Finite Group Databasebeta

← #33055513

Subgroups — #33055513

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
1#22618892C211yesmaximal, characteristic, complemented
2#22618887C211yesmaximal, characteristic, complemented
3C4×#14556613C211yesmaximal, characteristic
4#22608780C212yesmaximal, complemented
5#22608773C212yesmaximal, complemented
6#22608766C212yesmaximal, complemented
7#22608759C212yesmaximal, complemented
8#22608311C211yesmaximal, characteristic, complemented
9#22608311C211yesmaximal, characteristic, complemented
10C2×#17988309C211yesmaximal, characteristic, complemented
11C73×#1564C211yesmaximal, characteristic, complemented
12C2×#14556613C411yescharacteristic
13C4×#11924028C2211yescharacteristic
14C4×#11924028C2211yescharacteristic
15#17988714C2212yes—
16#17988711C2212yes—
17C4×Dic292C411yescharacteristic
18C4×Dic292C2211yescharacteristic
19C4×#11924017C411yescharacteristic
20C4×#11924017C411yescharacteristic
21C4×#11924017C2211yescharacteristic
22C4×#11924017C2211yescharacteristic
23C4×D584C411yescharacteristic
24C4×D584C2211yescharacteristic
25#17988664C2212yes—
26#17988655—22no—
27#17988652C2212yes—
28#17988650—22no—
29#17988638—21no—
30#17988638—21no—
31#17988625C411yescharacteristic
32#17988584C411yescharacteristic
33#17988584C411yescharacteristic
34C4×#11924011—21no—
35C2×C4×#10596831C2211yescharacteristic
36C2×C4×#10596831C2211yescharacteristic
37C4×C8×D73—21no—
38#17988317C2212yes—
39#17988317C2212yes—
40#17988314C2212yes—
41#17988314C2212yes—
42#17988309C2212yes—
43#17988309C2212yes—
44#431×C73C2211yescharacteristic
45#431×C73C2211yescharacteristic
46C2×C4×C8×C73C2211yescharacteristic
47#335×C73C2212yes—
48#334×C73C2212yes—
49C2×#11924028C2×C411yescharacteristic
50C2×#11924028C2×C411yescharacteristic
51#14556613—24no—
52C2×Dic292C2×C411yescharacteristic
53C2×#11924017C2×C411yescharacteristic
54C2×#11924017C2×C411yescharacteristic
55C2×D584C2×C411yescharacteristic
56C2×#11924011—22no—
57C22×#10596831—21no—
58C22×#10596831D412yes—
59C2×C8×D73—22no—
60C4×Dic146C2×C411yescharacteristic
61C4×Dic146C2×C411yescharacteristic
62C4×Dic146C2311yescharacteristic
63C4×Dic146C2311yescharacteristic
64C4×D292C2×C411yescharacteristic
65C4×D292C2×C411yescharacteristic
66C4×D292C2311yescharacteristic
67C4×D292C2311yescharacteristic
68C4×C22⋊D73—21no—
69C4×C22⋊D73—21no—
70C4×C22⋊D73—21no—
71C4×C22⋊D73—21no—
72#14556550C2×C411yescharacteristic
73#14556550C2×C411yescharacteristic
74#14556548—21no—
75#14556548—21no—
76C42×D73—21no—
77C42×D73—21no—
78#14556525C2×C411yescharacteristic
79#14556525C2×C411yescharacteristic
80#14556523C2×C411yescharacteristic
81#14556523C2×C411yescharacteristic
82#14556520—22no—
83#14556520—22no—
84#14556516C2×C411yescharacteristic
85#14556516C2×C411yescharacteristic
86#14556516C2×C411yescharacteristic
87#14556516C2×C411yescharacteristic
88#14556513C2×C411yescharacteristic
89#14556513C2×C411yescharacteristic
90#14556513C2×C411yescharacteristic
91#14556513C2×C411yescharacteristic
92#14556503—22no—
93#14556503—22no—
94#14556501—22no—
95C4×#10596831—21no—
96C4×#10596831D411yescharacteristic
97C4×#10596831D411yescharacteristic
98C4×#10596831C2312yes—
99C4×#10596831C2312yes—
100C8×Dic73—22no—
101C22×C8×C73C2×C411yescharacteristic
102C22×C8×C73C2×C411yescharacteristic
103C2×C42×C73C2311yescharacteristic
104C4⋊C8×C73C2312yes—
105C4⋊C8×C73C2312yes—
106C22⋊C8×C73—21no—
107C22⋊C8×C73—21no—
108C4×C8×C73—21no—
109C4×C8×C73—21no—
110C4×C8×C73C2×C411yescharacteristic
111C4×C8×C73C2×C411yescharacteristic
112C4×C8×C73C2311yescharacteristic
113C4×C8×C73C2311yescharacteristic
114#11924028—22no—
115#11924028—22no—
116#11924028—22no—
117#11924028M1614yescomplemented
118C2×Dic146C22×C411yescharacteristic
119C2×Dic146C22×C411yescharacteristic
120C2×D292C22×C411yescharacteristic
121C2×D292C22×C411yescharacteristic
122C2×C22⋊D73—21no—
123C2×C22⋊D73—21no—
124C2×C4×D73—21no—
125C2×C4×D73—21no—
126C2×C4×D73—21no—
127C2×C4×D73—21no—
128C2×C4×D73—21no—
129C2×C4×D73—21no—
130Dic292—22no—
131#11924017—22no—
132#11924017—22no—
133D584—22no—
134#11924011—44no—
135C2×#10596831—24no—
136C2×#10596831—22no—
137C2×#10596831—42no—
138C2×#10596831C2×D412yes—
139C2×#10596831C2×D412yes—
140C8×D73—44no—
141C4⋊Dic73C22×C411yescharacteristic
142C4⋊Dic73C22×C411yescharacteristic
143C4⋊Dic73C22×C411yescharacteristic
144C4⋊Dic73C22×C411yescharacteristic
145#11924007—21no—
146#11924007—21no—
147#11924007—21no—
148#11924007—21no—
149C22⋊Dic73—21no—
150C22⋊Dic73—21no—
151#11924005—21no—
152#11924005—21no—
153#11924005—21no—
154#11924005—21no—
155C4×Dic73—21no—
156C4×Dic73—21no—
157C4×Dic73—21no—
158C4×Dic73—21no—
159C4×Dic73—21no—
160C4×Dic73—21no—
161C22×C4×C73C22×C411yescharacteristic
162C22×C4×C73C2×D411yescharacteristic
163C22×C4×C73#4111yescharacteristic
164C2×C8×C73—22no—
165C2×C8×C73—22no—
166C2×C8×C73—22no—
167C2×C8×C73—22no—
168C2×C8×C73C22×C411yescharacteristic
169C2×C8×C73C22×C411yescharacteristic
170C2×C8×C73C22×C411yescharacteristic
171C2×C8×C73C22×C411yescharacteristic
172C42×C73C22×C411yescharacteristic
173C42×C73C2×D411yescharacteristic
174C42×C73#4111yescharacteristic
175C42×C73C2411yescharacteristic
176C23×C73C4×D411yescharacteristic
177C22×D73—21no—
178C22×D73—21no—
179C2×C4×C73—22no—
180C2×C4×C73—22no—
181C2×C4×C73C4×D411yescharacteristic
182C2×C4×C73C4×D411yescharacteristic
183C2×C4×C73C4×D411yescharacteristic
184C2×C4×C73C2×M1612yes—
185C2×C4×C73#13112yes—
186C2×C4×C73C23×C411yescharacteristic
187C2×C4×C73C22×D411yescharacteristic
188C2×C4×C73C2×#4111yescharacteristic
189C22⋊D73—24no—
190C22⋊D73—42no—
191C2×Dic73—21no—
192C2×Dic73—21no—
193C2×Dic73—21no—
194C2×Dic73—21no—
195C2×Dic73—21no—
196C2×Dic73—21no—
197D292—21no—
198D292—21no—
199D292—21no—
200D292C2×M1612yes—
201C4×D73—24no—
202C4×D73—42no—
203C4×D73—42no—
204C4×D73—42no—
205C4×D73—42no—
206C4×D73—42no—
207Dic146—21no—
208Dic146—21no—
209Dic146—21no—
210Dic146C2×M1612yes—
211C8×C73—21no—
212C8×C73—21no—
213C8×C73—21no—
214C8×C73—21no—
215C8×C73—42no—
216C8×C73—42no—
217#10596831—22no—
218#10596831—42no—
219#10596831#13514yescomplemented
220C22×C73—21no—
221C22×C73—21no—
222C22×C73#43512yes—
223C22×C73C2×C4×D411yescharacteristic
224C2×D73—22no—
225C2×D73—41no—
226C2×D73—41no—
227C2×D73—41no—
228C2×D73—41no—
229C2×D73—41no—
230C4×C73—21no—
231C4×C73—21no—
232C4×C73#43512yes—
233C4×C73#44412yes—
234C4×C73C22×M1611yescharacteristic
235C4×C73C2×#13111yescharacteristic
236C4×C73C2×#13512yes—
237Dic73—22no—
238Dic73—41no—
239Dic73—41no—
240Dic73—41no—
241Dic73—41no—
242Dic73—41no—
243#17863—731nomaximal, Sylow
244C2×C73—22no—
245C2×C73C2×#43511yescharacteristic
246C2×C73C4×#13511yescharacteristic
247C2×C73C2×#44411yescharacteristic
248D73—42no—
249D73—81no—
250#2965—731no—
251C8×#41—731no—
252C4×#135—731no—
253#1584—732no—
254#1583—732no—
255#1582—732no—
256#1581—732no—
257#1564D7311yescharacteristic, complemented
258#1476—731no—
259#1476—731no—
260C2×#322—731no—
261C73#1786311yescharacteristic, complemented, Sylow
262C2×#135—731no—
263C4×#41—731no—
264C4×#41—731no—
265#446—732no—
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#436—732no—
276#435—1462no—
277C8×D4—732no—
278C8×D4—1462no—
279#431—1461no—
280#431—1461no—
281#431C2×D7311yescharacteristic
282#431C2×D7311yescharacteristic
283#427—731no—
284#416—731no—
285#416—731no—
286C4×M16—1461no—
287C2×C4×C8—731no—
288C2×C4×C8—731no—
289C2×C4×C8—1461no—
290C2×C4×C8C2×D7311yescharacteristic
291#335C2×D7312yes—
292#334C2×D7312yes—
293Q8⋊C8—732no—
294Q8⋊C8—732no—
295D4⋊C8—732no—
296D4⋊C8—732no—
297#322—732no—
298#322—732no—
299C2×#41—731no—
300C2×#41—731no—
301#135—1464no—
302C2×Q16—731no—
303C2×SD16—731no—
304C2×SD16—731no—
305C2×D8—731no—
306C2×M16—1462no—
307C22×C8—732no—
308C22×C8—1462no—
309C22×C8—1461no—
310C22×C8C4×D7311yescharacteristic
311C22×C8C4×D7311yescharacteristic
312C4×Q8—731no—
313C4×Q8—731no—
314C4×Q8—731no—
315C4×Q8—731no—
316C4×D4—731no—
317C4×D4—731no—
318C4×D4—731no—
319C4×D4—731no—
320C4×D4—1461no—
321C4×D4—1461no—
322C4×D4—1461no—
323C4×D4—1461no—
324#117—731no—
325#117—731no—
326#117—1461no—
327#117—1461no—
328C2×C42—1461no—
329C2×C42—1461no—
330C2×C42C22×D7311yescharacteristic
331#107—731no—
332#107—731no—
333#106—731no—
334#106—731no—
335C4⋊C8—1462no—
336C4⋊C8—1462no—
337C4⋊C8C22×D7312yes—
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—21no—
349C22⋊C8—1462no—
350C22⋊C8—1462no—
351#97—1462no—
352C4×C8—21no—
353C4×C8—21no—
354C4×C8—731no—
355C4×C8—731no—
356C4×C8—732no—
357C4×C8—732no—
358C4×C8—1462no—
359C4×C8—1461no—
360C4×C8C4×D7311yescharacteristic
361C4×C8C4×D7311yescharacteristic
362C4×C8C22×D7311yescharacteristic
363C4×C8C22×D7311yescharacteristic
364#41—734no—
365#41—1462no—
366#41—1462no—
367#41—1462no—
368C2×Q8—731no—
369C2×Q8—731no—
370C2×D4—731no—
371C2×D4—731no—
372C2×D4—1461no—
373C2×D4—1461no—
374C22×C4—1461no—
375C22×C4—1461no—
376C22×C4—1461no—
377C22×C4—1461no—
378C22×C4—1461no—
379C22×C4—1461no—
380C22×C4C2×C4×D7311yescharacteristic
381C22×C4D4×D7311yescharacteristic
382C22×C4#1192402911yescharacteristic
383Q16—1462no—
384SD16—1462no—
385SD16—1462no—
386D8—1462no—
387M16—2924no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—2924no—
393C2×C8—2922no—
394C2×C8—732no—
395C2×C8—732no—
396C2×C8—1462no—
397C2×C8—1464no—
398C2×C8C2×C4×D7311yescharacteristic
399C2×C8C2×C4×D7311yescharacteristic
400C2×C8C2×C4×D7311yescharacteristic
401C2×C8C2×C4×D7311yescharacteristic
402C4⋊C4—731no—
403C4⋊C4—731no—
404C4⋊C4—731no—
405C4⋊C4—731no—
406C4⋊C4—1461no—
407C4⋊C4—1461no—
408C4⋊C4—1461no—
409C4⋊C4—1461no—
410C22⋊C4—1461no—
411C22⋊C4—1461no—
412C22⋊C4—1461no—
413C22⋊C4—1461no—
414C22⋊C4—1461no—
415C22⋊C4—1461no—
416C42—1461no—
417C42—1461no—
418C42—1461no—
419C42—1461no—
420C42—1461no—
421C42—1461no—
422C42C2×C4×D7311yescharacteristic
423C42D4×D7311yescharacteristic
424C42#1192402911yescharacteristic
425C42C23×D7311yescharacteristic
426C23—1461no—
427C23—1461no—
428C23#1455655311yescharacteristic
429Q8—732no—
430Q8—1461no—
431Q8—1461no—
432Q8—1461no—
433D4—2922no—
434D4—732no—
435D4—1464no—
436D4—1461no—
437D4—1461no—
438D4—1461no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—2922no—
442C2×C4—2922no—
443C2×C4—2922no—
444C2×C4—2922no—
445C2×C4—2922no—
446C2×C4—1461no—
447C2×C4—1461no—
448C2×C4—1461no—
449C2×C4—1461no—
450C2×C4—1461no—
451C2×C4—1461no—
452C2×C4—1464no—
453C2×C4#1455655311yescharacteristic
454C2×C4#1455655311yescharacteristic
455C2×C4#1455655311yescharacteristic
456C2×C4M16×D7312yes—
457C2×C4#1455659712yes—
458C2×C4C22×C4×D7311yescharacteristic
459C2×C4C2×D4×D7311yescharacteristic
460C2×C4C2×#1192402911yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—21no—
465C8—42no—
466C8—42no—
467C8—2922no—
468C8—734no—
469C8—1462no—
470C22—21no—
471C22—21no—
472C22—2921no—
473C22—2921no—
474C22—2921no—
475C22—2921no—
476C22—2921no—
477C22—1462no—
478C22#1798865712yes—
479C22C2×#1455655311yescharacteristic
480C4—21no—
481C4—21no—
482C4—2921no—
483C4—2921no—
484C4—2921no—
485C4—2921no—
486C4—2921no—
487C4—1462no—
488C4#1798865712yes—
489C4#1798870812yes—
490C4C2×M16×D7311yescharacteristic
491C4C2×#1455659711yescharacteristic
492C4#135×D7312yes—
493C2—22no—
494C2—2922no—
495C2—5841no—
496C2C2×#1798865711yescharacteristic
497C2#2261870211yescharacteristic
498C2C2×#1798870811yescharacteristic
subgroupG/H
centerC42D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1564C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556613C4
cosocleC42×C73C24
hypercenter#1564D73
hypocenterC73#17863
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C42 ◁ C2×C42 ◁ #1564

Complements

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

NH ≅ G/Ntype
1. #22618892C2semidirect
2. #22618887C2semidirect
4. #22608780C2semidirect
5. #22608773C2semidirect
6. #22608766C2semidirect
7. #22608759C2semidirect
8. #22608311C2semidirect
9. #22608311C2semidirect
10. C2×#17988309C2semidirect
11. C73×#1564C2semidirect
117. #11924028M16semidirect
219. #10596831#135semidirect
257. #1564D73semidirect
261. C73#17863semidirect

Maximal subgroups

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

subgroupindex
#226188922
#226188872
C4×#145566132
#226087802
#226087732
#226087662
#226087592
#226083112
C2×#179883092
C73×#15642
#1786373