Finite Group Databasebeta

← C73×#19683 #32929087

Subgroups — C73×#19683 #32929087

Order 18688.

724 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3209C211yesmaximal, characteristic, complemented
2C73×#3206C211yesmaximal, characteristic
3C73×#3167C211yesmaximal, characteristic
4C73×#3165C211yesmaximal, characteristic, complemented
5C2×#500×C73C211yesmaximal, characteristic, complemented
6C73×#3141C211yesmaximal, characteristic
7C73×#1723C212yesmaximal, complemented
8C73×#1722C212yesmaximal, complemented
9C73×#1718C212yesmaximal, complemented
10C73×#1714C212yesmaximal, complemented
11C73×#1451C211yesmaximal, characteristic, complemented
12#533×C73C2211yescharacteristic
13#532×C73C2211yescharacteristic
14C2×C4⋊Q8×C73C2211yescharacteristic
15C2×C4⋊Q8×C73C2211yescharacteristic
16#501×C73—22no—
17#500×C73—22no—
18#500×C73C2212yes—
19#500×C73C2212yes—
20#500×C73C2212yes—
21#498×C73C2212yes—
22#498×C73C2211yescharacteristic
23#494×C73C2212yes—
24#492×C73C2211yescharacteristic
25#492×C73C2212yes—
26#489×C73—22no—
27#487×C73C2212yes—
28#487×C73C2211yescharacteristic
29#487×C73C2212yes—
30#487×C73C2212yes—
31#486×C73C2212yes—
32#486×C73C2211yescharacteristic
33#465×C73—22no—
34#462×C73—22no—
35#461×C73—22no—
36#451×C73—22no—
37#448×C73—22no—
38C4×SD16×C73—22no—
39#431×C73C2211yescharacteristic
40#428×C73—22no—
41C2×#107×C73—22no—
42#417×C73—22no—
43C4×M16×C73C2211yescharacteristic
44C2×C4×C8×C73C2211yescharacteristic
45#322×C73C2212yes—
46#322×C73C2212yes—
47C2×#41×C73—21no—
48C22×Q8×C73—21no—
49C22×Q8×C73—21no—
50C2×Q16×C73—24no—
51C2×SD16×C73—22no—
52C2×SD16×C73—24no—
53C2×M16×C73D412yes—
54C22×C8×C73D412yes—
55C4⋊Q8×C73—21no—
56C4⋊Q8×C73—21no—
57C4⋊Q8×C73—21no—
58C4⋊Q8×C73—21no—
59C4⋊Q8×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73C2312yes—
62C4⋊Q8×C73C2311yescharacteristic
63C4⋊Q8×C73C2311yescharacteristic
64C4⋊Q8×C73C2311yescharacteristic
65C4⋊Q8×C73C2312yes—
66#127×C73C2311yescharacteristic
67#125×C73—21no—
68#124×C73—21no—
69C22⋊Q8×C73—21no—
70C22⋊Q8×C73—21no—
71#121×C73—21no—
72#121×C73—21no—
73C4×Q8×C73—21no—
74C4×Q8×C73—21no—
75C4×D4×C73—21no—
76C4×D4×C73—21no—
77#117×C73—21no—
78C2×C4⋊C4×C73—21no—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C42×C73C2311yescharacteristic
83#107×C73—24no—
84#107×C73—24no—
85#107×C73—24no—
86#106×C73—24no—
87#106×C73—22no—
88C4⋊C8×C73D412yes—
89#103×C73—24no—
90#103×C73—22no—
91#103×C73—24no—
92#103×C73—24no—
93#102×C73—24no—
94#102×C73—22no—
95C22⋊C8×C73D412yes—
96#97×C73D412yes—
97C4×C8×C73D412yes—
98C4×C8×C73C2311yescharacteristic
99C4×C8×C73C2311yescharacteristic
100C4×C8×C73C2312yes—
101C4×C8×C73C2312yes—
102#41×C73—44no—
103C2×Q8×C73—21no—
104C2×Q8×C73—21no—
105C2×Q8×C73—22no—
106C2×Q8×C73—22no—
107C2×Q8×C73—21no—
108C2×Q8×C73—41no—
109C2×Q8×C73—41no—
110C2×Q8×C73—82no—
111C2×Q8×C73—82no—
112C2×D4×C73—21no—
113C2×D4×C73—21no—
114C2×D4×C73—21no—
115C2×D4×C73—41no—
116C22×C4×C73—41no—
117C22×C4×C73—41no—
118C22×C4×C73—41no—
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73C2×D411yescharacteristic
122Q16×C73—84no—
123SD16×C73—44no—
124SD16×C73—84no—
125M16×C73—24no—
126C2×C8×C73—24no—
127C2×C8×C73C2×D412yes—
128C2×C8×C73C2×D414yes—
129C2×C8×C73C2×D414yes—
130C2×C8×C73C2×D412yes—
131C4⋊C4×C73—22no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—21no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—21no—
137C4⋊C4×C73—22no—
138C4⋊C4×C73—22no—
139C4⋊C4×C73—22no—
140C4⋊C4×C73—21no—
141C4⋊C4×C73—41no—
142C4⋊C4×C73—41no—
143C4⋊C4×C73—41no—
144C4⋊C4×C73—41no—
145C4⋊C4×C73—41no—
146C4⋊C4×C73—41no—
147C22⋊C4×C73—41no—
148C22⋊C4×C73—41no—
149C22⋊C4×C73—41no—
150C42×C73—41no—
151C42×C73C2×D411yescharacteristic
152C42×C73C2×D411yescharacteristic
153C42×C73C2×D411yescharacteristic
154C42×C73C2411yescharacteristic
155C23×C73—41no—
156C23×C73#12711yescharacteristic
157Q8×C73—42no—
158Q8×C73—42no—
159Q8×C73—44no—
160Q8×C73—42no—
161Q8×C73—44no—
162Q8×C73—82no—
163Q8×C73—82no—
164D4×C73—42no—
165D4×C73—42no—
166D4×C73—42no—
167D4×C73—82no—
168C2×C4×C73—22no—
169C2×C4×C73—22no—
170C2×C4×C73—41no—
171C2×C4×C73—41no—
172C2×C4×C73—41no—
173C2×C4×C73—42no—
174C2×C4×C73—42no—
175C2×C4×C73—82no—
176C2×C4×C73—82no—
177C2×C4×C73—82no—
178C2×C4×C73#12711yescharacteristic
179C2×C4×C73#12711yescharacteristic
180C2×C4×C73#12711yescharacteristic
181C2×C4×C73#13714yes—
182C2×C4×C73C22×D411yescharacteristic
183C2×C4×C73C22×D411yescharacteristic
184C2×C4×C73C22×D411yescharacteristic
185C8×C73—22no—
186C8×C73—24no—
187C8×C73—24no—
188C8×C73#13714yes—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73—81no—
192C22×C73—81no—
193C22×C73—81no—
194C22×C73#49712yes—
195C22×C73C2×#12711yescharacteristic
196C4×C73—21no—
197C4×C73—21no—
198C4×C73—81no—
199C4×C73—81no—
200C4×C73—82no—
201C4×C73—82no—
202C4×C73—81no—
203C4×C73#49712yes—
204C4×C73C2×#13712yes—
205C4×C73C2×#13712yes—
206C4×C73#57612yes—
207#19683C7311yesmaximal, characteristic, direct product factor, Sylow
208C2×C73—22no—
209C2×C73—161no—
210C2×C73C2×#49711yescharacteristic
211C2×C73#315211yescharacteristic
212C2×C73#315511yescharacteristic
213#3209C2×C7311yescharacteristic, complemented
214#3206C2×C7311yescharacteristic
215#3167C2×C7311yescharacteristic
216#3165C2×C7311yescharacteristic, complemented
217C2×#500C2×C7311yescharacteristic, complemented
218#3141C2×C7311yescharacteristic
219#1723C2×C7312yescomplemented
220#1722C2×C7312yescomplemented
221#1718C2×C7312yescomplemented
222#1714C2×C7312yescomplemented
223#1451C2×C7311yescharacteristic, complemented
224C73#1968311yescharacteristic, direct product factor, Sylow
225#533C22×C7311yescharacteristic
226#532C22×C7311yescharacteristic
227C2×C4⋊Q8C22×C7311yescharacteristic
228C2×C4⋊Q8C22×C7311yescharacteristic
229#501—22no—
230#500—22no—
231#500C22×C7312yes—
232#500C22×C7312yes—
233#500C22×C7312yes—
234#498C22×C7312yes—
235#498C22×C7311yescharacteristic
236#494C22×C7312yes—
237#492C22×C7311yescharacteristic
238#492C22×C7312yes—
239#489—22no—
240#487C22×C7312yes—
241#487C22×C7311yescharacteristic
242#487C22×C7312yes—
243#487C22×C7312yes—
244#486C22×C7312yes—
245#486C22×C7311yescharacteristic
246#465—22no—
247#462—22no—
248#461—22no—
249#451—22no—
250#448—22no—
251C4×SD16—22no—
252#431C22×C7311yescharacteristic
253#428—22no—
254C2×#107—22no—
255#417—22no—
256C4×M16C22×C7311yescharacteristic
257C2×C4×C8C22×C7311yescharacteristic
258#322C22×C7312yes—
259#322C22×C7312yes—
260C2×#41—21no—
261C22×Q8—21no—
262C22×Q8—21no—
263C2×Q16—24no—
264C2×SD16—22no—
265C2×SD16—24no—
266C2×M16D4×C7312yes—
267C22×C8D4×C7312yes—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8C23×C7312yes—
275C4⋊Q8C23×C7312yes—
276C4⋊Q8C23×C7311yescharacteristic
277C4⋊Q8C23×C7311yescharacteristic
278C4⋊Q8C23×C7311yescharacteristic
279#127C23×C7311yescharacteristic
280#125—21no—
281#124—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284#121—21no—
285#121—21no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×D4—21no—
289C4×D4—21no—
290#117—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C42C23×C7311yescharacteristic
296#107—24no—
297#107—24no—
298#107—24no—
299#106—24no—
300#106—22no—
301C4⋊C8D4×C7312yes—
302#103—24no—
303#103—22no—
304#103—24no—
305#103—24no—
306#102—24no—
307#102—22no—
308C22⋊C8D4×C7312yes—
309#97D4×C7312yes—
310C4×C8D4×C7312yes—
311C4×C8C23×C7311yescharacteristic
312C4×C8C23×C7311yescharacteristic
313C4×C8C23×C7312yes—
314C4×C8C23×C7312yes—
315#41—44no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—21no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—82no—
324C2×Q8—82no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4C2×D4×C7311yescharacteristic
333C22×C4C2×D4×C7311yescharacteristic
334C22×C4C2×D4×C7311yescharacteristic
335Q16—84no—
336SD16—44no—
337SD16—84no—
338M16—24no—
339C2×C8—24no—
340C2×C8C2×D4×C7312yes—
341C2×C8C2×D4×C7314yes—
342C2×C8C2×D4×C7314yes—
343C2×C8C2×D4×C7312yes—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C42—41no—
364C42C2×D4×C7311yescharacteristic
365C42C2×D4×C7311yescharacteristic
366C42C2×D4×C7311yescharacteristic
367C42C24×C7311yescharacteristic
368C23—41no—
369C23#127×C7311yescharacteristic
370Q8—42no—
371Q8—42no—
372Q8—44no—
373Q8—42no—
374Q8—44no—
375Q8—82no—
376Q8—82no—
377D4—42no—
378D4—42no—
379D4—42no—
380D4—82no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4#127×C7311yescharacteristic
392C2×C4#127×C7311yescharacteristic
393C2×C4#127×C7311yescharacteristic
394C2×C4#137×C7314yes—
395C2×C4C22×D4×C7311yescharacteristic
396C2×C4C22×D4×C7311yescharacteristic
397C2×C4C22×D4×C7311yescharacteristic
398C8—22no—
399C8—24no—
400C8—24no—
401C8#137×C7314yes—
402C22—21no—
403C22—21no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22#497×C7312yes—
408C22C2×#127×C7311yescharacteristic
409C4—21no—
410C4—21no—
411C4—81no—
412C4—81no—
413C4—82no—
414C4—82no—
415C4—81no—
416C4#497×C7312yes—
417C4C2×#137×C7312yes—
418C4C2×#137×C7312yes—
419C4#576×C7312yes—
420C2—22no—
421C2—161no—
422C2C2×#497×C7311yescharacteristic
423C2C73×#315211yescharacteristic
424C2C73×#315511yescharacteristic
subgroupG/H
center, socleC22×C73C2×#127
commutator, Frattini, cosocleC42C24×C73
prime radical#532×C73C22
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
1. C73×#3209C2semidirect
4. C73×#3165C2semidirect
5. C2×#500×C73C2semidirect
7. C73×#1723C2semidirect
8. C73×#1722C2semidirect
9. C73×#1718C2semidirect
10. C73×#1714C2semidirect
11. C73×#1451C2semidirect
207. #19683C73direct (N × H)
213. #3209C2×C73semidirect
216. #3165C2×C73semidirect
217. C2×#500C2×C73semidirect
219. #1723C2×C73semidirect
220. #1722C2×C73semidirect
221. #1718C2×C73semidirect
222. #1714C2×C73semidirect
223. #1451C2×C73semidirect
224. C73#19683direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#32092
C73×#32062
C73×#31672
C73×#31652
C2×#500×C732
C73×#31412
C73×#17232
C73×#17222
C73×#17182
C73×#17142
C73×#14512
#1968373