Finite Group Databasebeta

← C73×#33645 #32943049

Subgroups — C73×#33645 #32943049

Order 18688.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3050C211yesmaximal, characteristic
2C2×#427×C73C211yesmaximal, characteristic, complemented
3C73×#2237C212yesmaximal, complemented
4C73×#2236C211yesmaximal, characteristic, complemented
5C73×#2235C211yesmaximal, characteristic, complemented
6C73×#2234C212yesmaximal, complemented
7C73×#2233C211yesmaximal, characteristic, complemented
8C73×#2232C211yesmaximal, characteristic, complemented
9C73×#2158C212yesmaximal, complemented
10C73×#2138C212yesmaximal, complemented
11C2×C22⋊C16×C73C211yesmaximal, characteristic, complemented
12C2×#135×C73C2211yescharacteristic
13C23×C8×C73C2211yescharacteristic
14#525×C73—21no—
15C2×#117×C73—21no—
16C22×C16×C73C2212yes—
17#467×C73C2211yescharacteristic
18#467×C73C2211yescharacteristic
19#466×C73C2211yescharacteristic
20#466×C73C2211yescharacteristic
21#465×C73—21no—
22#465×C73—21no—
23#427×C73—21no—
24#427×C73—21no—
25#427×C73C2212yes—
26#427×C73C2211yescharacteristic
27#427×C73C2212yes—
28C2×#107×C73C2211yescharacteristic
29C2×#107×C73C2211yescharacteristic
30C2×#106×C73—21no—
31#416×C73—21no—
32#367×C73C2212yescomplemented
33#367×C73C2212yescomplemented
34#366×C73C2212yescomplemented
35#366×C73C2212yescomplemented
36#358×C73C2212yescomplemented
37#358×C73C2212yescomplemented
38#357×C73C2212yescomplemented
39#357×C73C2212yescomplemented
40C22⋊C16×C73C2211yescharacteristic
41C22⋊C16×C73C2211yescharacteristic
42C22⋊C16×C73C2212yes—
43C2×#41×C73—21no—
44C23×C4×C73D411yescharacteristic
45#135×C73—24no—
46C2×Q16×C73C2311yescharacteristic
47C2×SD16×C73—21no—
48C2×D8×C73C2311yescharacteristic
49C22×C8×C73—24no—
50C22×C8×C73D411yescharacteristic
51C22×C8×C73D411yescharacteristic
52C22×C8×C73D411yescharacteristic
53C22×C8×C73C2311yescharacteristic
54C22×C8×C73C2311yescharacteristic
55C22×C8×C73C2311yescharacteristic
56#123×C73—41no—
57C22⋊Q8×C73—21no—
58C22⋊Q8×C73—21no—
59#121×C73—21no—
60#121×C73—21no—
61#120×C73—41no—
62C4×D4×C73—41no—
63C4×D4×C73—41no—
64#117×C73—21no—
65#117×C73—22no—
66#117×C73—22no—
67#117×C73—41no—
68#117×C73—41no—
69C2×C4⋊C4×C73—21no—
70C2×C4⋊C4×C73—21no—
71C2×C22⋊C4×C73—41no—
72C2×C42×C73—41no—
73C2×C16×C73—24no—
74C2×C16×C73C2312yes—
75C2×C16×C73C2312yes—
76#107×C73—21no—
77#107×C73—21no—
78#107×C73C2312yes—
79#107×C73C2312yes—
80#107×C73C2311yescharacteristic
81#107×C73C2311yescharacteristic
82#106×C73—22no—
83#106×C73—22no—
84#106×C73—21no—
85#103×C73—21no—
86#103×C73—21no—
87#102×C73—21no—
88#102×C73—21no—
89C24×C73D811yescharacteristic
90#41×C73—42no—
91#41×C73—42no—
92C2×Q8×C73—21no—
93C2×D4×C73—21no—
94C2×D4×C73—41no—
95C2×D4×C73—41no—
96C2×D4×C73—41no—
97C22×C4×C73—22no—
98C22×C4×C73—22no—
99C22×C4×C73—41no—
100C22×C4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73D811yescharacteristic
103C22×C4×C73D811yescharacteristic
104C22×C4×C73D811yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108Q16×C73—22no—
109SD16×C73—42no—
110D8×C73—22no—
111C2×C8×C73—22no—
112C2×C8×C73—22no—
113C2×C8×C73—22no—
114C2×C8×C73—22no—
115C2×C8×C73—22no—
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73C2×D411yescharacteristic
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73#4114yes—
120C2×C8×C73C2411yescharacteristic
121C4⋊C4×C73—21no—
122C4⋊C4×C73—21no—
123C4⋊C4×C73—22no—
124C4⋊C4×C73—22no—
125C4⋊C4×C73—41no—
126C4⋊C4×C73—41no—
127C4⋊C4×C73—41no—
128C4⋊C4×C73—41no—
129C22⋊C4×C73—42no—
130C22⋊C4×C73—42no—
131C22⋊C4×C73—41no—
132C22⋊C4×C73—41no—
133C22⋊C4×C73—41no—
134C22⋊C4×C73—41no—
135C22⋊C4×C73—41no—
136C42×C73—42no—
137C42×C73—42no—
138C42×C73—41no—
139C16×C73—22no—
140C16×C73—22no—
141C23×C73—22no—
142C23×C73—22no—
143C23×C73—41no—
144C23×C73—41no—
145C23×C73C2×D811yescharacteristic
146C23×C73C2×D811yescharacteristic
147C23×C73C2×D811yescharacteristic
148Q8×C73—41no—
149Q8×C73—41no—
150D4×C73—41no—
151D4×C73—41no—
152D4×C73—82no—
153D4×C73—82no—
154D4×C73—82no—
155C2×C4×C73—22no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—22no—
161C2×C4×C73—22no—
162C2×C4×C73—42no—
163C2×C4×C73—42no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—82no—
168C2×C4×C73—82no—
169C2×C4×C73—82no—
170C2×C4×C73#12312yes—
171C2×C4×C73C2×D811yescharacteristic
172C2×C4×C73C2×D811yescharacteristic
173C2×C4×C73C2×D811yescharacteristic
174C2×C4×C73#13712yes—
175C2×C4×C73C22×D411yescharacteristic
176C8×C73—21no—
177C8×C73—21no—
178C8×C73#12312yes—
179C8×C73C2×#4112yes—
180C22×C73—21no—
181C22×C73—21no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—21no—
186C22×C73—21no—
187C22×C73—44no—
188C22×C73—81no—
189C22×C73—81no—
190C22×C73—81no—
191C22×C73#48012yes—
192C22×C73#50812yes—
193C22×C73C22×D811yescharacteristic
194C4×C73—21no—
195C4×C73—21no—
196C4×C73—81no—
197C4×C73—82no—
198C4×C73—81no—
199C4×C73—82no—
200C4×C73—81no—
201C4×C73#48012yes—
202C4×C73C2×#12311yescharacteristic
203C4×C73C2×#13711yescharacteristic
204#33645C7311yesmaximal, characteristic, direct product factor, Sylow
205C2×C73—22no—
206C2×C73—22no—
207C2×C73—41no—
208C2×C73—161no—
209C2×C73C2×#48011yescharacteristic
210C2×C73C2×#50811yescharacteristic
211C2×C73#341411yescharacteristic
212#3050C2×C7311yescharacteristic
213C2×#427C2×C7311yescharacteristic, complemented
214#2237C2×C7312yescomplemented
215#2236C2×C7311yescharacteristic, complemented
216#2235C2×C7311yescharacteristic, complemented
217#2234C2×C7312yescomplemented
218#2233C2×C7311yescharacteristic, complemented
219#2232C2×C7311yescharacteristic, complemented
220#2158C2×C7312yescomplemented
221#2138C2×C7312yescomplemented
222C2×C22⋊C16C2×C7311yescharacteristic, complemented
223C73#3364511yescharacteristic, direct product factor, Sylow
224C2×#135C22×C7311yescharacteristic
225C23×C8C22×C7311yescharacteristic
226#525—21no—
227C2×#117—21no—
228C22×C16C22×C7312yes—
229#467C22×C7311yescharacteristic
230#467C22×C7311yescharacteristic
231#466C22×C7311yescharacteristic
232#466C22×C7311yescharacteristic
233#465—21no—
234#465—21no—
235#427—21no—
236#427—21no—
237#427C22×C7312yes—
238#427C22×C7312yes—
239#427C22×C7311yescharacteristic
240C2×#107C22×C7311yescharacteristic
241C2×#107C22×C7311yescharacteristic
242C2×#106—21no—
243#416—21no—
244#367C22×C7312yescomplemented
245#367C22×C7312yescomplemented
246#366C22×C7312yescomplemented
247#366C22×C7312yescomplemented
248#358C22×C7312yescomplemented
249#358C22×C7312yescomplemented
250#357C22×C7312yescomplemented
251#357C22×C7312yescomplemented
252C22⋊C16C22×C7311yescharacteristic
253C22⋊C16C22×C7311yescharacteristic
254C22⋊C16C22×C7312yes—
255C2×#41—21no—
256C23×C4D4×C7311yescharacteristic
257#135—24no—
258C2×Q16C23×C7311yescharacteristic
259C2×SD16—21no—
260C2×D8C23×C7311yescharacteristic
261C22×C8—24no—
262C22×C8D4×C7311yescharacteristic
263C22×C8D4×C7311yescharacteristic
264C22×C8D4×C7311yescharacteristic
265C22×C8C23×C7311yescharacteristic
266C22×C8C23×C7311yescharacteristic
267C22×C8C23×C7311yescharacteristic
268#123—41no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271#121—21no—
272#121—21no—
273#120—41no—
274C4×D4—41no—
275C4×D4—41no—
276#117—22no—
277#117—22no—
278#117—21no—
279#117—41no—
280#117—41no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C22⋊C4—41no—
284C2×C42—41no—
285C2×C16—24no—
286C2×C16C23×C7312yes—
287C2×C16C23×C7312yes—
288#107—21no—
289#107—21no—
290#107C23×C7312yes—
291#107C23×C7312yes—
292#107C23×C7311yescharacteristic
293#107C23×C7311yescharacteristic
294#106—22no—
295#106—22no—
296#106—21no—
297#103—21no—
298#103—21no—
299#102—21no—
300#102—21no—
301C24D8×C7311yescharacteristic
302#41—42no—
303#41—42no—
304C2×Q8—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4D8×C7311yescharacteristic
315C22×C4D8×C7311yescharacteristic
316C22×C4D8×C7311yescharacteristic
317C22×C4C2×D4×C7311yescharacteristic
318C22×C4C2×D4×C7311yescharacteristic
319C22×C4C2×D4×C7311yescharacteristic
320Q16—22no—
321SD16—42no—
322D8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×D4×C7311yescharacteristic
329C2×C8C2×D4×C7311yescharacteristic
330C2×C8C2×D4×C7311yescharacteristic
331C2×C8#41×C7314yes—
332C2×C8C24×C7311yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—42no—
349C42—42no—
350C42—41no—
351C16—22no—
352C16—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23C2×D8×C7311yescharacteristic
358C23C2×D8×C7311yescharacteristic
359C23C2×D8×C7311yescharacteristic
360Q8—41no—
361Q8—41no—
362D4—41no—
363D4—41no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4#123×C7312yes—
383C2×C4C2×D8×C7311yescharacteristic
384C2×C4C2×D8×C7311yescharacteristic
385C2×C4C2×D8×C7311yescharacteristic
386C2×C4#137×C7312yes—
387C2×C4C22×D4×C7311yescharacteristic
388C8—21no—
389C8—21no—
390C8#123×C7312yes—
391C8C2×#41×C7312yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22#480×C7312yes—
404C22#508×C7312yes—
405C22C22×D8×C7311yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—82no—
410C4—81no—
411C4—82no—
412C4—81no—
413C4#480×C7312yes—
414C4C2×#123×C7311yescharacteristic
415C4C2×#137×C7311yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—161no—
420C2C2×#480×C7311yescharacteristic
421C2C2×#508×C7311yescharacteristic
422C2C73×#341411yescharacteristic
subgroupG/H
centerC2×C4×C73C2×D8
commutator, Frattini, cosocleC2×C8C24×C73
socleC22×C73C22×D8
prime radicalC73×#3050C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C73 ◁ C23×C4×C73 ◁ C23×C8×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#427×C73C2semidirect
3. C73×#2237C2semidirect
4. C73×#2236C2semidirect
5. C73×#2235C2semidirect
6. C73×#2234C2semidirect
7. C73×#2233C2semidirect
8. C73×#2232C2semidirect
9. C73×#2158C2semidirect
10. C73×#2138C2semidirect
11. C2×C22⋊C16×C73C2semidirect
32. #367×C73C22semidirect
33. #367×C73C22semidirect
34. #366×C73C22semidirect
35. #366×C73C22semidirect
36. #358×C73C22semidirect
37. #358×C73C22semidirect
38. #357×C73C22semidirect
39. #357×C73C22semidirect
204. #33645C73direct (N × H)
213. C2×#427C2×C73semidirect
214. #2237C2×C73semidirect
215. #2236C2×C73semidirect
216. #2235C2×C73semidirect
217. #2234C2×C73semidirect
218. #2233C2×C73semidirect
219. #2232C2×C73semidirect
220. #2158C2×C73semidirect
221. #2138C2×C73semidirect
222. C2×C22⋊C16C2×C73semidirect
223. C73#33645direct (N × H)
244. #367C22×C73semidirect
245. #367C22×C73semidirect
246. #366C22×C73semidirect
247. #366C22×C73semidirect
248. #358C22×C73semidirect
249. #358C22×C73semidirect
250. #357C22×C73semidirect
251. #357C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#30502
C2×#427×C732
C73×#22372
C73×#22362
C73×#22352
C73×#22342
C73×#22332
C73×#22322
C73×#21582
C73×#21382
C2×C22⋊C16×C732
#3364573