Finite Group Databasebeta

← C73×#18643 #32928047

Subgroups — C73×#18643 #32928047

Order 18688.

906 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3503C211yesmaximal, characteristic
2C73×#3230C211yesmaximal, characteristic, complemented
3C2×#487×C73C211yesmaximal, characteristic, complemented
4C73×#3079C211yesmaximal, characteristic
5C73×#1637C214yesmaximal, complemented
6C73×#1636C214yesmaximal, complemented
7C73×#1549C211yesmaximal, characteristic, complemented
8C73×#1493C211yesmaximal, characteristic, complemented
9C73×#1476C211yesmaximal, characteristic
10Q82×C73—21no—
11#557×C73C2212yes—
12#554×C73—21no—
13#554×C73—22no—
14D4×Q8×C73—22no—
15D4×Q8×C73C2212yes—
16#544×C73—22no—
17#544×C73—21no—
18#533×C73—21no—
19#533×C73C2211yescharacteristic
20C2×C4⋊Q8×C73—21no—
21C2×C4⋊Q8×C73C2211yescharacteristic
22C2×C4⋊Q8×C73C2211yescharacteristic
23#520×C73—21no—
24C4×#41×C73C2211yescharacteristic
25#500×C73—21no—
26#487×C73—21no—
27#487×C73—22no—
28#487×C73C2214yes—
29#484×C73—22no—
30#477×C73C2214yes—
31#477×C73C2212yes—
32#476×C73C2212yes—
33#475×C73—21no—
34#474×C73—21no—
35#431×C73C2211yescharacteristic
36#428×C73—21no—
37#423×C73C2211yescharacteristic
38#417×C73—21no—
39C2×#103×C73—22no—
40C2×C4×C8×C73C2211yescharacteristic
41#332×C73C2214yes—
42Q8⋊C8×C73C2212yes—
43Q8⋊C8×C73C2214yes—
44D4⋊C8×C73C2212yes—
45C2×#41×C73D411yescharacteristic
46C22×Q8×C73—22no—
47C22×Q8×C73—21no—
48C22×Q8×C73—21no—
49C2×M16×C73—21no—
50C22×C8×C73—21no—
51C4⋊Q8×C73—22no—
52C4⋊Q8×C73—22no—
53C4⋊Q8×C73—24no—
54C4⋊Q8×C73—22no—
55C4⋊Q8×C73—22no—
56C4⋊Q8×C73—21no—
57C4⋊Q8×C73—21no—
58C4⋊Q8×C73—21no—
59C4⋊Q8×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73D411yescharacteristic
62C4⋊Q8×C73D412yes—
63C4⋊Q8×C73D411yescharacteristic
64C4⋊Q8×C73C2312yes—
65C4⋊Q8×C73C2312yes—
66#125×C73—22no—
67#125×C73—24no—
68#125×C73—22no—
69#125×C73—21no—
70C22⋊Q8×C73—22no—
71C22⋊Q8×C73—24no—
72C22⋊Q8×C73—24no—
73C22⋊Q8×C73—21no—
74C22⋊Q8×C73—22no—
75C22⋊Q8×C73—24no—
76C22⋊Q8×C73D412yes—
77C4×Q8×C73—22no—
78C4×Q8×C73—22no—
79C4×Q8×C73—22no—
80C4×Q8×C73—21no—
81C4×Q8×C73D411yescharacteristic
82C4×Q8×C73C2311yescharacteristic
83C4×Q8×C73C2312yes—
84C4×D4×C73—24no—
85C4×D4×C73—22no—
86C4×D4×C73—21no—
87C4×D4×C73—21no—
88C4×D4×C73D411yescharacteristic
89C4×D4×C73C2311yescharacteristic
90#117×C73—22no—
91#117×C73—21no—
92#117×C73—21no—
93#117×C73D411yescharacteristic
94C2×C4⋊C4×C73—22no—
95C2×C4⋊C4×C73—22no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73D412yes—
100C2×C42×C73—21no—
101C2×C42×C73C2311yescharacteristic
102#107×C73—22no—
103#107×C73—24no—
104#106×C73—22no—
105C4⋊C8×C73—21no—
106C4⋊C8×C73C2312yes—
107C4⋊C8×C73C2312yes—
108#103×C73—22no—
109#103×C73—24no—
110#103×C73—28no—
111#102×C73—22no—
112C22⋊C8×C73—21no—
113C4×C8×C73—21no—
114C4×C8×C73—21no—
115C4×C8×C73C2312yes—
116#41×C73—22no—
117#41×C73Q1614yes—
118C2×Q8×C73—24no—
119C2×Q8×C73—21no—
120C2×Q8×C73—22no—
121C2×Q8×C73—22no—
122C2×Q8×C73—22no—
123C2×Q8×C73—44no—
124C2×Q8×C73—41no—
125C2×Q8×C73—82no—
126C2×Q8×C73—84no—
127C2×Q8×C73C2×D412yes—
128C2×Q8×C73C2×D411yescharacteristic
129C2×D4×C73—21no—
130C2×D4×C73C2×D411yescharacteristic
131C22×C4×C73—21no—
132C22×C4×C73—24no—
133C22×C4×C73—22no—
134C22×C4×C73—21no—
135C22×C4×C73—21no—
136C22×C4×C73—41no—
137C22×C4×C73C2×D411yescharacteristic
138C22×C4×C73C2×D411yescharacteristic
139C22×C4×C73C2×D411yescharacteristic
140M16×C73—42no—
141C2×C8×C73—22no—
142C2×C8×C73—22no—
143C2×C8×C73—22no—
144C2×C8×C73—42no—
145C4⋊C4×C73—24no—
146C4⋊C4×C73—24no—
147C4⋊C4×C73—24no—
148C4⋊C4×C73—24no—
149C4⋊C4×C73—21no—
150C4⋊C4×C73—22no—
151C4⋊C4×C73—21no—
152C4⋊C4×C73—22no—
153C4⋊C4×C73—21no—
154C4⋊C4×C73—21no—
155C4⋊C4×C73—22no—
156C4⋊C4×C73—22no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—22no—
159C4⋊C4×C73—22no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73—41no—
162C4⋊C4×C73—41no—
163C4⋊C4×C73C2×D414yes—
164C4⋊C4×C73C2×D411yescharacteristic
165C4⋊C4×C73C2×D411yescharacteristic
166C4⋊C4×C73C2×D412yes—
167C22⋊C4×C73—24no—
168C22⋊C4×C73—22no—
169C22⋊C4×C73—21no—
170C22⋊C4×C73—21no—
171C22⋊C4×C73—21no—
172C42×C73—24no—
173C42×C73—21no—
174C42×C73—21no—
175C42×C73—21no—
176C42×C73—22no—
177C42×C73C2×D411yescharacteristic
178C42×C73C2×D411yescharacteristic
179C42×C73C2×D411yescharacteristic
180C42×C73C2411yescharacteristic
181C23×C73—21no—
182C23×C73#12011yescharacteristic
183Q8×C73—21no—
184Q8×C73—24no—
185Q8×C73—44no—
186Q8×C73—44no—
187Q8×C73—42no—
188Q8×C73—82no—
189Q8×C73—84no—
190Q8×C73—82no—
191Q8×C73C2×Q1612yes—
192D4×C73—24no—
193D4×C73—21no—
194D4×C73C2×Q1612yes—
195C2×C4×C73—21no—
196C2×C4×C73—22no—
197C2×C4×C73—24no—
198C2×C4×C73—22no—
199C2×C4×C73—22no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—22no—
203C2×C4×C73—24no—
204C2×C4×C73—42no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73—82no—
208C2×C4×C73—84no—
209C2×C4×C73—82no—
210C2×C4×C73#12011yescharacteristic
211C2×C4×C73#12011yescharacteristic
212C2×C4×C73#12011yescharacteristic
213C2×C4×C73C2×Q1612yes—
214C2×C4×C73#13712yes—
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C22×D411yescharacteristic
217C2×C4×C73C22×D411yescharacteristic
218C8×C73—42no—
219C8×C73—42no—
220C8×C73—42no—
221C22×C73—21no—
222C22×C73—21no—
223C22×C73—22no—
224C22×C73—41no—
225C22×C73—41no—
226C22×C73#45112yes—
227C22×C73C2×#12011yescharacteristic
228C4×C73—22no—
229C4×C73—21no—
230C4×C73—21no—
231C4×C73—41no—
232C4×C73—42no—
233C4×C73—44no—
234C4×C73—41no—
235C4×C73—42no—
236C4×C73—82no—
237C4×C73#45112yes—
238C4×C73#45412yes—
239C4×C73C22×Q1611yescharacteristic
240C4×C73C2×#13711yescharacteristic
241C4×C73#57712yes—
242#18643C7311yesmaximal, characteristic, direct product factor, Sylow
243C2×C73—22no—
244C2×C73—42no—
245C2×C73C2×#45111yescharacteristic
246C2×C73#300911yescharacteristic
247C2×C73C2×#45411yescharacteristic
248#3503C2×C7311yescharacteristic
249#3230C2×C7311yescharacteristic, complemented
250C2×#487C2×C7311yescharacteristic, complemented
251#3079C2×C7311yescharacteristic
252#1637C2×C7314yescomplemented
253#1636C2×C7314yescomplemented
254#1549C2×C7311yescharacteristic, complemented
255#1493C2×C7311yescharacteristic, complemented
256#1476C2×C7311yescharacteristic
257C73#1864311yescharacteristic, direct product factor, Sylow
258Q82—21no—
259#557C22×C7312yes—
260#554—21no—
261#554—22no—
262D4×Q8—22no—
263D4×Q8C22×C7312yes—
264#544—22no—
265#544—21no—
266#533—21no—
267#533C22×C7311yescharacteristic
268C2×C4⋊Q8—21no—
269C2×C4⋊Q8C22×C7311yescharacteristic
270C2×C4⋊Q8C22×C7311yescharacteristic
271#520—21no—
272C4×#41C22×C7311yescharacteristic
273#500—21no—
274#487—22no—
275#487—21no—
276#487C22×C7314yes—
277#484—22no—
278#477C22×C7312yes—
279#477C22×C7314yes—
280#476C22×C7312yes—
281#475—21no—
282#474—21no—
283#431C22×C7311yescharacteristic
284#428—21no—
285#423C22×C7311yescharacteristic
286#417—21no—
287C2×#103—22no—
288C2×C4×C8C22×C7311yescharacteristic
289#332C22×C7314yes—
290Q8⋊C8C22×C7312yes—
291Q8⋊C8C22×C7314yes—
292D4⋊C8C22×C7312yes—
293C2×#41D4×C7311yescharacteristic
294C22×Q8—22no—
295C22×Q8—21no—
296C22×Q8—21no—
297C2×M16—21no—
298C22×C8—21no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—24no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8D4×C7311yescharacteristic
310C4⋊Q8D4×C7312yes—
311C4⋊Q8D4×C7311yescharacteristic
312C4⋊Q8C23×C7312yes—
313C4⋊Q8C23×C7312yes—
314#125—22no—
315#125—24no—
316#125—22no—
317#125—21no—
318C22⋊Q8—22no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—21no—
322C22⋊Q8—22no—
323C22⋊Q8—24no—
324C22⋊Q8D4×C7312yes—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—21no—
329C4×Q8D4×C7311yescharacteristic
330C4×Q8C23×C7311yescharacteristic
331C4×Q8C23×C7312yes—
332C4×D4—24no—
333C4×D4—22no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4D4×C7311yescharacteristic
337C4×D4C23×C7311yescharacteristic
338#117—22no—
339#117—21no—
340#117—21no—
341#117D4×C7311yescharacteristic
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4D4×C7312yes—
348C2×C42—21no—
349C2×C42C23×C7311yescharacteristic
350#107—22no—
351#107—24no—
352#106—22no—
353C4⋊C8—21no—
354C4⋊C8C23×C7312yes—
355C4⋊C8C23×C7312yes—
356#103—22no—
357#103—24no—
358#103—28no—
359#102—22no—
360C22⋊C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8C23×C7312yes—
364#41—22no—
365#41Q16×C7314yes—
366C2×Q8—24no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—44no—
372C2×Q8—41no—
373C2×Q8—82no—
374C2×Q8—84no—
375C2×Q8C2×D4×C7312yes—
376C2×Q8C2×D4×C7311yescharacteristic
377C2×D4—21no—
378C2×D4C2×D4×C7311yescharacteristic
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4C2×D4×C7311yescharacteristic
386C22×C4C2×D4×C7311yescharacteristic
387C22×C4C2×D4×C7311yescharacteristic
388M16—42no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—42no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4C2×D4×C7314yes—
412C4⋊C4C2×D4×C7311yescharacteristic
413C4⋊C4C2×D4×C7311yescharacteristic
414C4⋊C4C2×D4×C7312yes—
415C22⋊C4—24no—
416C22⋊C4—22no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C42—24no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—22no—
425C42C2×D4×C7311yescharacteristic
426C42C2×D4×C7311yescharacteristic
427C42C2×D4×C7311yescharacteristic
428C42C24×C7311yescharacteristic
429C23—21no—
430C23#120×C7311yescharacteristic
431Q8—21no—
432Q8—24no—
433Q8—44no—
434Q8—42no—
435Q8—44no—
436Q8—82no—
437Q8—84no—
438Q8—82no—
439Q8C2×Q16×C7312yes—
440D4—24no—
441D4—21no—
442D4C2×Q16×C7312yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—84no—
457C2×C4—82no—
458C2×C4#120×C7311yescharacteristic
459C2×C4#120×C7311yescharacteristic
460C2×C4#120×C7311yescharacteristic
461C2×C4C2×Q16×C7312yes—
462C2×C4#137×C7312yes—
463C2×C4C22×D4×C7311yescharacteristic
464C2×C4C22×D4×C7311yescharacteristic
465C2×C4C22×D4×C7311yescharacteristic
466C8—42no—
467C8—42no—
468C8—42no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22#451×C7312yes—
475C22C2×#120×C7311yescharacteristic
476C4—22no—
477C4—21no—
478C4—21no—
479C4—41no—
480C4—42no—
481C4—44no—
482C4—41no—
483C4—42no—
484C4—82no—
485C4#451×C7312yes—
486C4#454×C7312yes—
487C4C22×Q16×C7311yescharacteristic
488C4C2×#137×C7311yescharacteristic
489C4#577×C7312yes—
490C2—22no—
491C2—42no—
492C2C2×#451×C7311yescharacteristic
493C2C73×#300911yescharacteristic
494C2C2×#454×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radicalC2×#41×C73D4
LCSG ▷ C42 ▷ 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×#3230C2semidirect
3. C2×#487×C73C2semidirect
5. C73×#1637C2semidirect
6. C73×#1636C2semidirect
7. C73×#1549C2semidirect
8. C73×#1493C2semidirect
242. #18643C73direct (N × H)
249. #3230C2×C73semidirect
250. C2×#487C2×C73semidirect
252. #1637C2×C73semidirect
253. #1636C2×C73semidirect
254. #1549C2×C73semidirect
255. #1493C2×C73semidirect
257. C73#18643direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#35032
C73×#32302
C2×#487×C732
C73×#30792
C73×#16372
C73×#16362
C73×#15492
C73×#14932
C73×#14762
#1864373