Finite Group Databasebeta

← C73×#60621 #32970025

Subgroups — C73×#60621 #32970025

Order 18688.

2078 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544×C73C211yesmaximal, characteristic, complemented
2C73×#3449C211yesmaximal, characteristic
3C73×#3142C214yesmaximal, complemented
4C73×#3141C214yesmaximal, complemented
5C73×#3137C218yesmaximal, complemented
6C73×#3136C218yesmaximal, complemented
7C73×#2896C212yesmaximal, complemented
8C2×#417×C73C212yesmaximal, complemented
9C73×#2873C211yesmaximal, characteristic, complemented
10C22×#41×C73C2212yes—
11C22×M16×C73C2212yes—
12#544×C73—24no—
13#544×C73C2218yescomplemented
14#537×C73—21no—
15#537×C73—21no—
16#536×C73C2214yes—
17#535×C73C2214yes—
18#534×C73—22no—
19#532×C73C2214yes—
20C2×C4⋊Q8×C73—21no—
21C2×C4⋊Q8×C73C2212yescomplemented
22C2×#125×C73—21no—
23C2×#125×C73C2212yescomplemented
24#525×C73—24no—
25C2×C22⋊Q8×C73—22no—
26C2×#117×C73C2211yescharacteristic
27C2×#117×C73C2211yescharacteristic
28C22×C4⋊C4×C73C2211yescharacteristic
29#491×C73C2218yescomplemented
30#490×C73C2218yescomplemented
31#489×C73C22116yescomplemented
32#488×C73C22116yescomplemented
33#487×C73C2218yescomplemented
34#486×C73C2218yescomplemented
35#419×C73C2214yescomplemented
36#418×C73C2214yescomplemented
37#417×C73C2218yes—
38#417×C73C2214yescomplemented
39#417×C73C2218yescomplemented
40#417×C73C2214yescomplemented
41#416×C73C2218yes—
42C2×#103×C73C2214yescomplemented
43C2×#102×C73C2214yescomplemented
44#405×C73C2218yescomplemented
45C4×M16×C73C2214yescomplemented
46C2×#41×C73—22no—
47C2×#41×C73—22no—
48C2×#41×C73—28no—
49C2×#41×C73—28no—
50C2×#41×C73C2314yes—
51C2×#41×C73C2314yes—
52C22×Q8×C73—22no—
53C22×Q8×C73C2312yes—
54C22×D4×C73—22no—
55C22×D4×C73C2312yes—
56C23×C4×C73—21no—
57C23×C4×C73—22no—
58C23×C4×C73C2311yescharacteristic
59C2×M16×C73—28no—
60C2×M16×C73C2314yes—
61C2×M16×C73C2314yes—
62C22×C8×C73C2314yes—
63C4⋊Q8×C73—28no—
64C4⋊Q8×C73—24no—
65C4⋊Q8×C73C2314yes—
66C4⋊Q8×C73C2318yes—
67#127×C73C2314yes—
68#125×C73—28no—
69#125×C73—24no—
70#125×C73C2318yes—
71#124×C73C2318yes—
72#123×C73—28no—
73C22⋊Q8×C73—24no—
74C22⋊Q8×C73—24no—
75C22⋊Q8×C73—216no—
76#121×C73—24no—
77#121×C73—24no—
78#121×C73—24no—
79#121×C73—24no—
80#120×C73—28no—
81C4×D4×C73—24no—
82C4×D4×C73—24no—
83#117×C73—22no—
84#117×C73D414yes—
85#117×C73C2314yes—
86#117×C73C2314yes—
87C2×C4⋊C4×C73—21no—
88C2×C4⋊C4×C73—22no—
89C2×C4⋊C4×C73—22no—
90C2×C4⋊C4×C73—22no—
91C2×C4⋊C4×C73—22no—
92C2×C4⋊C4×C73D411yescharacteristic
93C2×C4⋊C4×C73D411yescharacteristic
94C2×C4⋊C4×C73C2314yes—
95C2×C4⋊C4×C73C2312yes—
96C2×C4⋊C4×C73C2311yescharacteristic
97C2×C4⋊C4×C73C2311yescharacteristic
98C2×C22⋊C4×C73—21no—
99C2×C22⋊C4×C73D412yes—
100C2×C42×C73—21no—
101C2×C42×C73C2312yes—
102#103×C73C23116yes—
103#103×C73C2318yes—
104#103×C73C2318yes—
105#102×C73C23116yes—
106#102×C73C2318yes—
107#102×C73C2318yes—
108#97×C73C2318yes—
109C4×C8×C73C2318yes—
110C24×C73—22no—
111C24×C73C2×D411yescharacteristic
112#41×C73—28no—
113#41×C73—28no—
114#41×C73—28no—
115#41×C73—28no—
116#41×C73—48no—
117#41×C73—48no—
118C2×Q8×C73—22no—
119C2×Q8×C73—28no—
120C2×Q8×C73—24no—
121C2×Q8×C73—24no—
122C2×Q8×C73—84no—
123C2×Q8×C73C2414yes—
124C2×D4×C73—24no—
125C2×D4×C73—24no—
126C2×D4×C73—22no—
127C2×D4×C73—24no—
128C2×D4×C73—24no—
129C2×D4×C73—24no—
130C2×D4×C73—24no—
131C2×D4×C73—48no—
132C2×D4×C73C2414yes—
133C22×C4×C73—21no—
134C22×C4×C73—21no—
135C22×C4×C73—22no—
136C22×C4×C73—22no—
137C22×C4×C73—24no—
138C22×C4×C73—24no—
139C22×C4×C73—24no—
140C22×C4×C73—48no—
141C22×C4×C73—82no—
142C22×C4×C73C2×D412yes—
143C22×C4×C73C2×D412yes—
144C22×C4×C73C2×D411yescharacteristic
145C22×C4×C73C2×D412yes—
146C22×C4×C73#4118yes—
147C22×C4×C73C2411yescharacteristic
148C22×C4×C73C2412yes—
149M16×C73—28no—
150M16×C73—28no—
151C2×C8×C73—28no—
152C2×C8×C73C2418yes—
153C4⋊C4×C73—24no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73—28no—
156C4⋊C4×C73—28no—
157C4⋊C4×C73—28no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—24no—
161C4⋊C4×C73—24no—
162C4⋊C4×C73C2×D414yes—
163C4⋊C4×C73C2×D414yes—
164C4⋊C4×C73C2414yes—
165C4⋊C4×C73C2412yes—
166C4⋊C4×C73C2412yes—
167C22⋊C4×C73—24no—
168C22⋊C4×C73—28no—
169C22⋊C4×C73—28no—
170C22⋊C4×C73C2×D418yes—
171C42×C73—24no—
172C42×C73C2×D414yes—
173C42×C73C2414yes—
174C23×C73—24no—
175C23×C73—41no—
176C23×C73—41no—
177C23×C73—44no—
178C23×C73—82no—
179C23×C73—82no—
180C23×C73#12414yes—
181C23×C73C22×D412yes—
182C23×C73C22×D411yescharacteristic
183Q8×C73—24no—
184Q8×C73—24no—
185Q8×C73—44no—
186Q8×C73—88no—
187D4×C73—24no—
188D4×C73—24no—
189D4×C73—48no—
190D4×C73—48no—
191D4×C73—44no—
192D4×C73—84no—
193D4×C73—84no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—24no—
197C2×C4×C73—24no—
198C2×C4×C73—42no—
199C2×C4×C73—42no—
200C2×C4×C73—42no—
201C2×C4×C73—44no—
202C2×C4×C73—42no—
203C2×C4×C73—44no—
204C2×C4×C73—48no—
205C2×C4×C73—48no—
206C2×C4×C73—84no—
207C2×C4×C73#12418yes—
208C2×C4×C73#12414yes—
209C2×C4×C73C22×D414yes—
210C2×C4×C73C22×D414yes—
211C2×C4×C73C22×D412yes—
212C2×C4×C73C22×D411yescharacteristic
213C2×C4×C73C2×#4114yes—
214C2×C4×C73C2×#4114yes—
215C2×C4×C73C2×#4114yes—
216C2×C4×C73C2511yescharacteristic
217C8×C73—28no—
218C22×C73—21no—
219C22×C73—22no—
220C22×C73—21no—
221C22×C73—22no—
222C22×C73—44no—
223C22×C73—44no—
224C22×C73—44no—
225C22×C73—44no—
226C22×C73—88no—
227C22×C73C2×#12414yes—
228C22×C73C2×#12412yes—
229C22×C73C23×D411yescharacteristic
230C4×C73—24no—
231C4×C73—24no—
232C4×C73—44no—
233C4×C73—42no—
234C4×C73—42no—
235C4×C73—44no—
236C4×C73C2×#12412yes—
237C4×C73C2×#12414yes—
238C4×C73C22×#4112yes—
239#60621C7311yesmaximal, characteristic, direct product factor, Sylow
240C2×C73—24no—
241C2×C73—22no—
242C2×C73—84no—
243C2×C73C22×#12411yescharacteristic
244C2×C73#358511yescharacteristic
245C2×C73#358611yescharacteristic
246C2×#544C2×C7311yescharacteristic, complemented
247#3449C2×C7311yescharacteristic
248#3142C2×C7314yescomplemented
249#3141C2×C7314yescomplemented
250#3137C2×C7318yescomplemented
251#3136C2×C7318yescomplemented
252#2896C2×C7312yescomplemented
253C2×#417C2×C7312yescomplemented
254#2873C2×C7311yescharacteristic, complemented
255C73#6062111yescharacteristic, direct product factor, Sylow
256C22×#41C22×C7312yes—
257C22×M16C22×C7312yes—
258#544—24no—
259#544C22×C7318yescomplemented
260#537—21no—
261#537—21no—
262#536C22×C7314yes—
263#535C22×C7314yes—
264#534—22no—
265#532C22×C7314yes—
266C2×C4⋊Q8—21no—
267C2×C4⋊Q8C22×C7312yescomplemented
268C2×#125—21no—
269C2×#125C22×C7312yescomplemented
270#525—24no—
271C2×C22⋊Q8—22no—
272C2×#117C22×C7311yescharacteristic
273C2×#117C22×C7311yescharacteristic
274C22×C4⋊C4C22×C7311yescharacteristic
275#491C22×C7318yescomplemented
276#490C22×C7318yescomplemented
277#489C22×C73116yescomplemented
278#488C22×C73116yescomplemented
279#487C22×C7318yescomplemented
280#486C22×C7318yescomplemented
281#419C22×C7314yescomplemented
282#418C22×C7314yescomplemented
283#417C22×C7318yes—
284#417C22×C7314yescomplemented
285#417C22×C7318yescomplemented
286#417C22×C7314yescomplemented
287#416C22×C7318yes—
288C2×#103C22×C7314yescomplemented
289C2×#102C22×C7314yescomplemented
290#405C22×C7318yescomplemented
291C4×M16C22×C7314yescomplemented
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—28no—
295C2×#41—28no—
296C2×#41C23×C7314yes—
297C2×#41C23×C7314yes—
298C22×Q8—22no—
299C22×Q8C23×C7312yes—
300C22×D4—22no—
301C22×D4C23×C7312yes—
302C23×C4—21no—
303C23×C4—22no—
304C23×C4C23×C7311yescharacteristic
305C2×M16—28no—
306C2×M16C23×C7314yes—
307C2×M16C23×C7314yes—
308C22×C8C23×C7314yes—
309C4⋊Q8—28no—
310C4⋊Q8—24no—
311C4⋊Q8C23×C7314yes—
312C4⋊Q8C23×C7318yes—
313#127C23×C7314yes—
314#125—28no—
315#125—24no—
316#125C23×C7318yes—
317#124C23×C7318yes—
318#123—28no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—216no—
322#121—24no—
323#121—24no—
324#121—24no—
325#121—24no—
326#120—28no—
327C4×D4—24no—
328C4×D4—24no—
329#117—22no—
330#117D4×C7314yes—
331#117C23×C7314yes—
332#117C23×C7314yes—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4D4×C7311yescharacteristic
339C2×C4⋊C4D4×C7311yescharacteristic
340C2×C4⋊C4C23×C7312yes—
341C2×C4⋊C4C23×C7314yes—
342C2×C4⋊C4C23×C7311yescharacteristic
343C2×C4⋊C4C23×C7311yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4D4×C7312yes—
346C2×C42—21no—
347C2×C42C23×C7312yes—
348#103C23×C73116yes—
349#103C23×C7318yes—
350#103C23×C7318yes—
351#102C23×C73116yes—
352#102C23×C7318yes—
353#102C23×C7318yes—
354#97C23×C7318yes—
355C4×C8C23×C7318yes—
356C24—22no—
357C24C2×D4×C7311yescharacteristic
358#41—28no—
359#41—28no—
360#41—28no—
361#41—28no—
362#41—48no—
363#41—48no—
364C2×Q8—22no—
365C2×Q8—28no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—84no—
369C2×Q8C24×C7314yes—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—48no—
378C2×D4C24×C7314yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—48no—
387C22×C4—82no—
388C22×C4C2×D4×C7312yes—
389C22×C4C2×D4×C7312yes—
390C22×C4C2×D4×C7311yescharacteristic
391C22×C4C2×D4×C7312yes—
392C22×C4#41×C7318yes—
393C22×C4C24×C7311yescharacteristic
394C22×C4C24×C7312yes—
395M16—28no—
396M16—28no—
397C2×C8—28no—
398C2×C8C24×C7318yes—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—28no—
402C4⋊C4—28no—
403C4⋊C4—28no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4C2×D4×C7314yes—
409C4⋊C4C2×D4×C7314yes—
410C4⋊C4C24×C7314yes—
411C4⋊C4C24×C7312yes—
412C4⋊C4C24×C7312yes—
413C22⋊C4—24no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4C2×D4×C7318yes—
417C42—24no—
418C42C2×D4×C7314yes—
419C42C24×C7314yes—
420C23—24no—
421C23—41no—
422C23—41no—
423C23—44no—
424C23—82no—
425C23—82no—
426C23#124×C7314yes—
427C23C22×D4×C7312yes—
428C23C22×D4×C7311yescharacteristic
429Q8—24no—
430Q8—24no—
431Q8—44no—
432Q8—88no—
433D4—24no—
434D4—24no—
435D4—48no—
436D4—48no—
437D4—44no—
438D4—84no—
439D4—84no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—84no—
453C2×C4#124×C7318yes—
454C2×C4#124×C7314yes—
455C2×C4C22×D4×C7314yes—
456C2×C4C22×D4×C7314yes—
457C2×C4C22×D4×C7312yes—
458C2×C4C22×D4×C7311yescharacteristic
459C2×C4C2×#41×C7314yes—
460C2×C4C2×#41×C7314yes—
461C2×C4C2×#41×C7314yes—
462C2×C4C25×C7311yescharacteristic
463C8—28no—
464C22—21no—
465C22—22no—
466C22—21no—
467C22—22no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—88no—
473C22C2×#124×C7314yes—
474C22C2×#124×C7312yes—
475C22C23×D4×C7311yescharacteristic
476C4—24no—
477C4—24no—
478C4—44no—
479C4—42no—
480C4—42no—
481C4—44no—
482C4C2×#124×C7312yes—
483C4C2×#124×C7314yes—
484C4C22×#41×C7312yes—
485C2—24no—
486C2—22no—
487C2—84no—
488C2C22×#124×C7311yescharacteristic
489C2C73×#358511yescharacteristic
490C2C73×#358611yescharacteristic
subgroupG/H
center, socleC22×C73C23×D4
commutator, Frattini, cosocleC2×C4C25×C73
prime radicalC73×#3449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×#117×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#544×C73C2semidirect
3. C73×#3142C2semidirect
4. C73×#3141C2semidirect
5. C73×#3137C2semidirect
6. C73×#3136C2semidirect
7. C73×#2896C2semidirect
8. C2×#417×C73C2semidirect
9. C73×#2873C2semidirect
13. #544×C73C22semidirect
21. C2×C4⋊Q8×C73C22semidirect
23. C2×#125×C73C22semidirect
29. #491×C73C22semidirect
30. #490×C73C22semidirect
31. #489×C73C22semidirect
32. #488×C73C22semidirect
33. #487×C73C22semidirect
34. #486×C73C22semidirect
35. #419×C73C22semidirect
36. #418×C73C22semidirect
38. #417×C73C22semidirect
39. #417×C73C22semidirect
40. #417×C73C22semidirect
42. C2×#103×C73C22semidirect
43. C2×#102×C73C22semidirect
44. #405×C73C22semidirect
45. C4×M16×C73C22semidirect
239. #60621C73direct (N × H)
246. C2×#544C2×C73semidirect
248. #3142C2×C73semidirect
249. #3141C2×C73semidirect
250. #3137C2×C73semidirect
251. #3136C2×C73semidirect
252. #2896C2×C73semidirect
253. C2×#417C2×C73semidirect
254. #2873C2×C73semidirect
255. C73#60621direct (N × H)
259. #544C22×C73semidirect
267. C2×C4⋊Q8C22×C73semidirect
269. C2×#125C22×C73semidirect
275. #491C22×C73semidirect
276. #490C22×C73semidirect
277. #489C22×C73semidirect
278. #488C22×C73semidirect
279. #487C22×C73semidirect
280. #486C22×C73semidirect
281. #419C22×C73semidirect
282. #418C22×C73semidirect
284. #417C22×C73semidirect
285. #417C22×C73semidirect
286. #417C22×C73semidirect
288. C2×#103C22×C73semidirect
289. C2×#102C22×C73semidirect
290. #405C22×C73semidirect
291. C4×M16C22×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#544×C732
C73×#34492
C73×#31422
C73×#31412
C73×#31372
C73×#31362
C73×#28962
C2×#417×C732
C73×#28732
#6062173