Finite Group Databasebeta

← #509×Dic73 #33239304

Subgroups — #509×Dic73 #33239304

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×Dic73C211yesmaximal, characteristic, complemented
2C2×D8×Dic73C211yesmaximal, characteristic, complemented
3#22611807C212yesmaximal, complemented
4#22611796C212yesmaximal, complemented
5SD32×Dic73C212yesmaximal, complemented
6D16×Dic73C212yesmaximal, complemented
7#22611719C211yesmaximal, characteristic, complemented
8#22611647C211yesmaximal, characteristic, complemented
9#22611643C211yesmaximal, characteristic, complemented
10M32×Dic73C211yesmaximal, characteristic, complemented
11C2×#509×C73C211yesmaximal, characteristic
12#41×Dic73—21no—
13C2×D4×Dic73—21no—
14C2×#14556531C2211yescharacteristic
15Q16×Dic73C2211yescharacteristic, complemented
16Q16×Dic73C2211yescharacteristic, complemented
17SD16×Dic73—21no—
18SD16×Dic73—21no—
19D8×Dic73—21no—
20D8×Dic73—21no—
21D8×Dic73C2212yes—
22D8×Dic73C2212yes—
23D8×Dic73C2211yescharacteristic, complemented
24D8×Dic73C2211yescharacteristic, complemented
25#17988625C2211yescharacteristic
26C2×#14556525C2211yescharacteristic
27#17988583—21no—
28C2×#14556512—21no—
29C2×C8×Dic73C2211yescharacteristic
30#17988415C2212yescomplemented
31#17988413C2212yescomplemented
32#17988391C2212yescomplemented
33#17988388C2212yes—
34#17988388C2212yes—
35#17988388C2212yescomplemented
36#17988362C2212yescomplemented
37C16×Dic73C2212yescomplemented
38C2×#135×C73C2211yescharacteristic
39C22×D8×C73C2211yescharacteristic
40#509×C73C418yescomplemented
41C2×SD32×C73C2212yes—
42C2×D16×C73C2212yes—
43C2×M32×C73C2211yescharacteristic
44C23×Dic73—41no—
45C22×#10596831D411yescharacteristic
46Q8×Dic73—21no—
47Q8×Dic73—21no—
48D4×Dic73—22no—
49D4×Dic73—22no—
50D4×Dic73—21no—
51D4×Dic73—21no—
52D4×Dic73—41no—
53D4×Dic73—41no—
54D4×Dic73—41no—
55D4×Dic73—41no—
56#14556550—21no—
57#14556550—41no—
58C2×C4⋊Dic73—21no—
59C2×C22⋊Dic73—41no—
60C2×C4×Dic73—41no—
61C2×C4×Dic73D411yescharacteristic
62#14556531—22no—
63C2×#11924003C2312yes—
64#14556525—21no—
65#14556525C2312yes—
66#14556525C2311yescharacteristic
67#14556525C2311yescharacteristic
68#14556523—21no—
69#14556515—21no—
70#14556515—21no—
71#14556512—22no—
72#14556512—22no—
73#14556512—21no—
74#14556512—21no—
75C8×Dic73D411yescharacteristic
76C8×Dic73D411yescharacteristic
77C8×Dic73C2311yescharacteristic
78C8×Dic73C2311yescharacteristic
79C2×#41×C73—21no—
80C22×D4×C73—21no—
81#135×C73—22no—
82#135×C73C2×C414yescomplemented
83C2×Q16×C73C2311yescharacteristic
84C2×SD16×C73—21no—
85C2×D8×C73—21no—
86C2×D8×C73—22no—
87C2×D8×C73C2×C414yescomplemented
88C2×D8×C73C2312yes—
89C2×D8×C73C2311yescharacteristic
90C22×C8×C73C2311yescharacteristic
91SD32×C73C2×C418yescomplemented
92D16×C73C2×C418yescomplemented
93M32×C73C2×C414yescomplemented
94C2×C16×C73C2312yes—
95C22×Dic73—41no—
96C22×Dic73—41no—
97C22×Dic73—41no—
98C22×Dic73—42no—
99C22×Dic73—42no—
100C22×Dic73—84no—
101C22×Dic73D811yescharacteristic
102C22×Dic73D811yescharacteristic
103C2×#10596831—22no—
104C2×#10596831C2×D411yescharacteristic
105C2×#10596831C2×D411yescharacteristic
106C4⋊Dic73—22no—
107C4⋊Dic73—21no—
108C4⋊Dic73—21no—
109C4⋊Dic73—41no—
110C4⋊Dic73—41no—
111C4⋊Dic73—41no—
112C22⋊Dic73—41no—
113C22⋊Dic73—42no—
114C22⋊Dic73—42no—
115C22⋊Dic73—41no—
116C22⋊Dic73—41no—
117C4×Dic73—41no—
118C4×Dic73—41no—
119C4×Dic73—41no—
120C4×Dic73D811yescharacteristic
121C4×Dic73D811yescharacteristic
122C4×Dic73C2×D411yescharacteristic
123C4×Dic73C2×D411yescharacteristic
124#11924003—22no—
125C24×C73—41no—
126#41×C73—24no—
127#41×C73—42no—
128C2×Q8×C73—21no—
129C2×D4×C73—24no—
130C2×D4×C73—22no—
131C2×D4×C73—21no—
132C2×D4×C73—41no—
133C2×D4×C73—41no—
134C2×D4×C73—42no—
135C22×C4×C73—41no—
136C22×C4×C73C2×D411yescharacteristic
137Q16×C73—21no—
138Q16×C73C22×C412yescomplemented
139SD16×C73—24no—
140D8×C73—24no—
141D8×C73—22no—
142D8×C73—21no—
143D8×C73C22×C414yes—
144D8×C73C22×C412yescomplemented
145C2×C8×C73C22×C412yes—
146C2×C8×C73C2×D411yescharacteristic
147C2×C8×C73#4112yes—
148C2×C8×C73C2411yescharacteristic
149C16×C73C22×C414yescomplemented
150C23×C73—41no—
151C23×C73—42no—
152C23×C73—44no—
153C23×C73—82no—
154C23×C73—82no—
155C23×C73C2×D811yescharacteristic
156Q8×C73—22no—
157Q8×C73—41no—
158D4×C73—24no—
159D4×C73—22no—
160D4×C73—44no—
161D4×C73—42no—
162D4×C73—41no—
163D4×C73—44no—
164C2×C4×C73—41no—
165C2×C4×C73—44no—
166C2×C4×C73C4×D412yes—
167C2×C4×C73C2×D811yescharacteristic
168C2×C4×C73#13512yes—
169C2×C4×C73C22×D411yescharacteristic
170C2×Dic73—22no—
171C2×Dic73—41no—
172C2×Dic73—41no—
173C2×Dic73—41no—
174C2×Dic73—42no—
175C2×Dic73—82no—
176C2×Dic73—84no—
177C2×Dic73—82no—
178C2×Dic73—82no—
179C2×Dic73—84no—
180C2×Dic73C2×D811yescharacteristic
181C2×Dic73C2×D811yescharacteristic
182C2×Dic73C2×D811yescharacteristic
183C2×Dic73C2×D811yescharacteristic
184C8×C73C4×D412yes—
185C8×C73C23×C411yescharacteristic
186C8×C73C2×#4111yescharacteristic
187#10596831—21no—
188#10596831—21no—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73—42no—
192C22×C73—44no—
193C22×C73—88no—
194C22×C73—82no—
195C22×C73—81no—
196C22×C73—81no—
197C22×C73—82no—
198C22×C73C4×D812yes—
199C22×C73C22×D811yescharacteristic
200C4×C73—42no—
201C4×C73C4×D812yes—
202C4×C73C2×C4×D411yescharacteristic
203C4×C73C2×#13511yescharacteristic
204Dic73—21no—
205Dic73—21no—
206Dic73—21no—
207Dic73—81no—
208Dic73—82no—
209Dic73—81no—
210Dic73#50912yesdirect product factor
211C4×#509—731nomaximal, Sylow
212C2×C73—22no—
213C2×C73—82no—
214C2×C73—84no—
215C2×C73C2×C4×D811yescharacteristic
216C2×C73C2×#50911yescharacteristic
217C2×C73#341411yescharacteristic
218C2×#509D7311yescharacteristic
219C4×#135—731no—
220C2×C4×D8—731no—
221#2177—732no—
222#2175—732no—
223C4×SD32—732no—
224C4×D16—732no—
225#2159—731no—
226#2140—731no—
227#2139—731no—
228C4×M32—731no—
229C73C4×#50911yescharacteristic, complemented, Sylow
230C2×#135C2×D7311yescharacteristic
231C22×D8C2×D7311yescharacteristic
232C4×#41—1461no—
233C2×C4×D4—1461no—
234#509Dic7318yesdirect product factor
235C2×SD32C2×D7312yes—
236C2×D16C2×D7312yes—
237C2×M32—731no—
238C2×M32C2×D7311yescharacteristic
239C4×Q16—731no—
240C4×Q16—731no—
241C4×SD16—1461no—
242C4×SD16—1461no—
243C4×D8—732no—
244C4×D8—732no—
245C4×D8—731no—
246C4×D8—731no—
247C4×D8—1461no—
248C4×D8—1461no—
249#427—731no—
250C2×#107—731no—
251#416—1461no—
252C2×#102—1461no—
253C2×C4×C8—731no—
254#367—732no—
255#366—732no—
256#358—732no—
257#357—732no—
258#357—732no—
259#357—732no—
260#346—732no—
261C4×C16—732no—
262C2×#41—21no—
263C22×D4—21no—
264C23×C4—2921no—
265#135—22no—
266#135C2×Dic7314yescomplemented
267C2×Q16C22×D7311yescharacteristic
268C2×SD16—21no—
269C2×D8—21no—
270C2×D8—22no—
271C2×D8C2×Dic7314yescomplemented
272C2×D8C22×D7312yes—
273C2×D8C22×D7311yescharacteristic
274C22×C8—731no—
275C22×C8C22×D7311yescharacteristic
276C4×Q8—1461no—
277C4×Q8—1461no—
278C4×D4—2921no—
279C4×D4—2921no—
280C4×D4—2921no—
281C4×D4—2921no—
282C4×D4—1462no—
283C4×D4—1462no—
284C4×D4—1461no—
285C4×D4—1461no—
286#117—2921no—
287#117—1461no—
288C2×C4⋊C4—1461no—
289C2×C22⋊C4—2921no—
290C2×C42—2921no—
291C2×C42—731no—
292SD32C2×Dic7318yescomplemented
293D16C2×Dic7318yescomplemented
294M32—1462no—
295M32C2×Dic7314yescomplemented
296C2×C16—732no—
297C2×C16C22×D7312yes—
298#107—731no—
299#107—731no—
300#107—732no—
301#107—1461no—
302#106—1461no—
303#103—1461no—
304#103—1461no—
305#102—1462no—
306#102—1462no—
307#102—1461no—
308#102—1461no—
309C4×C8—731no—
310C4×C8—731no—
311C4×C8—731no—
312C4×C8—731no—
313C24—41no—
314#41—24no—
315#41—42no—
316C2×Q8—21no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C22×C4—41no—
324C22×C4—2921no—
325C22×C4—2921no—
326C22×C4—2921no—
327C22×C4—2922no—
328C22×C4—2922no—
329C22×C4—5844no—
330C22×C4—731no—
331C22×C4—731no—
332C22×C4D4×D7311yescharacteristic
333Q16—21no—
334Q16C22×Dic7312yescomplemented
335SD16—24no—
336D8—24no—
337D8—22no—
338D8—21no—
339D8C22×Dic7314yes—
340D8C22×Dic7312yescomplemented
341C2×C8—731no—
342C2×C8—731no—
343C2×C8—1462no—
344C2×C8C22×Dic7312yes—
345C2×C8D4×D7311yescharacteristic
346C2×C8#1192402712yes—
347C2×C8C23×D7311yescharacteristic
348C4⋊C4—2921no—
349C4⋊C4—2921no—
350C4⋊C4—2921no—
351C4⋊C4—1462no—
352C4⋊C4—1461no—
353C4⋊C4—1461no—
354C22⋊C4—2921no—
355C22⋊C4—2922no—
356C22⋊C4—2922no—
357C22⋊C4—2921no—
358C22⋊C4—2921no—
359C42—2921no—
360C42—2921no—
361C42—2921no—
362C42—731no—
363C42—731no—
364C42—731no—
365C42—731no—
366C16—1462no—
367C16C22×Dic7314yescomplemented
368C23—44no—
369C23—41no—
370C23—42no—
371C23—82no—
372C23—82no—
373C23D8×D7311yescharacteristic
374Q8—22no—
375Q8—41no—
376D4—24no—
377D4—22no—
378D4—44no—
379D4—44no—
380D4—42no—
381D4—41no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—2921no—
385C2×C4—2921no—
386C2×C4—2922no—
387C2×C4—2921no—
388C2×C4—5842no—
389C2×C4—5844no—
390C2×C4—5842no—
391C2×C4—5844no—
392C2×C4—5842no—
393C2×C4—731no—
394C2×C4—731no—
395C2×C4—731no—
396C2×C4—731no—
397C2×C4—1462no—
398C2×C4D4×Dic7312yes—
399C2×C4D8×D7311yescharacteristic
400C2×C4#1455661012yes—
401C2×C4C2×D4×D7311yescharacteristic
402C8—1461no—
403C8—1461no—
404C8D4×Dic7312yes—
405C8C23×Dic7311yescharacteristic
406C8C2×#1192402711yescharacteristic
407C22—21no—
408C22—21no—
409C22—42no—
410C22—44no—
411C22—88no—
412C22—82no—
413C22—81no—
414C22—81no—
415C22—82no—
416C22D8×Dic7312yes—
417C22C2×D8×D7311yescharacteristic
418C4—42no—
419C4—5842no—
420C4—5841no—
421C4—5841no—
422C4—732no—
423C4—1461no—
424C4—1461no—
425C4—1461no—
426C4D8×Dic7312yes—
427C4C2×D4×Dic7311yescharacteristic
428C4C2×#1455661011yescharacteristic
429C2—22no—
430C2—84no—
431C2—82no—
432C2C2×D8×Dic7311yescharacteristic
433C2#509×D7311yescharacteristic
434C2#2262306311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
Fitting, prime radicalC2×#509×C73C2
socleC22×C73C22×D8
cosocleC2×C8×C73C24
hypercenterC2×#509D73
hypocenterC73C4×#509
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. #135×Dic73C2semidirect
2. C2×D8×Dic73C2semidirect
3. #22611807C2semidirect
4. #22611796C2semidirect
5. SD32×Dic73C2semidirect
6. D16×Dic73C2semidirect
7. #22611719C2semidirect
8. #22611647C2semidirect
9. #22611643C2semidirect
10. M32×Dic73C2semidirect
15. Q16×Dic73C22semidirect
16. Q16×Dic73C22semidirect
23. D8×Dic73C22semidirect
24. D8×Dic73C22semidirect
30. #17988415C22semidirect
31. #17988413C22semidirect
32. #17988391C22semidirect
35. #17988388C22semidirect
36. #17988362C22semidirect
37. C16×Dic73C22semidirect
40. #509×C73C4semidirect
82. #135×C73C2×C4semidirect
87. C2×D8×C73C2×C4semidirect
91. SD32×C73C2×C4semidirect
92. D16×C73C2×C4semidirect
93. M32×C73C2×C4semidirect
138. Q16×C73C22×C4semidirect
144. D8×C73C22×C4semidirect
149. C16×C73C22×C4semidirect
210. Dic73#509direct (N × H)
229. C73C4×#509semidirect
234. #509Dic73direct (N × H)
266. #135C2×Dic73semidirect
271. C2×D8C2×Dic73semidirect
292. SD32C2×Dic73semidirect
293. D16C2×Dic73semidirect
295. M32C2×Dic73semidirect
334. Q16C22×Dic73semidirect
340. D8C22×Dic73semidirect
367. C16C22×Dic73semidirect

Maximal subgroups

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

subgroupindex
#135×Dic732
C2×D8×Dic732
#226118072
#226117962
SD32×Dic732
D16×Dic732
#226117192
#226116472
#226116432
M32×Dic732
C2×#509×C732
C4×#50973