Finite Group Databasebeta

← #33064666

Subgroups — #33064666

Order 18688.

1234 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620872C211yesmaximal, characteristic, complemented
2#22619939C211yesmaximal, characteristic, complemented
3#22619736C211yesmaximal, characteristic, complemented
4#22609131C214yesmaximal, complemented
5#22609058C214yesmaximal, complemented
6#22608599C211yesmaximal, characteristic, complemented
7#22608356C211yesmaximal, characteristic, complemented
8#22608326C211yesmaximal, characteristic, complemented
9C2×#550×C73C211yesmaximal, characteristic
10#17989144C2211yescharacteristic
11#17988959C2212yescomplemented
12#17988933C2212yescomplemented
13#17988917—21no—
14#17988894—22no—
15#17988882C2212yescomplemented
16#17988877C2212yescomplemented
17#17988872C2212yescomplemented
18#17988792C2212yescomplemented
19#17988784—22no—
20#17988738—21no—
21#17988738—22no—
22C4×#11924016—22no—
23#17988640C2211yescharacteristic
24#17988640C2211yescharacteristic
25#17988627—21no—
26#17988611C2211yescharacteristic
27#17988596—21no—
28#17988331C2214yescomplemented
29Q8⋊#10596831C2214yescomplemented
30D4⋊#10596831C2214yescomplemented
31C22×#41×C73—21no—
32#550×C73—22no—
33#550×C73—22no—
34#550×C73—22no—
35#550×C73C2214yes—
36C2×#127×C73—21no—
37C2×#127×C73C2211yescharacteristic
38C2×#121×C73—22no—
39C2×#121×C73—21no—
40C2×C4×Q8×C73C2211yescharacteristic
41C2×C4×D4×C73—21no—
42C2×C4×D4×C73C2211yescharacteristic
43C2×#11924016—22no—
44C2×#11924016—24no—
45C2×#11924012—21no—
46#14556591C2311yescharacteristic
47#14556591C2311yescharacteristic
48#14556583—21no—
49#14556570—21no—
50#14556570—21no—
51C4×Dic146—21no—
52C4×Dic146—21no—
53#14556550—21no—
54#14556524—22no—
55#14556522—22no—
56#14556522—22no—
57C4⋊#10596831—21no—
58C4⋊#10596831—21no—
59C4⋊#10596831C2312yes—
60C4⋊#10596831C2312yes—
61#14556515—22no—
62#14556515—22no—
63#14556514—22no—
64#14556512—22no—
65#14556512—24no—
66C22⋊#10596831—21no—
67C22⋊#10596831—21no—
68C4×#10596831—21no—
69C4×#10596831C2312yes—
70C2×#41×C73—24no—
71C2×#41×C73—24no—
72C2×#41×C73—44no—
73C2×#41×C73—44no—
74C22×Q8×C73D411yescharacteristic
75C22×D4×C73—22no—
76C22×D4×C73—22no—
77C22×D4×C73—21no—
78C22×D4×C73—21no—
79C22×D4×C73D411yescharacteristic
80C22×D4×C73D412yes—
81C23×C4×C73—22no—
82C23×C4×C73—21no—
83#127×C73—24no—
84#127×C73—24no—
85#127×C73—21no—
86#127×C73—21no—
87#127×C73D412yes—
88#127×C73C2312yes—
89#121×C73—24no—
90#121×C73—28no—
91#121×C73—28no—
92#121×C73—24no—
93C4×Q8×C73—21no—
94C4×Q8×C73—21no—
95C4×Q8×C73D412yes—
96C4×Q8×C73C2312yes—
97C4×D4×C73—24no—
98C4×D4×C73—24no—
99C4×D4×C73—21no—
100C4×D4×C73—21no—
101C4×D4×C73D412yes—
102C4×D4×C73C2312yes—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73D411yescharacteristic
105C2×C4⋊C4×C73D411yescharacteristic
106C2×C22⋊C4×C73—22no—
107C2×C22⋊C4×C73—21no—
108C2×C42×C73—21no—
109C2×C42×C73C2311yescharacteristic
110C2×Dic146—21no—
111C2×Dic146—21no—
112#11924016—44no—
113#11924016—84no—
114#11924012—42no—
115C2×#10596831—22no—
116C2×#10596831—22no—
117C2×#10596831—22no—
118C4⋊Dic73—21no—
119C4⋊Dic73—21no—
120C4⋊Dic73—21no—
121C4⋊Dic73—21no—
122#11924007—41no—
123#11924007—41no—
124C22⋊Dic73—41no—
125C4×Dic73—41no—
126C24×C73—22no—
127C24×C73—21no—
128#41×C73—48no—
129#41×C73—416no—
130#41×C73—48no—
131C2×Q8×C73—21no—
132C2×Q8×C73—22no—
133C2×Q8×C73SD1614yescomplemented
134C2×Q8×C73C2×D412yes—
135C2×D4×C73—28no—
136C2×D4×C73—28no—
137C2×D4×C73—21no—
138C2×D4×C73—22no—
139C2×D4×C73—22no—
140C2×D4×C73—22no—
141C2×D4×C73—22no—
142C2×D4×C73—22no—
143C2×D4×C73—42no—
144C2×D4×C73—44no—
145C2×D4×C73—48no—
146C2×D4×C73—42no—
147C2×D4×C73—42no—
148C2×D4×C73—82no—
149C2×D4×C73—84no—
150C2×D4×C73C2×D412yes—
151C2×D4×C73C2×D414yes—
152C22×C4×C73—21no—
153C22×C4×C73—28no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—48no—
159C22×C4×C73—42no—
160C22×C4×C73—42no—
161C22×C4×C73C2×D411yescharacteristic
162C22×C4×C73C2×D411yescharacteristic
163C22×C4×C73C2×D411yescharacteristic
164C4⋊C4×C73—22no—
165C4⋊C4×C73—21no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—22no—
168C4⋊C4×C73C2×D412yes—
169C4⋊C4×C73C2×D412yes—
170C22⋊C4×C73—28no—
171C22⋊C4×C73—22no—
172C22⋊C4×C73—22no—
173C42×C73—22no—
174C42×C73—22no—
175C42×C73C2×D411yescharacteristic
176C42×C73C2×D411yescharacteristic
177C42×C73C2×D411yescharacteristic
178C42×C73C2411yescharacteristic
179C23×C73—22no—
180C23×C73—24no—
181C23×C73—42no—
182C23×C73—44no—
183C23×C73—44no—
184C23×C73—42no—
185C23×C73—44no—
186C23×C73—42no—
187C23×C73#12011yescharacteristic
188Q8×C73—22no—
189Q8×C73—24no—
190Q8×C73C2×SD1614yes—
191D4×C73—22no—
192D4×C73—22no—
193D4×C73—44no—
194D4×C73—42no—
195D4×C73—44no—
196D4×C73—416no—
197D4×C73—42no—
198D4×C73—42no—
199D4×C73—44no—
200D4×C73—44no—
201D4×C73—82no—
202D4×C73—88no—
203C2×C4×C73—22no—
204C2×C4×C73—24no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—41no—
209C2×C4×C73—416no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—44no—
215C2×C4×C73#12011yescharacteristic
216C2×C4×C73#12011yescharacteristic
217C2×C4×C73#12011yescharacteristic
218C2×C4×C73C2×SD1612yes—
219C2×C4×C73AGL(1,ℤ/8)12yes—
220C2×C4×C73C22×D411yescharacteristic
221C2×C4×C73C22×D411yescharacteristic
222C2×C4×C73C22×D411yescharacteristic
223C2×Dic73—41no—
224C2×Dic73—41no—
225Dic146—42no—
226Dic146—42no—
227#10596831—42no—
228#10596831—42no—
229#10596831—42no—
230C22×C73—21no—
231C22×C73—21no—
232C22×C73—44no—
233C22×C73—42no—
234C22×C73—42no—
235C22×C73—44no—
236C22×C73—44no—
237C22×C73—42no—
238C22×C73—84no—
239C22×C73—88no—
240C22×C73#45012yes—
241C22×C73C2×#12011yescharacteristic
242C4×C73—24no—
243C4×C73—21no—
244C4×C73—21no—
245C4×C73—21no—
246C4×C73—42no—
247C4×C73—42no—
248C4×C73#45012yes—
249C4×C73C22×SD1611yescharacteristic
250C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
251C4×C73#57512yes—
252Dic73—81no—
253Dic73—81no—
254#18767—731nomaximal, Sylow
255C2×C73—22no—
256C2×C73—82no—
257C2×C73—84no—
258C2×C73C2×#45011yescharacteristic
259C2×C73#300511yescharacteristic
260C2×C73#301911yescharacteristic
261C2×#550D7311yescharacteristic
262#3208—731no—
263#3101—731no—
264#3078—731no—
265#1642—734no—
266#1631—734no—
267#1545—731no—
268#1486—731no—
269#1479—731no—
270C73#1876711yescharacteristic, complemented, Sylow
271C22×#41—21no—
272#550—22no—
273#550—22no—
274#550—22no—
275#550C2×D7314yes—
276#533—731no—
277C2×#127—21no—
278C2×#127C2×D7311yescharacteristic
279C2×#121—22no—
280C2×#121—21no—
281C2×C4×Q8C2×D7311yescharacteristic
282C2×C4×D4—21no—
283C2×C4×D4C2×D7311yescharacteristic
284#492—732no—
285#486—732no—
286#483—1461no—
287#479—1462no—
288#477—732no—
289#476—732no—
290#475—732no—
291#461—732no—
292#460—1462no—
293#450—1461no—
294#450—1462no—
295C4×SD16—1462no—
296#431—731no—
297#431—731no—
298#428—1461no—
299#423—731no—
300#419—1461no—
301#331—734no—
302Q8⋊C8—734no—
303D4⋊C8—734no—
304C2×#41—24no—
305C2×#41—24no—
306C2×#41—44no—
307C2×#41—44no—
308C22×Q8C22⋊D7311yescharacteristic
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—21no—
312C22×D4—21no—
313C22×D4C22⋊D7311yescharacteristic
314C22×D4C22⋊D7312yes—
315C23×C4—22no—
316C23×C4—21no—
317C2×SD16—1462no—
318C2×SD16—1464no—
319C2×M16—1461no—
320C4⋊Q8—731no—
321C4⋊Q8—731no—
322#127—24no—
323#127—24no—
324#127—21no—
325#127—21no—
326#127C22⋊D7312yes—
327#127C22×D7312yes—
328#125—1461no—
329C22⋊Q8—1461no—
330C22⋊Q8—1461no—
331#121—24no—
332#121—28no—
333#121—28no—
334#121—24no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—1461no—
338C4×Q8—1461no—
339C4×Q8C22⋊D7312yes—
340C4×Q8C22×D7312yes—
341C4×D4—24no—
342C4×D4—24no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4C22⋊D7312yes—
346C4×D4C22×D7312yes—
347#117—1461no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C22⋊D7311yescharacteristic
350C2×C4⋊C4C22⋊D7311yescharacteristic
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C42—21no—
354C2×C42C22×D7311yescharacteristic
355#107—1462no—
356#106—1462no—
357#106—1462no—
358C4⋊C8—732no—
359C4⋊C8—732no—
360C4⋊C8—1461no—
361C4⋊C8—1461no—
362#103—1462no—
363#103—1462no—
364#103—1462no—
365#102—1462no—
366#102—1464no—
367C22⋊C8—1461no—
368C22⋊C8—1461no—
369C4×C8—732no—
370C4×C8—1461no—
371C24—22no—
372C24—21no—
373#41—416no—
374#41—48no—
375#41—48no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—1461no—
379C2×Q8—1461no—
380C2×Q8#1192401614yescomplemented
381C2×Q8C2×C22⋊D7312yes—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—42no—
391C2×D4—44no—
392C2×D4—48no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—82no—
396C2×D4—84no—
397C2×D4C2×C22⋊D7312yes—
398C2×D4C2×C22⋊D7314yes—
399C22×C4—21no—
400C22×C4—28no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4C2×C22⋊D7311yescharacteristic
409C22×C4C2×C22⋊D7311yescharacteristic
410C22×C4C2×C22⋊D7311yescharacteristic
411SD16—2924no—
412SD16—5844no—
413M16—2922no—
414C2×C8—1462no—
415C2×C8—1462no—
416C2×C8—1462no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—2921no—
422C4⋊C4—2921no—
423C4⋊C4—1461no—
424C4⋊C4—1461no—
425C4⋊C4—1461no—
426C4⋊C4—1461no—
427C4⋊C4C2×C22⋊D7312yes—
428C4⋊C4C2×C22⋊D7312yes—
429C22⋊C4—28no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—2921no—
433C42—22no—
434C42—22no—
435C42—2921no—
436C42C2×C22⋊D7311yescharacteristic
437C42C2×C22⋊D7311yescharacteristic
438C42C2×C22⋊D7311yescharacteristic
439C42C23×D7311yescharacteristic
440C23—22no—
441C23—24no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23#1455656011yescharacteristic
449Q8—24no—
450Q8—22no—
451Q8—2922no—
452Q8—2922no—
453Q8C2×#1192401614yes—
454D4—22no—
455D4—22no—
456D4—44no—
457D4—416no—
458D4—42no—
459D4—44no—
460D4—44no—
461D4—44no—
462D4—42no—
463D4—42no—
464D4—82no—
465D4—88no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—416no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—2921no—
479C2×C4—2921no—
480C2×C4#1455656011yescharacteristic
481C2×C4#1455656011yescharacteristic
482C2×C4#1455656011yescharacteristic
483C2×C4C2×#1192401612yes—
484C2×C4#1455661712yes—
485C2×C4C22×C22⋊D7311yescharacteristic
486C2×C4C22×C22⋊D7311yescharacteristic
487C2×C4C22×C22⋊D7311yescharacteristic
488C8—2922no—
489C8—2922no—
490C8—2922no—
491C22—21no—
492C22—21no—
493C22—42no—
494C22—44no—
495C22—44no—
496C22—42no—
497C22—44no—
498C22—42no—
499C22—84no—
500C22—88no—
501C22#1798873812yes—
502C22C2×#1455656011yescharacteristic
503C4—21no—
504C4—21no—
505C4—21no—
506C4—24no—
507C4—42no—
508C4—42no—
509C4—5841no—
510C4—5841no—
511C4#1798873812yes—
512C4C22×#1192401611yescharacteristic
513C4C2×#1455661711yescharacteristic
514C4#1798938512yes—
515C2—22no—
516C2—82no—
517C2—84no—
518C2C2×#1798873811yescharacteristic
519C2#2261917311yescharacteristic
520C2#2261928811yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC2×#550×C73C2
socleC22×C73C2×#120
hypercenterC2×#550D73
hypocenterC73#18767
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. #22620872C2semidirect
2. #22619939C2semidirect
3. #22619736C2semidirect
4. #22609131C2semidirect
5. #22609058C2semidirect
6. #22608599C2semidirect
7. #22608356C2semidirect
8. #22608326C2semidirect
11. #17988959C22semidirect
12. #17988933C22semidirect
15. #17988882C22semidirect
16. #17988877C22semidirect
17. #17988872C22semidirect
18. #17988792C22semidirect
28. #17988331C22semidirect
29. Q8⋊#10596831C22semidirect
30. D4⋊#10596831C22semidirect
133. C2×Q8×C73SD16semidirect
270. C73#18767semidirect
380. C2×Q8#11924016semidirect

Maximal subgroups

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

subgroupindex
#226208722
#226199392
#226197362
#226091312
#226090582
#226085992
#226083562
#226083262
C2×#550×C732
#1876773