Finite Group Databasebeta

← C73×#18781 #32928185

Subgroups — C73×#18781 #32928185

Order 18688.

910 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556×C73C211yesmaximal, characteristic
2C73×#3229C211yesmaximal, characteristic, complemented
3C73×#3104C211yesmaximal, characteristic, complemented
4C73×#3077C211yesmaximal, characteristic
5C73×#1652C214yesmaximal, complemented
6C73×#1638C214yesmaximal, complemented
7C73×#1546C211yesmaximal, characteristic, complemented
8C73×#1487C211yesmaximal, characteristic, complemented
9C73×#1478C211yesmaximal, characteristic
10#556×C73—22no—
11#556×C73—22no—
12#556×C73—22no—
13#556×C73C2214yes—
14#533×C73C2211yescharacteristic
15C2×C4⋊Q8×C73C2211yescharacteristic
16C2×#126×C73—22no—
17C2×#124×C73—21no—
18C2×C22⋊Q8×C73—21no—
19C2×C4×Q8×C73—21no—
20C2×C4×Q8×C73C2211yescharacteristic
21C2×C4×D4×C73C2211yescharacteristic
22C2×#117×C73—21no—
23#498×C73C2212yescomplemented
24#487×C73C2212yescomplemented
25#484×C73—21no—
26#483×C73—22no—
27#478×C73—22no—
28#477×C73C2212yescomplemented
29#476×C73C2212yes—
30#475×C73—22no—
31#474×C73C2212yes—
32#462×C73C2212yescomplemented
33#451×C73—21no—
34C4×Q16×C73—22no—
35#431×C73C2211yescharacteristic
36#431×C73C2211yescharacteristic
37#428×C73—21no—
38#423×C73C2211yescharacteristic
39#418×C73—21no—
40#332×C73C2214yescomplemented
41Q8⋊C8×C73C2214yescomplemented
42D4⋊C8×C73C2214yes—
43C22×Q8×C73—21no—
44C22×Q8×C73D411yescharacteristic
45C22×D4×C73D411yescharacteristic
46C23×C4×C73—21no—
47C2×Q16×C73—22no—
48C2×M16×C73—21no—
49C4⋊Q8×C73—21no—
50C4⋊Q8×C73—21no—
51C4⋊Q8×C73D412yes—
52C4⋊Q8×C73C2311yescharacteristic
53C4⋊Q8×C73C2311yescharacteristic
54C4⋊Q8×C73C2312yes—
55#126×C73—28no—
56#126×C73—28no—
57#125×C73—21no—
58#124×C73—24no—
59#124×C73—24no—
60C22⋊Q8×C73—24no—
61C22⋊Q8×C73—24no—
62C22⋊Q8×C73—21no—
63C22⋊Q8×C73—21no—
64C4×Q8×C73—24no—
65C4×Q8×C73—24no—
66C4×Q8×C73—21no—
67C4×Q8×C73—21no—
68C4×Q8×C73—21no—
69C4×Q8×C73—21no—
70C4×Q8×C73D412yes—
71C4×Q8×C73C2312yes—
72C4×D4×C73—21no—
73C4×D4×C73—21no—
74C4×D4×C73D412yes—
75C4×D4×C73C2312yes—
76#117×C73—24no—
77#117×C73—24no—
78#117×C73—21no—
79C2×C4⋊C4×C73—22no—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73D411yescharacteristic
83C2×C4⋊C4×C73D412yes—
84C2×C4⋊C4×C73D411yescharacteristic
85C2×C22⋊C4×C73—22no—
86C2×C22⋊C4×C73—22no—
87C2×C22⋊C4×C73—21no—
88C2×C42×C73—22no—
89C2×C42×C73—21no—
90C2×C42×C73C2311yescharacteristic
91#107×C73—22no—
92#107×C73—22no—
93#106×C73—22no—
94#106×C73—24no—
95C4⋊C8×C73—21no—
96C4⋊C8×C73—21no—
97C4⋊C8×C73C2312yes—
98C4⋊C8×C73C2312yes—
99#103×C73—22no—
100#103×C73—22no—
101#103×C73—24no—
102#102×C73—22no—
103#102×C73—22no—
104C22⋊C8×C73—21no—
105C22⋊C8×C73—21no—
106C4×C8×C73—21no—
107C4×C8×C73C2312yes—
108C24×C73—21no—
109C2×Q8×C73—21no—
110C2×Q8×C73—22no—
111C2×Q8×C73—22no—
112C2×Q8×C73—21no—
113C2×Q8×C73—21no—
114C2×Q8×C73—42no—
115C2×Q8×C73—42no—
116C2×Q8×C73—42no—
117C2×Q8×C73C2×D412yes—
118C2×D4×C73—21no—
119C2×D4×C73—22no—
120C2×D4×C73—22no—
121C2×D4×C73C2×D412yes—
122C22×C4×C73—21no—
123C22×C4×C73—21no—
124C22×C4×C73—21no—
125C22×C4×C73—22no—
126C22×C4×C73—22no—
127C22×C4×C73—22no—
128C22×C4×C73—44no—
129C22×C4×C73C2×D411yescharacteristic
130C22×C4×C73C2×D411yescharacteristic
131C22×C4×C73C2×D411yescharacteristic
132Q16×C73—44no—
133M16×C73—42no—
134C2×C8×C73—22no—
135C2×C8×C73—22no—
136C2×C8×C73—22no—
137C4⋊C4×C73—28no—
138C4⋊C4×C73—22no—
139C4⋊C4×C73—22no—
140C4⋊C4×C73—21no—
141C4⋊C4×C73—21no—
142C4⋊C4×C73—22no—
143C4⋊C4×C73—22no—
144C4⋊C4×C73—22no—
145C4⋊C4×C73—21no—
146C4⋊C4×C73—21no—
147C4⋊C4×C73—21no—
148C4⋊C4×C73—21no—
149C4⋊C4×C73—41no—
150C4⋊C4×C73—41no—
151C4⋊C4×C73C2×D414yes—
152C4⋊C4×C73C2×D412yes—
153C4⋊C4×C73C2×D412yes—
154C22⋊C4×C73—28no—
155C22⋊C4×C73—28no—
156C22⋊C4×C73—22no—
157C22⋊C4×C73—22no—
158C22⋊C4×C73—41no—
159C42×C73—28no—
160C42×C73—22no—
161C42×C73—22no—
162C42×C73—41no—
163C42×C73C2×D411yescharacteristic
164C42×C73C2×D411yescharacteristic
165C42×C73C2×D411yescharacteristic
166C42×C73C2411yescharacteristic
167C23×C73—22no—
168C23×C73—42no—
169C23×C73—42no—
170C23×C73—42no—
171C23×C73#12011yescharacteristic
172Q8×C73—22no—
173Q8×C73—22no—
174Q8×C73—42no—
175Q8×C73—42no—
176Q8×C73—42no—
177Q8×C73—42no—
178Q8×C73—44no—
179Q8×C73—42no—
180D4×C73—22no—
181D4×C73—22no—
182D4×C73—42no—
183C2×C4×C73—22no—
184C2×C4×C73—22no—
185C2×C4×C73—22no—
186C2×C4×C73—22no—
187C2×C4×C73—24no—
188C2×C4×C73—41no—
189C2×C4×C73—44no—
190C2×C4×C73—42no—
191C2×C4×C73—42no—
192C2×C4×C73—42no—
193C2×C4×C73—44no—
194C2×C4×C73—44no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73#12011yescharacteristic
198C2×C4×C73#12011yescharacteristic
199C2×C4×C73#12011yescharacteristic
200C2×C4×C73#13512yes—
201C2×C4×C73#13712yes—
202C2×C4×C73C22×D411yescharacteristic
203C2×C4×C73C22×D411yescharacteristic
204C2×C4×C73C22×D411yescharacteristic
205C8×C73—42no—
206C8×C73—42no—
207C8×C73—42no—
208C22×C73—21no—
209C22×C73—21no—
210C22×C73—42no—
211C22×C73—42no—
212C22×C73—42no—
213C22×C73—84no—
214C22×C73#45212yes—
215C22×C73C2×#12011yescharacteristic
216C4×C73—21no—
217C4×C73—21no—
218C4×C73—21no—
219C4×C73—44no—
220C4×C73—42no—
221C4×C73—42no—
222C4×C73—42no—
223C4×C73—81no—
224C4×C73—81no—
225C4×C73#45212yes—
226C4×C73C2×#13511yescharacteristic
227C4×C73C2×#13711yescharacteristic
228C4×C73#57512yes—
229#18781C7311yesmaximal, characteristic, direct product factor, Sylow
230C2×C73—22no—
231C2×C73—82no—
232C2×C73C2×#45211yescharacteristic
233C2×C73#300511yescharacteristic
234C2×C73#302011yescharacteristic
235C2×#556C2×C7311yescharacteristic
236#3229C2×C7311yescharacteristic, complemented
237#3104C2×C7311yescharacteristic, complemented
238#3077C2×C7311yescharacteristic
239#1652C2×C7314yescomplemented
240#1638C2×C7314yescomplemented
241#1546C2×C7311yescharacteristic, complemented
242#1487C2×C7311yescharacteristic, complemented
243#1478C2×C7311yescharacteristic
244C73#1878111yescharacteristic, direct product factor, Sylow
245#556—22no—
246#556—22no—
247#556—22no—
248#556C22×C7314yes—
249#533C22×C7311yescharacteristic
250C2×C4⋊Q8C22×C7311yescharacteristic
251C2×#126—22no—
252C2×#124—21no—
253C2×C22⋊Q8—21no—
254C2×C4×Q8—21no—
255C2×C4×Q8C22×C7311yescharacteristic
256C2×C4×D4C22×C7311yescharacteristic
257C2×#117—21no—
258#498C22×C7312yescomplemented
259#487C22×C7312yescomplemented
260#484—21no—
261#483—22no—
262#478—22no—
263#477C22×C7312yescomplemented
264#476C22×C7312yes—
265#475—22no—
266#474C22×C7312yes—
267#462C22×C7312yescomplemented
268#451—21no—
269C4×Q16—22no—
270#431C22×C7311yescharacteristic
271#431C22×C7311yescharacteristic
272#428—21no—
273#423C22×C7311yescharacteristic
274#418—21no—
275#332C22×C7314yescomplemented
276Q8⋊C8C22×C7314yescomplemented
277D4⋊C8C22×C7314yes—
278C22×Q8—21no—
279C22×Q8D4×C7311yescharacteristic
280C22×D4D4×C7311yescharacteristic
281C23×C4—21no—
282C2×Q16—22no—
283C2×M16—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8D4×C7312yes—
287C4⋊Q8C23×C7311yescharacteristic
288C4⋊Q8C23×C7311yescharacteristic
289C4⋊Q8C23×C7312yes—
290#126—28no—
291#126—28no—
292#125—21no—
293#124—24no—
294#124—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8D4×C7312yes—
306C4×Q8C23×C7312yes—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4D4×C7312yes—
310C4×D4C23×C7312yes—
311#117—24no—
312#117—24no—
313#117—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4D4×C7311yescharacteristic
318C2×C4⋊C4D4×C7311yescharacteristic
319C2×C4⋊C4D4×C7312yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42C23×C7311yescharacteristic
326#107—22no—
327#107—22no—
328#106—22no—
329#106—24no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8C23×C7312yes—
333C4⋊C8C23×C7312yes—
334#103—22no—
335#103—22no—
336#103—24no—
337#102—22no—
338#102—22no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C4×C8—21no—
342C4×C8C23×C7312yes—
343C24—21no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8C2×D4×C7312yes—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4C2×D4×C7312yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—44no—
364C22×C4C2×D4×C7311yescharacteristic
365C22×C4C2×D4×C7311yescharacteristic
366C22×C4C2×D4×C7311yescharacteristic
367Q16—44no—
368M16—42no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C4⋊C4—28no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4C2×D4×C7314yes—
387C4⋊C4C2×D4×C7312yes—
388C4⋊C4C2×D4×C7312yes—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—41no—
394C42—28no—
395C42—22no—
396C42—22no—
397C42—41no—
398C42C2×D4×C7311yescharacteristic
399C42C2×D4×C7311yescharacteristic
400C42C2×D4×C7311yescharacteristic
401C42C24×C7311yescharacteristic
402C23—22no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23#120×C7311yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—42no—
410Q8—42no—
411Q8—44no—
412Q8—42no—
413Q8—42no—
414Q8—42no—
415D4—22no—
416D4—22no—
417D4—42no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4#120×C7311yescharacteristic
433C2×C4#120×C7311yescharacteristic
434C2×C4#120×C7311yescharacteristic
435C2×C4#135×C7312yes—
436C2×C4#137×C7312yes—
437C2×C4C22×D4×C7311yescharacteristic
438C2×C4C22×D4×C7311yescharacteristic
439C2×C4C22×D4×C7311yescharacteristic
440C8—42no—
441C8—42no—
442C8—42no—
443C22—21no—
444C22—21no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—84no—
449C22#452×C7312yes—
450C22C2×#120×C7311yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—44no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4#452×C7312yes—
461C4C2×#135×C7311yescharacteristic
462C4C2×#137×C7311yescharacteristic
463C4#575×C7312yes—
464C2—22no—
465C2—82no—
466C2C2×#452×C7311yescharacteristic
467C2C73×#300511yescharacteristic
468C2C73×#302011yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radicalC22×D4×C73D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#3229C2semidirect
3. C73×#3104C2semidirect
5. C73×#1652C2semidirect
6. C73×#1638C2semidirect
7. C73×#1546C2semidirect
8. C73×#1487C2semidirect
23. #498×C73C22semidirect
24. #487×C73C22semidirect
28. #477×C73C22semidirect
32. #462×C73C22semidirect
40. #332×C73C22semidirect
41. Q8⋊C8×C73C22semidirect
229. #18781C73direct (N × H)
236. #3229C2×C73semidirect
237. #3104C2×C73semidirect
239. #1652C2×C73semidirect
240. #1638C2×C73semidirect
241. #1546C2×C73semidirect
242. #1487C2×C73semidirect
244. C73#18781direct (N × H)
258. #498C22×C73semidirect
259. #487C22×C73semidirect
263. #477C22×C73semidirect
267. #462C22×C73semidirect
275. #332C22×C73semidirect
276. Q8⋊C8C22×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#556×C732
C73×#32292
C73×#31042
C73×#30772
C73×#16522
C73×#16382
C73×#15462
C73×#14872
C73×#14782
#1878173