Finite Group Databasebeta

← #32982307

Subgroups — #32982307

Order 18688.

712 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618380C211yesmaximal, characteristic, complemented
2#22609875C211yesmaximal, characteristic, complemented
3#22608447C211yesmaximal, characteristic, complemented
4#22608291C211yesmaximal, characteristic, complemented
5#22607808C211yesmaximal, characteristic, complemented
6#22607778C211yesmaximal, characteristic, complemented
7C2×#449×C73C211yesmaximal, characteristic
8C2×#14556598—21no—
9C2×#14556518—21no—
10C2×#14556518—21no—
11#17988554—22no—
12#17988554—21no—
13#17988551—21no—
14#17988551C2211yescharacteristic
15#17988551C2211yescharacteristic
16C2×C22⋊#10596831—21no—
17C2×C22⋊#10596831C2211yescharacteristic
18C4×C22⋊Dic73C2211yescharacteristic
19#17988321—22no—
20#17988311—21no—
21#17988311—21no—
22C22×#41×C73C2211yescharacteristic
23C22×SD16×C73C411yescharacteristic, complemented
24C22×D8×C73C411yescharacteristic, complemented
25C2×#121×C73C2211yescharacteristic
26#449×C73—21no—
27#449×C73—21no—
28#449×C73—21no—
29#449×C73—21no—
30C2×#103×C73C411yescharacteristic, complemented
31C2×#102×C73C411yescharacteristic, complemented
32C2×C22⋊C8×C73C2211yescharacteristic
33#14556598—42no—
34#14556598—42no—
35C2×#11924012—22no—
36C2×#11924012—21no—
37C2×#11924012—21no—
38C22×#10596831—22no—
39C22×#10596831—21no—
40C2×C22⋊Dic73—21no—
41C2×C4×Dic73—21no—
42#14556518—42no—
43#14556518—42no—
44C22⋊#10596831—22no—
45C22⋊#10596831—22no—
46C22⋊#10596831—21no—
47C22⋊#10596831—21no—
48C22⋊#10596831—21no—
49C22⋊#10596831—21no—
50C22⋊#10596831D411yescharacteristic
51C22⋊#10596831D411yescharacteristic
52C22⋊#10596831D412yes—
53#14556498—21no—
54C2×#41×C73—21no—
55C2×#41×C73—21no—
56C2×#41×C73—42no—
57C2×#41×C73—42no—
58C2×#41×C73D411yescharacteristic
59C2×#41×C73D411yescharacteristic
60C2×#41×C73D411yescharacteristic
61C2×#41×C73D411yescharacteristic
62C22×Q8×C73C2×C411yescharacteristic
63C22×D4×C73—21no—
64C22×D4×C73—21no—
65C22×D4×C73C2×C411yescharacteristic
66C22×D4×C73C2×C411yescharacteristic
67C23×C4×C73—21no—
68C23×C4×C73C2311yescharacteristic
69C2×SD16×C73—21no—
70C2×SD16×C73—21no—
71C2×SD16×C73—42no—
72C2×D8×C73—21no—
73C2×D8×C73—21no—
74C2×D8×C73—42no—
75C22×C8×C73C2×C411yescharacteristic
76C22×C8×C73C2×C411yescharacteristic
77#121×C73—21no—
78#121×C73—21no—
79#121×C73—21no—
80#121×C73—21no—
81C2×C4⋊C4×C73C2×C411yescharacteristic
82C2×C22⋊C4×C73—21no—
83#103×C73—21no—
84#103×C73—21no—
85#102×C73—21no—
86#102×C73—21no—
87C22⋊C8×C73—21no—
88C22⋊C8×C73—21no—
89C22×Dic73—41no—
90#11924012—44no—
91#11924012—41no—
92#11924012—41no—
93#11924012—41no—
94#11924012—41no—
95C2×#10596831—21no—
96C2×#10596831—22no—
97C2×#10596831—21no—
98C2×#10596831—44no—
99C2×#10596831—41no—
100C2×#10596831—41no—
101C2×#10596831—41no—
102C22⋊Dic73—22no—
103C22⋊Dic73—41no—
104C4×Dic73—41no—
105C4×Dic73—41no—
106C24×C73—21no—
107C24×C73—21no—
108C24×C73#4111yescharacteristic
109#41×C73—22no—
110#41×C73—22no—
111#41×C73—22no—
112#41×C73—22no—
113#41×C73—41no—
114#41×C73—41no—
115#41×C73—41no—
116#41×C73—41no—
117#41×C73—44no—
118#41×C73—44no—
119C2×Q8×C73—21no—
120C2×Q8×C73—22no—
121C2×Q8×C73—41no—
122C2×Q8×C73C22⋊C411yescharacteristic
123C2×Q8×C73C22⋊C411yescharacteristic
124C2×D4×C73—21no—
125C2×D4×C73—22no—
126C2×D4×C73—21no—
127C2×D4×C73—21no—
128C2×D4×C73—21no—
129C2×D4×C73—21no—
130C2×D4×C73—21no—
131C2×D4×C73—21no—
132C2×D4×C73—41no—
133C2×D4×C73—41no—
134C2×D4×C73—41no—
135C2×D4×C73—44no—
136C2×D4×C73—41no—
137C2×D4×C73—41no—
138C2×D4×C73—82no—
139C2×D4×C73C22⋊C411yescharacteristic
140C2×D4×C73C22⋊C411yescharacteristic
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—41no—
148C22×C4×C73—41no—
149C22×C4×C73—44no—
150C22×C4×C73C22×C411yescharacteristic
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73#4111yescharacteristic
156SD16×C73—42no—
157SD16×C73—42no—
158D8×C73—42no—
159D8×C73—42no—
160C2×C8×C73—21no—
161C2×C8×C73—21no—
162C2×C8×C73—41no—
163C2×C8×C73—41no—
164C4⋊C4×C73—21no—
165C4⋊C4×C73—21no—
166C22⋊C4×C73—41no—
167C22⋊C4×C73—41no—
168C23×C73—21no—
169C23×C73—21no—
170C23×C73—21no—
171C23×C73—21no—
172C23×C73—41no—
173C23×C73—41no—
174C23×C73—42no—
175C23×C73—41no—
176C23×C73—41no—
177C23×C73—41no—
178C23×C73—41no—
179C23×C73—41no—
180C23×C73—81no—
181C23×C73—81no—
182C23×C73—81no—
183C23×C73C4×D411yescharacteristic
184C23×C73#12111yescharacteristic
185C23×C73#12311yescharacteristic
186Q8×C73—21no—
187Q8×C73—21no—
188Q8×C73—21no—
189Q8×C73—42no—
190Q8×C73#9912yes—
191D4×C73—24no—
192D4×C73—21no—
193D4×C73—21no—
194D4×C73—21no—
195D4×C73—42no—
196D4×C73—42no—
197D4×C73—42no—
198D4×C73—42no—
199D4×C73—41no—
200D4×C73—41no—
201D4×C73—81no—
202D4×C73—82no—
203D4×C73—82no—
204D4×C73#9912yes—
205C2×C4×C73—24no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—41no—
222C2×C4×C73—81no—
223C2×C4×C73C2×C22⋊C411yescharacteristic
224C2×C4×C73C4×D411yescharacteristic
225C2×C4×C73#12011yescharacteristic
226C2×C4×C73#12111yescharacteristic
227C2×Dic73—42no—
228C2×Dic73—81no—
229C2×Dic73—81no—
230C8×C73—41no—
231C8×C73—41no—
232#10596831—41no—
233#10596831—42no—
234#10596831—41no—
235#10596831—81no—
236C22×C73—21no—
237C22×C73—21no—
238C22×C73—21no—
239C22×C73—21no—
240C22×C73—21no—
241C22×C73—21no—
242C22×C73—22no—
243C22×C73—21no—
244C22×C73—21no—
245C22×C73—41no—
246C22×C73—41no—
247C22×C73—41no—
248C22×C73—41no—
249C22×C73—41no—
250C22×C73—81no—
251C22×C73—81no—
252C22×C73—82no—
253C22×C73—82no—
254C22×C73—81no—
255C22×C73—81no—
256C22×C73—81no—
257C22×C73—161no—
258C22×C73—161no—
259C22×C73#38611yescharacteristic
260C22×C73#44311yescharacteristic
261C22×C73#44411yescharacteristic
262C4×C73—21no—
263C4×C73—21no—
264C4×C73—22no—
265C4×C73—21no—
266C4×C73—41no—
267C4×C73—81no—
268C4×C73C2×#9911yescharacteristic
269C4×C73C2×#9911yescharacteristic
270Dic73—82no—
271#9241—731nomaximal, Sylow
272C2×C73—21no—
273C2×C73—21no—
274C2×C73—41no—
275C2×C73—41no—
276C2×C73—42no—
277C2×C73—81no—
278C2×C73—161no—
279C2×C73#187611yescharacteristic
280C2×C73#189811yescharacteristic
281C2×C73#190311yescharacteristic
282C2×#449D7311yescharacteristic
283#2880—731no—
284#1789—731no—
285#1507—731no—
286#1471—731no—
287#1293—731no—
288#1282—731no—
289C73#924111yescharacteristic, complemented, Sylow
290C22×#41C2×D7311yescharacteristic
291C22×SD16Dic7311yescharacteristic, complemented
292C22×D8Dic7311yescharacteristic, complemented
293C2×#131—1461no—
294C2×#121C2×D7311yescharacteristic
295#449—21no—
296#449—21no—
297#449—21no—
298#449—21no—
299C2×#104—1461no—
300C2×#104—1461no—
301C2×#103Dic7311yescharacteristic, complemented
302C2×#102Dic7311yescharacteristic, complemented
303#408—1462no—
304#408—1461no—
305#407—731no—
306#407—731no—
307#407—1461no—
308C2×C22⋊C8—731no—
309C2×C22⋊C8—1461no—
310C2×C22⋊C8C2×D7311yescharacteristic
311C4×C22⋊C4—731no—
312#327—1462no—
313#323—1461no—
314#323—1461no—
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—42no—
318C2×#41—42no—
319C2×#41C22⋊D7311yescharacteristic
320C2×#41C22⋊D7311yescharacteristic
321C2×#41C22⋊D7311yescharacteristic
322C2×#41C22⋊D7311yescharacteristic
323C22×Q8C2×Dic7311yescharacteristic
324C22×D4—21no—
325C22×D4—21no—
326C22×D4C2×Dic7311yescharacteristic
327C22×D4C2×Dic7311yescharacteristic
328C23×C4—21no—
329C23×C4C22×D7311yescharacteristic
330C2×SD16—21no—
331C2×SD16—21no—
332C2×SD16—42no—
333C2×D8—21no—
334C2×D8—21no—
335C2×D8—42no—
336#131—2922no—
337#131—2922no—
338C2×M16—1461no—
339C2×M16—1462no—
340C2×M16—1461no—
341C22×C8—1462no—
342C22×C8—1461no—
343C22×C8C2×Dic7311yescharacteristic
344C22×C8C2×Dic7311yescharacteristic
345#121—21no—
346#121—21no—
347#121—21no—
348#121—21no—
349C2×C4⋊C4C2×Dic7311yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1461no—
352C2×C42—1461no—
353#104—2922no—
354#104—2922no—
355#103—21no—
356#103—21no—
357#102—21no—
358#102—21no—
359C22⋊C8—21no—
360C22⋊C8—21no—
361C22⋊C8—731no—
362C22⋊C8—731no—
363C22⋊C8—732no—
364C22⋊C8—1461no—
365C22⋊C8—1461no—
366C22⋊C8—1462no—
367C22⋊C8—1462no—
368C22⋊C8—1461no—
369C22⋊C8—1461no—
370#95—1461no—
371C24—21no—
372C24—21no—
373C24#1192402711yescharacteristic
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—44no—
383#41—44no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—41no—
387C2×Q8C22⋊Dic7311yescharacteristic
388C2×Q8C22⋊Dic7311yescharacteristic
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—44no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—82no—
404C2×D4C22⋊Dic7311yescharacteristic
405C2×D4C22⋊Dic7311yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—44no—
415C22×C4—2921no—
416C22×C4C22×Dic7311yescharacteristic
417C22×C4D4×D7311yescharacteristic
418C22×C4D4×D7311yescharacteristic
419C22×C4C2×C22⋊D7311yescharacteristic
420C22×C4C2×C22⋊D7311yescharacteristic
421C22×C4#1192402711yescharacteristic
422SD16—42no—
423SD16—42no—
424D8—42no—
425D8—42no—
426M16—2924no—
427M16—2921no—
428M16—2921no—
429M16—2921no—
430M16—2921no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—2924no—
436C2×C8—2921no—
437C2×C8—2921no—
438C2×C8—2921no—
439C2×C8—1461no—
440C2×C8—1461no—
441C2×C8—1462no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—2921no—
447C22⋊C4—1462no—
448C42—2921no—
449C42—2921no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23D4×Dic7311yescharacteristic
466C23#1455656311yescharacteristic
467C23#1455657211yescharacteristic
468Q8—21no—
469Q8—21no—
470Q8—21no—
471Q8—42no—
472Q8#1455650612yes—
473D4—24no—
474D4—21no—
475D4—21no—
476D4—21no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—41no—
482D4—41no—
483D4—81no—
484D4—82no—
485D4—82no—
486D4#1455650612yes—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—81no—
505C2×C4—2922no—
506C2×C4—5841no—
507C2×C4—5841no—
508C2×C4C2×C22⋊Dic7311yescharacteristic
509C2×C4D4×Dic7311yescharacteristic
510C2×C4#1455655911yescharacteristic
511C2×C4#1455656311yescharacteristic
512C8—41no—
513C8—41no—
514C8—2921no—
515C8—2922no—
516C8—2921no—
517C8—5841no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—21no—
526C22—21no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—81no—
533C22—81no—
534C22—82no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—161no—
540C22—161no—
541C22#1798846711yescharacteristic
542C22#1798870411yescharacteristic
543C22#1798870911yescharacteristic
544C4—21no—
545C4—21no—
546C4—22no—
547C4—21no—
548C4—41no—
549C4—81no—
550C4—5842no—
551C4C2×#1455650611yescharacteristic
552C4C2×#1455650611yescharacteristic
553C2—21no—
554C2—21no—
555C2—41no—
556C2—41no—
557C2—42no—
558C2—81no—
559C2—161no—
560C2#2261026011yescharacteristic
561C2#2261038511yescharacteristic
562C2#2261041311yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
Fitting, prime radicalC2×#449×C73C2
socleC22×C73#386
cosocleC23×C4×C73C23
hypercenterC2×#449D73
hypocenterC73#9241
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#449

Complements

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

NH ≅ G/Ntype
1. #22618380C2semidirect
2. #22609875C2semidirect
3. #22608447C2semidirect
4. #22608291C2semidirect
5. #22607808C2semidirect
6. #22607778C2semidirect
23. C22×SD16×C73C4semidirect
24. C22×D8×C73C4semidirect
30. C2×#103×C73C4semidirect
31. C2×#102×C73C4semidirect
289. C73#9241semidirect
291. C22×SD16Dic73semidirect
292. C22×D8Dic73semidirect
301. C2×#103Dic73semidirect
302. C2×#102Dic73semidirect

Maximal subgroups

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

subgroupindex
#226183802
#226098752
#226084472
#226082912
#226078082
#226077782
C2×#449×C732
#924173