Finite Group Databasebeta

← C73×#25624 #32935028

Subgroups — C73×#25624 #32935028

Order 18688.

1126 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3507C211yesmaximal, characteristic, complemented
2C73×#3151C211yesmaximal, characteristic
3C73×#3138C211yesmaximal, characteristic, complemented
4C73×#2548C211yesmaximal, characteristic, complemented
5C73×#2003C214yesmaximal, complemented
6C73×#1986C211yesmaximal, characteristic, complemented
7C73×#1980C211yesmaximal, characteristic, complemented
8C73×#1883C214yesmaximal, complemented
9C4×#101×C73C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)×C73C2214yes—
11#559×C73—21no—
12#557×C73—21no—
13#556×C73—21no—
14#556×C73—22no—
15#554×C73—21no—
16#552×C73—22no—
17#539×C73—22no—
18#536×C73C2212yes—
19#533×C73C2211yescharacteristic
20#532×C73C2211yescharacteristic
21C2×#127×C73C2211yescharacteristic
22#529×C73—22no—
23C2×#124×C73C2212yes—
24#520×C73C2212yes—
25C2×C4×Q8×C73C2211yescharacteristic
26#496×C73—22no—
27#493×C73—21no—
28#492×C73—21no—
29#490×C73—22no—
30#486×C73—21no—
31#486×C73—21no—
32C2×#104×C73C2214yes—
33#418×C73C2214yes—
34C2×#101×C73C2212yes—
35C4×M16×C73C2211yescharacteristic
36C4×M16×C73C2211yescharacteristic
37#394×C73—22no—
38#390×C73—22no—
39#390×C73C2212yes—
40C4×C4⋊C4×C73C2211yescharacteristic
41#343×C73C2212yes—
42#339×C73C2214yes—
43C2×#41×C73C2312yes—
44C22×Q8×C73C2311yescharacteristic
45C22×D4×C73—21no—
46C22×D4×C73—21no—
47C22×D4×C73C2312yes—
48AGL(1,ℤ/8)×C73—28no—
49AGL(1,ℤ/8)×C73—44no—
50C2×SD16×C73—24no—
51C2×D8×C73—24no—
52C2×M16×C73C2312yes—
53C2×M16×C73C2312yes—
54C4⋊Q8×C73—22no—
55C4⋊Q8×C73D411yescharacteristic
56C4⋊Q8×C73D411yescharacteristic
57C4⋊Q8×C73D411yescharacteristic
58#127×C73—21no—
59#127×C73—21no—
60#127×C73—21no—
61#127×C73—21no—
62#127×C73D411yescharacteristic
63#126×C73—24no—
64#126×C73—24no—
65#125×C73—22no—
66#125×C73—24no—
67#125×C73D412yes—
68#124×C73—22no—
69#124×C73—22no—
70#124×C73—44no—
71#124×C73D412yes—
72#123×C73—24no—
73#123×C73—24no—
74C22⋊Q8×C73—24no—
75C22⋊Q8×C73—21no—
76C22⋊Q8×C73—21no—
77C22⋊Q8×C73—22no—
78C22⋊Q8×C73—22no—
79#121×C73—21no—
80#121×C73—21no—
81C4×Q8×C73—22no—
82C4×Q8×C73—22no—
83C4×Q8×C73—21no—
84C4×Q8×C73—21no—
85C4×Q8×C73—21no—
86C4×Q8×C73—21no—
87C4×Q8×C73—21no—
88C4×Q8×C73—21no—
89C4×D4×C73—22no—
90C4×D4×C73—22no—
91C4×D4×C73—24no—
92C4×D4×C73—21no—
93C4×D4×C73—21no—
94#117×C73—24no—
95#117×C73—22no—
96#117×C73C2312yes—
97C2×C4⋊C4×C73—22no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73D412yes—
100C2×C4⋊C4×C73C2311yescharacteristic
101C2×C22⋊C4×C73—24no—
102C2×C22⋊C4×C73—24no—
103C2×C42×C73D412yes—
104C2×C42×C73C2312yes—
105C2×C42×C73C2311yescharacteristic
106#104×C73—44no—
107#102×C73—24no—
108#102×C73—24no—
109#101×C73—24no—
110#97×C73—21no—
111#97×C73—21no—
112C4×C8×C73—21no—
113C4×C8×C73—21no—
114#95×C73—21no—
115C24×C73—22no—
116#41×C73—24no—
117#41×C73—24no—
118C2×Q8×C73—21no—
119C2×Q8×C73—21no—
120C2×Q8×C73—21no—
121C2×Q8×C73—21no—
122C2×Q8×C73—22no—
123C2×Q8×C73—22no—
124C2×Q8×C73C2×D412yes—
125C2×D4×C73—22no—
126C2×D4×C73—21no—
127C2×D4×C73—21no—
128C2×D4×C73—22no—
129C2×D4×C73—22no—
130C2×D4×C73—24no—
131C2×D4×C73—24no—
132C2×D4×C73—44no—
133C2×D4×C73—41no—
134C2×D4×C73—41no—
135C2×D4×C73—41no—
136C2×D4×C73—41no—
137C2×D4×C73—82no—
138C2×D4×C73C2×D412yes—
139C22×C4×C73—24no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73C2×D412yes—
143C22×C4×C73C2×D411yescharacteristic
144C22×C4×C73C2×D412yes—
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2411yescharacteristic
147SD16×C73—48no—
148D8×C73—48no—
149M16×C73—22no—
150M16×C73—22no—
151M16×C73—22no—
152M16×C73#4114yes—
153C2×C8×C73—22no—
154C2×C8×C73—22no—
155C4⋊C4×C73—24no—
156C4⋊C4×C73—24no—
157C4⋊C4×C73—24no—
158C4⋊C4×C73—22no—
159C4⋊C4×C73—22no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—22no—
165C4⋊C4×C73—21no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—21no—
168C4⋊C4×C73—42no—
169C4⋊C4×C73C2×D412yes—
170C4⋊C4×C73C2×D412yes—
171C22⋊C4×C73—24no—
172C22⋊C4×C73—24no—
173C22⋊C4×C73—24no—
174C22⋊C4×C73—22no—
175C22⋊C4×C73—22no—
176C22⋊C4×C73—22no—
177C22⋊C4×C73—48no—
178C22⋊C4×C73—84no—
179C42×C73—24no—
180C42×C73—22no—
181C42×C73—22no—
182C42×C73—22no—
183C42×C73—24no—
184C42×C73C2×D411yescharacteristic
185C42×C73C2×D411yescharacteristic
186C42×C73C2×D411yescharacteristic
187C42×C73C2×D411yescharacteristic
188C23×C73—22no—
189C23×C73—44no—
190C23×C73—44no—
191C23×C73—42no—
192C23×C73—42no—
193C23×C73—42no—
194C23×C73C22×D411yescharacteristic
195Q8×C73—24no—
196Q8×C73—42no—
197Q8×C73—42no—
198Q8×C73—42no—
199Q8×C73—81no—
200D4×C73—24no—
201D4×C73—42no—
202D4×C73—42no—
203D4×C73—44no—
204D4×C73—44no—
205D4×C73—42no—
206D4×C73—81no—
207D4×C73—82no—
208D4×C73—82no—
209D4×C73—82no—
210D4×C73—82no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—41no—
220C2×C4×C73—44no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73—84no—
226C2×C4×C73#12011yescharacteristic
227C2×C4×C73#12011yescharacteristic
228C2×C4×C73#12011yescharacteristic
229C2×C4×C73#12011yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C2×C4×C73C2×#4112yes—
233C2×C4×C73H(5,𝔽2)12yes—
234C8×C73—24no—
235C8×C73—42no—
236C22×C73—21no—
237C22×C73—21no—
238C22×C73—42no—
239C22×C73—42no—
240C22×C73—42no—
241C22×C73—84no—
242C22×C73—84no—
243C22×C73—82no—
244C22×C73—82no—
245C22×C73—82no—
246C22×C73C2×#12011yescharacteristic
247C22×C73#53211yescharacteristic
248C22×C73#53511yescharacteristic
249C4×C73—21no—
250C4×C73—21no—
251C4×C73—21no—
252C4×C73—42no—
253C4×C73—42no—
254C4×C73—42no—
255C4×C73—42no—
256C4×C73—42no—
257C4×C73—82no—
258C4×C73#45412yes—
259C4×C73#54612yes—
260C4×C73#55012yes—
261#25624C7311yesmaximal, characteristic, direct product factor, Sylow
262C2×C73—21no—
263C2×C73—21no—
264C2×C73—82no—
265C2×C73—162no—
266C2×C73#243111yescharacteristic
267C2×C73C2×#45411yescharacteristic
268C2×C73#302111yescharacteristic
269#3507C2×C7311yescharacteristic, complemented
270#3151C2×C7311yescharacteristic
271#3138C2×C7311yescharacteristic, complemented
272#2548C2×C7311yescharacteristic, complemented
273#2003C2×C7314yescomplemented
274#1986C2×C7311yescharacteristic, complemented
275#1980C2×C7311yescharacteristic, complemented
276#1883C2×C7314yescomplemented
277C4×#101C2×C7311yescharacteristic, complemented
278C73#2562411yescharacteristic, direct product factor, Sylow
279C2×AGL(1,ℤ/8)C22×C7314yes—
280#559—21no—
281#557—21no—
282#556—21no—
283#556—22no—
284#554—21no—
285#552—22no—
286#539—22no—
287#536C22×C7312yes—
288#533C22×C7311yescharacteristic
289#532C22×C7311yescharacteristic
290C2×#127C22×C7311yescharacteristic
291#529—22no—
292C2×#124C22×C7312yes—
293#520C22×C7312yes—
294C2×C4×Q8C22×C7311yescharacteristic
295#496—22no—
296#493—21no—
297#492—21no—
298#490—22no—
299#486—21no—
300#486—21no—
301C2×#104C22×C7314yes—
302#418C22×C7314yes—
303C2×#101C22×C7312yes—
304C4×M16C22×C7311yescharacteristic
305C4×M16C22×C7311yescharacteristic
306#394—22no—
307#390—22no—
308#390C22×C7312yes—
309C4×C4⋊C4C22×C7311yescharacteristic
310#343C22×C7312yes—
311#339C22×C7314yes—
312C2×#41C23×C7312yes—
313C22×Q8C23×C7311yescharacteristic
314C22×D4—21no—
315C22×D4—21no—
316C22×D4C23×C7312yes—
317AGL(1,ℤ/8)—28no—
318AGL(1,ℤ/8)—44no—
319C2×SD16—24no—
320C2×D8—24no—
321C2×M16C23×C7312yes—
322C2×M16C23×C7312yes—
323C4⋊Q8—22no—
324C4⋊Q8D4×C7311yescharacteristic
325C4⋊Q8D4×C7311yescharacteristic
326C4⋊Q8D4×C7311yescharacteristic
327#127—21no—
328#127—21no—
329#127—21no—
330#127—21no—
331#127D4×C7311yescharacteristic
332#126—24no—
333#126—24no—
334#125—22no—
335#125—24no—
336#125D4×C7312yes—
337#124—22no—
338#124—22no—
339#124—44no—
340#124D4×C7312yes—
341#123—24no—
342#123—24no—
343C22⋊Q8—24no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348#121—21no—
349#121—21no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—24no—
361C4×D4—21no—
362C4×D4—21no—
363#117—24no—
364#117—22no—
365#117C23×C7312yes—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4D4×C7312yes—
369C2×C4⋊C4C23×C7311yescharacteristic
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C42D4×C7312yes—
373C2×C42C23×C7312yes—
374C2×C42C23×C7311yescharacteristic
375#104—44no—
376#102—24no—
377#102—24no—
378#101—24no—
379#97—21no—
380#97—21no—
381C4×C8—21no—
382C4×C8—21no—
383#95—21no—
384C24—22no—
385#41—24no—
386#41—24no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8C2×D4×C7312yes—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—44no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—82no—
407C2×D4C2×D4×C7312yes—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4C2×D4×C7312yes—
412C22×C4C2×D4×C7311yescharacteristic
413C22×C4C2×D4×C7312yes—
414C22×C4C2×D4×C7311yescharacteristic
415C22×C4C24×C7311yescharacteristic
416SD16—48no—
417D8—48no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16#41×C7314yes—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—42no—
438C4⋊C4C2×D4×C7312yes—
439C4⋊C4C2×D4×C7312yes—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—48no—
447C22⋊C4—84no—
448C42—24no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42C2×D4×C7311yescharacteristic
454C42C2×D4×C7311yescharacteristic
455C42C2×D4×C7311yescharacteristic
456C42C2×D4×C7311yescharacteristic
457C23—22no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C22×D4×C7311yescharacteristic
464Q8—24no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—81no—
469D4—24no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—44no—
474D4—44no—
475D4—81no—
476D4—82no—
477D4—82no—
478D4—82no—
479D4—82no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4#120×C7311yescharacteristic
496C2×C4#120×C7311yescharacteristic
497C2×C4#120×C7311yescharacteristic
498C2×C4#120×C7311yescharacteristic
499C2×C4C22×D4×C7311yescharacteristic
500C2×C4C22×D4×C7311yescharacteristic
501C2×C4C2×#41×C7312yes—
502C2×C4H(5,𝔽2)×C7312yes—
503C8—24no—
504C8—42no—
505C22—21no—
506C22—21no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—84no—
511C22—84no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22C2×#120×C7311yescharacteristic
516C22#532×C7311yescharacteristic
517C22#535×C7311yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4#454×C7312yes—
528C4#546×C7312yes—
529C4#550×C7312yes—
530C2—21no—
531C2—21no—
532C2—82no—
533C2—162no—
534C2C73×#243111yescharacteristic
535C2C2×#454×C7311yescharacteristic
536C2C73×#302111yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC73×#3151C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3507C2semidirect
3. C73×#3138C2semidirect
4. C73×#2548C2semidirect
5. C73×#2003C2semidirect
6. C73×#1986C2semidirect
7. C73×#1980C2semidirect
8. C73×#1883C2semidirect
9. C4×#101×C73C2semidirect
261. #25624C73direct (N × H)
269. #3507C2×C73semidirect
271. #3138C2×C73semidirect
272. #2548C2×C73semidirect
273. #2003C2×C73semidirect
274. #1986C2×C73semidirect
275. #1980C2×C73semidirect
276. #1883C2×C73semidirect
277. C4×#101C2×C73semidirect
278. C73#25624direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#35072
C73×#31512
C73×#31382
C73×#25482
C73×#20032
C73×#19862
C73×#19802
C73×#18832
C4×#101×C732
#2562473