Finite Group Databasebeta

← #33950545

Subgroups — #33950545

Order 18688.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623321C211yesmaximal, characteristic
2#22618933C218yesmaximal, complemented
3#22618923C218yesmaximal, complemented
4#22618868C211yesmaximal, characteristic, complemented
5#22618861C211yesmaximal, characteristic, complemented
6#22618648C211yesmaximal, characteristic, complemented
7#22618642C211yesmaximal, characteristic, complemented
8#22618630C212yesmaximal, complemented
9#22618621C212yesmaximal, complemented
10#22618576C212yesmaximal, complemented
11#22618360C212yesmaximal, complemented
12C2×#17988543C211yesmaximal, characteristic, complemented
13C73×#2921C211yesmaximal, characteristic, complemented
14C2×#14556599C2212yes—
15C2×#14556599C2212yes—
16C22×#11924012C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18#17989208C414yes—
19#17989199C412yes—
20#17989192C412yes—
21#17989184C414yes—
22#17989175C414yes—
23#17989144C2211yescharacteristic
24#17989137C2211yescharacteristic
25#17989115C2218yes—
26#17989098C2212yes—
27C4×#11924028C2212yes—
28#17988716C2218yescomplemented
29#17988714C2214yescomplemented
30#17988711C2214yescomplemented
31#17988664C2214yescomplemented
32#17988662C2218yes—
33#17988661C2218yes—
34#17988657C2218yescomplemented
35#17988655C2218yes—
36#17988652C2214yescomplemented
37#17988650C2218yes—
38#17988649C2218yescomplemented
39#17988638C2218yescomplemented
40#17988615C2212yes—
41#17988615C2212yes—
42#17988555C2212yes—
43#17988555C2212yes—
44#17988545C2212yes—
45#17988545C2212yes—
46#17988543C2214yes—
47#17988543C2214yes—
48#17988541C2214yescomplemented
49#17988541C2214yescomplemented
50#17988539C2214yescomplemented
51C2×#14556502C2211yescharacteristic
52C2×#14556502C2211yescharacteristic
53#97×D73C2214yescomplemented
54C2×C4×#10596831C2211yescharacteristic
55C2×C4×#10596831C2211yescharacteristic
56C2×#117×C73C2211yescharacteristic
57#433×C73C2218yes—
58#423×C73C2211yescharacteristic
59#423×C73C2211yescharacteristic
60#407×C73C2212yes—
61C4×M16×C73C2212yes—
62C2×#11924028C2312yes—
63#14556599—28no—
64#14556599—28no—
65C2×#11924012—24no—
66C2×#11924012C2312yes—
67C2×#11924012C2312yes—
68C2×#11924011C2314yes—
69C2×#11924011C2314yes—
70C22×#10596831—24no—
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73C22×#10596831C2314yes—
74C22×#10596831C2311yescharacteristic
75C22×#10596831C2311yescharacteristic
76C2×C8×D73C2314yes—
77C2×C8×D73C2314yes—
78#14556591C2×C411yescharacteristic
79#14556591C2×C412yes—
80#14556588C2×C411yescharacteristic
81#14556586C2×C418yes—
82#14556584C2×C418yes—
83#14556583C2×C412yes—
84#14556581C2×C418yes—
85#14556579C2×C412yes—
86#14556577C2×C418yes—
87#14556576C2×C418yes—
88#14556574C2×C414yes—
89#14556573C2×C418yes—
90#14556570C2×C414yes—
91#14556570C2×C414yes—
92#14556569C2×C418yes—
93#14556565C2×C414yes—
94#14556565C2×C414yes—
95#14556564C2×C418yes—
96#14556560C2×C414yes—
97C4×Dic146C2×C414yes—
98C4×Dic146C2314yes—
99C4×D292C2×C414yes—
100C4×D292C2314yes—
101C4×C22⋊D73C2×C418yes—
102C4×C22⋊D73C2318yes—
103#14556550C2312yes—
104#14556548C2314yes—
105C42×D73C2314yes—
106#14556520C2318yes—
107#14556520C2318yes—
108#14556503C2318yes—
109#14556503C2318yes—
110#14556502C2314yes—
111#14556502C2312yes—
112#14556502C2312yes—
113#14556501C2314yes—
114#14556501C2314yes—
115C4×#10596831C2314yes—
116C4×#10596831C2312yes—
117C4×#10596831C2312yes—
118C8×Dic73C2314yes—
119C8×Dic73C2314yes—
120C23×C4×C73C2311yescharacteristic
121C2×M16×C73C2312yes—
122C2×M16×C73C2312yes—
123#117×C73C2×C412yes—
124#117×C73C2×C412yes—
125#117×C73C2314yes—
126C2×C4⋊C4×C73C2×C412yes—
127C2×C22⋊C4×C73C2×C412yes—
128C2×C42×C73C2311yescharacteristic
129C2×C42×C73C2311yescharacteristic
130C4⋊C8×C73C2314yes—
131C4⋊C8×C73C2314yes—
132C22⋊C8×C73C2318yes—
133#97×C73C2314yes—
134C4×C8×C73C2314yes—
135#11924028—24no—
136#11924028—24no—
137C2×Dic146C22×C412yes—
138C2×Dic146C22×C414yes—
139C2×D292C22×C412yes—
140C2×D292C22×C414yes—
141C2×C22⋊D73C22×C414yes—
142C2×C22⋊D73C22×C418yes—
143C2×C4×D73C22×C418yes—
144C2×C4×D73C2414yes—
145#11924012—24no—
146#11924012—24no—
147#11924011—28no—
148#11924011—28no—
149C2×#10596831—22no—
150C2×#10596831—22no—
151C2×#10596831—28no—
152C2×#10596831#4114yes—
153C2×#10596831#4114yes—
154C2×#10596831C2412yes—
155C2×#10596831C2412yes—
156C2×#10596831C2411yescharacteristic
157C2×#10596831C2412yes—
158C2×#10596831C2411yescharacteristic
159C8×D73—28no—
160C8×D73—28no—
161C4⋊Dic73C22×C414yes—
162C4⋊Dic73C22×C414yes—
163C4⋊Dic73C22×C414yes—
164#11924007C22×C418yes—
165#11924007C22×C418yes—
166#11924007C22×C418yes—
167C22⋊Dic73C22×C414yes—
168C22⋊Dic73C22×C418yes—
169#11924005C22×C418yes—
170#11924005C22×C418yes—
171#11924005C22×C418yes—
172C4×Dic73C22×C418yes—
173C4×Dic73C2414yes—
174C24×C73C22×C411yescharacteristic
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73C22×C412yes—
179C22×C4×C73C22×C412yes—
180C22×C4×C73C22×C412yes—
181C22×C4×C73C2412yes—
182C22×C4×C73C2411yescharacteristic
183M16×C73—24no—
184M16×C73—24no—
185C2×C8×C73C2414yes—
186C2×C8×C73C2414yes—
187C4⋊C4×C73C22×C414yes—
188C4⋊C4×C73C22×C414yes—
189C22⋊C4×C73C22×C414yes—
190C22⋊C4×C73C22×C414yes—
191C42×C73C22×C412yes—
192C42×C73C22×C411yescharacteristic
193C42×C73C22×C411yescharacteristic
194C42×C73C2412yes—
195C42×C73C2412yes—
196C23×C73—22no—
197C23×C73—42no—
198C23×C73C23×C412yes—
199C23×C73C23×C411yescharacteristic
200C22×D73C23×C414yes—
201C2×C4×C73—22no—
202C2×C4×C73—24no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—42no—
206C2×C4×C73—42no—
207C2×C4×C73C23×C414yes—
208C2×C4×C73C23×C412yes—
209C2×C4×C73C23×C412yes—
210C2×C4×C73C23×C412yes—
211C2×C4×C73C23×C411yescharacteristic
212C2×C4×C73C2×#4112yes—
213C2×C4×C73C2×#4112yes—
214C2×C4×C73C2511yescharacteristic
215C22⋊D73—28no—
216C22⋊D73—48no—
217C2×Dic73C23×C414yes—
218C2×Dic73C23×C418yes—
219D292—24no—
220D292—44no—
221C4×D73—24no—
222C4×D73—24no—
223C4×D73—48no—
224Dic146—24no—
225Dic146—44no—
226C8×C73—24no—
227C8×C73—24no—
228#10596831—22no—
229#10596831—22no—
230#10596831C2×#4112yes—
231#10596831C2×#4114yes—
232#10596831C2×#4112yes—
233C22×C73—22no—
234C22×C73—22no—
235C22×C73—24no—
236C22×C73—44no—
237C22×C73C4×#4112yes—
238C22×C73C24×C411yescharacteristic
239C2×D73—24no—
240C2×D73—28no—
241C4×C73—22no—
242C4×C73—22no—
243C4×C73—24no—
244C4×C73—22no—
245C4×C73C4×#4112yes—
246C4×C73C22×#4111yescharacteristic
247C4×C73C22×#4111yescharacteristic
248Dic73—24no—
249Dic73—28no—
250#60209—731nomaximal, Sylow
251C2×C73—22no—
252C2×C73—42no—
253C2×C73C2×C4×#4111yescharacteristic
254C2×C73#357312yes—
255D73—44no—
256#3455—731no—
257#2969—738no—
258#2968—738no—
259#2961—731no—
260#2960—731no—
261#2932—731no—
262#2931—731no—
263#2930—732no—
264#2929—732no—
265#2922—732no—
266#2921D7311yescharacteristic, complemented
267#2875—732no—
268C2×#405—731no—
269C73#6020911yescharacteristic, complemented, Sylow
270C2×#131—732no—
271C2×#131—732no—
272C22×M16—731no—
273C23×C8—731no—
274#542—734no—
275#541—732no—
276#540—732no—
277#539—734no—
278#538—734no—
279#533—731no—
280#532—731no—
281#529—738no—
282#525—732no—
283C4×#41—732no—
284C2×#117C2×D7311yescharacteristic
285#446—734no—
286#446—738no—
287C8×Q8—734no—
288#436—734no—
289#436—738no—
290#435—738no—
291#435—738no—
292#435—738no—
293C8×D4—734no—
294C8×D4—738no—
295#433—738no—
296#433C2×D7318yes—
297#431—738no—
298#424—732no—
299#424—732no—
300#423C2×D7311yescharacteristic
301#423C2×D7311yescharacteristic
302#408—732no—
303#408—732no—
304#407C2×D7312yes—
305#405—732no—
306#405—732no—
307#405—734no—
308#405—734no—
309#405—734no—
310#405—734no—
311C4×M16—734no—
312C4×M16C2×D7312yes—
313C2×#97—731no—
314C2×#97—731no—
315C2×#97—734no—
316C2×C4×C8—731no—
317C2×C4×C8—731no—
318C2×#41—732no—
319C23×C4C22×D7311yescharacteristic
320#131—1468no—
321#131—1468no—
322C2×M16—732no—
323C2×M16—732no—
324C2×M16—734no—
325C2×M16—734no—
326C2×M16—1464no—
327C2×M16C22×D7312yes—
328C2×M16C22×D7312yes—
329C22×C8—731no—
330C22×C8—731no—
331C22×C8—731no—
332C22×C8—734no—
333C22×C8—731no—
334C22×C8—734no—
335C22×C8—734no—
336C22×C8—1464no—
337C4⋊Q8—731no—
338C4⋊Q8—732no—
339#127—731no—
340#126—738no—
341#126—738no—
342#125—738no—
343#125—732no—
344#124—738no—
345#124—732no—
346#123—734no—
347#123—738no—
348#123—738no—
349C22⋊Q8—734no—
350C22⋊Q8—734no—
351C22⋊Q8—738no—
352#121—734no—
353#121—734no—
354#121—738no—
355#120—734no—
356C4×Q8—734no—
357C4×Q8—734no—
358C4×D4—738no—
359C4×D4—738no—
360C4×D4—734no—
361C4×D4—734no—
362#117—732no—
363#117—734no—
364#117C4×D7312yes—
365#117C4×D7312yes—
366#117C22×D7314yes—
367C2×C4⋊C4C4×D7312yes—
368C2×C22⋊C4C4×D7312yes—
369C2×C42—734no—
370C2×C42C22×D7311yescharacteristic
371C2×C42C22×D7311yescharacteristic
372C4⋊C8—738no—
373C4⋊C8—738no—
374C4⋊C8C22×D7314yes—
375C4⋊C8C22×D7314yes—
376C22⋊C8—738no—
377C22⋊C8—738no—
378C22⋊C8C22×D7318yes—
379#97—734no—
380#97—734no—
381#97—732no—
382#97—732no—
383#97—734no—
384#97C22×D7314yes—
385C4×C8—734no—
386C4×C8—734no—
387C4×C8—734no—
388C4×C8—732no—
389C4×C8—732no—
390C4×C8C22×D7314yes—
391C24C2×C4×D7311yescharacteristic
392#41—1464no—
393#41—1464no—
394C2×Q8—734no—
395C2×Q8—732no—
396C2×D4—738no—
397C2×D4—734no—
398C2×D4—734no—
399C2×D4—732no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—738no—
403C22×C4—734no—
404C22×C4C2×C4×D7311yescharacteristic
405C22×C4C2×C4×D7312yes—
406C22×C4C2×C4×D7312yes—
407C22×C4C2×C4×D7312yes—
408C22×C4C23×D7312yes—
409C22×C4C23×D7311yescharacteristic
410M16—24no—
411M16—24no—
412M16—1468no—
413M16—1468no—
414M16—1464no—
415M16—1464no—
416C2×C8—734no—
417C2×C8—734no—
418C2×C8—731no—
419C2×C8—732no—
420C2×C8—731no—
421C2×C8—732no—
422C2×C8—732no—
423C2×C8—1468no—
424C2×C8—1468no—
425C2×C8—1468no—
426C2×C8—1462no—
427C2×C8—1462no—
428C2×C8C23×D7314yes—
429C2×C8C23×D7314yes—
430C4⋊C4—738no—
431C4⋊C4—738no—
432C4⋊C4—738no—
433C4⋊C4—734no—
434C4⋊C4—734no—
435C4⋊C4—734no—
436C4⋊C4C2×C4×D7314yes—
437C4⋊C4C2×C4×D7314yes—
438C22⋊C4—734no—
439C22⋊C4—738no—
440C22⋊C4—738no—
441C22⋊C4—738no—
442C22⋊C4—738no—
443C22⋊C4C2×C4×D7314yes—
444C22⋊C4C2×C4×D7314yes—
445C42—738no—
446C42—734no—
447C42C2×C4×D7312yes—
448C42C2×C4×D7311yescharacteristic
449C42C2×C4×D7311yescharacteristic
450C42C23×D7312yes—
451C42C23×D7312yes—
452C23—22no—
453C23—42no—
454C23—734no—
455C23C22×C4×D7312yes—
456C23C22×C4×D7311yescharacteristic
457Q8—2924no—
458Q8—1464no—
459D4—2924no—
460D4—2928no—
461D4—1468no—
462D4—1464no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—2928no—
470C2×C4—738no—
471C2×C4—734no—
472C2×C4—1464no—
473C2×C4—1464no—
474C2×C4C22×C4×D7314yes—
475C2×C4C22×C4×D7312yes—
476C2×C4C22×C4×D7312yes—
477C2×C4C22×C4×D7312yes—
478C2×C4C22×C4×D7311yescharacteristic
479C2×C4#41×D7312yes—
480C2×C4#41×D7312yes—
481C2×C4C24×D7311yescharacteristic
482C8—24no—
483C8—24no—
484C8—734no—
485C8—732no—
486C8—732no—
487C8—1462no—
488C8—1462no—
489C22—22no—
490C22—24no—
491C22—22no—
492C22—44no—
493C22—1468no—
494C22—1464no—
495C22#1798905512yes—
496C22C23×C4×D7311yescharacteristic
497C4—24no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—1468no—
502C4—1464no—
503C4#1798905512yes—
504C4C2×#41×D7311yescharacteristic
505C4C2×#41×D7311yescharacteristic
506C2—22no—
507C2—42no—
508C2—2924no—
509C2C2×#1798905511yescharacteristic
510C2#2262464412yes—
subgroupG/H
center, FrattiniC2×C4C24×D73
commutator, socleC22×C73C24×C4
FittingC73×#2921C2
prime radical#22623321C2
cosocleC2×C4×C73C25
hypercenter#2921D73
hypocenterC73#60209
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #22618933C2semidirect
3. #22618923C2semidirect
4. #22618868C2semidirect
5. #22618861C2semidirect
6. #22618648C2semidirect
7. #22618642C2semidirect
8. #22618630C2semidirect
9. #22618621C2semidirect
10. #22618576C2semidirect
11. #22618360C2semidirect
12. C2×#17988543C2semidirect
13. C73×#2921C2semidirect
28. #17988716C22semidirect
29. #17988714C22semidirect
30. #17988711C22semidirect
31. #17988664C22semidirect
34. #17988657C22semidirect
36. #17988652C22semidirect
38. #17988649C22semidirect
39. #17988638C22semidirect
48. #17988541C22semidirect
49. #17988541C22semidirect
50. #17988539C22semidirect
53. #97×D73C22semidirect
266. #2921D73semidirect
269. C73#60209semidirect

Maximal subgroups

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

subgroupindex
#226233212
#226189332
#226189232
#226188682
#226188612
#226186482
#226186422
#226186302
#226186212
#226185762
#226183602
C2×#179885432
C73×#29212
#6020973