Finite Group Databasebeta

← C73×#19231 #32928635

Subgroups — C73×#19231 #32928635

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3245C211yesmaximal, characteristic
2C73×#3226C211yesmaximal, characteristic, complemented
3C73×#3152C211yesmaximal, characteristic, complemented
4C73×#2935C211yesmaximal, characteristic, complemented
5C73×#1689C212yesmaximal, complemented
6C73×#1685C212yesmaximal, complemented
7C73×#1669C212yesmaximal, complemented
8C73×#1665C212yesmaximal, complemented
9C73×#1565C211yesmaximal, characteristic, complemented
10C73×#1540C211yesmaximal, characteristic, complemented
11C2×#332×C73C211yesmaximal, characteristic, complemented
12C2×#137×C73—21no—
13C2×#137×C73—21no—
14C2×#131×C73—21no—
15#532×C73C2211yescharacteristic
16#532×C73C2211yescharacteristic
17C2×C4⋊Q8×C73C2211yescharacteristic
18#500×C73—21no—
19#497×C73—22no—
20#494×C73C2212yes—
21#492×C73C2212yes—
22#486×C73—21no—
23#480×C73—22no—
24#474×C73—21no—
25#474×C73C2212yes—
26#474×C73C2212yes—
27#474×C73C2212yes—
28#466×C73—22no—
29#459×C73—21no—
30#459×C73C2212yes—
31#436×C73C2212yes—
32#435×C73—22no—
33C8×D4×C73C2212yes—
34#431×C73C2211yescharacteristic
35C2×C4⋊C8×C73C2211yescharacteristic
36C2×C4⋊C8×C73C2211yescharacteristic
37#408×C73—21no—
38C4×M16×C73C2211yescharacteristic
39#335×C73C2212yes—
40#334×C73C2212yes—
41#332×C73C2212yes—
42#332×C73C2211yescharacteristic
43#332×C73C2212yes—
44#332×C73C2211yescharacteristic
45#331×C73C2211yescharacteristic
46#331×C73C2211yescharacteristic
47C2×#41×C73—21no—
48C2×#41×C73—21no—
49C22×Q8×C73—21no—
50#137×C73—44no—
51#137×C73—44no—
52C2×Q16×C73—22no—
53C2×Q16×C73—22no—
54C2×SD16×C73—22no—
55C2×SD16×C73—22no—
56C2×D8×C73—22no—
57#131×C73—44no—
58C2×M16×C73—22no—
59C2×M16×C73D411yescharacteristic
60C2×M16×C73D411yescharacteristic
61C22×C8×C73—21no—
62C22×C8×C73—21no—
63C22×C8×C73—22no—
64C4⋊Q8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73—21no—
67C4⋊Q8×C73D411yescharacteristic
68C4⋊Q8×C73C2312yes—
69C4⋊Q8×C73C2311yescharacteristic
70#127×C73D411yescharacteristic
71#127×C73C2311yescharacteristic
72#124×C73—21no—
73#124×C73—21no—
74#121×C73—21no—
75#121×C73—21no—
76#121×C73—21no—
77#121×C73D412yes—
78C4×D4×C73—21no—
79C4×D4×C73—21no—
80C4×D4×C73—21no—
81C4×D4×C73C2312yes—
82C2×C4⋊C4×C73—21no—
83C2×C4⋊C4×C73—21no—
84C2×C42×C73C2311yescharacteristic
85#107×C73—22no—
86#107×C73—22no—
87#107×C73—22no—
88C4⋊C8×C73—21no—
89C4⋊C8×C73—21no—
90C4⋊C8×C73C2311yescharacteristic
91C4⋊C8×C73C2311yescharacteristic
92C4⋊C8×C73C2312yes—
93C4⋊C8×C73C2311yescharacteristic
94C4⋊C8×C73C2311yescharacteristic
95#102×C73—22no—
96#102×C73—22no—
97#102×C73—22no—
98#102×C73—22no—
99C22⋊C8×C73—21no—
100C22⋊C8×C73—22no—
101C22⋊C8×C73—22no—
102#97×C73D412yes—
103C4×C8×C73—21no—
104C4×C8×C73C2312yes—
105#41×C73—42no—
106#41×C73—42no—
107#41×C73—42no—
108#41×C73—42no—
109C2×Q8×C73—21no—
110C2×Q8×C73—21no—
111C2×Q8×C73—22no—
112C2×Q8×C73—41no—
113C2×Q8×C73—42no—
114C2×Q8×C73—42no—
115C2×D4×C73—22no—
116C2×D4×C73—21no—
117C2×D4×C73—21no—
118C2×D4×C73—21no—
119C2×D4×C73—21no—
120C2×D4×C73—21no—
121C2×D4×C73—41no—
122C2×D4×C73C2×D412yes—
123C22×C4×C73—22no—
124C22×C4×C73—41no—
125C22×C4×C73—41no—
126C22×C4×C73C2×D411yescharacteristic
127C22×C4×C73C2×D411yescharacteristic
128C22×C4×C73#4111yescharacteristic
129Q16×C73—44no—
130Q16×C73—44no—
131SD16×C73—44no—
132SD16×C73—44no—
133D8×C73—82no—
134M16×C73—22no—
135M16×C73—44no—
136M16×C73D814yescomplemented
137C2×C8×C73—21no—
138C2×C8×C73—21no—
139C2×C8×C73—21no—
140C2×C8×C73—21no—
141C2×C8×C73—22no—
142C2×C8×C73—42no—
143C2×C8×C73—44no—
144C2×C8×C73—81no—
145C2×C8×C73C2×D412yes—
146C2×C8×C73C2×D412yes—
147C4⋊C4×C73—21no—
148C4⋊C4×C73—22no—
149C4⋊C4×C73—21no—
150C4⋊C4×C73—22no—
151C4⋊C4×C73—21no—
152C4⋊C4×C73—41no—
153C4⋊C4×C73—41no—
154C4⋊C4×C73C2×D412yes—
155C22⋊C4×C73—22no—
156C22⋊C4×C73—22no—
157C22⋊C4×C73—41no—
158C22⋊C4×C73—41no—
159C42×C73C2×D411yescharacteristic
160C42×C73C2×D411yescharacteristic
161C42×C73#4111yescharacteristic
162C42×C73C2411yescharacteristic
163C23×C73—22no—
164C23×C73—41no—
165C23×C73#12111yescharacteristic
166Q8×C73—42no—
167Q8×C73—42no—
168Q8×C73—41no—
169Q8×C73—42no—
170Q8×C73—41no—
171Q8×C73—82no—
172D4×C73—22no—
173D4×C73—22no—
174D4×C73—44no—
175D4×C73—41no—
176D4×C73—41no—
177D4×C73—41no—
178D4×C73—41no—
179D4×C73—42no—
180D4×C73—81no—
181D4×C73—81no—
182D4×C73—82no—
183C2×C4×C73—21no—
184C2×C4×C73—21no—
185C2×C4×C73—22no—
186C2×C4×C73—41no—
187C2×C4×C73—42no—
188C2×C4×C73—42no—
189C2×C4×C73—41no—
190C2×C4×C73—42no—
191C2×C4×C73—82no—
192C2×C4×C73—82no—
193C2×C4×C73#12111yescharacteristic
194C2×C4×C73#12111yescharacteristic
195C2×C4×C73#12111yescharacteristic
196C2×C4×C73C2×D812yes—
197C2×C4×C73AGL(1,ℤ/8)12yes—
198C2×C4×C73C22×D411yescharacteristic
199C2×C4×C73C22×D411yescharacteristic
200C2×C4×C73C2×#4111yescharacteristic
201C8×C73—22no—
202C8×C73—41no—
203C8×C73—41no—
204C8×C73—41no—
205C8×C73—42no—
206C8×C73—41no—
207C8×C73C2×D814yes—
208C22×C73—21no—
209C22×C73—21no—
210C22×C73—42no—
211C22×C73—42no—
212C22×C73—42no—
213C22×C73—81no—
214C22×C73—81no—
215C22×C73—81no—
216C22×C73#45912yes—
217C22×C73C2×#12111yescharacteristic
218C4×C73—21no—
219C4×C73—21no—
220C4×C73—21no—
221C4×C73—42no—
222C4×C73—81no—
223C4×C73—82no—
224C4×C73#45912yes—
225C4×C73C22×D811yescharacteristic
226C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C73#57711yescharacteristic
228C4×C73#57811yescharacteristic
229#19231C7311yesmaximal, characteristic, direct product factor, Sylow
230C2×C73—22no—
231C2×C73—82no—
232C2×C73—161no—
233C2×C73C2×#45911yescharacteristic
234C2×C73#306311yescharacteristic
235C2×C73#306911yescharacteristic
236#3245C2×C7311yescharacteristic
237#3226C2×C7311yescharacteristic, complemented
238#3152C2×C7311yescharacteristic, complemented
239#2935C2×C7311yescharacteristic, complemented
240#1689C2×C7312yescomplemented
241#1685C2×C7312yescomplemented
242#1669C2×C7312yescomplemented
243#1665C2×C7312yescomplemented
244#1565C2×C7311yescharacteristic, complemented
245#1540C2×C7311yescharacteristic, complemented
246C2×#332C2×C7311yescharacteristic, complemented
247C73#1923111yescharacteristic, direct product factor, Sylow
248C2×#137—21no—
249C2×#137—21no—
250C2×#131—21no—
251#532C22×C7311yescharacteristic
252#532C22×C7311yescharacteristic
253C2×C4⋊Q8C22×C7311yescharacteristic
254#500—21no—
255#497—22no—
256#494C22×C7312yes—
257#492C22×C7312yes—
258#486—21no—
259#480—22no—
260#474—21no—
261#474C22×C7312yes—
262#474C22×C7312yes—
263#474C22×C7312yes—
264#466—22no—
265#459—21no—
266#459C22×C7312yes—
267#436C22×C7312yes—
268#435—22no—
269C8×D4C22×C7312yes—
270#431C22×C7311yescharacteristic
271C2×C4⋊C8C22×C7311yescharacteristic
272C2×C4⋊C8C22×C7311yescharacteristic
273#408—21no—
274C4×M16C22×C7311yescharacteristic
275#335C22×C7312yes—
276#334C22×C7312yes—
277#332C22×C7312yes—
278#332C22×C7312yes—
279#332C22×C7311yescharacteristic
280#332C22×C7311yescharacteristic
281#331C22×C7311yescharacteristic
282#331C22×C7311yescharacteristic
283C2×#41—21no—
284C2×#41—21no—
285C22×Q8—21no—
286#137—44no—
287#137—44no—
288C2×Q16—22no—
289C2×Q16—22no—
290C2×SD16—22no—
291C2×SD16—22no—
292C2×D8—22no—
293#131—44no—
294C2×M16—22no—
295C2×M16D4×C7311yescharacteristic
296C2×M16D4×C7311yescharacteristic
297C22×C8—21no—
298C22×C8—21no—
299C22×C8—22no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8D4×C7311yescharacteristic
304C4⋊Q8C23×C7312yes—
305C4⋊Q8C23×C7311yescharacteristic
306#127D4×C7311yescharacteristic
307#127C23×C7311yescharacteristic
308#124—21no—
309#124—21no—
310#121—21no—
311#121—21no—
312#121—21no—
313#121D4×C7312yes—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4C23×C7312yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C42C23×C7311yescharacteristic
321#107—22no—
322#107—22no—
323#107—22no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8C23×C7311yescharacteristic
327C4⋊C8C23×C7311yescharacteristic
328C4⋊C8C23×C7311yescharacteristic
329C4⋊C8C23×C7311yescharacteristic
330C4⋊C8C23×C7312yes—
331#102—22no—
332#102—22no—
333#102—22no—
334#102—22no—
335C22⋊C8—21no—
336C22⋊C8—22no—
337C22⋊C8—22no—
338#97D4×C7312yes—
339C4×C8—21no—
340C4×C8C23×C7312yes—
341#41—42no—
342#41—42no—
343#41—42no—
344#41—42no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—41no—
358C2×D4C2×D4×C7312yes—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4C2×D4×C7311yescharacteristic
363C22×C4C2×D4×C7311yescharacteristic
364C22×C4#41×C7311yescharacteristic
365Q16—44no—
366Q16—44no—
367SD16—44no—
368SD16—44no—
369D8—82no—
370M16—22no—
371M16—44no—
372M16D8×C7314yescomplemented
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—22no—
378C2×C8—42no—
379C2×C8—44no—
380C2×C8—81no—
381C2×C8C2×D4×C7312yes—
382C2×C8C2×D4×C7312yes—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4C2×D4×C7312yes—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42C2×D4×C7311yescharacteristic
396C42C2×D4×C7311yescharacteristic
397C42#41×C7311yescharacteristic
398C42C24×C7311yescharacteristic
399C23—22no—
400C23—41no—
401C23#121×C7311yescharacteristic
402Q8—42no—
403Q8—42no—
404Q8—41no—
405Q8—42no—
406Q8—41no—
407Q8—82no—
408D4—22no—
409D4—22no—
410D4—44no—
411D4—42no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—81no—
417D4—81no—
418D4—82no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4#121×C7311yescharacteristic
430C2×C4#121×C7311yescharacteristic
431C2×C4#121×C7311yescharacteristic
432C2×C4C2×D8×C7312yes—
433C2×C4AGL(1,ℤ/8)×C7312yes—
434C2×C4C22×D4×C7311yescharacteristic
435C2×C4C22×D4×C7311yescharacteristic
436C2×C4C2×#41×C7311yescharacteristic
437C8—22no—
438C8—41no—
439C8—41no—
440C8—42no—
441C8—41no—
442C8—41no—
443C8C2×D8×C7314yes—
444C22—21no—
445C22—21no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#459×C7312yes—
453C22C2×#121×C7311yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—42no—
458C4—81no—
459C4—82no—
460C4#459×C7312yes—
461C4C22×D8×C7311yescharacteristic
462C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
463C4#577×C7311yescharacteristic
464C4#578×C7311yescharacteristic
465C2—22no—
466C2—82no—
467C2—161no—
468C2C2×#459×C7311yescharacteristic
469C2C73×#306311yescharacteristic
470C2C73×#306911yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3245C2
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×#3226C2semidirect
3. C73×#3152C2semidirect
4. C73×#2935C2semidirect
5. C73×#1689C2semidirect
6. C73×#1685C2semidirect
7. C73×#1669C2semidirect
8. C73×#1665C2semidirect
9. C73×#1565C2semidirect
10. C73×#1540C2semidirect
11. C2×#332×C73C2semidirect
136. M16×C73D8semidirect
229. #19231C73direct (N × H)
237. #3226C2×C73semidirect
238. #3152C2×C73semidirect
239. #2935C2×C73semidirect
240. #1689C2×C73semidirect
241. #1685C2×C73semidirect
242. #1669C2×C73semidirect
243. #1665C2×C73semidirect
244. #1565C2×C73semidirect
245. #1540C2×C73semidirect
246. C2×#332C2×C73semidirect
247. C73#19231direct (N × H)
372. M16D8×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#32452
C73×#32262
C73×#31522
C73×#29352
C73×#16892
C73×#16852
C73×#16692
C73×#16652
C73×#15652
C73×#15402
C2×#332×C732
#1923173