Finite Group Databasebeta

← #33249090

Subgroups — #33249090

Order 18688.

1926 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989142C211yesmaximal, characteristic
2#22613112C211yesmaximal, characteristic, complemented
3#22613111C211yesmaximal, characteristic, complemented
4#22613111C211yesmaximal, characteristic, complemented
5#22613010C214yesmaximal
6#22612893C214yesmaximal, complemented
7#22612786C214yesmaximal
8#22612734C214yesmaximal, complemented
9#22612660C214yesmaximal
10#22612576C214yesmaximal, complemented
11#22612328C211yesmaximal, characteristic
12#383×D73C211yesmaximal, characteristic
13C73×#2381C211yesmaximal, characteristic, complemented
14#17989142C4116yescomplemented
15C2×#14556591C2211yescharacteristic
16C2×#14556582C2212yes—
17C2×#14556571C2212yes—
18C2×#14556567C2212yes—
19C2×Q8×Dic73C2212yes—
20C2×#14556556C2212yes—
21C2×#14556549C2212yes—
22C2×C42×D73C2211yescharacteristic
23#17988494C2214yes—
24#17988491C2214yes—
25#17988491C2214yes—
26#17988485C2214yes—
27#17988483C2214yes—
28#17988474C2214yes—
29#17988473C2214yes—
30#17988473C2214yes—
31#17988472C2214yes—
32#17988472C2214yes—
33#17988472C2214yes—
34#17988470C2214yes—
35#17988470C2214yes—
36#17988470C2214yes—
37#17988468C2214yes—
38#17988461C2214yes—
39#17988458C2214yes—
40#17988455C2211yescharacteristic
41#17988455C2211yescharacteristic
42#17988454C2214yes—
43#17988454C2212yes—
44#17988454C2212yes—
45#17988454C2212yes—
46#17988451C2214yes—
47#17988451C2212yes—
48#17988449C2212yes—
49#17988446C2212yes—
50C4⋊C4×Dic73C2214yes—
51C4×#11924007C2214yes—
52#17988438C2214yes—
53#17988438C2212yes—
54C4×#11924005C2214yes—
55#17988434C2212yes—
56#17988434C2214yes—
57#17988433C2214yes—
58#17988432C2212yes—
59#95×D73C2212yes—
60#95×D73C2212yes—
61#95×D73C2212yes—
62C2×C4⋊Q8×C73C2211yescharacteristic
63#391×C73C2214yes—
64#387×C73C2214yes—
65#387×C73C2214yes—
66#383×C73C2211yescharacteristic
67#383×C73C2211yescharacteristic
68C22×Dic146C2312yes—
69C22×C4×D73C2311yescharacteristic
70C22×C4×D73C2314yes—
71C22×C4×D73C2311yescharacteristic
72C22×C4×D73C2311yescharacteristic
73#14556591C2×C418yescomplemented
74#14556582C2×C4116yescomplemented
75#14556571C2×C4116yes—
76#14556567C2×C4116yes—
77Q8×Dic73C2×C4116yescomplemented
78#14556556C2×C4116yescomplemented
79#14556549C2×C4116yes—
80C2×C4⋊Dic73C2314yes—
81C2×C4⋊Dic73C2312yes—
82C2×C4⋊Dic73C2312yes—
83C2×C4⋊Dic73C2314yes—
84C2×#11924007C2314yes—
85C2×#11924007C2314yes—
86C2×#11924007C2314yes—
87C2×#11924007C2314yes—
88C2×#11924005C2314yes—
89C2×#11924005C2314yes—
90C2×C4×Dic73C2314yes—
91C2×C4×Dic73C2314yes—
92C2×C4×Dic73C2314yes—
93C2×C4×Dic73C2314yes—
94C2×C4×Dic73C2311yescharacteristic
95C2×C4×Dic73C2314yes—
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98C42×D73C2×C418yes—
99#14556498C2314yes—
100#14556498C2314yes—
101#14556498C2314yes—
102#14556498C2312yes—
103#14556498C2312yes—
104#14556498C2312yes—
105#14556498C2314yes—
106#14556498C2312yes—
107#14556498C2314yes—
108#14556498C2314yes—
109#14556498C2312yes—
110#14556498C2314yes—
111#14556498C2314yes—
112#14556498C2314yes—
113#14556498C2312yes—
114#14556498C2314yes—
115C22×Q8×C73C2312yes—
116C4⋊Q8×C73C2×C418yescomplemented
117C2×C4⋊C4×C73C2314yes—
118C2×C4⋊C4×C73C2312yes—
119C2×C4⋊C4×C73C2312yes—
120C2×C42×C73C2314yes—
121C2×C42×C73C2311yescharacteristic
122#95×C73C2312yes—
123#95×C73C2312yes—
124#95×C73C2312yes—
125#95×C73C2312yes—
126#95×C73C2312yes—
127#95×C73C2312yes—
128#95×C73C2314yes—
129#95×C73C2314yes—
130C23×D73C2411yescharacteristic
131C2×Dic146—42no—
132C2×Dic146—42no—
133C2×Dic146C22×C418yes—
134C22×Dic73C2414yes—
135C22×Dic73C2414yes—
136C22×Dic73C2411yescharacteristic
137C22×Dic73C2411yescharacteristic
138C22×Dic73C2414yes—
139C22×Dic73C2411yescharacteristic
140C2×C4×D73—24no—
141C2×C4×D73—28no—
142C2×C4×D73—42no—
143C2×C4×D73—42no—
144C2×C4×D73—44no—
145C2×C4×D73—44no—
146C2×C4×D73C22×C414yes—
147C2×C4×D73C22×C414yes—
148C2×C4×D73C22×C414yes—
149C4⋊Dic73—44no—
150C4⋊Dic73C22×C418yes—
151C4⋊Dic73C22×C418yes—
152C4⋊Dic73C22×C4116yes—
153#11924007—44no—
154#11924007—44no—
155#11924007—44no—
156#11924007C22×C4116yes—
157#11924005—28no—
158#11924005C22×C4116yes—
159C4×Dic73—44no—
160C4×Dic73—44no—
161C4×Dic73—44no—
162C4×Dic73—44no—
163C4×Dic73C22×C414yes—
164C4×Dic73C22×C414yes—
165C4×Dic73C22×C414yes—
166C4×Dic73C22×C4116yes—
167C2×Q8×C73—42no—
168C2×Q8×C73—42no—
169C2×Q8×C73C22×C418yes—
170C22×C4×C73C2414yes—
171C22×C4×C73C2414yes—
172C22×C4×C73C2411yescharacteristic
173C22×C4×C73C2411yescharacteristic
174C22×C4×C73C2411yescharacteristic
175C22×C4×C73C2414yes—
176C4⋊C4×C73—44no—
177C4⋊C4×C73C22×C418yes—
178C4⋊C4×C73C22×C418yes—
179C42×C73—44no—
180C42×C73C22×C414yes—
181C23×C73C2511yescharacteristic
182C22×D73—22no—
183C22×D73—22no—
184C22×D73C23×C412yes—
185C22×D732−1+414yes—
186Q8×C73—48no—
187C2×C4×C73—22no—
188C2×C4×C73—24no—
189C2×C4×C73—24no—
190C2×C4×C73—22no—
191C2×C4×C73—22no—
192C2×C4×C73—24no—
193C2×C4×C73—24no—
194C2×C4×C73—24no—
195C2×C4×C73—24no—
196C2×C4×C73—24no—
197C2×C4×C73—24no—
198C2×C4×C73C23×C412yes—
199C2×C4×C73C23×C412yes—
200C2×C4×C73C23×C412yes—
201C2×C4×C73C23×C418yes—
202C2×Dic73—22no—
203C2×Dic73—22no—
204C2×Dic73—24no—
205C2×Dic73—22no—
206C2×Dic73—24no—
207C2×Dic73—24no—
208C2×Dic73—24no—
209C2×Dic73—24no—
210C2×Dic73—24no—
211C2×Dic73—24no—
212C2×Dic73—24no—
213C2×Dic73C23×C412yes—
214C2×Dic73C23×C412yes—
215C2×Dic73C23×C412yes—
216C2×Dic73C23×C418yes—
217C4×D73—28no—
218C4×D73—44no—
219C4×D73—44no—
220C4×D73—88no—
221Dic146—48no—
222C22×C73C24×C411yescharacteristic
223C22×C73C2×2−1+412yes—
224C22×C73#58512yes—
225C22×C73#58512yes—
226C2×D73—22no—
227C2×D73—22no—
228C2×D73—28no—
229C2×D73#51914yes—
230C4×C73—22no—
231C4×C73—22no—
232C4×C73—22no—
233C4×C73—28no—
234C4×C73—44no—
235C4×C73—44no—
236Dic73—22no—
237Dic73—28no—
238Dic73—22no—
239Dic73—22no—
240Dic73—44no—
241Dic73—44no—
242#34389—731nomaximal, Sylow
243C2×C73C2×#51911yescharacteristic
244C2×C73#342811yescharacteristic
245C2×C73#342811yescharacteristic
246C2×C73#356414yes—
247D73—24no—
248C2×#533—731no—
249#2381—731no—
250#2381—731no—
251#2381—731no—
252#2381D7311yescharacteristic, complemented
253#2368—734no—
254#2356—734no—
255#2343—734no—
256#2336—734no—
257#2329—734no—
258#2320—734no—
259#2283—731no—
260C2×#383—731no—
261C73#3438911yescharacteristic, complemented, Sylow
262#533—7316no—
263C2×C4⋊Q8—731no—
264C2×C4⋊Q8C2×D7311yescharacteristic
265C2×#125—732no—
266C2×C22⋊Q8—732no—
267C2×C22⋊Q8—732no—
268C2×C4×Q8—732no—
269C2×C4×Q8—732no—
270C2×#117—732no—
271C22×C42—731no—
272#391—734no—
273#391—734no—
274#391—734no—
275#391C2×D7314yes—
276#389—734no—
277#389—734no—
278#387—734no—
279#387—734no—
280#387—734no—
281#387—734no—
282#387—734no—
283#387—734no—
284#387—734no—
285#387—734no—
286#387—734no—
287#387—734no—
288#387C2×D7314yes—
289#387C2×D7314yes—
290#385—734no—
291#385—734no—
292#383—731no—
293#383—731no—
294#383—732no—
295#383—734no—
296#383—732no—
297#383—732no—
298#383C2×D7311yescharacteristic
299#383C2×D7311yescharacteristic
300#382—732no—
301#382—734no—
302#381—732no—
303#380—732no—
304C4×C4⋊C4—734no—
305C4×C4⋊C4—734no—
306C4×C4⋊C4—732no—
307C4×C4⋊C4—734no—
308C4×C22⋊C4—734no—
309C4×C22⋊C4—734no—
310C4×C22⋊C4—732no—
311#376—734no—
312#376—732no—
313C2×#95—732no—
314C2×#95—732no—
315C2×#95—732no—
316C22×Q8—732no—
317C22×Q8C22×D7312yes—
318C23×C4—731no—
319C23×C4—731no—
320C23×C4—734no—
321C23×C4—731no—
322C4⋊Q8—738no—
323C4⋊Q8C4×D7318yescomplemented
324#125—7316no—
325C22⋊Q8—7316no—
326C22⋊Q8—7316no—
327C4×Q8—7316no—
328C4×Q8—7316no—
329#117—7316no—
330C2×C4⋊C4—734no—
331C2×C4⋊C4—734no—
332C2×C4⋊C4—734no—
333C2×C4⋊C4—734no—
334C2×C4⋊C4—732no—
335C2×C4⋊C4—734no—
336C2×C4⋊C4—734no—
337C2×C4⋊C4—732no—
338C2×C4⋊C4C22×D7314yes—
339C2×C4⋊C4C22×D7312yes—
340C2×C4⋊C4C22×D7312yes—
341C2×C22⋊C4—734no—
342C2×C22⋊C4—734no—
343C2×C42—734no—
344C2×C42—734no—
345C2×C42—734no—
346C2×C42—734no—
347C2×C42—734no—
348C2×C42—731no—
349C2×C42—731no—
350C2×C42—731no—
351C2×C42—738no—
352C2×C42C22×D7314yes—
353C2×C42C22×D7311yescharacteristic
354#95—734no—
355#95—734no—
356#95—732no—
357#95—734no—
358#95—732no—
359#95—732no—
360#95—734no—
361#95—732no—
362#95—734no—
363#95—734no—
364#95—732no—
365#95—734no—
366#95—734no—
367#95—734no—
368#95—732no—
369#95—734no—
370#95C22×D7312yes—
371#95C22×D7312yes—
372#95C22×D7312yes—
373#95C22×D7312yes—
374#95C22×D7312yes—
375#95C22×D7314yes—
376#95C22×D7312yes—
377#95C22×D7314yes—
378C24—731no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—2922no—
382C2×Q8—2922no—
383C2×Q8—738no—
384C2×Q8C2×C4×D7318yes—
385C22×C4—2922no—
386C22×C4—2922no—
387C22×C4—2924no—
388C22×C4—2924no—
389C22×C4—734no—
390C22×C4—734no—
391C22×C4—731no—
392C22×C4—731no—
393C22×C4—734no—
394C22×C4—731no—
395C22×C4—734no—
396C22×C4—734no—
397C22×C4—734no—
398C22×C4—1468no—
399C22×C4—1464no—
400C22×C4C23×D7314yes—
401C22×C4C23×D7314yes—
402C22×C4C23×D7311yescharacteristic
403C22×C4C23×D7311yescharacteristic
404C22×C4C23×D7311yescharacteristic
405C22×C4C23×D7314yes—
406C4⋊C4—44no—
407C4⋊C4—2924no—
408C4⋊C4—2924no—
409C4⋊C4—2924no—
410C4⋊C4—2924no—
411C4⋊C4—7316no—
412C4⋊C4—738no—
413C4⋊C4—7316no—
414C4⋊C4—738no—
415C4⋊C4C2×C4×D7318yes—
416C4⋊C4C2×C4×D7318yes—
417C22⋊C4—7316no—
418C22⋊C4—1468no—
419C42—44no—
420C42—2924no—
421C42—2924no—
422C42—2924no—
423C42—2924no—
424C42—734no—
425C42—734no—
426C42—734no—
427C42—7316no—
428C42C2×C4×D7314yes—
429C23—732no—
430C23—734no—
431C23—1462no—
432C23—1462no—
433C23C24×D7311yescharacteristic
434Q8—48no—
435Q8—2928no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—2924no—
448C2×C4—2924no—
449C2×C4—5848no—
450C2×C4—732no—
451C2×C4—732no—
452C2×C4—732no—
453C2×C4—738no—
454C2×C4—1468no—
455C2×C4—1464no—
456C2×C4—1464no—
457C2×C4—1464no—
458C2×C4—1464no—
459C2×C4—1464no—
460C2×C4—1464no—
461C2×C4—1462no—
462C2×C4—1462no—
463C2×C4—1464no—
464C2×C4—1462no—
465C2×C4—1464no—
466C2×C4C22×C4×D7312yes—
467C2×C4C22×C4×D7312yes—
468C2×C4C22×C4×D7312yes—
469C2×C4C22×C4×D7318yes—
470C22—734no—
471C22—1468no—
472C22—1462no—
473C22—1462no—
474C22C23×C4×D7311yescharacteristic
475C222−1+4×D7312yes—
476C22#1798942212yes—
477C22#1798942412yes—
478C4—22no—
479C4—22no—
480C4—28no—
481C4—22no—
482C4—44no—
483C4—44no—
484C4—2924no—
485C4—2924no—
486C4—1462no—
487C4—1462no—
488C4—1462no—
489C4—1468no—
490C2—1464no—
491C2#519×D7311yescharacteristic
492C2#2262312911yescharacteristic
493C2#2262313311yescharacteristic
494C2#2262455314yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2381C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2381D73
hypocenterC73#34389
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. #22613112C2semidirect
3. #22613111C2semidirect
4. #22613111C2semidirect
6. #22612893C2semidirect
8. #22612734C2semidirect
10. #22612576C2semidirect
13. C73×#2381C2semidirect
14. #17989142C4semidirect
73. #14556591C2×C4semidirect
74. #14556582C2×C4semidirect
77. Q8×Dic73C2×C4semidirect
78. #14556556C2×C4semidirect
116. C4⋊Q8×C73C2×C4semidirect
252. #2381D73semidirect
261. C73#34389semidirect
323. C4⋊Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179891422
#226131122
#226131112
#226130102
#226128932
#226127862
#226127342
#226126602
#226125762
#226123282
#383×D732
C73×#23812
#3438973