Finite Group Databasebeta

← C73×#23045 #32932449

Subgroups — C73×#23045 #32932449

Order 18688.

844 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#473×C73C211yesmaximal, characteristic
2C73×#2950C211yesmaximal, characteristic, complemented
3C73×#2947C212yesmaximal
4C73×#2913C211yesmaximal, characteristic
5C73×#1969C211yesmaximal, characteristic, complemented
6C73×#1952C211yesmaximal, characteristic, complemented
7C73×#1930C212yesmaximal, complemented
8C73×#1882C212yesmaximal
9C73×#1857C212yesmaximal, complemented
10C73×#1772C211yesmaximal, characteristic, complemented
11C73×#1764C211yesmaximal, characteristic
12C2×#137×C73C2212yes—
13C22×Q16×C73C2211yescharacteristic
14C2×#131×C73C2211yescharacteristic
15#520×C73C2212yes—
16#519×C73C2212yes—
17#473×C73C418yescomplemented
18#441×C73—22no—
19#441×C73—22no—
20#440×C73—22no—
21C4×Q16×C73—21no—
22C4×Q16×C73—21no—
23C4×Q16×C73—22no—
24C4×SD16×C73—22no—
25C2×#108×C73C2211yescharacteristic
26#428×C73—22no—
27#428×C73C2212yes—
28#428×C73C2212yes—
29#427×C73—21no—
30#427×C73C2211yescharacteristic
31C2×#107×C73—21no—
32C2×#107×C73C2211yescharacteristic
33C2×#104×C73C2211yescharacteristic
34#419×C73C2212yes—
35#416×C73C2211yescharacteristic
36C2×#103×C73C2212yes—
37C2×#101×C73C2212yes—
38#405×C73C2212yes—
39#405×C73C2211yescharacteristic
40C2×C4×C8×C73C2211yescharacteristic
41#391×C73C2212yes—
42#343×C73C2212yes—
43#341×C73C2211yescharacteristic
44#339×C73C2211yescharacteristic
45#339×C73C2212yes—
46C2×#41×C73C2311yescharacteristic
47C22×Q8×C73C2312yes—
48#137×C73—24no—
49#137×C73C2×C418yes—
50C2×Q16×C73—22no—
51C2×Q16×C73—21no—
52C2×Q16×C73—21no—
53C2×Q16×C73—22no—
54C2×Q16×C73C2×C414yescomplemented
55C2×SD16×C73—22no—
56#131×C73—22no—
57#131×C73C2×C414yes—
58C2×M16×C73—21no—
59C2×M16×C73C2311yescharacteristic
60C2×M16×C73C2312yes—
61C2×M16×C73C2311yescharacteristic
62C22×C8×C73—21no—
63C22×C8×C73C2311yescharacteristic
64C22×C8×C73C2311yescharacteristic
65C4×Q8×C73—22no—
66C4×Q8×C73—22no—
67C4×Q8×C73—22no—
68C4×Q8×C73—22no—
69C4×Q8×C73—22no—
70C4×D4×C73—22no—
71C4×D4×C73—42no—
72#117×C73—22no—
73#117×C73—42no—
74#117×C73C2311yescharacteristic
75#117×C73C2312yes—
76C2×C4⋊C4×C73—22no—
77C2×C4⋊C4×C73C2312yes—
78C2×C42×C73C2311yescharacteristic
79#108×C73C2×C414yescomplemented
80#107×C73—21no—
81#107×C73—21no—
82#107×C73—22no—
83#107×C73—22no—
84#107×C73D411yescharacteristic
85#107×C73D411yescharacteristic
86#106×C73—22no—
87#106×C73—22no—
88#106×C73D412yes—
89#104×C73—22no—
90#103×C73—22no—
91#103×C73—22no—
92#103×C73—22no—
93#103×C73—22no—
94#103×C73D411yescharacteristic
95#103×C73D411yescharacteristic
96#102×C73D411yescharacteristic
97#102×C73D411yescharacteristic
98#101×C73C2×C418yescomplemented
99#97×C73—22no—
100#97×C73—21no—
101C4×C8×C73—22no—
102C4×C8×C73—21no—
103C4×C8×C73—22no—
104#95×C73—42no—
105#41×C73—21no—
106#41×C73—21no—
107#41×C73C22×C412yes—
108#41×C73#4112yes—
109C2×Q8×C73—22no—
110C2×Q8×C73—22no—
111C2×Q8×C73—22no—
112C2×Q8×C73—22no—
113C2×Q8×C73—22no—
114C2×Q8×C73C22×C414yes—
115C2×Q8×C73C2×D411yescharacteristic
116C2×D4×C73—21no—
117C2×D4×C73C2×D411yescharacteristic
118C22×C4×C73—21no—
119C22×C4×C73—21no—
120C22×C4×C73—22no—
121C22×C4×C73—22no—
122C22×C4×C73—42no—
123C22×C4×C73C2411yescharacteristic
124Q16×C73—28no—
125Q16×C73—24no—
126Q16×C73—24no—
127SD16×C73—28no—
128M16×C73—21no—
129M16×C73—22no—
130M16×C73—24no—
131M16×C73—21no—
132M16×C73—21no—
133M16×C73C22×C412yes—
134M16×C73C22×C414yes—
135C2×C8×C73—22no—
136C2×C8×C73—21no—
137C2×C8×C73—21no—
138C2×C8×C73—21no—
139C2×C8×C73—24no—
140C2×C8×C73—21no—
141C2×C8×C73C22×C412yes—
142C2×C8×C73C2×D411yescharacteristic
143C2×C8×C73C2×D411yescharacteristic
144C2×C8×C73C2×D411yescharacteristic
145C2×C8×C73C2×D411yescharacteristic
146C2×C8×C73#4112yes—
147C4⋊C4×C73—22no—
148C4⋊C4×C73—22no—
149C4⋊C4×C73—42no—
150C4⋊C4×C73—42no—
151C4⋊C4×C73—42no—
152C4⋊C4×C73C2×D411yescharacteristic
153C4⋊C4×C73C2×D411yescharacteristic
154C4⋊C4×C73C2×D412yes—
155C4⋊C4×C73C2×D412yes—
156C22⋊C4×C73—21no—
157C22⋊C4×C73—22no—
158C22⋊C4×C73—42no—
159C42×C73—22no—
160C42×C73—21no—
161C42×C73—21no—
162C42×C73—21no—
163C42×C73—42no—
164C42×C73—42no—
165C42×C73—42no—
166C23×C73—21no—
167C23×C73H(5,𝔽2)11yescharacteristic
168Q8×C73—21no—
169Q8×C73—24no—
170Q8×C73—24no—
171Q8×C73—42no—
172Q8×C73—42no—
173Q8×C73C4×D412yes—
174D4×C73—22no—
175D4×C73—22no—
176D4×C73—21no—
177D4×C73C4×D412yes—
178C2×C4×C73—21no—
179C2×C4×C73—22no—
180C2×C4×C73—22no—
181C2×C4×C73—22no—
182C2×C4×C73—24no—
183C2×C4×C73—22no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—22no—
187C2×C4×C73—41no—
188C2×C4×C73—42no—
189C2×C4×C73—41no—
190C2×C4×C73—42no—
191C2×C4×C73—42no—
192C2×C4×C73—82no—
193C2×C4×C73—162no—
194C2×C4×C73C23×C411yescharacteristic
195C2×C4×C73C22×D411yescharacteristic
196C2×C4×C73C22×D411yescharacteristic
197C2×C4×C73C2×#4111yescharacteristic
198C2×C4×C73C2×#4111yescharacteristic
199C2×C4×C732−1+411yescharacteristic
200C8×C73—22no—
201C8×C73—24no—
202C8×C73—21no—
203C8×C73—21no—
204C8×C73—41no—
205C8×C73C4×D412yes—
206C8×C73C4×D412yes—
207C22×C73—21no—
208C22×C73—21no—
209C22×C73—22no—
210C22×C73—41no—
211C22×C73—41no—
212C22×C73#52011yescharacteristic
213C22×C73D4211yescharacteristic
214C22×C73#55311yescharacteristic
215C4×C73—22no—
216C4×C73—42no—
217C4×C73—44no—
218C4×C73—41no—
219C4×C73—42no—
220C4×C73—42no—
221C4×C73—41no—
222C4×C73—82no—
223C4×C73C2×C4×D411yescharacteristic
224C4×C73C2×C4×D411yescharacteristic
225C4×C73#54711yescharacteristic
226C4×C73#54811yescharacteristic
227#23045C7311yesmaximal, characteristic, direct product factor, Sylow
228C2×C73—21no—
229C2×C73—21no—
230C2×C73—42no—
231C2×C73#236111yescharacteristic
232C2×C73#329011yescharacteristic
233C2×C73#329311yescharacteristic
234C2×#473C2×C7311yescharacteristic
235#2950C2×C7311yescharacteristic, complemented
236#2947C2×C7312yes—
237#2913C2×C7311yescharacteristic
238#1969C2×C7311yescharacteristic, complemented
239#1952C2×C7311yescharacteristic, complemented
240#1930C2×C7312yescomplemented
241#1882C2×C7312yes—
242#1857C2×C7312yescomplemented
243#1772C2×C7311yescharacteristic, complemented
244#1764C2×C7311yescharacteristic
245C73#2304511yescharacteristic, direct product factor, Sylow
246C2×#137C22×C7312yes—
247C22×Q16C22×C7311yescharacteristic
248C2×#131C22×C7311yescharacteristic
249#520C22×C7312yes—
250#519C22×C7312yes—
251#473C4×C7318yescomplemented
252#441—22no—
253#441—22no—
254#440—22no—
255C4×Q16—21no—
256C4×Q16—21no—
257C4×Q16—22no—
258C4×SD16—22no—
259C2×#108C22×C7311yescharacteristic
260#428—22no—
261#428C22×C7312yes—
262#428C22×C7312yes—
263#427—21no—
264#427C22×C7311yescharacteristic
265C2×#107—21no—
266C2×#107C22×C7311yescharacteristic
267C2×#104C22×C7311yescharacteristic
268#419C22×C7312yes—
269#416C22×C7311yescharacteristic
270C2×#103C22×C7312yes—
271C2×#101C22×C7312yes—
272#405C22×C7312yes—
273#405C22×C7311yescharacteristic
274C2×C4×C8C22×C7311yescharacteristic
275#391C22×C7312yes—
276#343C22×C7312yes—
277#341C22×C7311yescharacteristic
278#339C22×C7311yescharacteristic
279#339C22×C7312yes—
280C2×#41C23×C7311yescharacteristic
281C22×Q8C23×C7312yes—
282#137—24no—
283#137C2×C4×C7318yes—
284C2×Q16—22no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×Q16—22no—
288C2×Q16C2×C4×C7314yescomplemented
289C2×SD16—22no—
290#131—22no—
291#131C2×C4×C7314yes—
292C2×M16—21no—
293C2×M16C23×C7311yescharacteristic
294C2×M16C23×C7311yescharacteristic
295C2×M16C23×C7312yes—
296C22×C8—21no—
297C22×C8C23×C7311yescharacteristic
298C22×C8C23×C7311yescharacteristic
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×D4—22no—
305C4×D4—42no—
306#117—22no—
307#117—42no—
308#117C23×C7311yescharacteristic
309#117C23×C7312yes—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C23×C7312yes—
312C2×C42C23×C7311yescharacteristic
313#108C2×C4×C7314yescomplemented
314#107—21no—
315#107—21no—
316#107—22no—
317#107—22no—
318#107D4×C7311yescharacteristic
319#107D4×C7311yescharacteristic
320#106—22no—
321#106—22no—
322#106D4×C7312yes—
323#104—22no—
324#103—22no—
325#103—22no—
326#103—22no—
327#103—22no—
328#103D4×C7311yescharacteristic
329#103D4×C7311yescharacteristic
330#102D4×C7311yescharacteristic
331#102D4×C7311yescharacteristic
332#101C2×C4×C7318yescomplemented
333#97—22no—
334#97—21no—
335C4×C8—22no—
336C4×C8—21no—
337C4×C8—22no—
338#95—42no—
339#41—21no—
340#41—21no—
341#41C22×C4×C7312yes—
342#41#41×C7312yes—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8C22×C4×C7314yes—
349C2×Q8C2×D4×C7311yescharacteristic
350C2×D4—21no—
351C2×D4C2×D4×C7311yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4C24×C7311yescharacteristic
358Q16—28no—
359Q16—24no—
360Q16—24no—
361SD16—28no—
362M16—21no—
363M16—22no—
364M16—24no—
365M16—21no—
366M16—21no—
367M16C22×C4×C7312yes—
368M16C22×C4×C7314yes—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—24no—
374C2×C8—21no—
375C2×C8C22×C4×C7312yes—
376C2×C8C2×D4×C7311yescharacteristic
377C2×C8C2×D4×C7311yescharacteristic
378C2×C8C2×D4×C7311yescharacteristic
379C2×C8C2×D4×C7311yescharacteristic
380C2×C8#41×C7312yes—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4C2×D4×C7311yescharacteristic
387C4⋊C4C2×D4×C7311yescharacteristic
388C4⋊C4C2×D4×C7312yes—
389C4⋊C4C2×D4×C7312yes—
390C22⋊C4—21no—
391C22⋊C4—22no—
392C22⋊C4—42no—
393C42—22no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—42no—
398C42—42no—
399C42—42no—
400C23—21no—
401C23H(5,𝔽2)×C7311yescharacteristic
402Q8—21no—
403Q8—24no—
404Q8—24no—
405Q8—42no—
406Q8—42no—
407Q8C4×D4×C7312yes—
408D4—22no—
409D4—22no—
410D4—21no—
411D4C4×D4×C7312yes—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—162no—
428C2×C4C23×C4×C7311yescharacteristic
429C2×C4C22×D4×C7311yescharacteristic
430C2×C4C22×D4×C7311yescharacteristic
431C2×C4C2×#41×C7311yescharacteristic
432C2×C4C2×#41×C7311yescharacteristic
433C2×C42−1+4×C7311yescharacteristic
434C8—22no—
435C8—24no—
436C8—21no—
437C8—21no—
438C8—41no—
439C8C4×D4×C7312yes—
440C8C4×D4×C7312yes—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22#520×C7311yescharacteristic
447C22D42×C7311yescharacteristic
448C22#553×C7311yescharacteristic
449C4—22no—
450C4—42no—
451C4—42no—
452C4—41no—
453C4—42no—
454C4—41no—
455C4—44no—
456C4—82no—
457C4C2×C4×D4×C7311yescharacteristic
458C4C2×C4×D4×C7311yescharacteristic
459C4#547×C7311yescharacteristic
460C4#548×C7311yescharacteristic
461C2—21no—
462C2—21no—
463C2—42no—
464C2C73×#236111yescharacteristic
465C2C73×#329011yescharacteristic
466C2C73×#329311yescharacteristic
subgroupG/H
center, socleC22×C73D42
commutatorC2×C4C23×C4×C73
Frattini, cosocleC22×C4C24×C73
prime radicalC2×#41×C73C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C22×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#2950C2semidirect
5. C73×#1969C2semidirect
6. C73×#1952C2semidirect
7. C73×#1930C2semidirect
9. C73×#1857C2semidirect
10. C73×#1772C2semidirect
17. #473×C73C4semidirect
54. C2×Q16×C73C2×C4semidirect
79. #108×C73C2×C4semidirect
98. #101×C73C2×C4semidirect
227. #23045C73direct (N × H)
235. #2950C2×C73semidirect
238. #1969C2×C73semidirect
239. #1952C2×C73semidirect
240. #1930C2×C73semidirect
242. #1857C2×C73semidirect
243. #1772C2×C73semidirect
245. C73#23045direct (N × H)
251. #473C4×C73semidirect
288. C2×Q16C2×C4×C73semidirect
313. #108C2×C4×C73semidirect
332. #101C2×C4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#473×C732
C73×#29502
C73×#29472
C73×#29132
C73×#19692
C73×#19522
C73×#19302
C73×#18822
C73×#18572
C73×#17722
C73×#17642
#2304573