Finite Group Databasebeta

← #33063321

Subgroups — #33063321

Order 18688.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621056C211yesmaximal, characteristic, complemented
2C2×#17988939C211yesmaximal, characteristic, complemented
3#22619746C211yesmaximal, characteristic, complemented
4#22609096C214yesmaximal, complemented
5#22609082C214yesmaximal, complemented
6#22608603C211yesmaximal, characteristic, complemented
7#22608382C211yesmaximal, characteristic, complemented
8#22608313C211yesmaximal, characteristic, complemented
9C73×#3499C211yesmaximal, characteristic
10C2×#14556591C2211yescharacteristic
11#17988983—21no—
12#17988939—22no—
13#17988939C2214yes—
14#17988933—21no—
15#17988903—22no—
16#17988882C2212yes—
17#17988877C2212yes—
18#17988877C2214yes—
19#17988872—21no—
20#17988867—21no—
21#17988640C2211yescharacteristic
22#17988630—21no—
23#17988611C2211yescharacteristic
24#17988588—21no—
25C2×#14556514—22no—
26C2×C4×#10596831C2211yescharacteristic
27#17988333C2214yes—
28Q8⋊#10596831C2212yes—
29D4⋊#10596831C2212yes—
30D4⋊#10596831C2214yes—
31C2×2−1+4×C73—21no—
32D4×Q8×C73—21no—
33D4×Q8×C73C2212yes—
34#547×C73—22no—
35#547×C73—21no—
36#547×C73—22no—
37#547×C73C2212yes—
38#536×C73—22no—
39#536×C73—21no—
40#532×C73—21no—
41#532×C73C2211yescharacteristic
42C2×C4⋊Q8×C73—21no—
43C2×C4⋊Q8×C73C2211yescharacteristic
44C4×#41×C73C2211yescharacteristic
45C22×Dic146—21no—
46C2×#11924012—21no—
47C22×#10596831—21no—
48#14556591—21no—
49#14556591—21no—
50#14556591—21no—
51#14556591C2312yes—
52C2×C4⋊Dic73—21no—
53C2×C4⋊Dic73—21no—
54#14556524—22no—
55#14556522—22no—
56#14556522—24no—
57C4⋊#10596831—21no—
58C4⋊#10596831C2312yes—
59C4⋊#10596831C2312yes—
60#14556515—22no—
61#14556514—28no—
62#14556512—22no—
63#14556512—24no—
64C22⋊#10596831—21no—
65C4×#10596831—21no—
66C4×#10596831—21no—
67C4×#10596831C2312yes—
682−1+4×C73—44no—
692−1+4×C73—44no—
70C2×#41×C73—22no—
71C2×#41×C73—24no—
72C2×#41×C73—22no—
73C2×#41×C73—21no—
74C2×#41×C73D411yescharacteristic
75C22×Q8×C73—22no—
76C22×Q8×C73—22no—
77C22×Q8×C73—21no—
78C4⋊Q8×C73—22no—
79C4⋊Q8×C73—22no—
80C4⋊Q8×C73—22no—
81C4⋊Q8×C73—21no—
82C4⋊Q8×C73—21no—
83C4⋊Q8×C73D412yes—
84C4⋊Q8×C73D411yescharacteristic
85C4⋊Q8×C73C2312yes—
86#127×C73D411yescharacteristic
87#124×C73—22no—
88#124×C73—21no—
89#123×C73—24no—
90#123×C73—24no—
91#123×C73—24no—
92C22⋊Q8×C73—24no—
93C22⋊Q8×C73—22no—
94C22⋊Q8×C73—24no—
95C22⋊Q8×C73—22no—
96#121×C73—22no—
97#121×C73—22no—
98#121×C73—21no—
99#121×C73D412yes—
100C4×Q8×C73D411yescharacteristic
101C4×Q8×C73C2311yescharacteristic
102C4×D4×C73—22no—
103C4×D4×C73—22no—
104C4×D4×C73—21no—
105C4×D4×C73—21no—
106C4×D4×C73—21no—
107C4×D4×C73D411yescharacteristic
108C4×D4×C73C2311yescharacteristic
109C4×D4×C73C2312yes—
110#117×C73—21no—
111#117×C73D411yescharacteristic
112C2×C4⋊C4×C73—22no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—22no—
115C2×C4⋊C4×C73D412yes—
116C2×C42×C73—21no—
117C2×C42×C73C2311yescharacteristic
118C2×Dic146—22no—
119C2×Dic146—44no—
120C2×Dic146—41no—
121C22×Dic73—41no—
122#11924012—42no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C2×#10596831—42no—
127C4⋊Dic73—22no—
128C4⋊Dic73—22no—
129C4⋊Dic73—41no—
130C4⋊Dic73—41no—
131#41×C73—22no—
132#41×C73—48no—
133#41×C73—416no—
134#41×C73—44no—
135#41×C73—44no—
136#41×C73—42no—
137#41×C73—42no—
138#41×C73SD1614yes—
139C2×Q8×C73—24no—
140C2×Q8×C73—22no—
141C2×Q8×C73—21no—
142C2×Q8×C73—22no—
143C2×Q8×C73—48no—
144C2×Q8×C73—44no—
145C2×Q8×C73—44no—
146C2×Q8×C73—42no—
147C2×Q8×C73—42no—
148C2×Q8×C73C2×D411yescharacteristic
149C2×D4×C73—21no—
150C2×D4×C73—24no—
151C2×D4×C73—22no—
152C2×D4×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73C2×D412yes—
155C2×D4×C73C2×D411yescharacteristic
156C22×C4×C73—21no—
157C22×C4×C73—24no—
158C22×C4×C73—24no—
159C22×C4×C73—22no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—22no—
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C4⋊C4×C73—24no—
167C4⋊C4×C73—24no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—22no—
170C4⋊C4×C73—22no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—22no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73C2×D412yes—
175C4⋊C4×C73C2×D414yes—
176C4⋊C4×C73C2×D411yescharacteristic
177C4⋊C4×C73C2×D411yescharacteristic
178C22⋊C4×C73—24no—
179C22⋊C4×C73—24no—
180C22⋊C4×C73—21no—
181C22⋊C4×C73—21no—
182C22⋊C4×C73—22no—
183C22⋊C4×C73—21no—
184C22⋊C4×C73—22no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73C2×D411yescharacteristic
189C42×C73C2×D411yescharacteristic
190C42×C73C2×D411yescharacteristic
191C42×C73C2411yescharacteristic
192C23×C73—21no—
193C23×C73—22no—
194C23×C73#12011yescharacteristic
195Q8×C73—21no—
196Q8×C73—48no—
197Q8×C73—42no—
198Q8×C73—42no—
199Q8×C73—42no—
200Q8×C73—44no—
201Q8×C73C2×SD1612yes—
202D4×C73—24no—
203D4×C73—21no—
204D4×C73—24no—
205D4×C73—48no—
206D4×C73—44no—
207D4×C73—42no—
208D4×C73—42no—
209D4×C73C2×SD1612yes—
210C2×C4×C73—21no—
211C2×C4×C73—22no—
212C2×C4×C73—24no—
213C2×C4×C73—22no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—24no—
217C2×C4×C73—22no—
218C2×C4×C73—48no—
219C2×C4×C73—48no—
220C2×C4×C73—44no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73#12011yescharacteristic
226C2×C4×C73#12011yescharacteristic
227C2×C4×C73#12011yescharacteristic
228C2×C4×C73C2×SD1612yes—
229C2×C4×C73AGL(1,ℤ/8)12yes—
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C2×C4×C73C22×D411yescharacteristic
233C2×Dic73—42no—
234C2×Dic73—82no—
235Dic146—44no—
236Dic146—82no—
237#10596831—42no—
238#10596831—42no—
239#10596831—42no—
240C22×C73—21no—
241C22×C73—21no—
242C22×C73—22no—
243C22×C73—41no—
244C22×C73—41no—
245C22×C73—42no—
246C22×C73—42no—
247C22×C73—42no—
248C22×C73#45012yes—
249C22×C73C2×#12011yescharacteristic
250C4×C73—22no—
251C4×C73—21no—
252C4×C73—21no—
253C4×C73—44no—
254C4×C73—42no—
255C4×C73—41no—
256C4×C73—41no—
257C4×C73#45012yes—
258C4×C73#45412yes—
259C4×C73C22×SD1611yescharacteristic
260C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
261C4×C73#57812yes—
262Dic73—82no—
263#18641—731nomaximal, Sylow
264C2×C73—22no—
265C2×C73—42no—
266C2×C73—82no—
267C2×C73C2×#45011yescharacteristic
268C2×C73#301111yescharacteristic
269C2×C73C2×#45411yescharacteristic
270#3499D7311yescharacteristic
271#3227—731no—
272C2×#487—731no—
273#3079—731no—
274#1637—734no—
275#1635—734no—
276#1546—731no—
277#1492—731no—
278#1476—731no—
279C73#1864111yescharacteristic, complemented, Sylow
280C2×2−1+4—21no—
281D4×Q8—21no—
282D4×Q8C2×D7312yes—
283#547—22no—
284#547—21no—
285#547—22no—
286#547C2×D7312yes—
287#536—22no—
288#536—21no—
289#532—21no—
290#532C2×D7311yescharacteristic
291C2×C4⋊Q8—21no—
292C2×C4⋊Q8—731no—
293C2×C4⋊Q8C2×D7311yescharacteristic
294C4×#41C2×D7311yescharacteristic
295#498—1461no—
296#487—734no—
297#487—1462no—
298#486—1461no—
299#481—1462no—
300#477—732no—
301#476—732no—
302#476—734no—
303#475—1461no—
304#474—1461no—
305#431—731no—
306#428—1461no—
307#423—731no—
308#417—1461no—
309C2×#103—1462no—
310C2×C4×C8—731no—
311#332—734no—
312Q8⋊C8—732no—
313D4⋊C8—732no—
314D4⋊C8—734no—
3152−1+4—44no—
3162−1+4—44no—
317C2×#41—22no—
318C2×#41—24no—
319C2×#41—22no—
320C2×#41—21no—
321C2×#41C22⋊D7311yescharacteristic
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8—21no—
325C22×Q8—1461no—
326C2×M16—1461no—
327C22×C8—1461no—
328C4⋊Q8—22no—
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333C4⋊Q8—732no—
334C4⋊Q8—1461no—
335C4⋊Q8—1461no—
336C4⋊Q8—1461no—
337C4⋊Q8C22⋊D7311yescharacteristic
338C4⋊Q8C22⋊D7312yes—
339C4⋊Q8C22×D7312yes—
340#127C22⋊D7311yescharacteristic
341#124—22no—
342#124—21no—
343#123—24no—
344#123—24no—
345#123—24no—
346C22⋊Q8—24no—
347C22⋊Q8—22no—
348C22⋊Q8—24no—
349C22⋊Q8—22no—
350#121—22no—
351#121—22no—
352#121—21no—
353#121C22⋊D7312yes—
354C4×Q8C22⋊D7311yescharacteristic
355C4×Q8C22×D7311yescharacteristic
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4C22⋊D7311yescharacteristic
362C4×D4C22×D7311yescharacteristic
363C4×D4C22×D7312yes—
364#117—21no—
365#117C22⋊D7311yescharacteristic
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4C22⋊D7312yes—
372C2×C42—21no—
373C2×C42C22×D7311yescharacteristic
374#107—1462no—
375#106—1462no—
376#106—1464no—
377C4⋊C8—732no—
378C4⋊C8—732no—
379C4⋊C8—1461no—
380#103—1462no—
381#103—1468no—
382#102—1462no—
383#102—1464no—
384C22⋊C8—1461no—
385C4×C8—732no—
386C4×C8—1461no—
387C4×C8—1461no—
388#41—22no—
389#41—48no—
390#41—416no—
391#41—44no—
392#41—44no—
393#41—42no—
394#41—42no—
395#41#1192401614yes—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—2924no—
406C2×Q8—2921no—
407C2×Q8—1462no—
408C2×Q8C2×C22⋊D7311yescharacteristic
409C2×D4—21no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×C22⋊D7312yes—
415C2×D4C2×C22⋊D7311yescharacteristic
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—2921no—
424C22×C4C2×C22⋊D7311yescharacteristic
425C22×C4C2×C22⋊D7311yescharacteristic
426C22×C4C2×C22⋊D7311yescharacteristic
427M16—2922no—
428C2×C8—2922no—
429C2×C8—1462no—
430C2×C8—1462no—
431C2×C8—1462no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—2921no—
441C4⋊C4—2921no—
442C4⋊C4—1462no—
443C4⋊C4—1462no—
444C4⋊C4C2×C22⋊D7314yes—
445C4⋊C4C2×C22⋊D7312yes—
446C4⋊C4C2×C22⋊D7311yescharacteristic
447C4⋊C4C2×C22⋊D7311yescharacteristic
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—22no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42C2×C22⋊D7311yescharacteristic
459C42C2×C22⋊D7311yescharacteristic
460C42C2×C22⋊D7311yescharacteristic
461C42C23×D7311yescharacteristic
462C23—21no—
463C23—22no—
464C23#1455656011yescharacteristic
465Q8—21no—
466Q8—48no—
467Q8—42no—
468Q8—44no—
469Q8—42no—
470Q8—42no—
471Q8—2924no—
472Q8—5842no—
473Q8C2×#1192401612yes—
474D4—24no—
475D4—24no—
476D4—21no—
477D4—48no—
478D4—44no—
479D4—42no—
480D4—42no—
481D4C2×#1192401612yes—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—2922no—
498C2×C4—5842no—
499C2×C4#1455656011yescharacteristic
500C2×C4#1455656011yescharacteristic
501C2×C4#1455656011yescharacteristic
502C2×C4C2×#1192401612yes—
503C2×C4#1455661712yes—
504C2×C4C22×C22⋊D7311yescharacteristic
505C2×C4C22×C22⋊D7311yescharacteristic
506C2×C4C22×C22⋊D7311yescharacteristic
507C8—2922no—
508C8—2922no—
509C8—2922no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22#1798873812yes—
519C22C2×#1455656011yescharacteristic
520C4—21no—
521C4—21no—
522C4—22no—
523C4—41no—
524C4—41no—
525C4—44no—
526C4—42no—
527C4—5842no—
528C4#1798873812yes—
529C4#1798876012yes—
530C4C22×#1192401611yescharacteristic
531C4C2×#1455661711yescharacteristic
532C4#1798940212yes—
533C2—22no—
534C2—42no—
535C2—82no—
536C2C2×#1798873811yescharacteristic
537C2#2261922611yescharacteristic
538C2C2×#1798876011yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3499C2
socleC22×C73C2×#120
hypercenter#3499D73
hypocenterC73#18641
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. #22621056C2semidirect
2. C2×#17988939C2semidirect
3. #22619746C2semidirect
4. #22609096C2semidirect
5. #22609082C2semidirect
6. #22608603C2semidirect
7. #22608382C2semidirect
8. #22608313C2semidirect
279. C73#18641semidirect

Maximal subgroups

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

subgroupindex
#226210562
C2×#179889392
#226197462
#226090962
#226090822
#226086032
#226083822
#226083132
C73×#34992
#1864173