Finite Group Databasebeta

← C73×#59659 #32969063

Subgroups — C73×#59659 #32969063

Order 18688.

2244 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q82×C73C211yesmaximal, characteristic, complemented
2C2×#555×C73C211yesmaximal, characteristic
3C73×#2731C214yesmaximal, complemented
4C73×#2545C214yesmaximal, complemented
5C73×#2538C212yesmaximal, complemented
6C73×#2491C214yesmaximal, complemented
7C73×#2439C214yesmaximal, complemented
8C73×#2426C214yesmaximal, complemented
9C73×#2362C214yesmaximal, complemented
10C4×C4⋊Q8×C73C211yesmaximal, characteristic, complemented
11C4×#124×C73C212yesmaximal, complemented
12Q82×C73—28no—
13#555×C73—28no—
14C2×C4⋊Q8×C73C2214yes—
15C2×C4⋊Q8×C73C2214yes—
16C2×C4⋊Q8×C73C2211yescharacteristic
17C2×#124×C73C2214yes—
18C2×#124×C73C2212yes—
19C2×#124×C73C2214yes—
20C2×C22⋊Q8×C73C2214yes—
21C2×#121×C73C2214yes—
22C2×C4×Q8×C73C2212yes—
23C2×C4×Q8×C73C2214yes—
24C2×C4×Q8×C73C2212yes—
25C2×C4×D4×C73C2212yes—
26C2×#117×C73C2214yes—
27#396×C73C2218yes—
28#396×C73C2218yes—
29#394×C73C2218yes—
30#393×C73C2218yes—
31#391×C73C2214yes—
32#391×C73C2212yes—
33#391×C73C2212yes—
34#390×C73C2212yes—
35#390×C73C2214yes—
36#390×C73C2214yes—
37#389×C73C2214yes—
38#389×C73C2214yes—
39#388×C73C2214yes—
40#388×C73C2218yes—
41#388×C73C2218yes—
42#388×C73C2214yes—
43#387×C73C2218yes—
44#384×C73C2211yescharacteristic
45#382×C73C2212yes—
46#380×C73C2214yes—
47C4×C4⋊C4×C73C2214yes—
48C4×C4⋊C4×C73C2214yes—
49C4×C22⋊C4×C73C2214yes—
50C4×C22⋊C4×C73C2214yes—
51C43×C73C2211yescharacteristic
52C22×Q8×C73C2312yes—
53C22×Q8×C73C2314yes—
54C22×Q8×C73C2312yes—
55C22×D4×C73C2312yes—
56C22×D4×C73C2314yes—
57C23×C4×C73C2314yes—
58C4⋊Q8×C73—216no—
59C4⋊Q8×C73—216no—
60C4⋊Q8×C73D418yes—
61#124×C73—216no—
62#124×C73—28no—
63#124×C73—48no—
64C22⋊Q8×C73—216no—
65#121×C73—216no—
66C4×Q8×C73—216no—
67C4×Q8×C73—28no—
68C4×Q8×C73—44no—
69C4×D4×C73—28no—
70#117×C73—216no—
71C2×C4⋊C4×C73C2314yes—
72C2×C4⋊C4×C73C2318yes—
73C2×C4⋊C4×C73C2312yes—
74C2×C4⋊C4×C73C2312yes—
75C2×C4⋊C4×C73C2318yes—
76C2×C4⋊C4×C73C2314yes—
77C2×C4⋊C4×C73C2312yes—
78C2×C4⋊C4×C73C2314yes—
79C2×C4⋊C4×C73C2314yes—
80C2×C22⋊C4×C73C2314yes—
81C2×C22⋊C4×C73C2314yes—
82C2×C22⋊C4×C73C2314yes—
83C2×C22⋊C4×C73C2318yes—
84C2×C22⋊C4×C73C2318yes—
85C2×C22⋊C4×C73C2314yes—
86C2×C22⋊C4×C73C2314yes—
87C2×C22⋊C4×C73C2318yes—
88C2×C42×C73C2312yes—
89C2×C42×C73C2312yes—
90C2×C42×C73C2314yes—
91C2×C42×C73C2314yes—
92C2×C42×C73C2314yes—
93C2×C42×C73C2314yes—
94C2×C42×C73C2311yescharacteristic
95C2×C42×C73C2311yescharacteristic
96C2×C42×C73C2311yescharacteristic
97#95×C73C2314yes—
98#95×C73C2314yes—
99#95×C73C2314yes—
100#95×C73C2314yes—
101#95×C73C2318yes—
102#95×C73C2318yes—
103C24×C73C2414yes—
104C2×Q8×C73—24no—
105C2×Q8×C73—28no—
106C2×Q8×C73—24no—
107C2×Q8×C73—42no—
108C2×Q8×C73—42no—
109C2×Q8×C73—44no—
110C2×Q8×C73—44no—
111C2×Q8×C73C2×D418yes—
112C2×Q8×C73#4118yes—
113C2×D4×C73—24no—
114C2×D4×C73—28no—
115C2×D4×C73—24no—
116C2×D4×C73—24no—
117C2×D4×C73—44no—
118C2×D4×C73—44no—
119C22×C4×C73—28no—
120C22×C4×C73—28no—
121C22×C4×C73—28no—
122C22×C4×C73C2412yes—
123C22×C4×C73C2414yes—
124C22×C4×C73C2411yescharacteristic
125C22×C4×C73C2414yes—
126C22×C4×C73C2414yes—
127C22×C4×C73C2414yes—
128C22×C4×C73C2411yescharacteristic
129C22×C4×C73C2414yes—
130C22×C4×C73C2411yescharacteristic
131C22×C4×C73C2412yes—
132C4⋊C4×C73—24no—
133C4⋊C4×C73—216no—
134C4⋊C4×C73—28no—
135C4⋊C4×C73—24no—
136C4⋊C4×C73—28no—
137C4⋊C4×C73—216no—
138C4⋊C4×C73—42no—
139C4⋊C4×C73—44no—
140C4⋊C4×C73C2×D4116yes—
141C22⋊C4×C73—216no—
142C22⋊C4×C73—28no—
143C22⋊C4×C73—216no—
144C22⋊C4×C73—28no—
145C22⋊C4×C73—28no—
146C22⋊C4×C73—28no—
147C22⋊C4×C73—44no—
148C22⋊C4×C73—48no—
149C42×C73—28no—
150C42×C73—28no—
151C42×C73—28no—
152C42×C73—24no—
153C42×C73—22no—
154C42×C73—28no—
155C42×C73—24no—
156C42×C73—41no—
157C42×C73C2×D414yes—
158C23×C73—24no—
159C23×C73—24no—
160C23×C73—24no—
161C23×C73—24no—
162C23×C73—24no—
163C23×C73—24no—
164C23×C73—24no—
165C23×C73C2511yescharacteristic
166Q8×C73—216no—
167Q8×C73—48no—
168Q8×C73—88no—
169D4×C73—48no—
170D4×C73—88no—
171C2×C4×C73—24no—
172C2×C4×C73—21no—
173C2×C4×C73—24no—
174C2×C4×C73—22no—
175C2×C4×C73—24no—
176C2×C4×C73—21no—
177C2×C4×C73—24no—
178C2×C4×C73—24no—
179C2×C4×C73—24no—
180C2×C4×C73—22no—
181C2×C4×C73—22no—
182C2×C4×C73—28no—
183C2×C4×C73—24no—
184C2×C4×C73—24no—
185C2×C4×C73—21no—
186C2×C4×C73—24no—
187C2×C4×C73—22no—
188C2×C4×C73—24no—
189C2×C4×C73—22no—
190C2×C4×C73—24no—
191C2×C4×C73—416no—
192C2×C4×C73C22×D418yes—
193C2×C4×C73C22×D412yes—
194C2×C4×C73C22×D414yes—
195C2×C4×C73C2×#4118yes—
196C2×C4×C73C2×#4114yes—
197C2×C4×C73H(5,𝔽2)12yes—
198C2×C4×C732−1+414yes—
199C22×C73—44no—
200C22×C73—44no—
201C22×C73—44no—
202C22×C73—44no—
203C22×C73—44no—
204C22×C73—44no—
205C22×C73—44no—
206C22×C73C23×D411yescharacteristic
207C22×C73C22×#4112yes—
208C22×C73C2×H(5,𝔽2)11yescharacteristic
209C22×C73C2×2−1+412yes—
210C22×C73#58511yescharacteristic
211C4×C73—28no—
212C4×C73—28no—
213C4×C73—22no—
214C4×C73—22no—
215C4×C73—44no—
216C4×C73—41no—
217C4×C73—44no—
218C4×C73—44no—
219C4×C73—42no—
220C4×C73#54818yes—
221#59659C7311yesmaximal, characteristic, direct product factor, Sylow
222C2×C73—84no—
223C2×C73C2×#54812yes—
224C2×C73#348311yescharacteristic
225C2×C73#349611yescharacteristic
226C2×C73#351812yes—
227C2×C73#354411yescharacteristic
228C2×Q82C2×C7311yescharacteristic, complemented
229C2×#555C2×C7311yescharacteristic
230#2731C2×C7314yescomplemented
231#2545C2×C7314yescomplemented
232#2538C2×C7312yescomplemented
233#2491C2×C7314yescomplemented
234#2439C2×C7314yescomplemented
235#2426C2×C7314yescomplemented
236#2362C2×C7314yescomplemented
237C4×C4⋊Q8C2×C7311yescharacteristic, complemented
238C4×#124C2×C7312yescomplemented
239C73#5965911yescharacteristic, direct product factor, Sylow
240Q82—28no—
241#555—28no—
242C2×C4⋊Q8C22×C7314yes—
243C2×C4⋊Q8C22×C7314yes—
244C2×C4⋊Q8C22×C7311yescharacteristic
245C2×#124C22×C7312yes—
246C2×#124C22×C7314yes—
247C2×#124C22×C7314yes—
248C2×C22⋊Q8C22×C7314yes—
249C2×#121C22×C7314yes—
250C2×C4×Q8C22×C7314yes—
251C2×C4×Q8C22×C7312yes—
252C2×C4×Q8C22×C7312yes—
253C2×C4×D4C22×C7312yes—
254C2×#117C22×C7314yes—
255#396C22×C7318yes—
256#396C22×C7318yes—
257#394C22×C7318yes—
258#393C22×C7318yes—
259#391C22×C7312yes—
260#391C22×C7312yes—
261#391C22×C7314yes—
262#390C22×C7314yes—
263#390C22×C7314yes—
264#390C22×C7312yes—
265#389C22×C7314yes—
266#389C22×C7314yes—
267#388C22×C7318yes—
268#388C22×C7318yes—
269#388C22×C7314yes—
270#388C22×C7314yes—
271#387C22×C7318yes—
272#384C22×C7311yescharacteristic
273#382C22×C7312yes—
274#380C22×C7314yes—
275C4×C4⋊C4C22×C7314yes—
276C4×C4⋊C4C22×C7314yes—
277C4×C22⋊C4C22×C7314yes—
278C4×C22⋊C4C22×C7314yes—
279C43C22×C7311yescharacteristic
280C22×Q8C23×C7312yes—
281C22×Q8C23×C7314yes—
282C22×Q8C23×C7312yes—
283C22×D4C23×C7312yes—
284C22×D4C23×C7314yes—
285C23×C4C23×C7314yes—
286C4⋊Q8—216no—
287C4⋊Q8—216no—
288C4⋊Q8D4×C7318yes—
289#124—216no—
290#124—28no—
291#124—48no—
292C22⋊Q8—216no—
293#121—216no—
294C4×Q8—216no—
295C4×Q8—28no—
296C4×Q8—44no—
297C4×D4—28no—
298#117—216no—
299C2×C4⋊C4C23×C7312yes—
300C2×C4⋊C4C23×C7312yes—
301C2×C4⋊C4C23×C7318yes—
302C2×C4⋊C4C23×C7314yes—
303C2×C4⋊C4C23×C7314yes—
304C2×C4⋊C4C23×C7318yes—
305C2×C4⋊C4C23×C7312yes—
306C2×C4⋊C4C23×C7314yes—
307C2×C4⋊C4C23×C7314yes—
308C2×C22⋊C4C23×C7318yes—
309C2×C22⋊C4C23×C7314yes—
310C2×C22⋊C4C23×C7318yes—
311C2×C22⋊C4C23×C7314yes—
312C2×C22⋊C4C23×C7314yes—
313C2×C22⋊C4C23×C7314yes—
314C2×C22⋊C4C23×C7318yes—
315C2×C22⋊C4C23×C7314yes—
316C2×C42C23×C7314yes—
317C2×C42C23×C7314yes—
318C2×C42C23×C7314yes—
319C2×C42C23×C7311yescharacteristic
320C2×C42C23×C7314yes—
321C2×C42C23×C7312yes—
322C2×C42C23×C7312yes—
323C2×C42C23×C7311yescharacteristic
324C2×C42C23×C7311yescharacteristic
325#95C23×C7314yes—
326#95C23×C7314yes—
327#95C23×C7314yes—
328#95C23×C7314yes—
329#95C23×C7318yes—
330#95C23×C7318yes—
331C24C24×C7314yes—
332C2×Q8—24no—
333C2×Q8—28no—
334C2×Q8—24no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8C2×D4×C7318yes—
340C2×Q8#41×C7318yes—
341C2×D4—24no—
342C2×D4—28no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—44no—
346C2×D4—44no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4C24×C7312yes—
351C22×C4C24×C7314yes—
352C22×C4C24×C7311yescharacteristic
353C22×C4C24×C7314yes—
354C22×C4C24×C7314yes—
355C22×C4C24×C7314yes—
356C22×C4C24×C7311yescharacteristic
357C22×C4C24×C7314yes—
358C22×C4C24×C7311yescharacteristic
359C22×C4C24×C7312yes—
360C4⋊C4—24no—
361C4⋊C4—216no—
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—28no—
365C4⋊C4—216no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4C2×D4×C73116yes—
369C22⋊C4—216no—
370C22⋊C4—28no—
371C22⋊C4—216no—
372C22⋊C4—28no—
373C22⋊C4—28no—
374C22⋊C4—28no—
375C22⋊C4—44no—
376C22⋊C4—48no—
377C42—28no—
378C42—28no—
379C42—28no—
380C42—24no—
381C42—22no—
382C42—28no—
383C42—24no—
384C42—41no—
385C42C2×D4×C7314yes—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23C25×C7311yescharacteristic
394Q8—216no—
395Q8—48no—
396Q8—88no—
397D4—48no—
398D4—88no—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—416no—
420C2×C4C22×D4×C7318yes—
421C2×C4C22×D4×C7312yes—
422C2×C4C22×D4×C7314yes—
423C2×C4C2×#41×C7318yes—
424C2×C4C2×#41×C7314yes—
425C2×C4H(5,𝔽2)×C7312yes—
426C2×C42−1+4×C7314yes—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—44no—
434C22C23×D4×C7311yescharacteristic
435C22C22×#41×C7312yes—
436C22C2×H(5,𝔽2)×C7311yescharacteristic
437C22C2×2−1+4×C7312yes—
438C22#585×C7311yescharacteristic
439C4—28no—
440C4—28no—
441C4—22no—
442C4—22no—
443C4—44no—
444C4—41no—
445C4—44no—
446C4—44no—
447C4—42no—
448C4#548×C7318yes—
449C2—84no—
450C2C2×#548×C7312yes—
451C2C73×#348311yescharacteristic
452C2C73×#349611yescharacteristic
453C2C73×#351812yes—
454C2C73×#354411yescharacteristic
subgroupG/H
center, socleC23×C73C25
commutator, Frattini, cosocleC23C25×C73
prime radicalC2×#555×C73C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×Q82×C73C2semidirect
3. C73×#2731C2semidirect
4. C73×#2545C2semidirect
5. C73×#2538C2semidirect
6. C73×#2491C2semidirect
7. C73×#2439C2semidirect
8. C73×#2426C2semidirect
9. C73×#2362C2semidirect
10. C4×C4⋊Q8×C73C2semidirect
11. C4×#124×C73C2semidirect
221. #59659C73direct (N × H)
228. C2×Q82C2×C73semidirect
230. #2731C2×C73semidirect
231. #2545C2×C73semidirect
232. #2538C2×C73semidirect
233. #2491C2×C73semidirect
234. #2439C2×C73semidirect
235. #2426C2×C73semidirect
236. #2362C2×C73semidirect
237. C4×C4⋊Q8C2×C73semidirect
238. C4×#124C2×C73semidirect
239. C73#59659direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q82×C732
C2×#555×C732
C73×#27312
C73×#25452
C73×#25382
C73×#24912
C73×#24392
C73×#24262
C73×#23622
C4×C4⋊Q8×C732
C4×#124×C732
#5965973