Finite Group Databasebeta

← C73×#59419 #32968823

Subgroups — C73×#59419 #32968823

Order 18688.

2884 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
1C2×#550×C73C212yesmaximal, complemented
2C73×#2699C214yesmaximal, complemented
3C73×#2582C211yesmaximal, characteristic, complemented
4C73×#2543C214yesmaximal, complemented
5C73×#2537C211yesmaximal, characteristic, complemented
6C73×#2431C214yesmaximal, complemented
7C73×#2428C214yesmaximal, complemented
8C73×#2420C214yesmaximal, complemented
9C73×#2339C214yesmaximal, complemented
10C4×#127×C73C211yesmaximal, characteristic, complemented
11C4×#124×C73C212yesmaximal, complemented
12C22×#41×C73C2214yes—
13#550×C73—216no—
14C2×#127×C73C2214yes—
15C2×#127×C73C2211yescharacteristic
16C2×#124×C73C2212yes—
17C2×#121×C73C2218yes—
18C2×#121×C73C2214yes—
19C2×#121×C73C2214yes—
20C2×#121×C73C2214yes—
21C2×C4×Q8×C73C2212yes—
22C2×C4×D4×C73C2212yes—
23C2×C4×D4×C73C2214yes—
24C2×C4×D4×C73C2212yes—
25C2×C4×D4×C73C2214yes—
26#398×C73C2214yes—
27#397×C73C2214yes—
28#394×C73C2214yes—
29#394×C73C2218yes—
30#392×C73C2218yes—
31#392×C73C2214yes—
32#391×C73C2212yes—
33#391×C73C2214yes—
34#390×C73C2212yes—
35#390×C73C2212yes—
36#390×C73C2214yes—
37#389×C73C2214yes—
38#388×C73C2214yes—
39#388×C73C2214yes—
40#386×C73C2218yes—
41#386×C73C2218yes—
42#386×C73C2218yes—
43#384×C73C2211yescharacteristic
44#382×C73C2212yes—
45#381×C73C2218yes—
46#380×C73C2214yes—
47C4×C22⋊C4×C73C2214yes—
48C4×C22⋊C4×C73C2214yes—
49C2×#95×C73C2214yes—
50C43×C73C2211yescharacteristic
51C2×#41×C73—216no—
52C2×#41×C73—216no—
53C2×#41×C73—48no—
54C22×Q8×C73C2312yes—
55C22×D4×C73C2312yes—
56C22×D4×C73C2318yes—
57C22×D4×C73C2314yes—
58C22×D4×C73C2318yes—
59C22×D4×C73C2314yes—
60C22×D4×C73C2312yes—
61C23×C4×C73C2314yes—
62C23×C4×C73C2318yes—
63C23×C4×C73C2314yes—
64#127×C73—216no—
65#127×C73D418yescomplemented
66#124×C73—28no—
67#121×C73—216no—
68#121×C73—232no—
69#121×C73—48no—
70#121×C73—48no—
71C4×Q8×C73—28no—
72C4×D4×C73—216no—
73C4×D4×C73—28no—
74C4×D4×C73—48no—
75C4×D4×C73—44no—
76C2×C4⋊C4×C73C2312yes—
77C2×C4⋊C4×C73C2314yes—
78C2×C4⋊C4×C73C2312yes—
79C2×C4⋊C4×C73C2314yes—
80C2×C4⋊C4×C73C2312yes—
81C2×C4⋊C4×C73C2314yes—
82C2×C22⋊C4×C73C2314yes—
83C2×C22⋊C4×C73C2314yes—
84C2×C22⋊C4×C73C2314yes—
85C2×C22⋊C4×C73C2318yes—
86C2×C22⋊C4×C73C2314yes—
87C2×C22⋊C4×C73C2314yes—
88C2×C22⋊C4×C73C2318yes—
89C2×C22⋊C4×C73C2318yes—
90C2×C22⋊C4×C73C2314yes—
91C2×C42×C73C2312yes—
92C2×C42×C73C2314yes—
93C2×C42×C73C2312yes—
94C2×C42×C73C2311yescharacteristic
95C2×C42×C73C2311yescharacteristic
96C2×C42×C73C2311yescharacteristic
97#95×C73C2314yes—
98#95×C73C2314yes—
99#95×C73C2314yes—
100#95×C73C2314yes—
101#95×C73C2314yes—
102#95×C73C2318yes—
103#95×C73C2314yes—
104C24×C73C2414yes—
105C24×C73C2414yes—
106#41×C73—464no—
107C2×Q8×C73—24no—
108C2×Q8×C73—24no—
109C2×Q8×C73#4118yescomplemented
110C2×D4×C73—216no—
111C2×D4×C73—216no—
112C2×D4×C73—24no—
113C2×D4×C73—28no—
114C2×D4×C73—24no—
115C2×D4×C73—216no—
116C2×D4×C73—28no—
117C2×D4×C73—24no—
118C2×D4×C73—42no—
119C2×D4×C73—42no—
120C2×D4×C73—48no—
121C2×D4×C73—48no—
122C2×D4×C73—48no—
123C2×D4×C73—44no—
124C2×D4×C73—48no—
125C2×D4×C73C2×D4116yes—
126C2×D4×C73C2×D418yes—
127C22×C4×C73—216no—
128C22×C4×C73—28no—
129C22×C4×C73—216no—
130C22×C4×C73—28no—
131C22×C4×C73—28no—
132C22×C4×C73—28no—
133C22×C4×C73—48no—
134C22×C4×C73—48no—
135C22×C4×C73C2414yes—
136C22×C4×C73C2411yescharacteristic
137C22×C4×C73C2412yes—
138C22×C4×C73C2414yes—
139C22×C4×C73C2414yes—
140C22×C4×C73C2411yescharacteristic
141C22×C4×C73C2414yes—
142C22×C4×C73C2411yescharacteristic
143C22×C4×C73C2412yes—
144C4⋊C4×C73—24no—
145C4⋊C4×C73—28no—
146C4⋊C4×C73—24no—
147C4⋊C4×C73—28no—
148C4⋊C4×C73—42no—
149C4⋊C4×C73—44no—
150C22⋊C4×C73—216no—
151C22⋊C4×C73—28no—
152C22⋊C4×C73—28no—
153C22⋊C4×C73—28no—
154C22⋊C4×C73—28no—
155C22⋊C4×C73—44no—
156C22⋊C4×C73—48no—
157C22⋊C4×C73—44no—
158C22⋊C4×C73—48no—
159C42×C73—28no—
160C42×C73—24no—
161C42×C73—22no—
162C42×C73—24no—
163C42×C73—41no—
164C42×C73C2×D414yes—
165C23×C73—28no—
166C23×C73—24no—
167C23×C73—28no—
168C23×C73—24no—
169C23×C73—24no—
170C23×C73—24no—
171C23×C73—24no—
172C23×C73—24no—
173C23×C73—24no—
174C23×C73—24no—
175C23×C73—24no—
176C23×C73C22×D418yes—
177C23×C73C2511yescharacteristic
178Q8×C73—216no—
179D4×C73—232no—
180D4×C73—48no—
181D4×C73—432no—
182D4×C73—48no—
183D4×C73—816no—
184D4×C73—88no—
185C2×C4×C73—24no—
186C2×C4×C73—21no—
187C2×C4×C73—22no—
188C2×C4×C73—24no—
189C2×C4×C73—21no—
190C2×C4×C73—24no—
191C2×C4×C73—24no—
192C2×C4×C73—22no—
193C2×C4×C73—22no—
194C2×C4×C73—24no—
195C2×C4×C73—21no—
196C2×C4×C73—22no—
197C2×C4×C73—24no—
198C2×C4×C73—28no—
199C2×C4×C73—24no—
200C2×C4×C73—24no—
201C2×C4×C73—22no—
202C2×C4×C73—24no—
203C2×C4×C73—432no—
204C2×C4×C73—416no—
205C2×C4×C73—88no—
206C2×C4×C73C22×D412yes—
207C2×C4×C73C22×D414yes—
208C2×C4×C73C2×#4118yes—
209C2×C4×C73C2×#4114yes—
210C2×C4×C73H(5,𝔽2)12yes—
211C2×C4×C73H(5,𝔽2)14yes—
212C22×C73—216no—
213C22×C73—28no—
214C22×C73—44no—
215C22×C73—48no—
216C22×C73—44no—
217C22×C73—44no—
218C22×C73—44no—
219C22×C73—44no—
220C22×C73—44no—
221C22×C73—44no—
222C22×C73—44no—
223C22×C73—44no—
224C22×C73C23×D411yescharacteristic
225C22×C73C22×#4112yes—
226C22×C73C2×H(5,𝔽2)12yes—
227C22×C73C2×H(5,𝔽2)11yescharacteristic
228C22×C73#58511yescharacteristic
229C4×C73—28no—
230C4×C73—22no—
231C4×C73—22no—
232C4×C73—44no—
233C4×C73—41no—
234C4×C73—44no—
235C4×C73—42no—
236C4×C73—44no—
237C4×C73#54618yes—
238#59419C7311yesmaximal, characteristic, direct product factor, Sylow
239C2×C73—48no—
240C2×C73—84no—
241C2×C73C2×#54612yes—
242C2×C73#348311yescharacteristic
243C2×C73#349611yescharacteristic
244C2×C73#351712yes—
245C2×C73#354311yescharacteristic
246C2×#550C2×C7312yescomplemented
247#2699C2×C7314yescomplemented
248#2582C2×C7311yescharacteristic, complemented
249#2543C2×C7314yescomplemented
250#2537C2×C7311yescharacteristic, complemented
251#2431C2×C7314yescomplemented
252#2428C2×C7314yescomplemented
253#2420C2×C7314yescomplemented
254#2339C2×C7314yescomplemented
255C4×#127C2×C7311yescharacteristic, complemented
256C4×#124C2×C7312yescomplemented
257C73#5941911yescharacteristic, direct product factor, Sylow
258C22×#41C22×C7314yes—
259#550—216no—
260C2×#127C22×C7314yes—
261C2×#127C22×C7311yescharacteristic
262C2×#124C22×C7312yes—
263C2×#121C22×C7314yes—
264C2×#121C22×C7318yes—
265C2×#121C22×C7314yes—
266C2×#121C22×C7314yes—
267C2×C4×Q8C22×C7312yes—
268C2×C4×D4C22×C7314yes—
269C2×C4×D4C22×C7312yes—
270C2×C4×D4C22×C7314yes—
271C2×C4×D4C22×C7312yes—
272#398C22×C7314yes—
273#397C22×C7314yes—
274#394C22×C7318yes—
275#394C22×C7314yes—
276#392C22×C7318yes—
277#392C22×C7314yes—
278#391C22×C7314yes—
279#391C22×C7312yes—
280#390C22×C7314yes—
281#390C22×C7312yes—
282#390C22×C7312yes—
283#389C22×C7314yes—
284#388C22×C7314yes—
285#388C22×C7314yes—
286#386C22×C7318yes—
287#386C22×C7318yes—
288#386C22×C7318yes—
289#384C22×C7311yescharacteristic
290#382C22×C7312yes—
291#381C22×C7318yes—
292#380C22×C7314yes—
293C4×C22⋊C4C22×C7314yes—
294C4×C22⋊C4C22×C7314yes—
295C2×#95C22×C7314yes—
296C43C22×C7311yescharacteristic
297C2×#41—216no—
298C2×#41—216no—
299C2×#41—48no—
300C22×Q8C23×C7312yes—
301C22×D4C23×C7312yes—
302C22×D4C23×C7318yes—
303C22×D4C23×C7314yes—
304C22×D4C23×C7318yes—
305C22×D4C23×C7314yes—
306C22×D4C23×C7312yes—
307C23×C4C23×C7314yes—
308C23×C4C23×C7318yes—
309C23×C4C23×C7314yes—
310#127—216no—
311#127D4×C7318yescomplemented
312#124—28no—
313#121—216no—
314#121—232no—
315#121—48no—
316#121—48no—
317C4×Q8—28no—
318C4×D4—216no—
319C4×D4—28no—
320C4×D4—48no—
321C4×D4—44no—
322C2×C4⋊C4C23×C7312yes—
323C2×C4⋊C4C23×C7314yes—
324C2×C4⋊C4C23×C7312yes—
325C2×C4⋊C4C23×C7314yes—
326C2×C4⋊C4C23×C7312yes—
327C2×C4⋊C4C23×C7314yes—
328C2×C22⋊C4C23×C7318yes—
329C2×C22⋊C4C23×C7314yes—
330C2×C22⋊C4C23×C7314yes—
331C2×C22⋊C4C23×C7314yes—
332C2×C22⋊C4C23×C7314yes—
333C2×C22⋊C4C23×C7318yes—
334C2×C22⋊C4C23×C7314yes—
335C2×C22⋊C4C23×C7318yes—
336C2×C22⋊C4C23×C7314yes—
337C2×C42C23×C7312yes—
338C2×C42C23×C7311yescharacteristic
339C2×C42C23×C7314yes—
340C2×C42C23×C7312yes—
341C2×C42C23×C7311yescharacteristic
342C2×C42C23×C7311yescharacteristic
343#95C23×C7314yes—
344#95C23×C7314yes—
345#95C23×C7314yes—
346#95C23×C7314yes—
347#95C23×C7314yes—
348#95C23×C7314yes—
349#95C23×C7318yes—
350C24C24×C7314yes—
351C24C24×C7314yes—
352#41—464no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8#41×C7318yescomplemented
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—24no—
359C2×D4—28no—
360C2×D4—24no—
361C2×D4—216no—
362C2×D4—28no—
363C2×D4—24no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4C2×D4×C73116yes—
372C2×D4C2×D4×C7318yes—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4C24×C7314yes—
382C22×C4C24×C7311yescharacteristic
383C22×C4C24×C7312yes—
384C22×C4C24×C7314yes—
385C22×C4C24×C7314yes—
386C22×C4C24×C7311yescharacteristic
387C22×C4C24×C7314yes—
388C22×C4C24×C7311yescharacteristic
389C22×C4C24×C7312yes—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—24no—
393C4⋊C4—28no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C22⋊C4—216no—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—44no—
402C22⋊C4—48no—
403C22⋊C4—44no—
404C22⋊C4—48no—
405C42—28no—
406C42—24no—
407C42—22no—
408C42—24no—
409C42—41no—
410C42C2×D4×C7314yes—
411C23—28no—
412C23—24no—
413C23—28no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23C22×D4×C7318yes—
423C23C25×C7311yescharacteristic
424Q8—216no—
425D4—232no—
426D4—48no—
427D4—432no—
428D4—48no—
429D4—816no—
430D4—88no—
431C2×C4—24no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—432no—
450C2×C4—416no—
451C2×C4—88no—
452C2×C4C22×D4×C7312yes—
453C2×C4C22×D4×C7314yes—
454C2×C4C2×#41×C7318yes—
455C2×C4C2×#41×C7314yes—
456C2×C4H(5,𝔽2)×C7312yes—
457C2×C4H(5,𝔽2)×C7314yes—
458C22—216no—
459C22—28no—
460C22—44no—
461C22—48no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22C23×D4×C7311yescharacteristic
471C22C22×#41×C7312yes—
472C22C2×H(5,𝔽2)×C7312yes—
473C22C2×H(5,𝔽2)×C7311yescharacteristic
474C22#585×C7311yescharacteristic
475C4—28no—
476C4—22no—
477C4—22no—
478C4—44no—
479C4—41no—
480C4—44no—
481C4—42no—
482C4—44no—
483C4#546×C7318yes—
484C2—48no—
485C2—84no—
486C2C2×#546×C7312yes—
487C2C73×#348311yescharacteristic
488C2C73×#349611yescharacteristic
489C2C73×#351712yes—
490C2C73×#354311yescharacteristic
subgroupG/H
center, socleC23×C73C25
commutator, Frattini, cosocleC23C25×C73
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#550×C73C2semidirect
2. C73×#2699C2semidirect
3. C73×#2582C2semidirect
4. C73×#2543C2semidirect
5. C73×#2537C2semidirect
6. C73×#2431C2semidirect
7. C73×#2428C2semidirect
8. C73×#2420C2semidirect
9. C73×#2339C2semidirect
10. C4×#127×C73C2semidirect
11. C4×#124×C73C2semidirect
65. #127×C73D4semidirect
109. C2×Q8×C73#41semidirect
238. #59419C73direct (N × H)
246. C2×#550C2×C73semidirect
247. #2699C2×C73semidirect
248. #2582C2×C73semidirect
249. #2543C2×C73semidirect
250. #2537C2×C73semidirect
251. #2431C2×C73semidirect
252. #2428C2×C73semidirect
253. #2420C2×C73semidirect
254. #2339C2×C73semidirect
255. C4×#127C2×C73semidirect
256. C4×#124C2×C73semidirect
257. C73#59419direct (N × H)
311. #127D4×C73semidirect
355. C2×Q8#41×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#550×C732
C73×#26992
C73×#25822
C73×#25432
C73×#25372
C73×#24312
C73×#24282
C73×#24202
C73×#23392
C4×#127×C732
C4×#124×C732
#5941973