Finite Group Databasebeta

← C73×#18345 #32927749

Subgroups — C73×#18345 #32927749

Order 18688.

888 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3461C211yesmaximal, characteristic
2C73×#3239C212yesmaximal, complemented
3C73×#2568C211yesmaximal, characteristic, complemented
4C73×#1617C212yesmaximal, complemented
5C73×#1616C212yesmaximal
6C73×#1615C214yesmaximal, complemented
7C73×#1509C212yesmaximal, complemented
8C73×#1467C211yesmaximal, characteristic, complemented
9#544×C73C2211yescharacteristic
10#542×C73—22no—
11#541×C73—22no—
12#539×C73—21no—
13#539×C73C2212yes—
14#537×C73C2211yescharacteristic
15#536×C73—21no—
16#533×C73—22no—
17C2×C4⋊Q8×C73C2211yescharacteristic
18#529×C73—22no—
19#529×C73—21no—
20#529×C73C2212yes—
21C4×#41×C73—21no—
22C2×C4×Q8×C73C2211yescharacteristic
23#501×C73—22no—
24#489×C73—22no—
25#484×C73C2214yes—
26#481×C73C2214yes—
27#475×C73—22no—
28#474×C73—22no—
29#432×C73C2212yes—
30#400×C73C2212yes—
31#400×C73C2212yes—
32#391×C73—21no—
33#391×C73—21no—
34#391×C73—21no—
35#382×C73—21no—
36C4×C4⋊C4×C73C2211yescharacteristic
37#328×C73C2214yes—
38#327×C73C2214yes—
39#324×C73C2212yes—
40#324×C73C2212yes—
41C2×#41×C73—21no—
42C22×Q8×C73—21no—
43C22×Q8×C73D411yescharacteristic
44C4⋊Q8×C73—22no—
45C4⋊Q8×C73—22no—
46C4⋊Q8×C73—22no—
47C4⋊Q8×C73—22no—
48#126×C73—22no—
49#126×C73—22no—
50#126×C73—22no—
51#126×C73—22no—
52#125×C73—21no—
53#125×C73—22no—
54#125×C73—22no—
55#125×C73—22no—
56#125×C73D412yes—
57#124×C73—22no—
58#124×C73—21no—
59#124×C73D412yes—
60#123×C73—22no—
61#123×C73—22no—
62#123×C73—22no—
63#123×C73—22no—
64C22⋊Q8×C73—22no—
65C22⋊Q8×C73—22no—
66C22⋊Q8×C73—22no—
67C22⋊Q8×C73—22no—
68C22⋊Q8×C73—21no—
69C22⋊Q8×C73—22no—
70C22⋊Q8×C73C2312yes—
71#121×C73—22no—
72#121×C73—21no—
73#121×C73C2312yes—
74C4×Q8×C73—21no—
75C4×Q8×C73—21no—
76C4×Q8×C73—21no—
77C4×Q8×C73—22no—
78C4×Q8×C73—22no—
79C4×Q8×C73D412yes—
80C4×D4×C73—21no—
81C4×D4×C73—21no—
82C4×D4×C73—21no—
83C4×D4×C73—22no—
84C4×D4×C73—22no—
85C4×D4×C73D412yes—
86#117×C73—22no—
87#117×C73—22no—
88#117×C73—21no—
89#117×C73—22no—
90C2×C4⋊C4×C73—21no—
91C2×C4⋊C4×C73—21no—
92C2×C4⋊C4×C73D411yescharacteristic
93C2×C4⋊C4×C73C2312yes—
94C2×C4⋊C4×C73C2311yescharacteristic
95C2×C4⋊C4×C73C2311yescharacteristic
96C2×C42×C73—22no—
97C2×C42×C73—21no—
98C2×C42×C73D411yescharacteristic
99C2×C42×C73D411yescharacteristic
100C2×C42×C73C2311yescharacteristic
101#107×C73—24no—
102#106×C73—24no—
103C4⋊C8×C73—22no—
104#103×C73—24no—
105#102×C73—24no—
106C22⋊C8×C73C2314yes—
107#97×C73—22no—
108#95×C73—22no—
109#95×C73—22no—
110#95×C73—22no—
111#95×C73—22no—
112#95×C73C2312yes—
113#41×C73—42no—
114#41×C73—42no—
115C2×Q8×C73—21no—
116C2×Q8×C73—21no—
117C2×Q8×C73—22no—
118C2×Q8×C73—22no—
119C2×Q8×C73—41no—
120C2×Q8×C73—44no—
121C2×Q8×C73—41no—
122C2×Q8×C73—41no—
123C2×Q8×C73C2×D412yes—
124C2×D4×C73—21no—
125C2×D4×C73—22no—
126C2×D4×C73—21no—
127C2×D4×C73C2×D412yes—
128C22×C4×C73—22no—
129C22×C4×C73—22no—
130C22×C4×C73—22no—
131C22×C4×C73—22no—
132C22×C4×C73—22no—
133C22×C4×C73C2×D411yescharacteristic
134C22×C4×C73C2×D411yescharacteristic
135C22×C4×C73C2×D411yescharacteristic
136C22×C4×C73C2×D411yescharacteristic
137C22×C4×C73C2×D411yescharacteristic
138C22×C4×C73C2×D411yescharacteristic
139C22×C4×C73C2411yescharacteristic
140C2×C8×C73—24no—
141C4⋊C4×C73—21no—
142C4⋊C4×C73—22no—
143C4⋊C4×C73—21no—
144C4⋊C4×C73—22no—
145C4⋊C4×C73—22no—
146C4⋊C4×C73—21no—
147C4⋊C4×C73—21no—
148C4⋊C4×C73—22no—
149C4⋊C4×C73—22no—
150C4⋊C4×C73—22no—
151C4⋊C4×C73—22no—
152C4⋊C4×C73—22no—
153C4⋊C4×C73—22no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73—22no—
156C4⋊C4×C73—24no—
157C4⋊C4×C73—42no—
158C4⋊C4×C73—42no—
159C4⋊C4×C73C2×D412yes—
160C4⋊C4×C73C2×D412yes—
161C22⋊C4×C73—22no—
162C22⋊C4×C73—22no—
163C22⋊C4×C73—22no—
164C22⋊C4×C73—22no—
165C22⋊C4×C73—22no—
166C22⋊C4×C73—22no—
167C22⋊C4×C73—22no—
168C22⋊C4×C73—22no—
169C22⋊C4×C73—22no—
170C42×C73—22no—
171C42×C73—22no—
172C42×C73—21no—
173C42×C73—22no—
174C42×C73—21no—
175C42×C73—21no—
176C42×C73—21no—
177C42×C73—42no—
178C42×C73C2×D412yes—
179C42×C73C2×D412yes—
180C23×C73—22no—
181C23×C73C22×D411yescharacteristic
182Q8×C73—44no—
183Q8×C73—42no—
184Q8×C73—42no—
185Q8×C73—82no—
186Q8×C73—81no—
187Q8×C73—82no—
188D4×C73—42no—
189D4×C73—42no—
190D4×C73—42no—
191D4×C73—42no—
192D4×C73—81no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73—21no—
197C2×C4×C73—22no—
198C2×C4×C73—21no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—21no—
202C2×C4×C73—21no—
203C2×C4×C73—22no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—41no—
207C2×C4×C73—44no—
208C2×C4×C73—44no—
209C2×C4×C73—41no—
210C2×C4×C73—41no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—44no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—82no—
217C2×C4×C73#12011yescharacteristic
218C2×C4×C73#12012yes—
219C2×C4×C73#12011yescharacteristic
220C2×C4×C73#13712yes—
221C2×C4×C73C22×D412yes—
222C8×C73—44no—
223C22×C73—21no—
224C22×C73—21no—
225C22×C73—42no—
226C22×C73—42no—
227C22×C73—42no—
228C22×C73C2×#12011yescharacteristic
229C22×C73C2×#13711yescharacteristic
230C22×C73#57511yescharacteristic
231C4×C73—22no—
232C4×C73—22no—
233C4×C73—21no—
234C4×C73—22no—
235C4×C73—41no—
236C4×C73—42no—
237C4×C73—42no—
238C4×C73—41no—
239C4×C73—41no—
240C4×C73—42no—
241C4×C73—82no—
242C4×C73—82no—
243C4×C73#45412yes—
244#18345C7311yesmaximal, characteristic, direct product factor, Sylow
245C2×C73—21no—
246C2×C73—21no—
247C2×C73—82no—
248C2×C73#300711yescharacteristic
249C2×C73C2×#45411yescharacteristic
250C2×C73#302811yescharacteristic
251#3461C2×C7311yescharacteristic
252#3239C2×C7312yescomplemented
253#2568C2×C7311yescharacteristic, complemented
254#1617C2×C7312yescomplemented
255#1616C2×C7312yes—
256#1615C2×C7314yescomplemented
257#1509C2×C7312yescomplemented
258#1467C2×C7311yescharacteristic, complemented
259C73#1834511yescharacteristic, direct product factor, Sylow
260#544C22×C7311yescharacteristic
261#542—22no—
262#541—22no—
263#539—21no—
264#539C22×C7312yes—
265#537C22×C7311yescharacteristic
266#536—21no—
267#533—22no—
268C2×C4⋊Q8C22×C7311yescharacteristic
269#529—22no—
270#529—21no—
271#529C22×C7312yes—
272C4×#41—21no—
273C2×C4×Q8C22×C7311yescharacteristic
274#501—22no—
275#489—22no—
276#484C22×C7314yes—
277#481C22×C7314yes—
278#475—22no—
279#474—22no—
280#432C22×C7312yes—
281#400C22×C7312yes—
282#400C22×C7312yes—
283#391—21no—
284#391—21no—
285#391—21no—
286#382—21no—
287C4×C4⋊C4C22×C7311yescharacteristic
288#328C22×C7314yes—
289#327C22×C7314yes—
290#324C22×C7312yes—
291#324C22×C7312yes—
292C2×#41—21no—
293C22×Q8—21no—
294C22×Q8D4×C7311yescharacteristic
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299#126—22no—
300#126—22no—
301#126—22no—
302#126—22no—
303#125—21no—
304#125—22no—
305#125—22no—
306#125—22no—
307#125D4×C7312yes—
308#124—22no—
309#124—21no—
310#124D4×C7312yes—
311#123—22no—
312#123—22no—
313#123—22no—
314#123—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—21no—
320C22⋊Q8—22no—
321C22⋊Q8C23×C7312yes—
322#121—22no—
323#121—21no—
324#121C23×C7312yes—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8D4×C7312yes—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4D4×C7312yes—
337#117—22no—
338#117—22no—
339#117—21no—
340#117—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4D4×C7311yescharacteristic
344C2×C4⋊C4C23×C7312yes—
345C2×C4⋊C4C23×C7311yescharacteristic
346C2×C4⋊C4C23×C7311yescharacteristic
347C2×C42—22no—
348C2×C42—21no—
349C2×C42D4×C7311yescharacteristic
350C2×C42D4×C7311yescharacteristic
351C2×C42C23×C7311yescharacteristic
352#107—24no—
353#106—24no—
354C4⋊C8—22no—
355#103—24no—
356#102—24no—
357C22⋊C8C23×C7314yes—
358#97—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95C23×C7312yes—
364#41—42no—
365#41—42no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—44no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8C2×D4×C7312yes—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4C2×D4×C7312yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4C2×D4×C7311yescharacteristic
385C22×C4C2×D4×C7311yescharacteristic
386C22×C4C2×D4×C7311yescharacteristic
387C22×C4C2×D4×C7311yescharacteristic
388C22×C4C2×D4×C7311yescharacteristic
389C22×C4C2×D4×C7311yescharacteristic
390C22×C4C24×C7311yescharacteristic
391C2×C8—24no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4C2×D4×C7312yes—
411C4⋊C4C2×D4×C7312yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—22no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—42no—
429C42C2×D4×C7312yes—
430C42C2×D4×C7312yes—
431C23—22no—
432C23C22×D4×C7311yescharacteristic
433Q8—44no—
434Q8—42no—
435Q8—42no—
436Q8—82no—
437Q8—81no—
438Q8—82no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—81no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4#120×C7311yescharacteristic
469C2×C4#120×C7312yes—
470C2×C4#120×C7311yescharacteristic
471C2×C4#137×C7312yes—
472C2×C4C22×D4×C7312yes—
473C8—44no—
474C22—21no—
475C22—21no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22C2×#120×C7311yescharacteristic
480C22C2×#137×C7311yescharacteristic
481C22#575×C7311yescharacteristic
482C4—22no—
483C4—22no—
484C4—21no—
485C4—22no—
486C4—41no—
487C4—42no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—82no—
493C4—82no—
494C4#454×C7312yes—
495C2—21no—
496C2—21no—
497C2—82no—
498C2C73×#300711yescharacteristic
499C2C2×#454×C7311yescharacteristic
500C2C73×#302811yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC22×C4C24×C73
prime radical#537×C73C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#3239C2semidirect
3. C73×#2568C2semidirect
4. C73×#1617C2semidirect
6. C73×#1615C2semidirect
7. C73×#1509C2semidirect
8. C73×#1467C2semidirect
244. #18345C73direct (N × H)
252. #3239C2×C73semidirect
253. #2568C2×C73semidirect
254. #1617C2×C73semidirect
256. #1615C2×C73semidirect
257. #1509C2×C73semidirect
258. #1467C2×C73semidirect
259. C73#18345direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#34612
C73×#32392
C73×#25682
C73×#16172
C73×#16162
C73×#16152
C73×#15092
C73×#14672
#1834573