Finite Group Databasebeta

← #33119338

Subgroups — #33119338

Order 18688.

1130 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988903C211yesmaximal, characteristic, complemented
2C2×#17988898C211yesmaximal, characteristic, complemented
3#22612551C211yesmaximal, characteristic, complemented
4#22611266C212yesmaximal, complemented
5#22611162C212yesmaximal, complemented
6#22610254C211yesmaximal, characteristic, complemented
7#22610254C211yesmaximal, characteristic, complemented
8#22610192C212yesmaximal, complemented
9#22610178C212yesmaximal, complemented
10#22610002C211yesmaximal, characteristic, complemented
11C2×#540×C73C211yesmaximal, characteristic
12C2×#14556550C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14#17988903—24no—
15#17988898—24no—
16C2×#14556524C2212yes—
17C2×#14556522C2212yes—
18C2×#14556512C2212yes—
19C2×#14556512C2212yes—
20C2×#14556512C2212yes—
21C2×#14556512C2212yes—
22C2×C22⋊#10596831C2211yescharacteristic
23C2×C22⋊#10596831C2211yescharacteristic
24#17988485C2212yes—
25#17988485C2212yes—
26#17988459—21no—
27#17988459—21no—
28#17988456—21no—
29#17988452—21no—
30#17988349C2212yes—
31#17988349C2212yes—
32#17988342C2212yes—
33#17988342C2212yes—
34#540×C73—24no—
35#540×C73—24no—
36C2×#127×C73C2211yescharacteristic
37C2×#125×C73C2211yescharacteristic
38C2×#123×C73—21no—
39C2×#123×C73—21no—
40C2×#121×C73—21no—
41C2×#121×C73—21no—
42C2×#121×C73C2211yescharacteristic
43C2×#121×C73C2211yescharacteristic
44C2×C4×D4×C73C2211yescharacteristic
45C2×C4×D4×C73C2211yescharacteristic
46C2×#117×C73C2211yescharacteristic
47C23×Dic73—21no—
48C22×#10596831C2312yes—
49C22×#10596831C2312yes—
50#14556550—24no—
51C2×C4⋊Dic73—22no—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—21no—
54C2×C4⋊Dic73C2312yes—
55C2×C4⋊Dic73C2312yes—
56C2×#11924007—22no—
57C2×#11924007—22no—
58C2×C22⋊Dic73—21no—
59C2×C4×Dic73—21no—
60#14556524—24no—
61#14556522—24no—
62#14556512—24no—
63#14556512—24no—
64#14556512—24no—
65#14556512—24no—
66C22⋊#10596831—22no—
67C22⋊#10596831—22no—
68#14556498—22no—
69#14556498—22no—
70C22×D4×C73—21no—
71C22×D4×C73—21no—
72C22×D4×C73—21no—
73C22×D4×C73—21no—
74C22×D4×C73C2311yescharacteristic
75C22×D4×C73C2311yescharacteristic
76C23×C4×C73—21no—
77C23×C4×C73—21no—
78C23×C4×C73C2311yescharacteristic
79#127×C73—22no—
80#127×C73—22no—
81#125×C73—22no—
82#125×C73—22no—
83#123×C73—28no—
84#123×C73—28no—
85#121×C73—28no—
86#121×C73—28no—
87#121×C73—22no—
88#121×C73—22no—
89#121×C73D414yes—
90#121×C73D414yes—
91C4×D4×C73—22no—
92C4×D4×C73—22no—
93C4×D4×C73—22no—
94C4×D4×C73—22no—
95#117×C73—22no—
96#117×C73—22no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73D411yescharacteristic
100C2×C4⋊C4×C73D411yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73D411yescharacteristic
108C2×C22⋊C4×C73D411yescharacteristic
109C2×C42×C73C2311yescharacteristic
110C2×C42×C73C2311yescharacteristic
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—42no—
114C22×Dic73—82no—
115C2×#10596831—22no—
116C2×#10596831—22no—
117C2×#10596831—22no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—22no—
121C4⋊Dic73—22no—
122C4⋊Dic73—22no—
123C4⋊Dic73—22no—
124C4⋊Dic73—42no—
125C4⋊Dic73#4114yes—
126#11924007—82no—
127#11924007—82no—
128C22⋊Dic73—24no—
129C4×Dic73—24no—
130C24×C73—21no—
131C24×C73—21no—
132C24×C73C2×D411yescharacteristic
133C2×D4×C73—24no—
134C2×D4×C73—24no—
135C2×D4×C73—24no—
136C2×D4×C73—24no—
137C2×D4×C73—21no—
138C2×D4×C73—21no—
139C2×D4×C73—22no—
140C2×D4×C73—22no—
141C2×D4×C73—22no—
142C2×D4×C73—22no—
143C2×D4×C73—82no—
144C2×D4×C73—82no—
145C2×D4×C73—82no—
146C2×D4×C73—82no—
147C2×D4×C73C2×D412yes—
148C2×D4×C73C2×D412yes—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—24no—
154C22×C4×C73—24no—
155C22×C4×C73—82no—
156C22×C4×C73—82no—
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73C2×D411yescharacteristic
159C22×C4×C73C2×D411yescharacteristic
160C22×C4×C73C2×D411yescharacteristic
161C22×C4×C73C2×D411yescharacteristic
162C22×C4×C73C2×D412yes—
163C22×C4×C73C2×D412yes—
164C22×C4×C73C2411yescharacteristic
165C4⋊C4×C73—24no—
166C4⋊C4×C73—24no—
167C4⋊C4×C73—22no—
168C4⋊C4×C73—22no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—21no—
171C4⋊C4×C73C2×D412yes—
172C4⋊C4×C73C2×D412yes—
173C22⋊C4×C73—24no—
174C22⋊C4×C73—24no—
175C22⋊C4×C73—22no—
176C22⋊C4×C73—22no—
177C22⋊C4×C73—24no—
178C22⋊C4×C73—24no—
179C42×C73—21no—
180C42×C73—21no—
181C42×C73—21no—
182C42×C73—21no—
183C23×C73—21no—
184C23×C73—21no—
185C23×C73—22no—
186C23×C73—22no—
187C23×C73—22no—
188C23×C73—22no—
189C23×C73—41no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—42no—
196C23×C73—42no—
197C23×C73#12012yes—
198C23×C73C22×D411yescharacteristic
199D4×C73—24no—
200D4×C73—24no—
201D4×C73—42no—
202D4×C73—42no—
203D4×C73—84no—
204D4×C73—84no—
205D4×C73—84no—
206D4×C73—84no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—24no—
210C2×C4×C73—24no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—84no—
224C2×C4×C73—84no—
225C2×C4×C73#12012yes—
226C2×C4×C73C22×D411yescharacteristic
227C2×C4×C73C22×D411yescharacteristic
228C2×C4×C73C2×#4112yes—
229C2×C4×C732−1+412yes—
230C2×Dic73—24no—
231C2×Dic73—42no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—84no—
237#10596831—42no—
238#10596831—42no—
239C22×C73—22no—
240C22×C73—22no—
241C22×C73—22no—
242C22×C73—42no—
243C22×C73—42no—
244C22×C73—42no—
245C22×C73—42no—
246C22×C73—42no—
247C22×C73—42no—
248C22×C73—42no—
249C22×C73—42no—
250C22×C73—82no—
251C22×C73—82no—
252C22×C73—82no—
253C22×C73—82no—
254C22×C73C2×#12011yescharacteristic
255C22×C73#52511yescharacteristic
256C22×C73#53611yescharacteristic
257C22×C73#57612yes—
258C22×C73#57712yes—
259C4×C73—22no—
260C4×C73—41no—
261C4×C73—41no—
262C4×C73—42no—
263C4×C73—41no—
264C4×C73—41no—
265C4×C73—42no—
266C4×C73#54712yes—
267C4×C73#54712yes—
268Dic73—44no—
269Dic73—82no—
270#24069—731nomaximal, Sylow
271C2×C73—42no—
272C2×C73—82no—
273C2×C73—82no—
274C2×C73#240611yescharacteristic
275C2×C73#300811yescharacteristic
276C2×C73#300911yescharacteristic
277C2×C73#311912yes—
278C2×C73#324112yes—
279C2×#540D7311yescharacteristic
280C2×#481—731no—
281C2×#480—731no—
282#2317—731no—
283#2059—732no—
284#2041—732no—
285#1875—731no—
286#1875—731no—
287#1864—732no—
288#1862—732no—
289#1818—731no—
290C73#2406911yescharacteristic, complemented, Sylow
291#540—24no—
292#540—24no—
293C2×#127C2×D7311yescharacteristic
294C2×#125C2×D7311yescharacteristic
295C2×#123—21no—
296C2×#123—21no—
297C2×#121—21no—
298C2×#121—21no—
299C2×#121C2×D7311yescharacteristic
300C2×#121C2×D7311yescharacteristic
301C2×C4×D4C2×D7311yescharacteristic
302C2×C4×D4C2×D7311yescharacteristic
303C2×#117—731no—
304C2×#117C2×D7311yescharacteristic
305C22×C4⋊C4—731no—
306#481—1464no—
307#480—1464no—
308C2×#107—732no—
309C2×#106—732no—
310C2×#102—732no—
311C2×#102—732no—
312C2×#102—732no—
313C2×#102—732no—
314C2×C22⋊C8—731no—
315C2×C22⋊C8—731no—
316#389—732no—
317#389—732no—
318#385—1461no—
319#385—1461no—
320#384—1461no—
321#382—1461no—
322#340—732no—
323#340—732no—
324#336—732no—
325#336—732no—
326C22×D4—21no—
327C22×D4—21no—
328C22×D4—21no—
329C22×D4—21no—
330C22×D4C22×D7311yescharacteristic
331C22×D4C22×D7311yescharacteristic
332C23×C4—21no—
333C23×C4—21no—
334C23×C4—1461no—
335C23×C4C22×D7311yescharacteristic
336C22×C8—732no—
337C22×C8—732no—
338#127—22no—
339#127—22no—
340#125—22no—
341#125—22no—
342#123—28no—
343#123—28no—
344#121—28no—
345#121—28no—
346#121—22no—
347#121—22no—
348#121C22⋊D7314yes—
349#121C22⋊D7314yes—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354#117—22no—
355#117—22no—
356#117—1464no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—732no—
360C2×C4⋊C4—732no—
361C2×C4⋊C4—1461no—
362C2×C4⋊C4—1462no—
363C2×C4⋊C4—1462no—
364C2×C4⋊C4—1462no—
365C2×C4⋊C4—1462no—
366C2×C4⋊C4C22⋊D7311yescharacteristic
367C2×C4⋊C4C22⋊D7311yescharacteristic
368C2×C4⋊C4C22×D7311yescharacteristic
369C2×C4⋊C4C22×D7311yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1461no—
375C2×C22⋊C4C22⋊D7311yescharacteristic
376C2×C22⋊C4C22⋊D7311yescharacteristic
377C2×C42—1461no—
378C2×C42C22×D7311yescharacteristic
379C2×C42C22×D7311yescharacteristic
380#107—1464no—
381#106—1464no—
382#102—1464no—
383#102—1464no—
384#102—1464no—
385#102—1464no—
386C22⋊C8—1462no—
387C22⋊C8—1462no—
388#95—1462no—
389#95—1462no—
390C24—21no—
391C24—21no—
392C24C2×C22⋊D7311yescharacteristic
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4C2×C22⋊D7312yes—
408C2×D4C2×C22⋊D7312yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—2922no—
418C22×C4—5842no—
419C22×C4—1462no—
420C22×C4—1462no—
421C22×C4C2×C22⋊D7311yescharacteristic
422C22×C4C2×C22⋊D7311yescharacteristic
423C22×C4C2×C22⋊D7311yescharacteristic
424C22×C4C2×C22⋊D7311yescharacteristic
425C22×C4C2×C22⋊D7311yescharacteristic
426C22×C4C2×C22⋊D7312yes—
427C22×C4C2×C22⋊D7312yes—
428C22×C4C23×D7311yescharacteristic
429C2×C8—1462no—
430C2×C8—1462no—
431C2×C8—1462no—
432C2×C8—1462no—
433C2×C8—1462no—
434C2×C8—1462no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—2922no—
442C4⋊C4—5842no—
443C4⋊C4—5842no—
444C4⋊C4—734no—
445C4⋊C4—1462no—
446C4⋊C4—1462no—
447C4⋊C4—1462no—
448C4⋊C4C2×C22⋊D7312yes—
449C4⋊C4C2×C22⋊D7312yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—1464no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—1464no—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23#1455656012yes—
477C23C22×C22⋊D7311yescharacteristic
478D4—24no—
479D4—24no—
480D4—42no—
481D4—42no—
482D4—84no—
483D4—84no—
484D4—84no—
485D4—84no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4—2922no—
505C2×C4—2922no—
506C2×C4—2922no—
507C2×C4—2922no—
508C2×C4—2922no—
509C2×C4—5844no—
510C2×C4—1464no—
511C2×C4#1455656012yes—
512C2×C4C22×C22⋊D7311yescharacteristic
513C2×C4C22×C22⋊D7311yescharacteristic
514C2×C4#41×D7312yes—
515C2×C4#1455663812yes—
516C8—2922no—
517C8—2922no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22C2×#1455656011yescharacteristic
534C22#1798909411yescharacteristic
535C22#1798915811yescharacteristic
536C22#1798939012yes—
537C22#1798939812yes—
538C4—22no—
539C4—41no—
540C4—41no—
541C4—42no—
542C4—41no—
543C4—41no—
544C4—42no—
545C4—2924no—
546C4—5842no—
547C4#1798923612yes—
548C4#1798923612yes—
549C2—42no—
550C2—82no—
551C2—82no—
552C2#2261326211yescharacteristic
553C2#2261919911yescharacteristic
554C2#2261920811yescharacteristic
555C2#2262012412yes—
556C2#2262117812yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#540×C73C2
socleC23×C73C22×D4
hypercenterC2×#540D73
hypocenterC73#24069
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#17988903C2semidirect
2. C2×#17988898C2semidirect
3. #22612551C2semidirect
4. #22611266C2semidirect
5. #22611162C2semidirect
6. #22610254C2semidirect
7. #22610254C2semidirect
8. #22610192C2semidirect
9. #22610178C2semidirect
10. #22610002C2semidirect
290. C73#24069semidirect

Maximal subgroups

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

subgroupindex
C2×#179889032
C2×#179888982
#226125512
#226112662
#226111622
#226102542
#226101922
#226101782
#226100022
C2×#540×C732
#2406973