Finite Group Databasebeta

← C73×#59660 #32969064

Subgroups — C73×#59660 #32969064

Order 18688.

2300 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554×C73C212yesmaximal, complemented
2C73×#2728C214yesmaximal, complemented
3C73×#2582C211yesmaximal, characteristic, complemented
4C73×#2537C211yesmaximal, characteristic
5C73×#2536C214yesmaximal, complemented
6C73×#2489C214yesmaximal, complemented
7C73×#2442C214yesmaximal, complemented
8C73×#2427C214yesmaximal, complemented
9C73×#2361C214yesmaximal, complemented
10C4×C4⋊Q8×C73C211yesmaximal, characteristic, complemented
11C4×#124×C73C212yesmaximal, complemented
12#554×C73—216no—
13C2×C4⋊Q8×C73C2211yescharacteristic
14C2×#125×C73C2214yes—
15C2×#124×C73C2212yes—
16C2×C22⋊Q8×C73C2218yes—
17C2×C22⋊Q8×C73C2214yes—
18C2×#121×C73C2214yes—
19C2×C4×Q8×C73C2212yes—
20C2×C4×Q8×C73C2212yes—
21C2×C4×D4×C73C2212yes—
22C2×C4×D4×C73C2214yes—
23C2×C4×D4×C73C2214yes—
24C22×C4⋊C4×C73C2214yes—
25#400×C73C2218yes—
26#398×C73C2214yes—
27#398×C73C2214yes—
28#397×C73C2218yes—
29#394×C73C2214yes—
30#394×C73C2214yes—
31#391×C73C2212yes—
32#391×C73C2212yes—
33#391×C73C2214yes—
34#390×C73C2212yes—
35#389×C73C2214yes—
36#389×C73C2214yes—
37#388×C73C2214yes—
38#388×C73C2218yes—
39#388×C73C2214yes—
40#386×C73C2218yes—
41#385×C73C2218yes—
42#384×C73C2211yescharacteristic
43#382×C73C2212yes—
44#382×C73C2214yes—
45#380×C73C2214yes—
46#380×C73C2218yes—
47C4×C4⋊C4×C73C2214yes—
48C4×C22⋊C4×C73C2214yes—
49C4×C22⋊C4×C73C2214yes—
50C43×C73C2211yescharacteristic
51C22×Q8×C73C2312yes—
52C22×Q8×C73C2312yes—
53C22×D4×C73C2314yes—
54C22×D4×C73C2312yes—
55C23×C4×C73C2314yes—
56C23×C4×C73C2318yes—
57C4⋊Q8×C73D418yescomplemented
58#125×C73—216no—
59#124×C73—28no—
60C22⋊Q8×C73—232no—
61C22⋊Q8×C73—216no—
62#121×C73—216no—
63C4×Q8×C73—28no—
64C4×Q8×C73—44no—
65C4×D4×C73—216no—
66C4×D4×C73—28no—
67C4×D4×C73—48no—
68C2×C4⋊C4×C73—216no—
69C2×C4⋊C4×C73C2314yes—
70C2×C4⋊C4×C73C2312yes—
71C2×C4⋊C4×C73C2318yes—
72C2×C4⋊C4×C73C2312yes—
73C2×C4⋊C4×C73C2318yes—
74C2×C4⋊C4×C73C2314yes—
75C2×C4⋊C4×C73C2314yes—
76C2×C4⋊C4×C73C2312yes—
77C2×C4⋊C4×C73C2314yes—
78C2×C4⋊C4×C73C2314yes—
79C2×C22⋊C4×C73C2314yes—
80C2×C22⋊C4×C73C2314yes—
81C2×C22⋊C4×C73C2314yes—
82C2×C22⋊C4×C73C2318yes—
83C2×C22⋊C4×C73C2318yes—
84C2×C22⋊C4×C73C2314yes—
85C2×C22⋊C4×C73C2314yes—
86C2×C42×C73C2312yes—
87C2×C42×C73C2312yes—
88C2×C42×C73C2314yes—
89C2×C42×C73C2314yes—
90C2×C42×C73C2311yescharacteristic
91C2×C42×C73C2311yescharacteristic
92C2×C42×C73C2311yescharacteristic
93#95×C73C2314yes—
94#95×C73C2314yes—
95#95×C73C2314yes—
96#95×C73C2314yes—
97#95×C73C2318yes—
98#95×C73C2314yes—
99#95×C73C2318yes—
100#95×C73C2314yes—
101C24×C73C2414yes—
102C2×Q8×C73—24no—
103C2×Q8×C73—24no—
104C2×Q8×C73—24no—
105C2×Q8×C73—42no—
106C2×Q8×C73—42no—
107C2×Q8×C73C2×D418yes—
108C2×D4×C73—216no—
109C2×D4×C73—24no—
110C2×D4×C73—24no—
111C2×D4×C73—44no—
112C2×D4×C73#4118yes—
113C22×C4×C73—28no—
114C22×C4×C73—28no—
115C22×C4×C73—28no—
116C22×C4×C73—216no—
117C22×C4×C73—48no—
118C22×C4×C73—48no—
119C22×C4×C73C2412yes—
120C22×C4×C73C2414yes—
121C22×C4×C73C2414yes—
122C22×C4×C73C2414yes—
123C22×C4×C73C2411yescharacteristic
124C22×C4×C73C2414yes—
125C22×C4×C73C2414yes—
126C22×C4×C73C2411yescharacteristic
127C22×C4×C73C2411yescharacteristic
128C22×C4×C73C2412yes—
129C4⋊C4×C73—24no—
130C4⋊C4×C73—216no—
131C4⋊C4×C73—24no—
132C4⋊C4×C73—28no—
133C4⋊C4×C73—216no—
134C4⋊C4×C73—28no—
135C4⋊C4×C73—44no—
136C4⋊C4×C73—42no—
137C4⋊C4×C73—44no—
138C4⋊C4×C73C2×D4116yes—
139C22⋊C4×C73—216no—
140C22⋊C4×C73—28no—
141C22⋊C4×C73—28no—
142C22⋊C4×C73—216no—
143C22⋊C4×C73—28no—
144C22⋊C4×C73—28no—
145C22⋊C4×C73—44no—
146C42×C73—28no—
147C42×C73—24no—
148C42×C73—28no—
149C42×C73—22no—
150C42×C73—24no—
151C42×C73—41no—
152C42×C73C2×D414yes—
153C23×C73—24no—
154C23×C73—24no—
155C23×C73—24no—
156C23×C73—24no—
157C23×C73—24no—
158C23×C73—24no—
159C23×C73C2×#4118yes—
160C23×C73C2511yescharacteristic
161Q8×C73—48no—
162Q8×C73—48no—
163D4×C73—216no—
164D4×C73—416no—
165C2×C4×C73—21no—
166C2×C4×C73—24no—
167C2×C4×C73—22no—
168C2×C4×C73—21no—
169C2×C4×C73—24no—
170C2×C4×C73—24no—
171C2×C4×C73—24no—
172C2×C4×C73—24no—
173C2×C4×C73—22no—
174C2×C4×C73—22no—
175C2×C4×C73—24no—
176C2×C4×C73—24no—
177C2×C4×C73—28no—
178C2×C4×C73—21no—
179C2×C4×C73—24no—
180C2×C4×C73—24no—
181C2×C4×C73—22no—
182C2×C4×C73—24no—
183C2×C4×C73—24no—
184C2×C4×C73—22no—
185C2×C4×C73—24no—
186C2×C4×C73—416no—
187C2×C4×C73—816no—
188C2×C4×C73C22×D412yes—
189C2×C4×C73C22×D418yes—
190C2×C4×C73C22×D414yes—
191C2×C4×C73C2×#4114yes—
192C2×C4×C73H(5,𝔽2)12yes—
193C2×C4×C732−1+414yes—
194C22×C73—28no—
195C22×C73—28no—
196C22×C73—28no—
197C22×C73—44no—
198C22×C73—44no—
199C22×C73—44no—
200C22×C73—44no—
201C22×C73C23×D411yescharacteristic
202C22×C73C22×#4112yes—
203C22×C73C2×H(5,𝔽2)11yescharacteristic
204C22×C73C2×2−1+412yes—
205C22×C73#58511yescharacteristic
206C4×C73—28no—
207C4×C73—22no—
208C4×C73—22no—
209C4×C73—44no—
210C4×C73—44no—
211C4×C73—41no—
212C4×C73—44no—
213C4×C73—44no—
214C4×C73—42no—
215C4×C73#54818yes—
216#59660C7311yesmaximal, characteristic, direct product factor, Sylow
217C2×C73—48no—
218C2×C73C2×#54812yes—
219C2×C73#348111yescharacteristic
220C2×C73#349611yescharacteristic
221C2×C73#351512yes—
222C2×C73#354411yescharacteristic
223C2×#554C2×C7312yescomplemented
224#2728C2×C7314yescomplemented
225#2582C2×C7311yescharacteristic, complemented
226#2537C2×C7311yescharacteristic
227#2536C2×C7314yescomplemented
228#2489C2×C7314yescomplemented
229#2442C2×C7314yescomplemented
230#2427C2×C7314yescomplemented
231#2361C2×C7314yescomplemented
232C4×C4⋊Q8C2×C7311yescharacteristic, complemented
233C4×#124C2×C7312yescomplemented
234C73#5966011yescharacteristic, direct product factor, Sylow
235#554—216no—
236C2×C4⋊Q8C22×C7311yescharacteristic
237C2×#125C22×C7314yes—
238C2×#124C22×C7312yes—
239C2×C22⋊Q8C22×C7314yes—
240C2×C22⋊Q8C22×C7318yes—
241C2×#121C22×C7314yes—
242C2×C4×Q8C22×C7312yes—
243C2×C4×Q8C22×C7312yes—
244C2×C4×D4C22×C7314yes—
245C2×C4×D4C22×C7312yes—
246C2×C4×D4C22×C7314yes—
247C22×C4⋊C4C22×C7314yes—
248#400C22×C7318yes—
249#398C22×C7314yes—
250#398C22×C7314yes—
251#397C22×C7318yes—
252#394C22×C7314yes—
253#394C22×C7314yes—
254#391C22×C7312yes—
255#391C22×C7314yes—
256#391C22×C7312yes—
257#390C22×C7312yes—
258#389C22×C7314yes—
259#389C22×C7314yes—
260#388C22×C7318yes—
261#388C22×C7314yes—
262#388C22×C7314yes—
263#386C22×C7318yes—
264#385C22×C7318yes—
265#384C22×C7311yescharacteristic
266#382C22×C7312yes—
267#382C22×C7314yes—
268#380C22×C7318yes—
269#380C22×C7314yes—
270C4×C4⋊C4C22×C7314yes—
271C4×C22⋊C4C22×C7314yes—
272C4×C22⋊C4C22×C7314yes—
273C43C22×C7311yescharacteristic
274C22×Q8C23×C7312yes—
275C22×Q8C23×C7312yes—
276C22×D4C23×C7314yes—
277C22×D4C23×C7312yes—
278C23×C4C23×C7314yes—
279C23×C4C23×C7318yes—
280C4⋊Q8D4×C7318yescomplemented
281#125—216no—
282#124—28no—
283C22⋊Q8—232no—
284C22⋊Q8—216no—
285#121—216no—
286C4×Q8—28no—
287C4×Q8—44no—
288C4×D4—216no—
289C4×D4—28no—
290C4×D4—48no—
291C2×C4⋊C4—216no—
292C2×C4⋊C4C23×C7312yes—
293C2×C4⋊C4C23×C7318yes—
294C2×C4⋊C4C23×C7312yes—
295C2×C4⋊C4C23×C7314yes—
296C2×C4⋊C4C23×C7314yes—
297C2×C4⋊C4C23×C7318yes—
298C2×C4⋊C4C23×C7314yes—
299C2×C4⋊C4C23×C7314yes—
300C2×C4⋊C4C23×C7312yes—
301C2×C4⋊C4C23×C7314yes—
302C2×C22⋊C4C23×C7314yes—
303C2×C22⋊C4C23×C7318yes—
304C2×C22⋊C4C23×C7314yes—
305C2×C22⋊C4C23×C7318yes—
306C2×C22⋊C4C23×C7314yes—
307C2×C22⋊C4C23×C7314yes—
308C2×C22⋊C4C23×C7314yes—
309C2×C42C23×C7314yes—
310C2×C42C23×C7314yes—
311C2×C42C23×C7311yescharacteristic
312C2×C42C23×C7312yes—
313C2×C42C23×C7312yes—
314C2×C42C23×C7311yescharacteristic
315C2×C42C23×C7311yescharacteristic
316#95C23×C7314yes—
317#95C23×C7314yes—
318#95C23×C7314yes—
319#95C23×C7314yes—
320#95C23×C7318yes—
321#95C23×C7314yes—
322#95C23×C7318yes—
323#95C23×C7314yes—
324C24C24×C7314yes—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8C2×D4×C7318yes—
331C2×D4—216no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—44no—
335C2×D4#41×C7318yes—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4C24×C7312yes—
343C22×C4C24×C7314yes—
344C22×C4C24×C7314yes—
345C22×C4C24×C7314yes—
346C22×C4C24×C7311yescharacteristic
347C22×C4C24×C7314yes—
348C22×C4C24×C7314yes—
349C22×C4C24×C7311yescharacteristic
350C22×C4C24×C7311yescharacteristic
351C22×C4C24×C7312yes—
352C4⋊C4—24no—
353C4⋊C4—216no—
354C4⋊C4—24no—
355C4⋊C4—28no—
356C4⋊C4—216no—
357C4⋊C4—28no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4C2×D4×C73116yes—
362C22⋊C4—216no—
363C22⋊C4—28no—
364C22⋊C4—28no—
365C22⋊C4—216no—
366C22⋊C4—28no—
367C22⋊C4—28no—
368C22⋊C4—44no—
369C42—28no—
370C42—24no—
371C42—28no—
372C42—22no—
373C42—24no—
374C42—41no—
375C42C2×D4×C7314yes—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23C2×#41×C7318yes—
383C23C25×C7311yescharacteristic
384Q8—48no—
385Q8—48no—
386D4—216no—
387D4—416no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—416no—
410C2×C4—816no—
411C2×C4C22×D4×C7312yes—
412C2×C4C22×D4×C7318yes—
413C2×C4C22×D4×C7314yes—
414C2×C4C2×#41×C7314yes—
415C2×C4H(5,𝔽2)×C7312yes—
416C2×C42−1+4×C7314yes—
417C22—28no—
418C22—28no—
419C22—28no—
420C22—44no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22C23×D4×C7311yescharacteristic
425C22C22×#41×C7312yes—
426C22C2×H(5,𝔽2)×C7311yescharacteristic
427C22C2×2−1+4×C7312yes—
428C22#585×C7311yescharacteristic
429C4—28no—
430C4—22no—
431C4—22no—
432C4—44no—
433C4—44no—
434C4—44no—
435C4—41no—
436C4—44no—
437C4—42no—
438C4#548×C7318yes—
439C2—48no—
440C2C2×#548×C7312yes—
441C2C73×#348111yescharacteristic
442C2C73×#349611yescharacteristic
443C2C73×#351512yes—
444C2C73×#354411yescharacteristic
subgroupG/H
center, socleC23×C73C25
commutator, Frattini, cosocleC23C25×C73
prime radicalC73×#2537C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#554×C73C2semidirect
2. C73×#2728C2semidirect
3. C73×#2582C2semidirect
5. C73×#2536C2semidirect
6. C73×#2489C2semidirect
7. C73×#2442C2semidirect
8. C73×#2427C2semidirect
9. C73×#2361C2semidirect
10. C4×C4⋊Q8×C73C2semidirect
11. C4×#124×C73C2semidirect
57. C4⋊Q8×C73D4semidirect
216. #59660C73direct (N × H)
223. C2×#554C2×C73semidirect
224. #2728C2×C73semidirect
225. #2582C2×C73semidirect
227. #2536C2×C73semidirect
228. #2489C2×C73semidirect
229. #2442C2×C73semidirect
230. #2427C2×C73semidirect
231. #2361C2×C73semidirect
232. C4×C4⋊Q8C2×C73semidirect
233. C4×#124C2×C73semidirect
234. C73#59660direct (N × H)
280. C4⋊Q8D4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#554×C732
C73×#27282
C73×#25822
C73×#25372
C73×#25362
C73×#24892
C73×#24422
C73×#24272
C73×#23612
C4×C4⋊Q8×C732
C4×#124×C732
#5966073