Finite Group Databasebeta

← #33240626

Subgroups — #33240626

Order 18688.

546 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620443C211yesmaximal, characteristic, complemented
2#135×Dic73C211yesmaximal, characteristic
3#22612097C211yesmaximal, characteristic, complemented
4#22612093C211yesmaximal, characteristic, complemented
5#22612090C211yesmaximal, characteristic, complemented
6#22612086C211yesmaximal, characteristic, complemented
7#22612081C211yesmaximal, characteristic, complemented
8#22612076C211yesmaximal, characteristic, complemented
9#22612071C211yesmaximal, characteristic, complemented
10#22612066C211yesmaximal, characteristic, complemented
11#22611715C211yesmaximal, characteristic, complemented
12#22611713C211yesmaximal, characteristic, complemented
13C2×#17988405C211yesmaximal, characteristic, complemented
14#22611638C211yesmaximal, characteristic
15C73×#2138C211yesmaximal, characteristic
16#17989140—21no—
17#41×Dic73—21no—
18C22×#11924003C2211yescharacteristic
19#17988991C2211yescharacteristic
20#17988991C2211yescharacteristic
21#17988988C2211yescharacteristic
22#17988988C2211yescharacteristic
23#17988987—21no—
24#17988984—21no—
25Q16×Dic73C2211yescharacteristic
26Q16×Dic73C2211yescharacteristic
27SD16×Dic73—21no—
28SD16×Dic73—21no—
29D8×Dic73C2211yescharacteristic
30D8×Dic73C2211yescharacteristic
31#17988625C2211yescharacteristic
32#17988625C2211yescharacteristic
33#17988623—21no—
34#17988583—21no—
35C2×C8×Dic73C2211yescharacteristic
36#17988415C2211yescharacteristic
37#17988415C2211yescharacteristic
38#17988414C2211yescharacteristic
39#17988414C2211yescharacteristic
40#17988413C2211yescharacteristic
41#17988413C2211yescharacteristic
42#17988412C2211yescharacteristic
43#17988412C2211yescharacteristic
44#17988405C2211yescharacteristic
45#17988405C2211yescharacteristic
46#17988405C2211yescharacteristic
47#17988405C2211yescharacteristic
48#17988391C2211yescharacteristic
49#17988391C2211yescharacteristic
50#17988388C2211yescharacteristic
51#17988388C2211yescharacteristic
52C2×#135×C73C2211yescharacteristic
53C22×C16×C73C2211yescharacteristic
54#427×C73C2211yescharacteristic
55#358×C73C2211yescharacteristic
56#358×C73C2211yescharacteristic
57#357×C73C2211yescharacteristic
58#357×C73C2211yescharacteristic
59C22×#10596831D411yescharacteristic
60#14556590—21no—
61#14556590—21no—
62#14556582—21no—
63#14556582—21no—
64#14556568—41no—
65#14556568—41no—
66Q8×Dic73—21no—
67Q8×Dic73—21no—
68C4×Dic146—41no—
69C4×Dic146—41no—
70D4×Dic73—21no—
71D4×Dic73—21no—
72D4×Dic73—41no—
73D4×Dic73—41no—
74#14556550—21no—
75#14556550—21no—
76#14556550—41no—
77C2×C4×Dic73—41no—
78C2×C4×Dic73D411yescharacteristic
79C2×#11924003—22no—
80C2×#11924003C2311yescharacteristic
81C2×#11924003C2311yescharacteristic
82#14556525C2311yescharacteristic
83#14556525C2311yescharacteristic
84#14556525C2311yescharacteristic
85#14556525C2311yescharacteristic
86#14556524—21no—
87#14556524—21no—
88#14556523—21no—
89#14556523—21no—
90#14556522—21no—
91#14556522—21no—
92#14556515—21no—
93#14556515—21no—
94#14556512—21no—
95#14556512—21no—
96C8×Dic73D411yescharacteristic
97C8×Dic73D411yescharacteristic
98C8×Dic73C2311yescharacteristic
99C8×Dic73C2311yescharacteristic
100C2×#41×C73—21no—
101#135×C73—22no—
102#135×C73Q814yescomplemented
103C2×Q16×C73C2311yescharacteristic
104C2×SD16×C73—21no—
105C2×D8×C73C2311yescharacteristic
106C22×C8×C73C2311yescharacteristic
107#117×C73—21no—
108C2×C16×C73—22no—
109C2×C16×C73C2311yescharacteristic
110C2×C16×C73C2311yescharacteristic
111#107×C73C2311yescharacteristic
112#107×C73C2311yescharacteristic
113#106×C73—21no—
114C2×Dic146—41no—
115C2×Dic146—41no—
116C22×Dic73—41no—
117C22×Dic73—41no—
118C22×Dic73D811yescharacteristic
119C22×Dic73D811yescharacteristic
120C2×#10596831—22no—
121C2×#10596831C2×D411yescharacteristic
122C2×#10596831C2×D411yescharacteristic
123C4⋊Dic73—21no—
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126C4⋊Dic73—21no—
127C4⋊Dic73—41no—
128C4⋊Dic73—41no—
129C4⋊Dic73—41no—
130C4⋊Dic73—41no—
131#11924007—41no—
132#11924007—41no—
133#11924007—41no—
134#11924007—41no—
135C22⋊Dic73—41no—
136C22⋊Dic73—41no—
137C22⋊Dic73—41no—
138C22⋊Dic73—41no—
139C4×Dic73—41no—
140C4×Dic73—41no—
141C4×Dic73—41no—
142C4×Dic73—41no—
143C4×Dic73D811yescharacteristic
144C4×Dic73D811yescharacteristic
145C4×Dic73C2×D411yescharacteristic
146C4×Dic73C2×D411yescharacteristic
147#11924003—21no—
148#11924003—21no—
149#41×C73—24no—
150#41×C73—42no—
151C2×Q8×C73—21no—
152C2×D4×C73—21no—
153C2×D4×C73—41no—
154C22×C4×C73—41no—
155C22×C4×C73C2×D411yescharacteristic
156Q16×C73—21no—
157Q16×C73C2×Q812yes—
158SD16×C73—24no—
159D8×C73—21no—
160D8×C73C2×Q812yes—
161C2×C8×C73C2×D411yescharacteristic
162C2×C8×C73C2×Q812yes—
163C2×C8×C73#4112yes—
164C2×C8×C73C2411yescharacteristic
165C4⋊C4×C73—21no—
166C4⋊C4×C73—21no—
167C22⋊C4×C73—41no—
168C42×C73—41no—
169C16×C73—21no—
170C16×C73—21no—
171C23×C73—41no—
172C23×C73C2×D811yescharacteristic
173Q8×C73—22no—
174Q8×C73—41no—
175D4×C73—22no—
176D4×C73—44no—
177D4×C73—41no—
178C2×C4×C73—41no—
179C2×C4×C73—44no—
180C2×C4×C73—41no—
181C2×C4×C73—41no—
182C2×C4×C73C22⋊Q812yes—
183C2×C4×C73C2×D811yescharacteristic
184C2×C4×C73#13712yes—
185C2×C4×C73C22×D411yescharacteristic
186C2×Dic73—22no—
187C2×Dic73—41no—
188C2×Dic73—41no—
189C2×Dic73—41no—
190C2×Dic73—41no—
191C2×Dic73—41no—
192C2×Dic73—82no—
193C2×Dic73—82no—
194C2×Dic73C2×D811yescharacteristic
195C2×Dic73C2×D811yescharacteristic
196C2×Dic73C2×D811yescharacteristic
197C2×Dic73C2×D811yescharacteristic
198Dic146—82no—
199Dic146—82no—
200C8×C73C22⋊Q812yes—
201C8×C73C22×Q811yescharacteristic
202C8×C73C2×#4111yescharacteristic
203#10596831—21no—
204#10596831—21no—
205C22×C73—21no—
206C22×C73—21no—
207C22×C73—42no—
208C22×C73—81no—
209C22×C73—81no—
210C22×C73#47412yes—
211C22×C73C22×D811yescharacteristic
212C4×C73—42no—
213C4×C73—81no—
214C4×C73—81no—
215C4×C73#47412yes—
216C4×C73C2×C22⋊Q811yescharacteristic
217C4×C73C2×#13711yescharacteristic
218Dic73—21no—
219Dic73—21no—
220Dic73—21no—
221Dic73—81no—
222Dic73—81no—
223Dic73—81no—
224Dic73—81no—
225Dic73#50812yes—
226#33626—731nomaximal, Sylow
227C2×C73—22no—
228C2×C73—82no—
229C2×C73C2×#47411yescharacteristic
230C2×C73C2×#50811yescharacteristic
231C2×C73#341411yescharacteristic
232#3160—731no—
233C4×#135—731no—
234#2229—731no—
235#2229—731no—
236#2228—731no—
237#2228—731no—
238#2227—731no—
239#2226—731no—
240#2225—731no—
241#2224—731no—
242#2158—731no—
243#2158—731no—
244C2×C4⋊C16—731no—
245#2138—731no—
246#2138D7311yescharacteristic
247C73#3362611yescharacteristic, complemented, Sylow
248C2×#135C2×D7311yescharacteristic
249#533—1461no—
250C4×#41—1461no—
251C22×C16—731no—
252C22×C16C2×D7311yescharacteristic
253#500—731no—
254#500—731no—
255#499—731no—
256#499—731no—
257#499—1461no—
258#498—1461no—
259C4×Q16—731no—
260C4×Q16—731no—
261C4×SD16—1461no—
262C4×SD16—1461no—
263C4×D8—731no—
264C4×D8—731no—
265#427—731no—
266#427—731no—
267#427—1461no—
268#427C2×D7311yescharacteristic
269#416—1461no—
270C2×C4×C8—731no—
271#367—731no—
272#367—731no—
273#367—731no—
274#367—731no—
275#366—731no—
276#366—731no—
277#366—731no—
278#366—731no—
279C4⋊C16—731no—
280C4⋊C16—731no—
281C4⋊C16—731no—
282C4⋊C16—731no—
283#358—731no—
284#358—731no—
285#358C2×D7311yescharacteristic
286#358C2×D7311yescharacteristic
287#357—731no—
288#357—731no—
289#357C2×D7311yescharacteristic
290#357C2×D7311yescharacteristic
291C2×#41—21no—
292#135—22no—
293#135Dic14614yescomplemented
294C2×Q16C22×D7311yescharacteristic
295C2×SD16—21no—
296C2×D8C22×D7311yescharacteristic
297C22×C8—731no—
298C22×C8C22×D7311yescharacteristic
299C4⋊Q8—1461no—
300C4⋊Q8—1461no—
301#125—1461no—
302#125—1461no—
303C22⋊Q8—2921no—
304C22⋊Q8—2921no—
305C4×Q8—2921no—
306C4×Q8—2921no—
307C4×Q8—1461no—
308C4×Q8—1461no—
309C4×D4—2921no—
310C4×D4—2921no—
311C4×D4—1461no—
312C4×D4—1461no—
313#117—21no—
314#117—2921no—
315#117—1461no—
316#117—1461no—
317C2×C42—2921no—
318C2×C42—731no—
319C2×C16—22no—
320C2×C16—731no—
321C2×C16—731no—
322C2×C16—1462no—
323C2×C16C22×D7311yescharacteristic
324C2×C16C22×D7311yescharacteristic
325#107—731no—
326#107—731no—
327#107—731no—
328#107—731no—
329#107—1461no—
330#107—1461no—
331#107C22×D7311yescharacteristic
332#107C22×D7311yescharacteristic
333#106—21no—
334#106—1461no—
335#106—1461no—
336#106—1461no—
337#106—1461no—
338#103—1461no—
339#103—1461no—
340#102—1461no—
341#102—1461no—
342C4×C8—731no—
343C4×C8—731no—
344C4×C8—731no—
345C4×C8—731no—
346#41—24no—
347#41—42no—
348C2×Q8—21no—
349C2×Q8—2921no—
350C2×Q8—2921no—
351C2×D4—21no—
352C2×D4—41no—
353C22×C4—41no—
354C22×C4—2921no—
355C22×C4—2921no—
356C22×C4—731no—
357C22×C4—731no—
358C22×C4D4×D7311yescharacteristic
359Q16—21no—
360Q16C2×Dic14612yes—
361SD16—24no—
362D8—21no—
363D8C2×Dic14612yes—
364C2×C8—731no—
365C2×C8—731no—
366C2×C8—1462no—
367C2×C8D4×D7311yescharacteristic
368C2×C8C2×Dic14612yes—
369C2×C8#1192402712yes—
370C2×C8C23×D7311yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—2921no—
374C4⋊C4—2921no—
375C4⋊C4—2921no—
376C4⋊C4—2921no—
377C4⋊C4—2921no—
378C4⋊C4—2921no—
379C4⋊C4—2921no—
380C4⋊C4—2921no—
381C4⋊C4—1461no—
382C4⋊C4—1461no—
383C4⋊C4—1461no—
384C4⋊C4—1461no—
385C22⋊C4—41no—
386C22⋊C4—2921no—
387C22⋊C4—2921no—
388C22⋊C4—2921no—
389C22⋊C4—2921no—
390C42—41no—
391C42—2921no—
392C42—2921no—
393C42—2921no—
394C42—2921no—
395C42—731no—
396C42—731no—
397C42—731no—
398C42—731no—
399C16—21no—
400C16—21no—
401C16—1461no—
402C16—1461no—
403C23—41no—
404C23D8×D7311yescharacteristic
405Q8—22no—
406Q8—41no—
407Q8—5842no—
408Q8—5842no—
409D4—22no—
410D4—44no—
411D4—41no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—2921no—
417C2×C4—2921no—
418C2×C4—2921no—
419C2×C4—2921no—
420C2×C4—2921no—
421C2×C4—5842no—
422C2×C4—5842no—
423C2×C4—731no—
424C2×C4—731no—
425C2×C4—731no—
426C2×C4—731no—
427C2×C4—1462no—
428C2×C4#1455656812yes—
429C2×C4D8×D7311yescharacteristic
430C2×C4#1455662012yes—
431C2×C4C2×D4×D7311yescharacteristic
432C8—1461no—
433C8—1461no—
434C8#1455656812yes—
435C8C22×Dic14611yescharacteristic
436C8C2×#1192402711yescharacteristic
437C22—21no—
438C22—21no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22#1798886512yes—
443C22C2×D8×D7311yescharacteristic
444C4—42no—
445C4—81no—
446C4—81no—
447C4—5841no—
448C4—5841no—
449C4—5841no—
450C4—5841no—
451C4—732no—
452C4—1461no—
453C4—1461no—
454C4—1461no—
455C4#1798886512yes—
456C4C2×#1455656811yescharacteristic
457C4C2×#1455662011yescharacteristic
458C2—22no—
459C2—82no—
460C2C2×#1798886511yescharacteristic
461C2#508×D7311yescharacteristic
462C2#2262306411yescharacteristic
subgroupG/H
centerC2×C4D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2138C2
socleC22×C73C22×D8
prime radicalC2×#135×C73C22
hypercenter#2138D73
hypocenterC73#33626
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #22620443C2semidirect
3. #22612097C2semidirect
4. #22612093C2semidirect
5. #22612090C2semidirect
6. #22612086C2semidirect
7. #22612081C2semidirect
8. #22612076C2semidirect
9. #22612071C2semidirect
10. #22612066C2semidirect
11. #22611715C2semidirect
12. #22611713C2semidirect
13. C2×#17988405C2semidirect
102. #135×C73Q8semidirect
247. C73#33626semidirect
293. #135Dic146semidirect

Maximal subgroups

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

subgroupindex
#226204432
#135×Dic732
#226120972
#226120932
#226120902
#226120862
#226120812
#226120762
#226120712
#226120662
#226117152
#226117132
C2×#179884052
#226116382
C73×#21382
#3362673