Finite Group Databasebeta

← #33095670

Subgroups — #33095670

Order 18688.

1166 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989118C211yesmaximal, characteristic
2C2×#17988637C211yesmaximal, characteristic, complemented
3C2×#17988452C211yesmaximal, characteristic, complemented
4#22610894C212yesmaximal, complemented
5#22610741C212yesmaximal, complemented
6#22610249C211yesmaximal, characteristic
7#22610238C211yesmaximal, characteristic, complemented
8#22610236C211yesmaximal, characteristic, complemented
9#22609945C212yesmaximal, complemented
10#22609939C212yesmaximal, complemented
11C73×#1874C211yesmaximal, characteristic
12#17989118—24no—
13#17989118C418yescomplemented
14C2×#14556584—21no—
15C2×#14556582C2211yescharacteristic
16C2×#14556578C2211yescharacteristic
17C2×#14556572—21no—
18C2×#14556570C2211yescharacteristic
19C2×#14556556C2211yescharacteristic
20C2×D4×Dic73—21no—
21C2×D4×Dic73C2211yescharacteristic
22C2×#14556547—21no—
23C22×C4⋊Dic73C2211yescharacteristic
24C22×C4×Dic73C2211yescharacteristic
25#17988637—24no—
26C2×#14556520C2212yes—
27C2×#14556516C2212yes—
28C2×#14556514C2212yes—
29C2×#14556512C2212yes—
30C2×C22⋊#10596831C2211yescharacteristic
31C2×C8×Dic73C2212yes—
32#17988452—21no—
33#17988452—21no—
34#17988452C2212yes—
35#17988452C2212yes—
36C2×#14556498—21no—
37C2×#14556498—21no—
38#17988350C2212yes—
39#17988349C2212yes—
40#17988349C2212yes—
41C2×#121×C73C2211yescharacteristic
42C22×C4⋊C4×C73C2211yescharacteristic
43C2×#102×C73C2212yes—
44C2×C22⋊C8×C73C2211yescharacteristic
45#340×C73C2212yes—
46C22×Dic146C2311yescharacteristic
47C23×Dic73—21no—
48C23×Dic73—21no—
49C23×Dic73D411yescharacteristic
50C23×Dic73D411yescharacteristic
51C22×#10596831C2312yes—
52#14556584—28no—
53#14556582—22no—
54#14556582C2×C414yescomplemented
55#14556578—22no—
56#14556578C2×C414yescomplemented
57#14556572—28no—
58#14556570—22no—
59#14556570C2×C414yes—
60#14556556—22no—
61#14556556C2×C414yescomplemented
62D4×Dic73—28no—
63D4×Dic73—22no—
64D4×Dic73C2×C414yescomplemented
65#14556547—28no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—42no—
69C2×C4⋊Dic73C2312yes—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×C22⋊Dic73—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73C2×C414yes—
80C2×C4×Dic73D411yescharacteristic
81C2×C4×Dic73D411yescharacteristic
82C2×C4×Dic73D414yes—
83C2×C4×Dic73C2311yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85#14556520—24no—
86#14556516—42no—
87#14556514—42no—
88#14556512—42no—
89C22⋊#10596831—22no—
90C8×Dic73—24no—
91#14556498—22no—
92#14556498—22no—
93#14556498—22no—
94#14556498—22no—
95C22×D4×C73—21no—
96C22×D4×C73C2311yescharacteristic
97C23×C4×C73—21no—
98C23×C4×C73C2311yescharacteristic
99C22×C8×C73C2312yes—
100#121×C73—22no—
101#121×C73C2×C414yes—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—42no—
104C2×C4⋊C4×C73C2312yes—
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C22⋊C4×C73—21no—
107#102×C73—42no—
108C22⋊C8×C73—22no—
109C2×Dic146—22no—
110C2×Dic146—22no—
111C2×Dic146—21no—
112C2×Dic146C22×C412yes—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—24no—
118C22×Dic73—22no—
119C22×Dic73—41no—
120C22×Dic73—41no—
121C22×Dic73—42no—
122C22×Dic73—42no—
123C22×Dic73—42no—
124C22×Dic73—42no—
125C22×Dic73—42no—
126C22×Dic73C22⋊C414yes—
127C22×Dic73C22⋊C414yes—
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C22×Dic73C2×D411yescharacteristic
131C22×Dic73C2×D411yescharacteristic
132C2×#10596831—22no—
133C2×#10596831—22no—
134C2×#10596831—22no—
135C4⋊Dic73—21no—
136C4⋊Dic73—24no—
137C4⋊Dic73—22no—
138C4⋊Dic73—22no—
139C4⋊Dic73—42no—
140C4⋊Dic73C22×C412yes—
141#11924007—24no—
142#11924007—24no—
143#11924007—24no—
144C22⋊Dic73—24no—
145C22⋊Dic73—24no—
146C22⋊Dic73—24no—
147C22⋊Dic73—24no—
148C4×Dic73—24no—
149C4×Dic73C22⋊C414yes—
150C4×Dic73C22⋊C414yes—
151C4×Dic73C22×C412yes—
152C4×Dic73C22×C412yes—
153C4×Dic73C2×D412yes—
154C4×Dic73C2×D412yes—
155C24×C73—21no—
156C24×C73C2×D411yescharacteristic
157C2×D4×C73—24no—
158C2×D4×C73—21no—
159C2×D4×C73—22no—
160C2×D4×C73—22no—
161C2×D4×C73—42no—
162C2×D4×C73—42no—
163C2×D4×C73C22×C412yes—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—21no—
167C22×C4×C73—22no—
168C22×C4×C73—42no—
169C22×C4×C73—42no—
170C22×C4×C73—42no—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D412yes—
174C22×C4×C73C2411yescharacteristic
175C2×C8×C73—22no—
176C2×C8×C73—22no—
177C2×C8×C73—22no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—22no—
181C4⋊C4×C73—42no—
182C4⋊C4×C73C22×C412yes—
183C22⋊C4×C73—24no—
184C23×C73—21no—
185C23×C73—21no—
186C23×C73—21no—
187C23×C73—21no—
188C23×C73—22no—
189C23×C73—41no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73C2×C22⋊C412yes—
196C23×C73#13512yes—
197C23×C73#13512yes—
198C23×C73C22×D411yescharacteristic
199D4×C73—24no—
200D4×C73—48no—
201D4×C73—42no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—22no—
207C2×C4×C73—41no—
208C2×C4×C73—41no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—84no—
213C2×C4×C73C2×C22⋊C412yes—
214C2×C4×C73C23×C411yescharacteristic
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73H(5,𝔽2)12yes—
217C2×C4×C732−1+412yes—
218C2×Dic73—21no—
219C2×Dic73—21no—
220C2×Dic73—21no—
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—28no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—44no—
230C2×Dic73—48no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73—84no—
239C2×Dic73C2×C22⋊C412yes—
240C2×Dic73C2×C22⋊C412yes—
241C2×Dic73C2×C22⋊C412yes—
242C2×Dic73C2×C22⋊C412yes—
243Dic146—24no—
244Dic146—42no—
245C8×C73—42no—
246#10596831—42no—
247C22×C73—22no—
248C22×C73—22no—
249C22×C73—22no—
250C22×C73—22no—
251C22×C73—22no—
252C22×C73—22no—
253C22×C73—42no—
254C22×C73—42no—
255C22×C73—42no—
256C22×C73—81no—
257C22×C73—81no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73#41614yes—
261C22×C73C22×C22⋊C411yescharacteristic
262C22×C73#53511yescharacteristic
263C22×C73#53611yescharacteristic
264C22×C73C2×#13511yescharacteristic
265C22×C73C2×#13511yescharacteristic
266C22×C73#57511yescharacteristic
267C22×C73#57511yescharacteristic
268C4×C73—22no—
269C4×C73—42no—
270C4×C73—82no—
271C4×C73#52012yes—
272C4×C73#53712yes—
273Dic73—22no—
274Dic73—22no—
275Dic73—22no—
276Dic73—22no—
277Dic73—42no—
278Dic73—42no—
279Dic73—82no—
280#21861—731nomaximal, Sylow
281C2×C73—24no—
282C2×C73—82no—
283C2×C73#231111yescharacteristic
284C2×C73C2×#41611yescharacteristic
285C2×C73#289611yescharacteristic
286C2×C73#321111yescharacteristic
287C2×C73#321111yescharacteristic
288C2×C73#323211yescharacteristic
289C2×C73#323211yescharacteristic
290C2×#529—731no—
291C2×#431—731no—
292C2×#382—731no—
293#1991—732no—
294#1958—732no—
295#1874—731no—
296#1874D7311yescharacteristic
297#1872—731no—
298#1872—731no—
299#1805—732no—
300#1804—732no—
301C73#2186111yescharacteristic, complemented, Sylow
302#529—738no—
303#529—1464no—
304C2×#126—1461no—
305C2×#125—731no—
306C2×#124—731no—
307C2×#123—1461no—
308C2×C22⋊Q8—731no—
309C2×#121C2×D7311yescharacteristic
310C2×C4×Q8—731no—
311C2×C4×D4—731no—
312C2×C4×D4—1461no—
313C2×#117—1461no—
314C22×C4⋊C4—731no—
315C22×C4⋊C4C2×D7311yescharacteristic
316C22×C42—731no—
317#431—1464no—
318C2×C4⋊C8—732no—
319C2×#103—732no—
320C2×#103—732no—
321C2×#102—732no—
322C2×#102C2×D7312yes—
323C2×C22⋊C8—731no—
324C2×C22⋊C8C2×D7311yescharacteristic
325C2×C4×C8—732no—
326#382—732no—
327#382—732no—
328#382—1461no—
329#382—1461no—
330C2×#95—1461no—
331C2×#95—1461no—
332#340—732no—
333#340—732no—
334#340—732no—
335#340C2×D7312yes—
336C22×Q8—731no—
337C22×D4—21no—
338C22×D4C22×D7311yescharacteristic
339C23×C4—21no—
340C23×C4—731no—
341C23×C4—731no—
342C23×C4—1461no—
343C23×C4—1461no—
344C23×C4C22×D7311yescharacteristic
345C22×C8—732no—
346C22×C8C22×D7312yes—
347#126—1468no—
348#125—734no—
349#125—1462no—
350#124—734no—
351#124—1462no—
352#123—1468no—
353C22⋊Q8—734no—
354C22⋊Q8—1462no—
355#121—22no—
356#121C4×D7314yes—
357C4×Q8—734no—
358C4×Q8—1462no—
359C4×D4—734no—
360C4×D4—1462no—
361C4×D4—1468no—
362#117—1468no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—2922no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—732no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4—1461no—
370C2×C4⋊C4—1461no—
371C2×C4⋊C4—1461no—
372C2×C4⋊C4—1461no—
373C2×C4⋊C4C22×D7312yes—
374C2×C4⋊C4C22×D7311yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1461no—
377C2×C22⋊C4—1461no—
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C42—734no—
381C2×C42—731no—
382C2×C42—731no—
383C2×C42—734no—
384C2×C42—731no—
385C2×C42—731no—
386C2×C42—1461no—
387C4⋊C8—1464no—
388#103—2922no—
389#103—2922no—
390#102—42no—
391#102—2922no—
392C22⋊C8—22no—
393C22⋊C8—1462no—
394C4×C8—1464no—
395#95—1462no—
396#95—1462no—
397#95—1462no—
398#95—1462no—
399C24—21no—
400C24D4×D7311yescharacteristic
401C2×Q8—732no—
402C2×Q8—1462no—
403C2×Q8—1462no—
404C2×Q8—1461no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×C4×D7312yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—2922no—
420C22×C4—2922no—
421C22×C4—2921no—
422C22×C4—2921no—
423C22×C4—2922no—
424C22×C4—2922no—
425C22×C4—2922no—
426C22×C4—734no—
427C22×C4—734no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—1462no—
433C22×C4—1464no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4C2×C4×D7312yes—
439C22×C4D4×D7311yescharacteristic
440C22×C4D4×D7312yes—
441C22×C4C23×D7311yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—1462no—
446C2×C8—1462no—
447C2×C8—1462no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—2922no—
453C4⋊C4—732no—
454C4⋊C4—1464no—
455C4⋊C4—1464no—
456C4⋊C4—1464no—
457C4⋊C4—1462no—
458C4⋊C4—1462no—
459C4⋊C4—1464no—
460C4⋊C4—1461no—
461C4⋊C4C2×C4×D7312yes—
462C22⋊C4—24no—
463C22⋊C4—1464no—
464C22⋊C4—1464no—
465C22⋊C4—1464no—
466C22⋊C4—1464no—
467C42—734no—
468C42—734no—
469C42—732no—
470C42—732no—
471C42—732no—
472C42—732no—
473C42—1464no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C22⋊C4×D7312yes—
486C23#1455661012yes—
487C23#1455661212yes—
488C23C2×D4×D7311yescharacteristic
489Q8—2922no—
490Q8—1464no—
491D4—24no—
492D4—48no—
493D4—42no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—84no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2921no—
509C2×C4—2928no—
510C2×C4—2922no—
511C2×C4—2922no—
512C2×C4—2922no—
513C2×C4—2924no—
514C2×C4—5844no—
515C2×C4—732no—
516C2×C4—732no—
517C2×C4—732no—
518C2×C4—732no—
519C2×C4—1462no—
520C2×C4—1462no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1468no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4C22⋊C4×D7312yes—
531C2×C4C22×C4×D7311yescharacteristic
532C2×C4C2×D4×D7311yescharacteristic
533C2×C4#1455663612yes—
534C2×C4#1455663812yes—
535C8—42no—
536C8—2922no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22#1798858014yes—
551C22C2×C22⋊C4×D7311yescharacteristic
552C22#1798915211yescharacteristic
553C22#1798915911yescharacteristic
554C22C2×#1455661011yescharacteristic
555C22C2×#1455661211yescharacteristic
556C22#1798938311yescharacteristic
557C22#1798938611yescharacteristic
558C4—22no—
559C4—42no—
560C4—82no—
561C4—2922no—
562C4—2922no—
563C4—5842no—
564C4—1462no—
565C4—1462no—
566C4—1462no—
567C4—1462no—
568C4#1798906612yes—
569C4#1798916412yes—
570C2—24no—
571C2—82no—
572C2#2261251811yescharacteristic
573C2C2×#1798858011yescharacteristic
574C2#2261844711yescharacteristic
575C2#2262089311yescharacteristic
576C2#2262089511yescharacteristic
577C2#2262109611yescharacteristic
578C2#2262109811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1874C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
cosocleC22×C4×C73C24
hypercenter#1874D73
hypocenterC73#21861
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#17988637C2semidirect
3. C2×#17988452C2semidirect
4. #22610894C2semidirect
5. #22610741C2semidirect
7. #22610238C2semidirect
8. #22610236C2semidirect
9. #22609945C2semidirect
10. #22609939C2semidirect
13. #17989118C4semidirect
54. #14556582C2×C4semidirect
56. #14556578C2×C4semidirect
61. #14556556C2×C4semidirect
64. D4×Dic73C2×C4semidirect
301. C73#21861semidirect

Maximal subgroups

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

subgroupindex
C2×#179891182
C2×#179886372
C2×#179884522
#226108942
#226107412
#226102492
#226102382
#226102362
#226099452
#226099392
C73×#18742
#2186173