Finite Group Databasebeta

← C73×#25772 #32935176

Subgroups — C73×#25772 #32935176

Order 18688.

1618 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
1C73×#3496C211yesmaximal, characteristic
2C2×#458×C73C214yesmaximal, complemented
3C73×#2501C212yesmaximal, complemented
4C73×#2486C211yesmaximal, characteristic, complemented
5C73×#2022C214yesmaximal, complemented
6C73×#1964C212yesmaximal, complemented
7C4×#99×C73C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C73C2212yes—
9#547×C73—22no—
10#546×C73—24no—
11D42×C73—22no—
12#537×C73—21no—
13#534×C73—21no—
14#534×C73C2212yes—
15#532×C73—21no—
16C2×#123×C73C2212yes—
17C2×#123×C73C2218yescomplemented
18C2×#121×C73—21no—
19C2×#121×C73C2212yes—
20C2×C4×D4×C73C2211yescharacteristic
21C22×C4⋊C4×C73C2211yescharacteristic, complemented
22#458×C73—216no—
23C2×#99×C73C2212yes—
24C2×#99×C73C2218yescomplemented
25#399×C73—24no—
26#398×C73—21no—
27#397×C73—21no—
28#390×C73—22no—
29#389×C73—21no—
30#384×C73—22no—
31#381×C73C2212yescomplemented
32#380×C73—21no—
33#380×C73C2211yescharacteristic, complemented
34C4×C22⋊C4×C73C2212yescomplemented
35#342×C73C2212yescomplemented
36H(5,𝔽2)×C73—28no—
37H(5,𝔽2)×C73—44no—
38C2×#41×C73—22no—
39C2×#41×C73—24no—
40C22×D4×C73—22no—
41C22×D4×C73—24no—
42C22×D4×C73—21no—
43C22×D4×C73D412yes—
44C22×D4×C73C2314yes—
45C22×D4×C73C2311yescharacteristic
46C23×C4×C73D411yescharacteristic
47C23×C4×C73C2312yes—
48C23×C4×C73C2311yescharacteristic
49C4⋊Q8×C73—21no—
50#127×C73—21no—
51#124×C73—22no—
52#123×C73—24no—
53#123×C73—24no—
54#123×C73—28no—
55#123×C73—216no—
56#123×C73—48no—
57C22⋊Q8×C73—24no—
58#121×C73—22no—
59#121×C73—22no—
60#121×C73—24no—
61#121×C73—22no—
62#121×C73—22no—
63#121×C73—22no—
64#121×C73—22no—
65#121×C73—22no—
66#121×C73—22no—
67#120×C73—24no—
68#120×C73—24no—
69C4×D4×C73—22no—
70C4×D4×C73—22no—
71C4×D4×C73—22no—
72C4×D4×C73—22no—
73C2×C4⋊C4×C73—21no—
74C2×C4⋊C4×C73—21no—
75C2×C4⋊C4×C73—21no—
76C2×C4⋊C4×C73—22no—
77C2×C4⋊C4×C73—28no—
78C2×C4⋊C4×C73—22no—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—22no—
81C2×C4⋊C4×C73D412yes—
82C2×C4⋊C4×C73D411yescharacteristic
83C2×C22⋊C4×C73—22no—
84C2×C22⋊C4×C73—22no—
85C2×C22⋊C4×C73—28no—
86C2×C22⋊C4×C73D411yescharacteristic
87C2×C22⋊C4×C73D414yes—
88C2×C22⋊C4×C73C2312yes—
89C2×C22⋊C4×C73C2311yescharacteristic
90C2×C22⋊C4×C73C2314yes—
91C2×C42×C73—22no—
92C2×C42×C73D411yescharacteristic
93#99×C73—216no—
94#99×C73D418yescomplemented
95#95×C73—22no—
96#95×C73—22no—
97#95×C73—21no—
98#95×C73—22no—
99C24×C73—22no—
100C24×C73C2×D414yes—
101C24×C73C2×D411yescharacteristic
102C24×C73C2411yescharacteristic
103#41×C73—416no—
104#41×C73—44no—
105#41×C73—44no—
106C2×Q8×C73—22no—
107C2×D4×C73—22no—
108C2×D4×C73—22no—
109C2×D4×C73—24no—
110C2×D4×C73—22no—
111C2×D4×C73—28no—
112C2×D4×C73—24no—
113C2×D4×C73—24no—
114C2×D4×C73—28no—
115C2×D4×C73—22no—
116C2×D4×C73—22no—
117C2×D4×C73—22no—
118C2×D4×C73—22no—
119C2×D4×C73—44no—
120C2×D4×C73—416no—
121C2×D4×C73—44no—
122C2×D4×C73—44no—
123C2×D4×C73—82no—
124C2×D4×C73C2×D414yes—
125C22×C4×C73—22no—
126C22×C4×C73—24no—
127C22×C4×C73—24no—
128C22×C4×C73—21no—
129C22×C4×C73—21no—
130C22×C4×C73—22no—
131C22×C4×C73—22no—
132C22×C4×C73—24no—
133C22×C4×C73—24no—
134C22×C4×C73—24no—
135C22×C4×C73—42no—
136C22×C4×C73—42no—
137C22×C4×C73C2×D414yes—
138C22×C4×C73C2×D412yes—
139C22×C4×C73C2×D411yescharacteristic
140C22×C4×C73C2×D411yescharacteristic
141C22×C4×C73C2×D411yescharacteristic
142C22×C4×C73C2×D411yescharacteristic
143C22×C4×C73C2×D412yes—
144C22×C4×C73C2×D411yescharacteristic
145C4⋊C4×C73—22no—
146C4⋊C4×C73—22no—
147C4⋊C4×C73—22no—
148C4⋊C4×C73—22no—
149C4⋊C4×C73—22no—
150C4⋊C4×C73—416no—
151C4⋊C4×C73—42no—
152C4⋊C4×C73—41no—
153C4⋊C4×C73—42no—
154C4⋊C4×C73—41no—
155C4⋊C4×C73—44no—
156C22⋊C4×C73—24no—
157C22⋊C4×C73—24no—
158C22⋊C4×C73—22no—
159C22⋊C4×C73—24no—
160C22⋊C4×C73—24no—
161C22⋊C4×C73—22no—
162C22⋊C4×C73—24no—
163C22⋊C4×C73—24no—
164C22⋊C4×C73—416no—
165C22⋊C4×C73—44no—
166C22⋊C4×C73C2×D418yescomplemented
167C42×C73—21no—
168C42×C73—21no—
169C42×C73—44no—
170C23×C73—21no—
171C23×C73—21no—
172C23×C73—24no—
173C23×C73—28no—
174C23×C73—41no—
175C23×C73—41no—
176C23×C73—41no—
177C23×C73—41no—
178C23×C73—44no—
179C23×C73—44no—
180C23×C73—44no—
181C23×C73—82no—
182C23×C73—82no—
183C23×C73#12712yes—
184C23×C73C22×D411yescharacteristic
185C23×C73C22×D412yes—
186C23×C73C22×D412yes—
187C23×C73H(5,𝔽2)12yes—
188Q8×C73—44no—
189D4×C73—28no—
190D4×C73—44no—
191D4×C73—48no—
192D4×C73—44no—
193D4×C73—44no—
194D4×C73—48no—
195D4×C73—44no—
196D4×C73—82no—
197D4×C73—82no—
198D4×C73—82no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—24no—
202C2×C4×C73—22no—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—24no—
206C2×C4×C73—28no—
207C2×C4×C73—28no—
208C2×C4×C73—42no—
209C2×C4×C73—41no—
210C2×C4×C73—48no—
211C2×C4×C73—44no—
212C2×C4×C73—44no—
213C2×C4×C73—44no—
214C2×C4×C73—42no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—48no—
224C2×C4×C73—81no—
225C2×C4×C73#12011yescharacteristic
226C2×C4×C73#12012yes—
227C2×C4×C73#12011yescharacteristic
228C2×C4×C73#12712yes—
229C22×C73—21no—
230C22×C73—22no—
231C22×C73—21no—
232C22×C73—22no—
233C22×C73—22no—
234C22×C73—42no—
235C22×C73—42no—
236C22×C73—42no—
237C22×C73—41no—
238C22×C73—41no—
239C22×C73—44no—
240C22×C73—44no—
241C22×C73—44no—
242C22×C73—81no—
243C22×C73—88no—
244C22×C73—164no—
245C22×C73C2×#12011yescharacteristic
246C22×C73C2×#12711yescharacteristic
247C22×C73#53511yescharacteristic
248C22×C73D4214yes—
249C4×C73—22no—
250C4×C73—21no—
251C4×C73—22no—
252C4×C73—48no—
253C4×C73—44no—
254C4×C73—41no—
255C4×C73—41no—
256C4×C73—82no—
257C4×C73—82no—
258C4×C73#45812yescomplemented
259#25772C7311yesmaximal, characteristic, direct product factor, Sylow
260C2×C73—24no—
261C2×C73—21no—
262C2×C73—21no—
263C2×C73—42no—
264C2×C73—84no—
265C2×C73#243311yescharacteristic
266C2×C73C2×#45811yescharacteristic
267C2×C73#302811yescharacteristic
268#3496C2×C7311yescharacteristic
269C2×#458C2×C7314yescomplemented
270#2501C2×C7312yescomplemented
271#2486C2×C7311yescharacteristic, complemented
272#2022C2×C7314yescomplemented
273#1964C2×C7312yescomplemented
274C4×#99C2×C7311yescharacteristic, complemented
275C73#2577211yescharacteristic, direct product factor, Sylow
276C2×H(5,𝔽2)C22×C7312yes—
277#547—22no—
278#546—24no—
279D42—22no—
280#537—21no—
281#534—21no—
282#534C22×C7312yes—
283#532—21no—
284C2×#123C22×C7312yes—
285C2×#123C22×C7318yescomplemented
286C2×#121—21no—
287C2×#121C22×C7312yes—
288C2×C4×D4C22×C7311yescharacteristic
289C22×C4⋊C4C22×C7311yescharacteristic, complemented
290#458—216no—
291C2×#99C22×C7312yes—
292C2×#99C22×C7318yescomplemented
293#399—24no—
294#398—21no—
295#397—21no—
296#390—22no—
297#389—21no—
298#384—22no—
299#381C22×C7312yescomplemented
300#380—21no—
301#380C22×C7311yescharacteristic, complemented
302C4×C22⋊C4C22×C7312yescomplemented
303#342C22×C7312yescomplemented
304H(5,𝔽2)—28no—
305H(5,𝔽2)—44no—
306C2×#41—22no—
307C2×#41—24no—
308C22×D4—22no—
309C22×D4—24no—
310C22×D4—21no—
311C22×D4D4×C7312yes—
312C22×D4C23×C7314yes—
313C22×D4C23×C7311yescharacteristic
314C23×C4D4×C7311yescharacteristic
315C23×C4C23×C7312yes—
316C23×C4C23×C7311yescharacteristic
317C4⋊Q8—21no—
318#127—21no—
319#124—22no—
320#123—24no—
321#123—24no—
322#123—28no—
323#123—216no—
324#123—48no—
325C22⋊Q8—24no—
326#121—22no—
327#121—22no—
328#121—24no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—22no—
333#121—22no—
334#121—22no—
335#120—24no—
336#120—24no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4D4×C7311yescharacteristic
350C2×C4⋊C4D4×C7312yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—28no—
354C2×C22⋊C4D4×C7311yescharacteristic
355C2×C22⋊C4D4×C7314yes—
356C2×C22⋊C4C23×C7312yes—
357C2×C22⋊C4C23×C7311yescharacteristic
358C2×C22⋊C4C23×C7314yes—
359C2×C42—22no—
360C2×C42D4×C7311yescharacteristic
361#99—216no—
362#99D4×C7318yescomplemented
363#95—22no—
364#95—22no—
365#95—21no—
366#95—22no—
367C24—22no—
368C24C2×D4×C7314yes—
369C24C2×D4×C7311yescharacteristic
370C24C24×C7311yescharacteristic
371#41—416no—
372#41—44no—
373#41—44no—
374C2×Q8—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—28no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—28no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—44no—
388C2×D4—416no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—82no—
392C2×D4C2×D4×C7314yes—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C2×D4×C7314yes—
406C22×C4C2×D4×C7312yes—
407C22×C4C2×D4×C7311yescharacteristic
408C22×C4C2×D4×C7311yescharacteristic
409C22×C4C2×D4×C7311yescharacteristic
410C22×C4C2×D4×C7311yescharacteristic
411C22×C4C2×D4×C7312yes—
412C22×C4C2×D4×C7311yescharacteristic
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—416no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—44no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—22no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—416no—
433C22⋊C4—44no—
434C22⋊C4C2×D4×C7318yescomplemented
435C42—21no—
436C42—21no—
437C42—44no—
438C23—21no—
439C23—21no—
440C23—24no—
441C23—28no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—82no—
450C23—82no—
451C23#127×C7312yes—
452C23C22×D4×C7311yescharacteristic
453C23C22×D4×C7312yes—
454C23C22×D4×C7312yes—
455C23H(5,𝔽2)×C7312yes—
456Q8—44no—
457D4—28no—
458D4—44no—
459D4—48no—
460D4—44no—
461D4—44no—
462D4—48no—
463D4—44no—
464D4—82no—
465D4—82no—
466D4—82no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—48no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—48no—
492C2×C4—81no—
493C2×C4#120×C7311yescharacteristic
494C2×C4#120×C7312yes—
495C2×C4#120×C7311yescharacteristic
496C2×C4#127×C7312yes—
497C22—21no—
498C22—22no—
499C22—21no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—81no—
511C22—88no—
512C22—164no—
513C22C2×#120×C7311yescharacteristic
514C22C2×#127×C7311yescharacteristic
515C22#535×C7311yescharacteristic
516C22D42×C7314yes—
517C4—21no—
518C4—22no—
519C4—22no—
520C4—48no—
521C4—41no—
522C4—41no—
523C4—44no—
524C4—82no—
525C4—82no—
526C4#458×C7312yescomplemented
527C2—24no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—84no—
532C2C73×#243311yescharacteristic
533C2C2×#458×C7311yescharacteristic
534C2C73×#302811yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC24C24×C73
prime radicalC73×#3496C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#458×C73C2semidirect
3. C73×#2501C2semidirect
4. C73×#2486C2semidirect
5. C73×#2022C2semidirect
6. C73×#1964C2semidirect
7. C4×#99×C73C2semidirect
17. C2×#123×C73C22semidirect
21. C22×C4⋊C4×C73C22semidirect
24. C2×#99×C73C22semidirect
31. #381×C73C22semidirect
33. #380×C73C22semidirect
34. C4×C22⋊C4×C73C22semidirect
35. #342×C73C22semidirect
94. #99×C73D4semidirect
166. C22⋊C4×C73C2×D4semidirect
258. C4×C73#458semidirect
259. #25772C73direct (N × H)
269. C2×#458C2×C73semidirect
270. #2501C2×C73semidirect
271. #2486C2×C73semidirect
272. #2022C2×C73semidirect
273. #1964C2×C73semidirect
274. C4×#99C2×C73semidirect
275. C73#25772direct (N × H)
285. C2×#123C22×C73semidirect
289. C22×C4⋊C4C22×C73semidirect
292. C2×#99C22×C73semidirect
299. #381C22×C73semidirect
301. #380C22×C73semidirect
302. C4×C22⋊C4C22×C73semidirect
303. #342C22×C73semidirect
362. #99D4×C73semidirect
434. C22⋊C4C2×D4×C73semidirect
526. C4#458×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#34962
C2×#458×C732
C73×#25012
C73×#24862
C73×#20222
C73×#19642
C4×#99×C732
#2577273