Finite Group Databasebeta

← C73×#32811 #32942215

Subgroups — C73×#32811 #32942215

Order 18688.

742 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
1C73×#2862C211yesmaximal, characteristic, complemented
2C73×#2861C211yesmaximal, characteristic
3C73×#2056C212yesmaximal, complemented
4C73×#2055C212yesmaximal
5C73×#2026C212yesmaximal, complemented
6C73×#2024C211yesmaximal, characteristic, complemented
7C73×#2023C211yesmaximal, characteristic
8C73×#2016C211yesmaximal, characteristic, complemented
9C73×#2015C212yesmaximal
10C73×#2013C211yesmaximal, characteristic
11C73×#1831C211yesmaximal, characteristic, complemented
12C22×Q16×C73C2212yes—
13C22×SD16×C73C2212yes—
14C2×C4⋊Q8×C73C2211yescharacteristic
15C2×C4⋊Q8×C73C2211yescharacteristic
16C2×#125×C73C2211yescharacteristic
17C2×#124×C73C2211yescharacteristic
18C2×C4⋊C8×C73C2212yes—
19C2×#103×C73C2212yes—
20C2×#103×C73C2212yes—
21C2×#103×C73C2212yes—
22C2×#102×C73C2212yes—
23#391×C73—21no—
24#391×C73—21no—
25#391×C73C2212yes—
26#391×C73C2211yescharacteristic
27#391×C73C2212yes—
28#391×C73C2211yescharacteristic
29#390×C73C2211yescharacteristic
30#390×C73C2211yescharacteristic
31#388×C73—21no—
32#388×C73—21no—
33#388×C73—21no—
34#388×C73—21no—
35#382×C73—21no—
36#382×C73—21no—
37C43×C73C2211yescharacteristic
38#340×C73C2212yes—
39#340×C73C2212yes—
40#336×C73C2211yescharacteristic
41#336×C73C2212yes—
42#336×C73C2211yescharacteristic
43C22×Q8×C73—21no—
44C22×Q8×C73C2311yescharacteristic
45C22×Q8×C73C2312yes—
46C22×D4×C73C2311yescharacteristic
47C2×Q16×C73—42no—
48C2×Q16×C73—42no—
49C2×Q16×C73—42no—
50C2×SD16×C73—42no—
51C2×SD16×C73—42no—
52C2×SD16×C73—42no—
53C22×C8×C73C2312yes—
54C22×C8×C73C2312yes—
55C4⋊Q8×C73—42no—
56C4⋊Q8×C73—42no—
57#125×C73—42no—
58#124×C73—42no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73C2311yescharacteristic
65C2×C4⋊C4×C73C2311yescharacteristic
66C2×C4⋊C4×C73C2312yes—
67C2×C22⋊C4×C73—21no—
68C2×C22⋊C4×C73—21no—
69C2×C22⋊C4×C73—21no—
70C2×C22⋊C4×C73—21no—
71C2×C22⋊C4×C73—21no—
72C2×C22⋊C4×C73—21no—
73C2×C42×C73D411yescharacteristic
74C2×C42×C73D411yescharacteristic
75C2×C42×C73D411yescharacteristic
76C2×C42×C73D411yescharacteristic
77C2×C42×C73C2311yescharacteristic
78C2×C42×C73C2311yescharacteristic
79C2×C42×C73C2311yescharacteristic
80C4⋊C8×C73—42no—
81#103×C73—42no—
82#103×C73—42no—
83#103×C73—42no—
84#102×C73—42no—
85#95×C73—22no—
86#95×C73—22no—
87#95×C73—22no—
88#95×C73—22no—
89#95×C73—21no—
90#95×C73—21no—
91#95×C73—21no—
92#95×C73—21no—
93C24×C73—21no—
94C2×Q8×C73—22no—
95C2×Q8×C73—22no—
96C2×Q8×C73—22no—
97C2×Q8×C73—22no—
98C2×Q8×C73—22no—
99C2×Q8×C73—22no—
100C2×Q8×C73—22no—
101C2×Q8×C73—21no—
102C2×Q8×C73—21no—
103C2×Q8×C73—22no—
104C2×Q8×C73—42no—
105C2×Q8×C73—42no—
106C2×Q8×C73—42no—
107C2×D4×C73—22no—
108C2×D4×C73—22no—
109C2×D4×C73—21no—
110C2×D4×C73—21no—
111C22×C4×C73—21no—
112C22×C4×C73—22no—
113C22×C4×C73—22no—
114C22×C4×C73—21no—
115C22×C4×C73—21no—
116C22×C4×C73C2×D411yescharacteristic
117C22×C4×C73C2×D411yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73C2×D411yescharacteristic
122C22×C4×C73C2411yescharacteristic
123Q16×C73—48no—
124SD16×C73—48no—
125C2×C8×C73—22no—
126C2×C8×C73—22no—
127C2×C8×C73—22no—
128C2×C8×C73—22no—
129C2×C8×C73—22no—
130C2×C8×C73—22no—
131C4⋊C4×C73—22no—
132C4⋊C4×C73—22no—
133C4⋊C4×C73—22no—
134C4⋊C4×C73—22no—
135C4⋊C4×C73—42no—
136C4⋊C4×C73—42no—
137C4⋊C4×C73—42no—
138C4⋊C4×C73—42no—
139C4⋊C4×C73—42no—
140C22⋊C4×C73—42no—
141C22⋊C4×C73—42no—
142C22⋊C4×C73—42no—
143C22⋊C4×C73—42no—
144C22⋊C4×C73—42no—
145C22⋊C4×C73—42no—
146C42×C73—22no—
147C42×C73—22no—
148C42×C73—22no—
149C42×C73—21no—
150C42×C73—22no—
151C42×C73—22no—
152C42×C73—21no—
153C42×C73—41no—
154C23×C73—41no—
155C23×C73—41no—
156C23×C73—41no—
157C23×C73—41no—
158C23×C73—41no—
159C23×C73—41no—
160C23×C73—41no—
161C23×C73C22×D411yescharacteristic
162Q8×C73—42no—
163Q8×C73—42no—
164Q8×C73—42no—
165Q8×C73—42no—
166Q8×C73—42no—
167Q8×C73—42no—
168Q8×C73—84no—
169D4×C73—42no—
170D4×C73—42no—
171C2×C4×C73—21no—
172C2×C4×C73—21no—
173C2×C4×C73—21no—
174C2×C4×C73—21no—
175C2×C4×C73—21no—
176C2×C4×C73—21no—
177C2×C4×C73—21no—
178C2×C4×C73—21no—
179C2×C4×C73—21no—
180C2×C4×C73—21no—
181C2×C4×C73—21no—
182C2×C4×C73—21no—
183C2×C4×C73—21no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—41no—
187C2×C4×C73—41no—
188C2×C4×C73—41no—
189C2×C4×C73—42no—
190C2×C4×C73—42no—
191C2×C4×C73—42no—
192C2×C4×C73—41no—
193C2×C4×C73—41no—
194C2×C4×C73—42no—
195C2×C4×C73—41no—
196C2×C4×C73—42no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—41no—
200C2×C4×C73—42no—
201C2×C4×C73#13512yes—
202C2×C4×C73#13712yes—
203C2×C4×C73#13712yes—
204C2×C4×C73H(5,𝔽2)11yescharacteristic
205C2×C4×C73H(5,𝔽2)12yes—
206C2×C4×C73H(5,𝔽2)12yes—
207C2×C4×C732−1+411yescharacteristic
208C8×C73—42no—
209C8×C73—42no—
210C22×C73—81no—
211C22×C73—81no—
212C22×C73—81no—
213C22×C73—81no—
214C22×C73—81no—
215C22×C73—81no—
216C22×C73—81no—
217C22×C73#53511yescharacteristic
218C22×C73#53511yescharacteristic
219C22×C73#53711yescharacteristic
220C22×C73C2×#13511yescharacteristic
221C22×C73C2×#13711yescharacteristic
222C22×C73C2×#13711yescharacteristic
223C22×C73#57511yescharacteristic
224C4×C73—22no—
225C4×C73—22no—
226C4×C73—22no—
227C4×C73—41no—
228C4×C73—41no—
229C4×C73—41no—
230C4×C73—81no—
231C4×C73—82no—
232C4×C73—81no—
233C4×C73—82no—
234C4×C73—81no—
235C4×C73H(3,𝔽4)12yes—
236C4×C73#56212yes—
237#32811C7311yesmaximal, characteristic, direct product factor, Sylow
238C2×C73—161no—
239C2×C73#281911yescharacteristic
240C2×C73#321011yescharacteristic
241C2×C73#321111yescharacteristic
242C2×C73#321611yescharacteristic
243C2×C73#321811yescharacteristic
244C2×C73#324811yescharacteristic
245C2×C73#325211yescharacteristic
246#2862C2×C7311yescharacteristic, complemented
247#2861C2×C7311yescharacteristic
248#2056C2×C7312yescomplemented
249#2055C2×C7312yes—
250#2026C2×C7312yescomplemented
251#2024C2×C7311yescharacteristic, complemented
252#2023C2×C7311yescharacteristic
253#2016C2×C7311yescharacteristic, complemented
254#2015C2×C7312yes—
255#2013C2×C7311yescharacteristic
256#1831C2×C7311yescharacteristic, complemented
257C73#3281111yescharacteristic, direct product factor, Sylow
258C22×Q16C22×C7312yes—
259C22×SD16C22×C7312yes—
260C2×C4⋊Q8C22×C7311yescharacteristic
261C2×C4⋊Q8C22×C7311yescharacteristic
262C2×#125C22×C7311yescharacteristic
263C2×#124C22×C7311yescharacteristic
264C2×C4⋊C8C22×C7312yes—
265C2×#103C22×C7312yes—
266C2×#103C22×C7312yes—
267C2×#103C22×C7312yes—
268C2×#102C22×C7312yes—
269#391—21no—
270#391—21no—
271#391C22×C7312yes—
272#391C22×C7311yescharacteristic
273#391C22×C7311yescharacteristic
274#391C22×C7312yes—
275#390C22×C7311yescharacteristic
276#390C22×C7311yescharacteristic
277#388—21no—
278#388—21no—
279#388—21no—
280#388—21no—
281#382—21no—
282#382—21no—
283C43C22×C7311yescharacteristic
284#340C22×C7312yes—
285#340C22×C7312yes—
286#336C22×C7311yescharacteristic
287#336C22×C7312yes—
288#336C22×C7311yescharacteristic
289C22×Q8—21no—
290C22×Q8C23×C7311yescharacteristic
291C22×Q8C23×C7312yes—
292C22×D4C23×C7311yescharacteristic
293C2×Q16—42no—
294C2×Q16—42no—
295C2×Q16—42no—
296C2×SD16—42no—
297C2×SD16—42no—
298C2×SD16—42no—
299C22×C8C23×C7312yes—
300C22×C8C23×C7312yes—
301C4⋊Q8—42no—
302C4⋊Q8—42no—
303#125—42no—
304#124—42no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4C23×C7311yescharacteristic
311C2×C4⋊C4C23×C7311yescharacteristic
312C2×C4⋊C4C23×C7312yes—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C42D4×C7311yescharacteristic
320C2×C42D4×C7311yescharacteristic
321C2×C42D4×C7311yescharacteristic
322C2×C42D4×C7311yescharacteristic
323C2×C42C23×C7311yescharacteristic
324C2×C42C23×C7311yescharacteristic
325C2×C42C23×C7311yescharacteristic
326C4⋊C8—42no—
327#103—42no—
328#103—42no—
329#103—42no—
330#102—42no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339C24—21no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—21no—
356C2×D4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4C2×D4×C7311yescharacteristic
363C22×C4C2×D4×C7311yescharacteristic
364C22×C4C2×D4×C7311yescharacteristic
365C22×C4C2×D4×C7311yescharacteristic
366C22×C4C2×D4×C7311yescharacteristic
367C22×C4C2×D4×C7311yescharacteristic
368C22×C4C24×C7311yescharacteristic
369Q16—48no—
370SD16—48no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—21no—
396C42—22no—
397C42—22no—
398C42—21no—
399C42—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C22×D4×C7311yescharacteristic
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414Q8—84no—
415D4—42no—
416D4—42no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4#135×C7312yes—
448C2×C4#137×C7312yes—
449C2×C4#137×C7312yes—
450C2×C4H(5,𝔽2)×C7311yescharacteristic
451C2×C4H(5,𝔽2)×C7312yes—
452C2×C4H(5,𝔽2)×C7312yes—
453C2×C42−1+4×C7311yescharacteristic
454C8—42no—
455C8—42no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22#535×C7311yescharacteristic
464C22#535×C7311yescharacteristic
465C22#537×C7311yescharacteristic
466C22C2×#135×C7311yescharacteristic
467C22C2×#137×C7311yescharacteristic
468C22C2×#137×C7311yescharacteristic
469C22#575×C7311yescharacteristic
470C4—22no—
471C4—22no—
472C4—22no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—82no—
478C4—81no—
479C4—81no—
480C4—82no—
481C4H(3,𝔽4)×C7312yes—
482C4#562×C7312yes—
483C2—161no—
484C2C73×#281911yescharacteristic
485C2C73×#321011yescharacteristic
486C2C73×#321111yescharacteristic
487C2C73×#321611yescharacteristic
488C2C73×#321811yescharacteristic
489C2C73×#324811yescharacteristic
490C2C73×#325211yescharacteristic
subgroupG/H
center, socleC23×C73C22×D4
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC22×D4×C73C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C43×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#2862C2semidirect
3. C73×#2056C2semidirect
5. C73×#2026C2semidirect
6. C73×#2024C2semidirect
8. C73×#2016C2semidirect
11. C73×#1831C2semidirect
237. #32811C73direct (N × H)
246. #2862C2×C73semidirect
248. #2056C2×C73semidirect
250. #2026C2×C73semidirect
251. #2024C2×C73semidirect
253. #2016C2×C73semidirect
256. #1831C2×C73semidirect
257. C73#32811direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#28622
C73×#28612
C73×#20562
C73×#20552
C73×#20262
C73×#20242
C73×#20232
C73×#20162
C73×#20152
C73×#20132
C73×#18312
#3281173