Finite Group Databasebeta

← #33258724

Subgroups — #33258724

Order 18688.

2042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618237C214yesmaximal, complemented
2#22615903C214yesmaximal, complemented
3#22614434C214yesmaximal, complemented
4#22614432C214yesmaximal, complemented
5#22614074C214yesmaximal, complemented
6#22614066C214yesmaximal, complemented
7#22612545C211yesmaximal, characteristic
8#22612543C211yesmaximal, characteristic
9#22612543C211yesmaximal, characteristic
10#22612543C211yesmaximal, characteristic
11C2×#17988460C211yesmaximal, characteristic
12C2×C4×C22⋊Dic73C211yesmaximal, characteristic
13C2×#385×C73C211yesmaximal, characteristic
14C2×#14556581C2214yescomplemented
15C22×#11924007C2212yes—
16C22×C22⋊Dic73C2211yescharacteristic
17C22×C22⋊Dic73C2211yescharacteristic
18C22×C22⋊Dic73C2211yescharacteristic
19C22×C22⋊Dic73C2211yescharacteristic
20C22×C4×Dic73C2212yes—
21#17988528C2214yescomplemented
22#17988528C2214yescomplemented
23#17988527C2214yescomplemented
24#17988525C2214yescomplemented
25#17988521C2214yescomplemented
26#17988520C2214yescomplemented
27#17988518C2214yescomplemented
28#17988517C2214yescomplemented
29#17988484C2214yescomplemented
30#17988482C2214yescomplemented
31#17988473C2214yescomplemented
32#17988472C2214yescomplemented
33#17988469C2214yescomplemented
34#17988468C2214yescomplemented
35#17988462C2214yes—
36#17988462C2214yes—
37#17988460C2212yes—
38#17988460C2214yes—
39#17988460C2214yes—
40#17988460C2214yes—
41#17988460C2212yes—
42#17988460C2212yes—
43#17988460C2212yes—
44#17988459C2212yes—
45#17988459C2214yes—
46#17988459C2214yes—
47#17988459C2212yes—
48#17988451C2214yescomplemented
49#17988447C2212yes—
50#17988447C2212yes—
51#17988445C2212yes—
52#17988445C2214yes—
53#17988445C2214yes—
54#17988445C2212yes—
55C4×C22⋊Dic73C2214yes—
56C4×C22⋊Dic73C2214yes—
57C2×#14556498C2212yes—
58C2×#14556498C2212yes—
59C24×C4×C73C2211yescharacteristic
60C22×C4⋊C4×C73C2212yes—
61C22×C22⋊C4×C73C2211yescharacteristic
62C22×C22⋊C4×C73C2211yescharacteristic
63#385×C73C2214yes—
64#385×C73C2212yes—
65#385×C73C2212yes—
66C2×#95×C73C2212yes—
67C23×Dic73C2311yescharacteristic
68C23×Dic73C2312yes—
69C23×Dic73C2312yes—
70C23×Dic73C2311yescharacteristic
71C23×Dic73C2312yes—
72#14556581—48no—
73C2×C4⋊Dic73C2314yes—
74C2×C4⋊Dic73C2314yes—
75C2×#11924007—44no—
76C2×#11924007C2314yes—
77C2×#11924007C2314yes—
78C2×#11924007C2314yes—
79C2×#11924007C2314yes—
80C2×#11924007C2314yes—
81C2×#11924007C2314yes—
82C2×#11924007C2314yes—
83C2×#11924007C2314yes—
84C2×C22⋊Dic73—24no—
85C2×C22⋊Dic73—24no—
86C2×C22⋊Dic73—24no—
87C2×C22⋊Dic73Q818yes—
88C2×C22⋊Dic73C2312yes—
89C2×C22⋊Dic73C2312yes—
90C2×C22⋊Dic73C2312yes—
91C2×C22⋊Dic73C2312yes—
92C2×C22⋊Dic73C2312yes—
93C2×C22⋊Dic73C2312yes—
94C2×C22⋊Dic73C2312yes—
95C2×C22⋊Dic73C2312yes—
96C2×C4×Dic73—44no—
97C2×C4×Dic73C2314yes—
98C2×C4×Dic73C2314yes—
99#14556498C2314yes—
100#14556498C2314yes—
101#14556498C2314yes—
102#14556498C2314yes—
103#14556498C2314yes—
104#14556498C2314yes—
105#14556498C2314yes—
106#14556498C2314yes—
107#14556498C2314yes—
108#14556498C2314yes—
109#14556498C2314yes—
110#14556498C2314yes—
111C25×C73C2311yescharacteristic
112C23×C4×C73—24no—
113C23×C4×C73—44no—
114C23×C4×C73C2311yescharacteristic
115C23×C4×C73C2312yes—
116C23×C4×C73C2312yes—
117C23×C4×C73C2311yescharacteristic
118C23×C4×C73C2311yescharacteristic
119C23×C4×C73C2311yescharacteristic
120C23×C4×C73C2312yes—
121C2×C4⋊C4×C73—44no—
122C2×C4⋊C4×C73C2314yes—
123C2×C4⋊C4×C73C2314yes—
124C2×C22⋊C4×C73—24no—
125C2×C22⋊C4×C73—24no—
126C2×C22⋊C4×C73C2312yes—
127C2×C22⋊C4×C73C2312yes—
128C2×C22⋊C4×C73C2312yes—
129C2×C22⋊C4×C73C2312yes—
130#95×C73C2314yes—
131#95×C73C2314yes—
132C22×Dic73—24no—
133C22×Dic73—24no—
134C22×Dic73—22no—
135C22×Dic73—22no—
136C22×Dic73—42no—
137C22×Dic73—42no—
138C22×Dic73—42no—
139C22×Dic73—42no—
140C22×Dic73—42no—
141C22×Dic73—42no—
142C22×Dic73—42no—
143C22×Dic73—42no—
144C22×Dic73C2×Q818yes—
145C22×Dic73C2412yes—
146C22×Dic73C2414yes—
147C22×Dic73C2414yes—
148C22×Dic73C2412yes—
149C22×Dic73C2414yes—
150C4⋊Dic73—44no—
151C4⋊Dic73—44no—
152#11924007—28no—
153#11924007—44no—
154#11924007—44no—
155#11924007—44no—
156#11924007—44no—
157#11924007—44no—
158#11924007—44no—
159#11924007—44no—
160C22⋊Dic73—24no—
161C22⋊Dic73—24no—
162C22⋊Dic73—24no—
163C22⋊Dic73—24no—
164C22⋊Dic73—24no—
165C22⋊Dic73—24no—
166C22⋊Dic73C2×Q818yes—
167C22⋊Dic73C2×Q818yes—
168C4×Dic73—28no—
169C4×Dic73—44no—
170C24×C73—42no—
171C24×C73—42no—
172C24×C73C2×Q814yes—
173C24×C73#4114yes—
174C24×C73#4114yes—
175C24×C73C2411yescharacteristic
176C24×C73C2411yescharacteristic
177C24×C73C2411yescharacteristic
178C22×C4×C73—22no—
179C22×C4×C73—24no—
180C22×C4×C73—24no—
181C22×C4×C73—24no—
182C22×C4×C73—24no—
183C22×C4×C73—22no—
184C22×C4×C73—22no—
185C22×C4×C73—22no—
186C22×C4×C73—24no—
187C22×C4×C73—24no—
188C22×C4×C73—42no—
189C22×C4×C73—42no—
190C22×C4×C73—42no—
191C22×C4×C73—42no—
192C22×C4×C73—42no—
193C22×C4×C73—42no—
194C22×C4×C73—42no—
195C22×C4×C73—416no—
196C22×C4×C73C2412yes—
197C22×C4×C73C2414yes—
198C22×C4×C73C2412yes—
199C22×C4×C73C2412yes—
200C22×C4×C73C2412yes—
201C4⋊C4×C73—28no—
202C4⋊C4×C73—44no—
203C22⋊C4×C73—24no—
204C22⋊C4×C73—24no—
205C22⋊C4×C73—24no—
206C22⋊C4×C73—24no—
207C23×C73—22no—
208C23×C73—22no—
209C23×C73—22no—
210C23×C73—22no—
211C23×C73—22no—
212C23×C73—22no—
213C23×C73—44no—
214C23×C73—48no—
215C23×C73—44no—
216C23×C73—48no—
217C23×C73#125116yescomplemented
218C23×C73C22×Q812yes—
219C23×C73C22×Q812yes—
220C23×C73C22×Q812yes—
221C23×C73C2×#4112yes—
222C23×C73C2×#4112yes—
223C23×C73C2×#4112yes—
224C23×C73C2×#4112yes—
225C23×C73C2×#4112yes—
226C23×C73C2×#4112yes—
227C23×C73C2511yescharacteristic
228C2×C4×C73—28no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—24no—
234C2×C4×C73—22no—
235C2×C4×C73—24no—
236C2×C4×C73—24no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—24no—
240C2×C4×C73—24no—
241C2×C4×C73—48no—
242C2×C4×C73—48no—
243C2×C4×C73—48no—
244C2×C4×C73—44no—
245C2×C4×C73—82no—
246C2×C4×C73—82no—
247C2×C4×C73H(5,𝔽2)14yes—
248C2×Dic73—24no—
249C2×Dic73—24no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—24no—
253C2×Dic73—24no—
254C2×Dic73—24no—
255C2×Dic73—24no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—24no—
260C2×Dic73—48no—
261C2×Dic73—48no—
262C2×Dic73—48no—
263C2×Dic73—82no—
264C2×Dic73—82no—
265C2×Dic73C22×Q818yes—
266C22×C73—24no—
267C22×C73—24no—
268C22×C73—24no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—24no—
276C22×C73—24no—
277C22×C73—24no—
278C22×C73—416no—
279C22×C73C2×#12514yes—
280C22×C73C2×#12514yes—
281C22×C73C2×#12514yes—
282C22×C73C23×Q811yescharacteristic
283C22×C73C22×#4111yescharacteristic
284C22×C73C22×#4111yescharacteristic
285C22×C73C2×H(5,𝔽2)12yes—
286C22×C73#58512yes—
287C4×C73—24no—
288C4×C73—44no—
289C4×C73—42no—
290C4×C73—42no—
291C4×C73—42no—
292Dic73—28no—
293Dic73—44no—
294Dic73—44no—
295Dic73—42no—
296Dic73—42no—
297#34994—731nomaximal, Sylow
298C2×C73—24no—
299C2×C73—24no—
300C2×C73—24no—
301C2×C73C22×#12511yescharacteristic
302C2×C73#345912yes—
303C2×C73#349011yescharacteristic
304C2×C73#349111yescharacteristic
305C2×C73#350112yes—
306#2851—734no—
307#2646—734no—
308#2505—734no—
309#2505—734no—
310#2473—734no—
311#2473—734no—
312#2315—731no—
313#2315—731no—
314#2315—731no—
315#2315—731no—
316C2×#385—731no—
317C2×#385D7311yescharacteristic
318C2×C4×C22⋊C4—731no—
319C73#3499411yescharacteristic, complemented, Sylow
320C24×C4C2×D7311yescharacteristic
321C2×#125—734no—
322C22×C4⋊C4—732no—
323C22×C4⋊C4C2×D7312yes—
324C22×C22⋊C4—731no—
325C22×C22⋊C4—731no—
326C22×C22⋊C4—731no—
327C22×C22⋊C4—731no—
328C22×C22⋊C4C2×D7311yescharacteristic
329C22×C22⋊C4C2×D7311yescharacteristic
330C22×C42—732no—
331#400—734no—
332#400—734no—
333#400—734no—
334#400—734no—
335#398—734no—
336#398—734no—
337#398—734no—
338#398—734no—
339#389—734no—
340#389—734no—
341#387—734no—
342#387—734no—
343#387—734no—
344#387—734no—
345#385—732no—
346#385—734no—
347#385—734no—
348#385—732no—
349#385—732no—
350#385—732no—
351#385—734no—
352#385—734no—
353#385—734no—
354#385—732no—
355#385—734no—
356#385—732no—
357#385—734no—
358#385C2×D7312yes—
359#385C2×D7312yes—
360#385C2×D7314yes—
361#382—734no—
362#380—732no—
363#380—732no—
364#380—734no—
365#380—734no—
366#380—732no—
367#380—732no—
368C4×C22⋊C4—734no—
369C4×C22⋊C4—734no—
370C2×#95—732no—
371C2×#95—732no—
372C2×#95C2×D7312yes—
373C25C22×D7311yescharacteristic
374C23×C4—24no—
375C23×C4—44no—
376C23×C4—732no—
377C23×C4—731no—
378C23×C4—732no—
379C23×C4—731no—
380C23×C4—732no—
381C23×C4C22×D7311yescharacteristic
382C23×C4C22×D7312yes—
383C23×C4C22×D7312yes—
384C23×C4C22×D7311yescharacteristic
385C23×C4C22×D7312yes—
386C23×C4C22×D7311yescharacteristic
387C23×C4C22×D7311yescharacteristic
388#125—2928no—
389C2×C4⋊C4—44no—
390C2×C4⋊C4—2924no—
391C2×C4⋊C4—734no—
392C2×C4⋊C4—734no—
393C2×C4⋊C4—734no—
394C2×C4⋊C4—734no—
395C2×C4⋊C4—734no—
396C2×C4⋊C4—734no—
397C2×C4⋊C4—734no—
398C2×C4⋊C4—734no—
399C2×C4⋊C4—734no—
400C2×C4⋊C4—734no—
401C2×C4⋊C4C22×D7314yes—
402C2×C4⋊C4C22×D7314yes—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—738no—
406C2×C22⋊C4—732no—
407C2×C22⋊C4—732no—
408C2×C22⋊C4—732no—
409C2×C22⋊C4—732no—
410C2×C22⋊C4—732no—
411C2×C22⋊C4—732no—
412C2×C22⋊C4—732no—
413C2×C22⋊C4—732no—
414C2×C22⋊C4—1464no—
415C2×C22⋊C4—1464no—
416C2×C22⋊C4—1464no—
417C2×C22⋊C4C22×D7312yes—
418C2×C22⋊C4C22×D7312yes—
419C2×C22⋊C4C22×D7312yes—
420C2×C22⋊C4C22×D7312yes—
421C2×C42—2924no—
422C2×C42—734no—
423C2×C42—734no—
424#95—734no—
425#95—734no—
426#95—734no—
427#95—734no—
428#95—734no—
429#95—734no—
430#95—734no—
431#95—734no—
432#95—734no—
433#95—734no—
434#95—734no—
435#95—734no—
436#95C22×D7314yes—
437#95C22×D7314yes—
438C24—42no—
439C24—42no—
440C24Q8×D7314yes—
441C24#1192402714yes—
442C24#1192402814yes—
443C24C23×D7311yescharacteristic
444C24C23×D7311yescharacteristic
445C24C23×D7311yescharacteristic
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—416no—
464C22×C4—2922no—
465C22×C4—2922no—
466C22×C4—2922no—
467C22×C4—2922no—
468C22×C4—2922no—
469C22×C4—2922no—
470C22×C4—2922no—
471C22×C4—2922no—
472C22×C4—738no—
473C22×C4—734no—
474C22×C4—732no—
475C22×C4—734no—
476C22×C4—734no—
477C22×C4—732no—
478C22×C4—1464no—
479C22×C4—1464no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4C23×D7312yes—
483C22×C4C23×D7312yes—
484C22×C4C23×D7314yes—
485C22×C4C23×D7312yes—
486C22×C4C23×D7312yes—
487C4⋊C4—28no—
488C4⋊C4—44no—
489C4⋊C4—2924no—
490C4⋊C4—2924no—
491C4⋊C4—2924no—
492C4⋊C4—2924no—
493C4⋊C4—2924no—
494C4⋊C4—2924no—
495C4⋊C4—2924no—
496C4⋊C4—2924no—
497C4⋊C4—2924no—
498C4⋊C4—1468no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—738no—
504C22⋊C4—738no—
505C22⋊C4—1464no—
506C22⋊C4—1464no—
507C22⋊C4—1464no—
508C22⋊C4—1464no—
509C22⋊C4—1464no—
510C22⋊C4—1464no—
511C42—2924no—
512C42—1468no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—44no—
520C23—48no—
521C23—44no—
522C23—48no—
523C23#14556581116yescomplemented
524C23C2×Q8×D7312yes—
525C23C2×Q8×D7312yes—
526C23C2×Q8×D7312yes—
527C23C2×#1192402712yes—
528C23C2×#1192402712yes—
529C23C2×#1192402712yes—
530C23C2×#1192402812yes—
531C23C2×#1192402812yes—
532C23C2×#1192402812yes—
533C23C24×D7311yescharacteristic
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—2928no—
554C2×C4—2928no—
555C2×C4—2928no—
556C2×C4—5842no—
557C2×C4—5842no—
558C2×C4—738no—
559C2×C4—1462no—
560C2×C4—1462no—
561C2×C4—1464no—
562C2×C4—1464no—
563C2×C4—1464no—
564C2×C4—1464no—
565C2×C4—1464no—
566C2×C4—1464no—
567C2×C4—1464no—
568C2×C4—1464no—
569C2×C4—1464no—
570C2×C4—1464no—
571C2×C4#1455663614yes—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—416no—
585C22C2×#1455658114yes—
586C22C2×#1455658114yes—
587C22C2×#1455658114yes—
588C22C22×Q8×D7311yescharacteristic
589C22C22×#1192402711yescharacteristic
590C22C22×#1192402811yescharacteristic
591C22C2×#1455663612yes—
592C22#1798942312yes—
593C4—24no—
594C4—44no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—2924no—
599C4—2924no—
600C4—2922no—
601C4—2922no—
602C4—1468no—
603C2—24no—
604C2—24no—
605C2—24no—
606C2C22×#1455658111yescharacteristic
607C2#2262334012yes—
608C2#2262361811yescharacteristic
609C2#2262362711yescharacteristic
610C2#2262372712yes—
subgroupG/H
center, FrattiniC23C24×D73
commutator, socle, cosocleC23×C73C25
FittingC2×#385×C73C2
prime radicalC25×C73C23
hypercenterC2×#385D73
hypocenterC73#34994
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. #22618237C2semidirect
2. #22615903C2semidirect
3. #22614434C2semidirect
4. #22614432C2semidirect
5. #22614074C2semidirect
6. #22614066C2semidirect
14. C2×#14556581C22semidirect
21. #17988528C22semidirect
22. #17988528C22semidirect
23. #17988527C22semidirect
24. #17988525C22semidirect
25. #17988521C22semidirect
26. #17988520C22semidirect
27. #17988518C22semidirect
28. #17988517C22semidirect
29. #17988484C22semidirect
30. #17988482C22semidirect
31. #17988473C22semidirect
32. #17988472C22semidirect
33. #17988469C22semidirect
34. #17988468C22semidirect
48. #17988451C22semidirect
217. C23×C73#125semidirect
319. C73#34994semidirect
523. C23#14556581semidirect

Maximal subgroups

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

subgroupindex
#226182372
#226159032
#226144342
#226144322
#226140742
#226140662
#226125452
#226125432
C2×#179884602
C2×C4×C22⋊Dic732
C2×#385×C732
#3499473