Finite Group Databasebeta

← C73×#33823 #32943227

Subgroups — C73×#33823 #32943227

Order 18688.

500 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×#3176C211yesmaximal, characteristic, complemented
2C73×#3175C211yesmaximal, characteristic, complemented
3C73×#2229C211yesmaximal, characteristic, complemented
4C73×#2228C211yesmaximal, characteristic, complemented
5C73×#2226C211yesmaximal, characteristic, complemented
6C73×#2224C211yesmaximal, characteristic, complemented
7C73×#2220C211yesmaximal, characteristic, complemented
8C73×#2218C211yesmaximal, characteristic, complemented
9C73×#2217C211yesmaximal, characteristic, complemented
10C73×#2215C211yesmaximal, characteristic, complemented
11C73×#2210C211yesmaximal, characteristic, complemented
12C73×#2210C211yesmaximal, characteristic, complemented
13C73×#2207C211yesmaximal, characteristic, complemented
14C73×#2206C211yesmaximal, characteristic, complemented
15C73×#2151C211yesmaximal, characteristic, complemented
16#542×C73—21no—
17#540×C73—21no—
18C2×SD32×C73C2211yescharacteristic
19C2×SD32×C73C2211yescharacteristic
20C2×D16×C73C2211yescharacteristic
21C2×D16×C73C2211yescharacteristic
22C2×M32×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24#500×C73C2211yescharacteristic
25#499×C73C2211yescharacteristic
26#495×C73C2211yescharacteristic
27#493×C73C2211yescharacteristic
28#490×C73—21no—
29#489×C73—21no—
30#469×C73—21no—
31#468×C73—21no—
32#467×C73C2211yescharacteristic
33#466×C73C2211yescharacteristic
34#466×C73C2211yescharacteristic
35#466×C73C2211yescharacteristic
36#442×C73—21no—
37#440×C73—21no—
38C4×Q16×C73C2211yescharacteristic
39C4×D8×C73C2211yescharacteristic
40C4×D8×C73C2211yescharacteristic
41C4×D8×C73C2211yescharacteristic
42#405×C73C2211yescharacteristic
43#367×C73C2211yescharacteristic
44#367×C73C2211yescharacteristic
45#366×C73C2211yescharacteristic
46#366×C73C2211yescharacteristic
47C4⋊C16×C73C2211yescharacteristic
48C4⋊C16×C73C2211yescharacteristic
49C4⋊C16×C73C2211yescharacteristic
50C4⋊C16×C73C2211yescharacteristic
51#358×C73C2211yescharacteristic
52#358×C73C2211yescharacteristic
53#357×C73C2211yescharacteristic
54#357×C73C2211yescharacteristic
55#357×C73C2211yescharacteristic
56#357×C73C2211yescharacteristic
57#357×C73C2211yescharacteristic
58#357×C73C2211yescharacteristic
59C2×Q16×C73C2311yescharacteristic
60C2×SD16×C73—21no—
61C2×D8×C73—21no—
62C2×D8×C73C2311yescharacteristic
63C2×D8×C73C2311yescharacteristic
64C2×D8×C73C2311yescharacteristic
65C2×M16×C73D411yescharacteristic
66C22×C8×C73C2311yescharacteristic
67C4⋊Q8×C73—21no—
68#127×C73—21no—
69#126×C73—41no—
70#125×C73—21no—
71#124×C73—21no—
72#124×C73—41no—
73#123×C73—41no—
74#123×C73—41no—
75#123×C73—41no—
76C22⋊Q8×C73—21no—
77C22⋊Q8×C73—41no—
78#121×C73—21no—
79#121×C73—21no—
80#121×C73—21no—
81#121×C73—41no—
82#121×C73—41no—
83C4×Q8×C73—21no—
84C4×D4×C73—21no—
85C4×D4×C73—21no—
86C4×D4×C73—21no—
87#117×C73D411yescharacteristic
88SD32×C73—41no—
89SD32×C73—41no—
90D16×C73—41no—
91D16×C73—41no—
92M32×C73—41no—
93C2×C16×C73—41no—
94C2×C16×C73C2311yescharacteristic
95C2×C16×C73C2311yescharacteristic
96C2×C16×C73C2311yescharacteristic
97C2×C16×C73C2311yescharacteristic
98#107×C73—21no—
99#107×C73C2311yescharacteristic
100#107×C73C2311yescharacteristic
101#107×C73C2311yescharacteristic
102#107×C73C2311yescharacteristic
103#106×C73—21no—
104#103×C73—21no—
105#103×C73—21no—
106#102×C73—21no—
107#102×C73—21no—
108#102×C73—21no—
109#102×C73—21no—
110#102×C73—21no—
111#102×C73—21no—
112#97×C73D411yescharacteristic
113#97×C73D411yescharacteristic
114C4×C8×C73C2311yescharacteristic
115C4×C8×C73C2311yescharacteristic
116C2×Q8×C73—21no—
117C2×Q8×C73—41no—
118C2×D4×C73—21no—
119C2×D4×C73—21no—
120C2×D4×C73—21no—
121C2×D4×C73—41no—
122C2×D4×C73—41no—
123C2×D4×C73—41no—
124C2×D4×C73—41no—
125C2×D4×C73—41no—
126C22×C4×C73—41no—
127C22×C4×C73—41no—
128C22×C4×C73—41no—
129C22×C4×C73C2×D411yescharacteristic
130Q16×C73—21no—
131Q16×C73—21no—
132SD16×C73—42no—
133D8×C73—21no—
134D8×C73—21no—
135D8×C73—21no—
136D8×C73—21no—
137D8×C73—21no—
138D8×C73—21no—
139D8×C73—42no—
140M16×C73—41no—
141C2×C8×C73—21no—
142C2×C8×C73—21no—
143C2×C8×C73C2×D411yescharacteristic
144C2×C8×C73C2×D411yescharacteristic
145C2×C8×C73C2×D411yescharacteristic
146C2×C8×C73C2411yescharacteristic
147C4⋊C4×C73—21no—
148C4⋊C4×C73—21no—
149C4⋊C4×C73—21no—
150C4⋊C4×C73—21no—
151C4⋊C4×C73—41no—
152C4⋊C4×C73—41no—
153C4⋊C4×C73—41no—
154C4⋊C4×C73—41no—
155C4⋊C4×C73—41no—
156C4⋊C4×C73D811yescharacteristic
157C4⋊C4×C73D811yescharacteristic
158C22⋊C4×C73—41no—
159C22⋊C4×C73—41no—
160C22⋊C4×C73—41no—
161C22⋊C4×C73—41no—
162C22⋊C4×C73—41no—
163C22⋊C4×C73—41no—
164C22⋊C4×C73—41no—
165C22⋊C4×C73—41no—
166C22⋊C4×C73—41no—
167C22⋊C4×C73D811yescharacteristic
168C22⋊C4×C73D811yescharacteristic
169C42×C73—41no—
170C42×C73C2×D411yescharacteristic
171C42×C73C2×D411yescharacteristic
172C16×C73—21no—
173C16×C73—21no—
174C16×C73—21no—
175C16×C73—21no—
176C23×C73—41no—
177C23×C73—41no—
178C23×C73—41no—
179C23×C73C2×D811yescharacteristic
180Q8×C73—41no—
181Q8×C73—41no—
182Q8×C73—161no—
183D4×C73—41no—
184D4×C73—41no—
185D4×C73—41no—
186D4×C73—41no—
187D4×C73—41no—
188D4×C73—41no—
189D4×C73—161no—
190D4×C73—161no—
191D4×C73—161no—
192D4×C73—161no—
193D4×C73—161no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—161no—
202C2×C4×C73—161no—
203C2×C4×C73—161no—
204C2×C4×C73C2×D811yescharacteristic
205C2×C4×C73C2×D811yescharacteristic
206C2×C4×C73C2×D811yescharacteristic
207C2×C4×C73C2×D811yescharacteristic
208C2×C4×C73C2×D811yescharacteristic
209C2×C4×C73C22×D411yescharacteristic
210C8×C73—21no—
211C8×C73—21no—
212C8×C73—21no—
213C8×C73H(5,𝔽2)11yescharacteristic
214C8×C732−1+411yescharacteristic
215C22×C73—21no—
216C22×C73—21no—
217C22×C73—21no—
218C22×C73—81no—
219C22×C73—81no—
220C22×C73—81no—
221C22×C73—81no—
222C22×C73—81no—
223C22×C73—81no—
224C22×C73—81no—
225C22×C73—81no—
226C22×C73—81no—
227C22×C73C22×D811yescharacteristic
228C4×C73—21no—
229C4×C73—21no—
230C4×C73—21no—
231C4×C73—21no—
232C4×C73—21no—
233C4×C73—81no—
234C4×C73—81no—
235C4×C73—81no—
236C4×C73—81no—
237C4×C73—81no—
238C4×C73#53711yescharacteristic
239C4×C73#57511yescharacteristic
240#33823C7311yesmaximal, characteristic, direct product factor, Sylow
241C2×C73—41no—
242C2×C73—161no—
243C2×C73—161no—
244C2×C73—161no—
245C2×C73#324511yescharacteristic
246C2×C73#341511yescharacteristic
247C2×C73#341611yescharacteristic
248#3176C2×C7311yescharacteristic, complemented
249#3175C2×C7311yescharacteristic, complemented
250#2229C2×C7311yescharacteristic, complemented
251#2228C2×C7311yescharacteristic, complemented
252#2226C2×C7311yescharacteristic, complemented
253#2224C2×C7311yescharacteristic, complemented
254#2220C2×C7311yescharacteristic, complemented
255#2218C2×C7311yescharacteristic, complemented
256#2217C2×C7311yescharacteristic, complemented
257#2215C2×C7311yescharacteristic, complemented
258#2210C2×C7311yescharacteristic, complemented
259#2210C2×C7311yescharacteristic, complemented
260#2207C2×C7311yescharacteristic, complemented
261#2206C2×C7311yescharacteristic, complemented
262#2151C2×C7311yescharacteristic, complemented
263C73#3382311yescharacteristic, direct product factor, Sylow
264#542—21no—
265#540—21no—
266C2×SD32C22×C7311yescharacteristic
267C2×SD32C22×C7311yescharacteristic
268C2×D16C22×C7311yescharacteristic
269C2×D16C22×C7311yescharacteristic
270C2×M32C22×C7311yescharacteristic
271C22×C16C22×C7311yescharacteristic
272#500C22×C7311yescharacteristic
273#499C22×C7311yescharacteristic
274#495C22×C7311yescharacteristic
275#493C22×C7311yescharacteristic
276#490—21no—
277#489—21no—
278#469—21no—
279#468—21no—
280#467C22×C7311yescharacteristic
281#466C22×C7311yescharacteristic
282#466C22×C7311yescharacteristic
283#466C22×C7311yescharacteristic
284#442—21no—
285#440—21no—
286C4×Q16C22×C7311yescharacteristic
287C4×D8C22×C7311yescharacteristic
288C4×D8C22×C7311yescharacteristic
289C4×D8C22×C7311yescharacteristic
290#405C22×C7311yescharacteristic
291#367C22×C7311yescharacteristic
292#367C22×C7311yescharacteristic
293#366C22×C7311yescharacteristic
294#366C22×C7311yescharacteristic
295C4⋊C16C22×C7311yescharacteristic
296C4⋊C16C22×C7311yescharacteristic
297C4⋊C16C22×C7311yescharacteristic
298C4⋊C16C22×C7311yescharacteristic
299#358C22×C7311yescharacteristic
300#358C22×C7311yescharacteristic
301#357C22×C7311yescharacteristic
302#357C22×C7311yescharacteristic
303#357C22×C7311yescharacteristic
304#357C22×C7311yescharacteristic
305#357C22×C7311yescharacteristic
306#357C22×C7311yescharacteristic
307C2×Q16C23×C7311yescharacteristic
308C2×SD16—21no—
309C2×D8—21no—
310C2×D8C23×C7311yescharacteristic
311C2×D8C23×C7311yescharacteristic
312C2×D8C23×C7311yescharacteristic
313C2×M16D4×C7311yescharacteristic
314C22×C8C23×C7311yescharacteristic
315C4⋊Q8—21no—
316#127—21no—
317#126—41no—
318#125—21no—
319#124—21no—
320#124—41no—
321#123—41no—
322#123—41no—
323#123—41no—
324C22⋊Q8—21no—
325C22⋊Q8—41no—
326#121—21no—
327#121—21no—
328#121—21no—
329#121—41no—
330#121—41no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335#117D4×C7311yescharacteristic
336SD32—41no—
337SD32—41no—
338D16—41no—
339D16—41no—
340M32—41no—
341C2×C16—41no—
342C2×C16C23×C7311yescharacteristic
343C2×C16C23×C7311yescharacteristic
344C2×C16C23×C7311yescharacteristic
345C2×C16C23×C7311yescharacteristic
346#107—21no—
347#107C23×C7311yescharacteristic
348#107C23×C7311yescharacteristic
349#107C23×C7311yescharacteristic
350#107C23×C7311yescharacteristic
351#106—21no—
352#103—21no—
353#103—21no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—21no—
358#102—21no—
359#102—21no—
360#97D4×C7311yescharacteristic
361#97D4×C7311yescharacteristic
362C4×C8C23×C7311yescharacteristic
363C4×C8C23×C7311yescharacteristic
364C2×Q8—21no—
365C2×Q8—41no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4C2×D4×C7311yescharacteristic
378Q16—21no—
379Q16—21no—
380SD16—42no—
381D8—21no—
382D8—21no—
383D8—21no—
384D8—21no—
385D8—21no—
386D8—21no—
387D8—42no—
388M16—41no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8C2×D4×C7311yescharacteristic
392C2×C8C2×D4×C7311yescharacteristic
393C2×C8C2×D4×C7311yescharacteristic
394C2×C8C24×C7311yescharacteristic
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4D8×C7311yescharacteristic
405C4⋊C4D8×C7311yescharacteristic
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4D8×C7311yescharacteristic
416C22⋊C4D8×C7311yescharacteristic
417C42—41no—
418C42C2×D4×C7311yescharacteristic
419C42C2×D4×C7311yescharacteristic
420C16—21no—
421C16—21no—
422C16—21no—
423C16—21no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23C2×D8×C7311yescharacteristic
428Q8—41no—
429Q8—41no—
430Q8—161no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—161no—
438D4—161no—
439D4—161no—
440D4—161no—
441D4—161no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—161no—
450C2×C4—161no—
451C2×C4—161no—
452C2×C4C2×D8×C7311yescharacteristic
453C2×C4C2×D8×C7311yescharacteristic
454C2×C4C2×D8×C7311yescharacteristic
455C2×C4C2×D8×C7311yescharacteristic
456C2×C4C2×D8×C7311yescharacteristic
457C2×C4C22×D4×C7311yescharacteristic
458C8—21no—
459C8—21no—
460C8—21no—
461C8H(5,𝔽2)×C7311yescharacteristic
462C82−1+4×C7311yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22C22×D8×C7311yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4#537×C7311yescharacteristic
487C4#575×C7311yescharacteristic
488C2—41no—
489C2—161no—
490C2—161no—
491C2—161no—
492C2C73×#324511yescharacteristic
493C2C73×#341511yescharacteristic
494C2C73×#341611yescharacteristic
subgroupG/H
center, socleC22×C73C22×D8
commutator, Frattini, cosocleC2×C8C24×C73
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ #117×C73 ◁ #405×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3176C2semidirect
2. C73×#3175C2semidirect
3. C73×#2229C2semidirect
4. C73×#2228C2semidirect
5. C73×#2226C2semidirect
6. C73×#2224C2semidirect
7. C73×#2220C2semidirect
8. C73×#2218C2semidirect
9. C73×#2217C2semidirect
10. C73×#2215C2semidirect
11. C73×#2210C2semidirect
12. C73×#2210C2semidirect
13. C73×#2207C2semidirect
14. C73×#2206C2semidirect
15. C73×#2151C2semidirect
240. #33823C73direct (N × H)
248. #3176C2×C73semidirect
249. #3175C2×C73semidirect
250. #2229C2×C73semidirect
251. #2228C2×C73semidirect
252. #2226C2×C73semidirect
253. #2224C2×C73semidirect
254. #2220C2×C73semidirect
255. #2218C2×C73semidirect
256. #2217C2×C73semidirect
257. #2215C2×C73semidirect
258. #2210C2×C73semidirect
259. #2210C2×C73semidirect
260. #2207C2×C73semidirect
261. #2206C2×C73semidirect
262. #2151C2×C73semidirect
263. C73#33823direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#31762
C73×#31752
C73×#22292
C73×#22282
C73×#22262
C73×#22242
C73×#22202
C73×#22182
C73×#22172
C73×#22152
C73×#22102
C73×#22072
C73×#22062
C73×#21512
#3382373