Finite Group Databasebeta

← #33055554

Subgroups — #33055554

Order 18688.

688 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#11924028C211yesmaximal, characteristic, complemented
2C4×#14556611C211yesmaximal, characteristic
3#22608807C212yesmaximal, complemented
4#22608800C212yesmaximal, complemented
5#22608793C212yesmaximal, complemented
6#22608786C212yesmaximal, complemented
7#22608677C211yesmaximal, characteristic, complemented
8#22608313C211yesmaximal, characteristic, complemented
9#22608311C211yesmaximal, characteristic, complemented
10C2×#17988309C211yesmaximal, characteristic, complemented
11C73×#2965C211yesmaximal, characteristic, complemented
12C2×#14556611C411yescharacteristic
13C4×#11924028C2211yescharacteristic
14C8×Dic146C2212yes—
15C4×#11924018C411yescharacteristic
16C4×#11924018C2211yescharacteristic
17C4×#11924015C411yescharacteristic
18C4×#11924015C2211yescharacteristic
19C4×#11924016C411yescharacteristic
20C4×#11924016C2211yescharacteristic
21C4×#11924013C411yescharacteristic
22C4×#11924013C2211yescharacteristic
23C8×D292C2212yes—
24C8×C22⋊D73—22no—
25#17988640C2211yescharacteristic
26#17988638—21no—
27#17988623C411yescharacteristic
28#17988583C411yescharacteristic
29#17988581C411yescharacteristic
30C2×C4×#10596831C2211yescharacteristic
31C2×C4×#10596831C2211yescharacteristic
32C4×C8×D73—21no—
33#17988339C2212yes—
34#17988337C2212yes—
35Q8⋊#10596831C2212yes—
36#17988317C2212yes—
37D4⋊#10596831C2212yes—
38#17988314C2212yes—
39#17988309C2212yes—
40#17988309C2212yes—
41C4×#41×C73C2211yescharacteristic
42#446×C73C2212yes—
43#436×C73C2212yes—
44#435×C73—22no—
45#431×C73—21no—
46#431×C73C2211yescharacteristic
47C4×M16×C73—21no—
48C2×C4×C8×C73C2211yescharacteristic
49C2×#11924028C2×C411yescharacteristic
50#14556611—24no—
51C2×#11924018C2×C411yescharacteristic
52C2×#11924016C2×C411yescharacteristic
53C2×#11924015C2×C411yescharacteristic
54C2×#11924013C2×C411yescharacteristic
55C22×#10596831—21no—
56C22×#10596831C2×C411yescharacteristic
57C22×#10596831C2×C411yescharacteristic
58C2×C8×D73—22no—
59C4×Dic146C2×C411yescharacteristic
60C4×Dic146C2311yescharacteristic
61C4×D292C2×C411yescharacteristic
62C4×D292C2311yescharacteristic
63C4×C22⋊D73—21no—
64C4×C22⋊D73—21no—
65#14556550C2×C411yescharacteristic
66#14556548—21no—
67C42×D73—21no—
68#14556524C2×C411yescharacteristic
69#14556524C2×C411yescharacteristic
70#14556522C2×C411yescharacteristic
71#14556522C2×C411yescharacteristic
72C4⋊#10596831C2312yes—
73#14556520—22no—
74#14556515C2×C411yescharacteristic
75#14556515C2×C411yescharacteristic
76#14556514C2×C411yescharacteristic
77#14556514C2×C411yescharacteristic
78#14556512C2×C411yescharacteristic
79#14556512C2×C411yescharacteristic
80#14556511C2×C411yescharacteristic
81#14556511C2×C411yescharacteristic
82C22⋊#10596831—21no—
83#14556503—22no—
84C4×#10596831—21no—
85C4×#10596831—21no—
86C4×#10596831C2×C411yescharacteristic
87C4×#10596831C2×C411yescharacteristic
88C4×#10596831C2312yes—
89C4×#10596831C2311yescharacteristic
90C4×#10596831C2311yescharacteristic
91C8×Dic73—22no—
92C2×#41×C73C2×C411yescharacteristic
93C2×M16×C73—22no—
94C22×C8×C73D412yes—
95C4×Q8×C73C2×C411yescharacteristic
96C4×Q8×C73C2311yescharacteristic
97C4×D4×C73—21no—
98C4×D4×C73—21no—
99C4×D4×C73C2×C411yescharacteristic
100C4×D4×C73C2311yescharacteristic
101#117×C73—21no—
102#117×C73C2×C411yescharacteristic
103C2×C42×C73—21no—
104C2×C42×C73C2311yescharacteristic
105C4⋊C8×C73—22no—
106C4⋊C8×C73C2312yes—
107C22⋊C8×C73—21no—
108C22⋊C8×C73—22no—
109#97×C73—22no—
110C4×C8×C73—21no—
111C4×C8×C73D411yescharacteristic
112C4×C8×C73D411yescharacteristic
113C4×C8×C73C2312yes—
114#11924028—22no—
115#11924028M1614yescomplemented
116C2×Dic146C22×C411yescharacteristic
117C2×D292C22×C411yescharacteristic
118C2×C22⋊D73—21no—
119C2×C4×D73—21no—
120C2×C4×D73—21no—
121C2×C4×D73—21no—
122#11924018—22no—
123#11924016—22no—
124#11924015—22no—
125#11924013—22no—
126C2×#10596831—24no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C2×#10596831—22no—
131C2×#10596831C22×C411yescharacteristic
132C2×#10596831C22×C411yescharacteristic
133C2×#10596831C22×C411yescharacteristic
134C2×#10596831C22×C411yescharacteristic
135C8×D73—44no—
136C4⋊Dic73C22×C411yescharacteristic
137C4⋊Dic73C22×C411yescharacteristic
138#11924007—21no—
139#11924007—21no—
140C22⋊Dic73—21no—
141#11924005—21no—
142#11924005—21no—
143C4×Dic73—21no—
144C4×Dic73—21no—
145C4×Dic73—21no—
146#41×C73—22no—
147#41×C73—22no—
148C2×Q8×C73C22×C411yescharacteristic
149C2×D4×C73—21no—
150C2×D4×C73C22×C411yescharacteristic
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73C22×C411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73#4111yescharacteristic
157M16×C73—44no—
158C2×C8×C73—22no—
159C2×C8×C73—42no—
160C2×C8×C73C2×D412yes—
161C2×C8×C73C2×D412yes—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73C22×C411yescharacteristic
165C4⋊C4×C73C22×C411yescharacteristic
166C22⋊C4×C73—21no—
167C22⋊C4×C73—21no—
168C22⋊C4×C73—21no—
169C42×C73—21no—
170C42×C73—21no—
171C42×C73—21no—
172C42×C73C22×C411yescharacteristic
173C42×C73C2×D411yescharacteristic
174C42×C73#4111yescharacteristic
175C42×C73C2411yescharacteristic
176C23×C73—21no—
177C23×C73C4×D411yescharacteristic
178C22×D73—21no—
179Q8×C73—21no—
180Q8×C73—21no—
181D4×C73—21no—
182D4×C73—21no—
183D4×C73—42no—
184C2×C4×C73—22no—
185C2×C4×C73—22no—
186C2×C4×C73—21no—
187C2×C4×C73—21no—
188C2×C4×C73—21no—
189C2×C4×C73—42no—
190C2×C4×C73—42no—
191C2×C4×C73—42no—
192C2×C4×C73C4×D411yescharacteristic
193C2×C4×C73C4×D411yescharacteristic
194C2×C4×C73C4×D411yescharacteristic
195C2×C4×C73C2×M1612yes—
196C2×C4×C73#13112yes—
197C2×C4×C73C23×C411yescharacteristic
198C2×C4×C73C22×D411yescharacteristic
199C2×C4×C73C2×#4111yescharacteristic
200C22⋊D73—24no—
201C2×Dic73—21no—
202C2×Dic73—21no—
203C2×Dic73—21no—
204D292—21no—
205D292C2×M1612yes—
206C4×D73—24no—
207C4×D73—42no—
208C4×D73—42no—
209Dic146—21no—
210Dic146C2×M1612yes—
211C8×C73—22no—
212C8×C73—22no—
213C8×C73—42no—
214#10596831—24no—
215#10596831—21no—
216#10596831—21no—
217#10596831—21no—
218#10596831—21no—
219#10596831—42no—
220C22×C73—21no—
221C22×C73—21no—
222C22×C73—41no—
223C22×C73—41no—
224C22×C73—41no—
225C22×C73#43512yes—
226C22×C73C2×C4×D411yescharacteristic
227C2×D73—22no—
228C2×D73—41no—
229C2×D73—41no—
230C4×C73—21no—
231C4×C73—21no—
232C4×C73—41no—
233C4×C73—41no—
234C4×C73—41no—
235C4×C73#43512yes—
236C4×C73#44312yes—
237C4×C73C22×M1611yescharacteristic
238C4×C73C2×#13111yescharacteristic
239C4×C73#57512yes—
240Dic73—22no—
241Dic73—41no—
242Dic73—41no—
243#17868—731nomaximal, Sylow
244C2×C73—22no—
245C2×C73—81no—
246C2×C73C2×#43511yescharacteristic
247C2×C73#294311yescharacteristic
248C2×C73C2×#44311yescharacteristic
249D73—42no—
250#2965D7311yescharacteristic, complemented
251C8×#41—731no—
252C4×#135—731no—
253#1588—732no—
254#1587—732no—
255#1586—732no—
256#1585—732no—
257#1564—731no—
258#1476—731no—
259#1476—731no—
260C2×#322—731no—
261C73#1786811yescharacteristic, complemented, Sylow
262C2×#135—731no—
263C4×#41—731no—
264C4×#41C2×D7311yescharacteristic
265#446C2×D7312yes—
266C8×Q8—732no—
267C4×Q16—731no—
268C4×Q16—731no—
269C4×SD16—731no—
270C4×SD16—731no—
271C4×SD16—731no—
272C4×SD16—731no—
273C4×D8—731no—
274C4×D8—731no—
275#436C2×D7312yes—
276#435—22no—
277C8×D4—732no—
278C8×D4—1462no—
279#431—21no—
280#431—731no—
281#431—1461no—
282#431C2×D7311yescharacteristic
283#427—731no—
284#416—731no—
285#416—731no—
286C4×M16—21no—
287C2×C4×C8—731no—
288C2×C4×C8—731no—
289C2×C4×C8—1461no—
290C2×C4×C8C2×D7311yescharacteristic
291#335—732no—
292#334—732no—
293Q8⋊C8—732no—
294Q8⋊C8—732no—
295D4⋊C8—732no—
296D4⋊C8—732no—
297#322—732no—
298#322—732no—
299C2×#41—731no—
300C2×#41C4×D7311yescharacteristic
301#135—1464no—
302C2×Q16—731no—
303C2×SD16—731no—
304C2×SD16—731no—
305C2×D8—731no—
306C2×M16—22no—
307C22×C8—731no—
308C22×C8—731no—
309C22×C8—1462no—
310C22×C8—1461no—
311C22×C8C22⋊D7312yes—
312C4×Q8—731no—
313C4×Q8—731no—
314C4×Q8C4×D7311yescharacteristic
315C4×Q8C22×D7311yescharacteristic
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—731no—
319C4×D4—731no—
320C4×D4—1461no—
321C4×D4—1461no—
322C4×D4C4×D7311yescharacteristic
323C4×D4C22×D7311yescharacteristic
324#117—21no—
325#117—731no—
326#117—1461no—
327#117C4×D7311yescharacteristic
328C2×C42—21no—
329C2×C42—1461no—
330C2×C42C22×D7311yescharacteristic
331#107—731no—
332#107—731no—
333#106—731no—
334#106—731no—
335C4⋊C8—22no—
336C4⋊C8—732no—
337C4⋊C8—1462no—
338C4⋊C8C22×D7312yes—
339#103—731no—
340#103—731no—
341#103—731no—
342#103—731no—
343#102—731no—
344#102—731no—
345#102—731no—
346#102—731no—
347C22⋊C8—21no—
348C22⋊C8—22no—
349C22⋊C8—1461no—
350C22⋊C8—1462no—
351#97—22no—
352C4×C8—21no—
353C4×C8—731no—
354C4×C8—731no—
355C4×C8—731no—
356C4×C8—731no—
357C4×C8—732no—
358C4×C8—1462no—
359C4×C8—1461no—
360C4×C8—1461no—
361C4×C8C22⋊D7311yescharacteristic
362C4×C8C22⋊D7311yescharacteristic
363C4×C8C22×D7312yes—
364#41—22no—
365#41—22no—
366#41—734no—
367#41—1462no—
368C2×Q8—731no—
369C2×Q8C2×C4×D7311yescharacteristic
370C2×D4—21no—
371C2×D4—731no—
372C2×D4—1461no—
373C2×D4C2×C4×D7311yescharacteristic
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—1461no—
378C22×C4—1461no—
379C22×C4—1461no—
380C22×C4C2×C4×D7311yescharacteristic
381C22×C4C2×C22⋊D7311yescharacteristic
382C22×C4#1192402811yescharacteristic
383Q16—1462no—
384SD16—1462no—
385SD16—1462no—
386D8—1462no—
387M16—44no—
388C2×C8—22no—
389C2×C8—42no—
390C2×C8—2924no—
391C2×C8—731no—
392C2×C8—731no—
393C2×C8—731no—
394C2×C8—731no—
395C2×C8—1462no—
396C2×C8—1464no—
397C2×C8—1462no—
398C2×C8—1462no—
399C2×C8—1462no—
400C2×C8C2×C22⋊D7312yes—
401C2×C8C2×C22⋊D7312yes—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—731no—
405C4⋊C4—731no—
406C4⋊C4—1461no—
407C4⋊C4—1461no—
408C4⋊C4C2×C4×D7311yescharacteristic
409C4⋊C4C2×C4×D7311yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—1461no—
414C22⋊C4—1461no—
415C22⋊C4—1461no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—1461no—
420C42—1461no—
421C42—1461no—
422C42C2×C4×D7311yescharacteristic
423C42C2×C22⋊D7311yescharacteristic
424C42#1192402811yescharacteristic
425C42C23×D7311yescharacteristic
426C23—21no—
427C23—1461no—
428C23C4×C22⋊D7311yescharacteristic
429Q8—21no—
430Q8—21no—
431Q8—732no—
432Q8—1461no—
433D4—21no—
434D4—21no—
435D4—42no—
436D4—732no—
437D4—1464no—
438D4—1461no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—2922no—
448C2×C4—2922no—
449C2×C4—1461no—
450C2×C4—1461no—
451C2×C4—1461no—
452C2×C4—1464no—
453C2×C4C4×C22⋊D7311yescharacteristic
454C2×C4C4×C22⋊D7311yescharacteristic
455C2×C4C4×C22⋊D7311yescharacteristic
456C2×C4M16×D7312yes—
457C2×C4#1455659912yes—
458C2×C4C22×C4×D7311yescharacteristic
459C2×C4C22×C22⋊D7311yescharacteristic
460C2×C4C2×#1192402811yescharacteristic
461C8—22no—
462C8—22no—
463C8—42no—
464C8—2922no—
465C8—1464no—
466C8—1461no—
467C8—1461no—
468C8—1461no—
469C8—1461no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—2921no—
476C22—2921no—
477C22—1462no—
478C22#1798865612yes—
479C22C2×C4×C22⋊D7311yescharacteristic
480C4—21no—
481C4—21no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—2921no—
486C4—2921no—
487C4—1462no—
488C4#1798865612yes—
489C4#1798870212yes—
490C4C2×M16×D7311yescharacteristic
491C4C2×#1455659911yescharacteristic
492C4#1798938512yes—
493C2—22no—
494C2—81no—
495C2—2922no—
496C2C2×#1798865611yescharacteristic
497C2#2261872911yescharacteristic
498C2C2×#1798870211yescharacteristic
subgroupG/H
centerC42C2×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#2965C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556611C4
cosocleC42×C73C24
hypercenter#2965D73
hypocenterC73#17868
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C42 ◁ C2×C4×C8 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. C8×#11924028C2semidirect
3. #22608807C2semidirect
4. #22608800C2semidirect
5. #22608793C2semidirect
6. #22608786C2semidirect
7. #22608677C2semidirect
8. #22608313C2semidirect
9. #22608311C2semidirect
10. C2×#17988309C2semidirect
11. C73×#2965C2semidirect
115. #11924028M16semidirect
250. #2965D73semidirect
261. C73#17868semidirect

Maximal subgroups

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

subgroupindex
C8×#119240282
C4×#145566112
#226088072
#226088002
#226087932
#226087862
#226086772
#226083132
#226083112
C2×#179883092
C73×#29652
#1786873