×C4#121×C7311yescharacteristic
479C2×C4#127×C7311yescharacteristic
480C2×C4C2×D8×C7311yescharacteristic
481C2×C4C2×SD16×C7311yescharacteristic
482C2×C4#135×C7311yescharacteristic
483C2×C4#135×C7311yescharacteristic
484C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
485C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
486C2×C4AGL(1,ℤ/8)×C7311yescharacteristic
487C2×C4#137×C7311yescharacteristic
488C8—42no—
489C8—42no—
490C8—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22#392×C7311yescharacteristic
506C22#447×C7311yescharacteristic
507C22#449×C7311yescharacteristic
508C22#450×C7311yescharacteristic
509C22#452×C7311yescharacteristic
510C22#455×C7311yescharacteristic
511C22#456×C7311yescharacteristic
512C4—21no—
513C4—21no—
514C4—81no—
515C4—82no—
516C4—81no—
517C4—82no—
518C4#447×C7311yescharacteristic
519C4#448×C7311yescharacteristic
520C4#449×C7311yescharacteristic
521C4#449×C7311yescharacteristic
522C4#465×C7311yescharacteristic
523C4#466×C7311yescharacteristic
524C4#468×C7311yescharacteristic
525C4#469×C7311yescharacteristic
526C2—82no—
527C2—82no—
528C2C73×#163911yescharacteristic
529C2C73×#164611yescharacteristic
530C2C73×#164711yescharacteristic
531C2C73×#165411yescharacteristic
532C2C73×#199911yescharacteristic
533C2C73×#200011yescharacteristic
534C2C73×#200711yescharacteristic
subgroupG/H
center, socleC23×C73#120
commutator, Frattini, cosocleC2×C42C23×C73
prime radicalC73×#2537C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#474×C73C2semidirect
2. C2×#461×C73C2semidirect
4. C73×#1990C2semidirect
5. C2×D4⋊C8×C73C2semidirect
6. C2×D4⋊C8×C73C2semidirect
7. C73×#1296C2semidirect
46. C4⋊Q8×C73D4semidirect
74. C4⋊C8×C73D4semidirect
75. C4⋊C8×C73D4semidirect
103. C2×D4×C73SD16semidirect
258. #10231C73direct (N × H)
268. C2×#474C2×C73semidirect
269. C2×#461C2×C73semidirect
271. #1990C2×C73semidirect
272. C2×D4⋊C8C2×C73semidirect
273. C2×D4⋊C8C2×C73semidirect
274. #1296C2×C73semidirect
275. C73#10231direct (N × H)
314. C4⋊Q8D4×C73semidirect
342. C4⋊C8D4×C73semidirect
343. C4⋊C8D4×C73semidirect
371. C2×D4SD16×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#474×C732
C2×#461×C732
C73×#25372
C73×#19902
C2×D4⋊C8×C732
C73×#12962
#1023173