Finite Group Databasebeta

← C73×#35119 #32944523

Subgroups — C73×#35119 #32944523

Order 18688.

2262 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529×C73C212yesmaximal, complemented
2C73×#2854C214yesmaximal, complemented
3C73×#2659C211yesmaximal, characteristic, complemented
4C73×#2656C211yesmaximal, characteristic
5C73×#2639C212yesmaximal, complemented
6C73×#2611C214yesmaximal, complemented
7C73×#2610C214yesmaximal, complemented
8C73×#2604C214yesmaximal, complemented
9C73×#2603C214yesmaximal, complemented
10C73×#2325C214yesmaximal, complemented
11C2×#376×C73C211yesmaximal, characteristic, complemented
12#529×C73—216no—
13C2×#126×C73C2214yes—
14C2×#125×C73C2212yes—
15C2×#124×C73C2212yes—
16C2×#123×C73C2214yes—
17C2×C22⋊Q8×C73C2212yes—
18C2×#121×C73C2214yes—
19C2×#121×C73C2212yes—
20C2×C4×Q8×C73C2212yes—
21C2×C4×D4×C73C2214yes—
22C2×C4×D4×C73C2212yes—
23C2×#117×C73C2214yes—
24C22×C42×C73C2212yes—
25C22×C42×C73C2211yescharacteristic
26#400×C73C2214yes—
27#399×C73C2214yes—
28#398×C73C2214yes—
29#397×C73C2214yes—
30#395×C73C2214yes—
31#394×C73C2214yes—
32#393×C73C2214yes—
33#392×C73C2214yes—
34#390×C73C2214yes—
35#390×C73C2214yes—
36#389×C73C2214yes—
37#389×C73C2214yes—
38#388×C73C2218yes—
39#388×C73C2218yes—
40#388×C73C2218yes—
41#387×C73C2218yes—
42#386×C73C2218yes—
43#386×C73C2218yes—
44#385×C73C2218yes—
45#380×C73C2214yes—
46#376×C73C2214yes—
47#376×C73C2214yes—
48C2×#95×C73C2212yes—
49C2×#95×C73C2211yescharacteristic
50C2×#95×C73C2211yescharacteristic
51C22×Q8×C73C2312yes—
52C22×D4×C73C2314yes—
53C22×D4×C73C2314yes—
54C22×D4×C73C2312yes—
55C23×C4×C73C2311yescharacteristic
56C23×C4×C73C2314yes—
57C23×C4×C73C2312yes—
58C23×C4×C73C2311yescharacteristic
59C23×C4×C73C2311yescharacteristic
60C23×C4×C73C2312yes—
61#126×C73—216no—
62#125×C73—28no—
63#124×C73—28no—
64#123×C73—216no—
65C22⋊Q8×C73—28no—
66#121×C73—28no—
67#121×C73—216no—
68C4×Q8×C73—28no—
69C4×D4×C73—216no—
70C4×D4×C73—28no—
71#117×C73—216no—
72C2×C4⋊C4×C73C2314yes—
73C2×C4⋊C4×C73C2314yes—
74C2×C4⋊C4×C73C2314yes—
75C2×C4⋊C4×C73C2314yes—
76C2×C4⋊C4×C73C2314yes—
77C2×C4⋊C4×C73C2312yes—
78C2×C4⋊C4×C73C2312yes—
79C2×C4⋊C4×C73C2314yes—
80C2×C4⋊C4×C73C2314yes—
81C2×C22⋊C4×C73C2314yes—
82C2×C22⋊C4×C73C2318yes—
83C2×C22⋊C4×C73C2314yes—
84C2×C22⋊C4×C73C2314yes—
85C2×C22⋊C4×C73C2314yes—
86C2×C22⋊C4×C73C2314yes—
87C2×C22⋊C4×C73C2318yes—
88C2×C22⋊C4×C73C2318yes—
89C2×C42×C73—28no—
90C2×C42×C73D418yescomplemented
91C2×C42×C73C2314yes—
92C2×C42×C73C2314yes—
93C2×C42×C73C2314yes—
94C2×C42×C73C2312yes—
95C2×C42×C73C2312yes—
96#95×C73C2318yes—
97#95×C73C2318yes—
98#95×C73C2318yes—
99#95×C73C2318yes—
100#95×C73C2314yes—
101#95×C73C2314yes—
102C24×C73C2414yes—
103C24×C73C2411yescharacteristic
104C2×Q8×C73—24no—
105C2×Q8×C73—44no—
106C2×D4×C73—28no—
107C2×D4×C73—216no—
108C2×D4×C73—24no—
109C2×D4×C73—28no—
110C2×D4×C73—44no—
111C2×D4×C73—44no—
112C2×D4×C73—44no—
113C22×C4×C73—24no—
114C22×C4×C73—24no—
115C22×C4×C73—28no—
116C22×C4×C73—42no—
117C22×C4×C73—42no—
118C22×C4×C73—41no—
119C22×C4×C73—41no—
120C22×C4×C73—44no—
121C22×C4×C73—44no—
122C22×C4×C73—42no—
123C22×C4×C73—42no—
124C22×C4×C73—42no—
125C22×C4×C73C2×D414yes—
126C22×C4×C73C2×D414yes—
127C22×C4×C73C2×D414yes—
128C22×C4×C73#4118yes—
129C22×C4×C73C2414yes—
130C22×C4×C73C2414yes—
131C22×C4×C73C2414yes—
132C22×C4×C73C2414yes—
133C22×C4×C73C2412yes—
134C22×C4×C73C2412yes—
135C22×C4×C73C2411yescharacteristic
136C22×C4×C73C2411yescharacteristic
137C22×C4×C73C2414yes—
138C4⋊C4×C73—28no—
139C4⋊C4×C73—28no—
140C4⋊C4×C73—24no—
141C4⋊C4×C73—28no—
142C4⋊C4×C73—24no—
143C4⋊C4×C73—28no—
144C4⋊C4×C73—28no—
145C4⋊C4×C73—44no—
146C4⋊C4×C73—44no—
147C22⋊C4×C73—28no—
148C22⋊C4×C73—28no—
149C22⋊C4×C73—28no—
150C22⋊C4×C73—28no—
151C22⋊C4×C73—28no—
152C22⋊C4×C73—216no—
153C22⋊C4×C73—48no—
154C22⋊C4×C73—48no—
155C42×C73—28no—
156C42×C73—28no—
157C42×C73—28no—
158C42×C73C2×D418yes—
159C42×C73C2×D418yes—
160C23×C73—21no—
161C23×C73—22no—
162C23×C73—21no—
163C23×C73—24no—
164C23×C73—28no—
165C23×C73—24no—
166C23×C73—28no—
167C23×C73—24no—
168C23×C73C22×D412yes—
169C23×C73C2×#4114yes—
170C23×C73C2511yescharacteristic
171Q8×C73—48no—
172D4×C73—48no—
173D4×C73—416no—
174D4×C73—88no—
175C2×C4×C73—28no—
176C2×C4×C73—22no—
177C2×C4×C73—22no—
178C2×C4×C73—24no—
179C2×C4×C73—24no—
180C2×C4×C73—22no—
181C2×C4×C73—22no—
182C2×C4×C73—24no—
183C2×C4×C73—24no—
184C2×C4×C73—24no—
185C2×C4×C73—24no—
186C2×C4×C73—24no—
187C2×C4×C73—28no—
188C2×C4×C73—24no—
189C2×C4×C73—24no—
190C2×C4×C73—24no—
191C2×C4×C73—28no—
192C2×C4×C73—24no—
193C2×C4×C73—44no—
194C2×C4×C73—44no—
195C2×C4×C73—48no—
196C2×C4×C73—416no—
197C2×C4×C73—48no—
198C2×C4×C73C22×D414yes—
199C2×C4×C73C22×D412yes—
200C2×C4×C73C22×D412yes—
201C2×C4×C73C22×D414yes—
202C2×C4×C73C2×#4118yes—
203C22×C73—22no—
204C22×C73—22no—
205C22×C73—24no—
206C22×C73—24no—
207C22×C73—44no—
208C22×C73—44no—
209C22×C73—44no—
210C22×C73—48no—
211C22×C73—48no—
212C22×C73#53214yes—
213C22×C73C23×D411yescharacteristic
214C22×C73C22×#4112yes—
215C22×C73#58511yescharacteristic
216C22×C73#58512yes—
217C22×C73#58511yescharacteristic
218C4×C73—22no—
219C4×C73—22no—
220C4×C73—24no—
221C4×C73—24no—
222C4×C73—28no—
223C4×C73—44no—
224C4×C73—44no—
225C4×C73—44no—
226C4×C73—44no—
227#35119C7311yesmaximal, characteristic, direct product factor, Sylow
228C2×C73—24no—
229C2×C73—84no—
230C2×C73C2×#53211yescharacteristic
231C2×C73#344911yescharacteristic
232C2×C73#344911yescharacteristic
233C2×C73#349112yes—
234C2×C73#349312yes—
235C2×#529C2×C7312yescomplemented
236#2854C2×C7314yescomplemented
237#2659C2×C7311yescharacteristic, complemented
238#2656C2×C7311yescharacteristic
239#2639C2×C7312yescomplemented
240#2611C2×C7314yescomplemented
241#2610C2×C7314yescomplemented
242#2604C2×C7314yescomplemented
243#2603C2×C7314yescomplemented
244#2325C2×C7314yescomplemented
245C2×#376C2×C7311yescharacteristic, complemented
246C73#3511911yescharacteristic, direct product factor, Sylow
247#529—216no—
248C2×#126C22×C7314yes—
249C2×#125C22×C7312yes—
250C2×#124C22×C7312yes—
251C2×#123C22×C7314yes—
252C2×C22⋊Q8C22×C7312yes—
253C2×#121C22×C7314yes—
254C2×#121C22×C7312yes—
255C2×C4×Q8C22×C7312yes—
256C2×C4×D4C22×C7314yes—
257C2×C4×D4C22×C7312yes—
258C2×#117C22×C7314yes—
259C22×C42C22×C7312yes—
260C22×C42C22×C7311yescharacteristic
261#400C22×C7314yes—
262#399C22×C7314yes—
263#398C22×C7314yes—
264#397C22×C7314yes—
265#395C22×C7314yes—
266#394C22×C7314yes—
267#393C22×C7314yes—
268#392C22×C7314yes—
269#390C22×C7314yes—
270#390C22×C7314yes—
271#389C22×C7314yes—
272#389C22×C7314yes—
273#388C22×C7318yes—
274#388C22×C7318yes—
275#388C22×C7318yes—
276#387C22×C7318yes—
277#386C22×C7318yes—
278#386C22×C7318yes—
279#385C22×C7318yes—
280#380C22×C7314yes—
281#376C22×C7314yes—
282#376C22×C7314yes—
283C2×#95C22×C7312yes—
284C2×#95C22×C7311yescharacteristic
285C2×#95C22×C7311yescharacteristic
286C22×Q8C23×C7312yes—
287C22×D4C23×C7314yes—
288C22×D4C23×C7312yes—
289C22×D4C23×C7314yes—
290C23×C4C23×C7311yescharacteristic
291C23×C4C23×C7314yes—
292C23×C4C23×C7312yes—
293C23×C4C23×C7311yescharacteristic
294C23×C4C23×C7311yescharacteristic
295C23×C4C23×C7312yes—
296#126—216no—
297#125—28no—
298#124—28no—
299#123—216no—
300C22⋊Q8—28no—
301#121—28no—
302#121—216no—
303C4×Q8—28no—
304C4×D4—216no—
305C4×D4—28no—
306#117—216no—
307C2×C4⋊C4C23×C7314yes—
308C2×C4⋊C4C23×C7314yes—
309C2×C4⋊C4C23×C7312yes—
310C2×C4⋊C4C23×C7314yes—
311C2×C4⋊C4C23×C7312yes—
312C2×C4⋊C4C23×C7314yes—
313C2×C4⋊C4C23×C7314yes—
314C2×C4⋊C4C23×C7314yes—
315C2×C4⋊C4C23×C7314yes—
316C2×C22⋊C4C23×C7314yes—
317C2×C22⋊C4C23×C7314yes—
318C2×C22⋊C4C23×C7314yes—
319C2×C22⋊C4C23×C7314yes—
320C2×C22⋊C4C23×C7314yes—
321C2×C22⋊C4C23×C7318yes—
322C2×C22⋊C4C23×C7318yes—
323C2×C22⋊C4C23×C7318yes—
324C2×C42—28no—
325C2×C42D4×C7318yescomplemented
326C2×C42C23×C7314yes—
327C2×C42C23×C7314yes—
328C2×C42C23×C7314yes—
329C2×C42C23×C7312yes—
330C2×C42C23×C7312yes—
331#95C23×C7318yes—
332#95C23×C7314yes—
333#95C23×C7314yes—
334#95C23×C7318yes—
335#95C23×C7318yes—
336#95C23×C7318yes—
337C24C24×C7314yes—
338C24C24×C7311yescharacteristic
339C2×Q8—24no—
340C2×Q8—44no—
341C2×D4—28no—
342C2×D4—216no—
343C2×D4—24no—
344C2×D4—28no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4C2×D4×C7314yes—
361C22×C4C2×D4×C7314yes—
362C22×C4C2×D4×C7314yes—
363C22×C4#41×C7318yes—
364C22×C4C24×C7314yes—
365C22×C4C24×C7314yes—
366C22×C4C24×C7314yes—
367C22×C4C24×C7314yes—
368C22×C4C24×C7312yes—
369C22×C4C24×C7312yes—
370C22×C4C24×C7311yescharacteristic
371C22×C4C24×C7311yescharacteristic
372C22×C4C24×C7314yes—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C22⋊C4—28no—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—28no—
387C22⋊C4—216no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C42—28no—
391C42—28no—
392C42—28no—
393C42C2×D4×C7318yes—
394C42C2×D4×C7318yes—
395C23—21no—
396C23—22no—
397C23—21no—
398C23—24no—
399C23—28no—
400C23—24no—
401C23—28no—
402C23—24no—
403C23C22×D4×C7312yes—
404C23C2×#41×C7314yes—
405C23C25×C7311yescharacteristic
406Q8—48no—
407D4—48no—
408D4—416no—
409D4—88no—
410C2×C4—28no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—48no—
431C2×C4—416no—
432C2×C4—48no—
433C2×C4C22×D4×C7314yes—
434C2×C4C22×D4×C7312yes—
435C2×C4C22×D4×C7312yes—
436C2×C4C22×D4×C7314yes—
437C2×C4C2×#41×C7318yes—
438C22—24no—
439C22—24no—
440C22—22no—
441C22—22no—
442C22—48no—
443C22—48no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22#532×C7314yes—
448C22C23×D4×C7311yescharacteristic
449C22C22×#41×C7312yes—
450C22#585×C7311yescharacteristic
451C22#585×C7312yes—
452C22#585×C7311yescharacteristic
453C4—28no—
454C4—22no—
455C4—22no—
456C4—24no—
457C4—24no—
458C4—44no—
459C4—44no—
460C4—44no—
461C4—44no—
462C2—24no—
463C2—84no—
464C2C2×#532×C7311yescharacteristic
465C2C73×#344911yescharacteristic
466C2C73×#344911yescharacteristic
467C2C73×#349112yes—
468C2C73×#349312yes—
subgroupG/H
center, socleC23×C73C25
commutator, Frattini, cosocleC23C25×C73
prime radicalC73×#2656C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#529×C73C2semidirect
2. C73×#2854C2semidirect
3. C73×#2659C2semidirect
5. C73×#2639C2semidirect
6. C73×#2611C2semidirect
7. C73×#2610C2semidirect
8. C73×#2604C2semidirect
9. C73×#2603C2semidirect
10. C73×#2325C2semidirect
11. C2×#376×C73C2semidirect
90. C2×C42×C73D4semidirect
227. #35119C73direct (N × H)
235. C2×#529C2×C73semidirect
236. #2854C2×C73semidirect
237. #2659C2×C73semidirect
239. #2639C2×C73semidirect
240. #2611C2×C73semidirect
241. #2610C2×C73semidirect
242. #2604C2×C73semidirect
243. #2603C2×C73semidirect
244. #2325C2×C73semidirect
245. C2×#376C2×C73semidirect
246. C73#35119direct (N × H)
325. C2×C42D4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#529×C732
C73×#28542
C73×#26592
C73×#26562
C73×#26392
C73×#26112
C73×#26102
C73×#26042
C73×#26032
C73×#23252
C2×#376×C732
#3511973