Finite Group Databasebeta

← C73×#19242 #32928646

Subgroups — C73×#19242 #32928646

Order 18688.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3249C211yesmaximal, characteristic, complemented
2C73×#3208C211yesmaximal, characteristic
3C73×#3138C211yesmaximal, characteristic
4C73×#2962C211yesmaximal, characteristic, complemented
5C73×#1702C212yesmaximal, complemented
6C73×#1698C212yesmaximal, complemented
7C73×#1682C212yesmaximal, complemented
8C73×#1678C212yesmaximal, complemented
9C73×#1565C211yesmaximal, characteristic, complemented
10C73×#1542C211yesmaximal, characteristic, complemented
11C2×#331×C73C211yesmaximal, characteristic
12#533×C73C2211yescharacteristic
13#533×C73C2211yescharacteristic
14C2×#127×C73C2211yescharacteristic
15#492×C73—21no—
16#490×C73—22no—
17#486×C73—21no—
18#486×C73C2212yes—
19#486×C73C2212yes—
20#477×C73—21no—
21#477×C73C2212yescomplemented
22#467×C73—22no—
23#462×C73—21no—
24#462×C73C2212yescomplemented
25#460×C73C2212yes—
26#460×C73C2212yes—
27#450×C73—22no—
28#446×C73—22no—
29#446×C73C2212yescomplemented
30C8×Q8×C73C2212yescomplemented
31#431×C73C2211yescharacteristic
32#424×C73—21no—
33C2×C4⋊C8×C73C2211yescharacteristic
34C2×C4⋊C8×C73C2211yescharacteristic
35#418×C73—21no—
36#418×C73—21no—
37#405×C73—21no—
38C4×M16×C73C2211yescharacteristic
39#335×C73C2212yescomplemented
40#334×C73C2212yescomplemented
41#333×C73C2211yescharacteristic, complemented
42#333×C73C2211yescharacteristic, complemented
43#332×C73C2211yescharacteristic, complemented
44#332×C73C2211yescharacteristic, complemented
45#331×C73C2212yes—
46#331×C73C2212yes—
47C22×D4×C73—21no—
48C22×D4×C73—21no—
49C22×D4×C73—21no—
50C2×Q16×C73—22no—
51C2×SD16×C73—22no—
52C2×SD16×C73—22no—
53C2×M16×C73D411yescharacteristic
54C2×M16×C73D411yescharacteristic
55C22×C8×C73—21no—
56C22×C8×C73—21no—
57C4⋊Q8×C73D411yescharacteristic
58C4⋊Q8×C73D411yescharacteristic
59C4⋊Q8×C73C2311yescharacteristic
60C4⋊Q8×C73C2311yescharacteristic
61#127×C73—21no—
62#127×C73—21no—
63#127×C73—21no—
64#127×C73C2312yes—
65#125×C73—21no—
66#125×C73—21no—
67C22⋊Q8×C73—21no—
68C22⋊Q8×C73—21no—
69C22⋊Q8×C73—21no—
70C22⋊Q8×C73D412yes—
71C4×Q8×C73—21no—
72C4×Q8×C73—21no—
73C4×Q8×C73—21no—
74C4×Q8×C73C2312yes—
75#117×C73—21no—
76#117×C73—21no—
77C2×C42×C73C2311yescharacteristic
78#107×C73—22no—
79C4⋊C8×C73—22no—
80C4⋊C8×C73—22no—
81C4⋊C8×C73—21no—
82C4⋊C8×C73—21no—
83C4⋊C8×C73C2311yescharacteristic
84C4⋊C8×C73C2311yescharacteristic
85C4⋊C8×C73C2312yes—
86C4⋊C8×C73C2311yescharacteristic
87C4⋊C8×C73C2311yescharacteristic
88#103×C73—22no—
89#103×C73—22no—
90#102×C73—22no—
91#102×C73—22no—
92#102×C73—22no—
93#102×C73—22no—
94#102×C73—22no—
95#102×C73—22no—
96C22⋊C8×C73—21no—
97#97×C73—22no—
98#97×C73D412yes—
99C4×C8×C73—22no—
100C4×C8×C73—21no—
101C4×C8×C73C2312yes—
102C24×C73—41no—
103C2×Q8×C73—21no—
104C2×Q8×C73—21no—
105C2×Q8×C73—21no—
106C2×Q8×C73C2×D412yes—
107C2×D4×C73—22no—
108C2×D4×C73—22no—
109C2×D4×C73—22no—
110C2×D4×C73—41no—
111C2×D4×C73—41no—
112C2×D4×C73—41no—
113C2×D4×C73—42no—
114C2×D4×C73—42no—
115C2×D4×C73—81no—
116C2×D4×C73—81no—
117C2×D4×C73—82no—
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73#4111yescharacteristic
121Q16×C73—82no—
122SD16×C73—82no—
123SD16×C73—82no—
124M16×C73—22no—
125M16×C73—22no—
126C2×C8×C73—21no—
127C2×C8×C73—21no—
128C2×C8×C73—21no—
129C2×C8×C73—21no—
130C2×C8×C73—22no—
131C2×C8×C73—42no—
132C2×C8×C73—81no—
133C2×C8×C73C2×D412yes—
134C2×C8×C73C2×D412yes—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—21no—
137C4⋊C4×C73—21no—
138C4⋊C4×C73—22no—
139C4⋊C4×C73—22no—
140C4⋊C4×C73—21no—
141C4⋊C4×C73—21no—
142C4⋊C4×C73—21no—
143C4⋊C4×C73—21no—
144C4⋊C4×C73—41no—
145C4⋊C4×C73—41no—
146C4⋊C4×C73C2×D412yes—
147C22⋊C4×C73—22no—
148C22⋊C4×C73—41no—
149C42×C73—22no—
150C42×C73—41no—
151C42×C73C2×D411yescharacteristic
152C42×C73C2×D411yescharacteristic
153C42×C73#4111yescharacteristic
154C42×C73C2411yescharacteristic
155C23×C73—42no—
156C23×C73—82no—
157C23×C73—82no—
158C23×C73—82no—
159C23×C73#12111yescharacteristic
160Q8×C73—22no—
161Q8×C73—22no—
162Q8×C73—41no—
163Q8×C73—41no—
164Q8×C73—81no—
165Q8×C73—81no—
166D4×C73—42no—
167D4×C73—42no—
168D4×C73—42no—
169D4×C73—42no—
170D4×C73—82no—
171D4×C73—82no—
172D4×C73—82no—
173D4×C73—82no—
174C2×C4×C73—21no—
175C2×C4×C73—21no—
176C2×C4×C73—22no—
177C2×C4×C73—22no—
178C2×C4×C73—41no—
179C2×C4×C73—41no—
180C2×C4×C73—41no—
181C2×C4×C73#12111yescharacteristic
182C2×C4×C73#12111yescharacteristic
183C2×C4×C73#12111yescharacteristic
184C2×C4×C73#13512yes—
185C2×C4×C73#13712yes—
186C2×C4×C73C22×D411yescharacteristic
187C2×C4×C73C22×D411yescharacteristic
188C2×C4×C73C2×#4111yescharacteristic
189C8×C73—22no—
190C8×C73—22no—
191C8×C73—41no—
192C8×C73—41no—
193C8×C73—42no—
194C8×C73—41no—
195C8×C73—41no—
196C22×C73—21no—
197C22×C73—21no—
198C22×C73—82no—
199C22×C73—82no—
200C22×C73—82no—
201C22×C73—164no—
202C22×C73#46412yes—
203C22×C73C2×#12111yescharacteristic
204C4×C73—21no—
205C4×C73—21no—
206C4×C73—21no—
207C4×C73—42no—
208C4×C73—42no—
209C4×C73—81no—
210C4×C73—81no—
211C4×C73#46412yes—
212C4×C73C2×#13511yescharacteristic
213C4×C73C2×#13711yescharacteristic
214C4×C73#57611yescharacteristic
215C4×C73#57711yescharacteristic
216#19242C7311yesmaximal, characteristic, direct product factor, Sylow
217C2×C73—22no—
218C2×C73—162no—
219C2×C73C2×#46411yescharacteristic
220C2×C73#306211yescharacteristic
221C2×C73#306811yescharacteristic
222#3249C2×C7311yescharacteristic, complemented
223#3208C2×C7311yescharacteristic
224#3138C2×C7311yescharacteristic
225#2962C2×C7311yescharacteristic, complemented
226#1702C2×C7312yescomplemented
227#1698C2×C7312yescomplemented
228#1682C2×C7312yescomplemented
229#1678C2×C7312yescomplemented
230#1565C2×C7311yescharacteristic, complemented
231#1542C2×C7311yescharacteristic, complemented
232C2×#331C2×C7311yescharacteristic
233C73#1924211yescharacteristic, direct product factor, Sylow
234#533C22×C7311yescharacteristic
235#533C22×C7311yescharacteristic
236C2×#127C22×C7311yescharacteristic
237#492—21no—
238#490—22no—
239#486—21no—
240#486C22×C7312yes—
241#486C22×C7312yes—
242#477—21no—
243#477C22×C7312yescomplemented
244#467—22no—
245#462—21no—
246#462C22×C7312yescomplemented
247#460C22×C7312yes—
248#460C22×C7312yes—
249#450—22no—
250#446—22no—
251#446C22×C7312yescomplemented
252C8×Q8C22×C7312yescomplemented
253#431C22×C7311yescharacteristic
254#424—21no—
255C2×C4⋊C8C22×C7311yescharacteristic
256C2×C4⋊C8C22×C7311yescharacteristic
257#418—21no—
258#418—21no—
259#405—21no—
260C4×M16C22×C7311yescharacteristic
261#335C22×C7312yescomplemented
262#334C22×C7312yescomplemented
263#333C22×C7311yescharacteristic, complemented
264#333C22×C7311yescharacteristic, complemented
265#332C22×C7311yescharacteristic, complemented
266#332C22×C7311yescharacteristic, complemented
267#331C22×C7312yes—
268#331C22×C7312yes—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272C2×Q16—22no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×M16D4×C7311yescharacteristic
276C2×M16D4×C7311yescharacteristic
277C22×C8—21no—
278C22×C8—21no—
279C4⋊Q8D4×C7311yescharacteristic
280C4⋊Q8D4×C7311yescharacteristic
281C4⋊Q8C23×C7311yescharacteristic
282C4⋊Q8C23×C7311yescharacteristic
283#127—21no—
284#127—21no—
285#127—21no—
286#127C23×C7312yes—
287#125—21no—
288#125—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8D4×C7312yes—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8C23×C7312yes—
297#117—21no—
298#117—21no—
299C2×C42C23×C7311yescharacteristic
300#107—22no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8C23×C7311yescharacteristic
306C4⋊C8C23×C7311yescharacteristic
307C4⋊C8C23×C7312yes—
308C4⋊C8C23×C7311yescharacteristic
309C4⋊C8C23×C7311yescharacteristic
310#103—22no—
311#103—22no—
312#102—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316#102—22no—
317#102—22no—
318C22⋊C8—21no—
319#97—22no—
320#97D4×C7312yes—
321C4×C8—22no—
322C4×C8—21no—
323C4×C8C23×C7312yes—
324C24—41no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8C2×D4×C7312yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4C2×D4×C7311yescharacteristic
341C22×C4C2×D4×C7311yescharacteristic
342C22×C4#41×C7311yescharacteristic
343Q16—82no—
344SD16—82no—
345SD16—82no—
346M16—22no—
347M16—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—22no—
353C2×C8—42no—
354C2×C8—81no—
355C2×C8C2×D4×C7312yes—
356C2×C8C2×D4×C7312yes—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4C2×D4×C7312yes—
369C22⋊C4—22no—
370C22⋊C4—41no—
371C42—22no—
372C42—41no—
373C42C2×D4×C7311yescharacteristic
374C42C2×D4×C7311yescharacteristic
375C42#41×C7311yescharacteristic
376C42C24×C7311yescharacteristic
377C23—42no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23#121×C7311yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—41no—
385Q8—41no—
386Q8—81no—
387Q8—81no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4#121×C7311yescharacteristic
404C2×C4#121×C7311yescharacteristic
405C2×C4#121×C7311yescharacteristic
406C2×C4#135×C7312yes—
407C2×C4#137×C7312yes—
408C2×C4C22×D4×C7311yescharacteristic
409C2×C4C22×D4×C7311yescharacteristic
410C2×C4C2×#41×C7311yescharacteristic
411C8—22no—
412C8—22no—
413C8—41no—
414C8—41no—
415C8—42no—
416C8—41no—
417C8—41no—
418C22—21no—
419C22—21no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—164no—
424C22#464×C7312yes—
425C22C2×#121×C7311yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—81no—
432C4—81no—
433C4#464×C7312yes—
434C4C2×#135×C7311yescharacteristic
435C4C2×#137×C7311yescharacteristic
436C4#576×C7311yescharacteristic
437C4#577×C7311yescharacteristic
438C2—22no—
439C2—162no—
440C2C2×#464×C7311yescharacteristic
441C2C73×#306211yescharacteristic
442C2C73×#306811yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC42C24×C73
prime radicalC2×#127×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×#3249C2semidirect
4. C73×#2962C2semidirect
5. C73×#1702C2semidirect
6. C73×#1698C2semidirect
7. C73×#1682C2semidirect
8. C73×#1678C2semidirect
9. C73×#1565C2semidirect
10. C73×#1542C2semidirect
21. #477×C73C22semidirect
24. #462×C73C22semidirect
29. #446×C73C22semidirect
30. C8×Q8×C73C22semidirect
39. #335×C73C22semidirect
40. #334×C73C22semidirect
41. #333×C73C22semidirect
42. #333×C73C22semidirect
43. #332×C73C22semidirect
44. #332×C73C22semidirect
216. #19242C73direct (N × H)
222. #3249C2×C73semidirect
225. #2962C2×C73semidirect
226. #1702C2×C73semidirect
227. #1698C2×C73semidirect
228. #1682C2×C73semidirect
229. #1678C2×C73semidirect
230. #1565C2×C73semidirect
231. #1542C2×C73semidirect
233. C73#19242direct (N × H)
243. #477C22×C73semidirect
246. #462C22×C73semidirect
251. #446C22×C73semidirect
252. C8×Q8C22×C73semidirect
261. #335C22×C73semidirect
262. #334C22×C73semidirect
263. #333C22×C73semidirect
264. #333C22×C73semidirect
265. #332C22×C73semidirect
266. #332C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#32492
C73×#32082
C73×#31382
C73×#29622
C73×#17022
C73×#16982
C73×#16822
C73×#16782
C73×#15652
C73×#15422
C2×#331×C732
#1924273