Finite Group Databasebeta

← #33013916

Subgroups — #33013916

Order 18688.

1750 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988486C212yesmaximal, complemented
2C2×#17988480C212yesmaximal
3C2×#17988447C212yesmaximal
4#22608196C212yesmaximal, complemented
5#22608184C212yesmaximal, complemented
6#22608176C211yesmaximal, characteristic
7#22608176C211yesmaximal, characteristic
8#22608165C212yesmaximal, complemented
9C2×#385×C73C211yesmaximal, characteristic
10C22×C4⋊Dic73C2212yes—
11C22×C4⋊Dic73C2211yescharacteristic
12C22×C4⋊Dic73C2212yes—
13C22×C4⋊Dic73C2212yes—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×C22⋊Dic73C2212yes—
16C22×C22⋊Dic73C2212yes—
17C22×C4×Dic73C2212yes—
18#17988486—28no—
19#17988480—28no—
20#17988447—28no—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C24×C4×C73C2211yescharacteristic
30C22×C4⋊C4×C73C2212yes—
31C22×C22⋊C4×C73C2211yescharacteristic
32C22×C22⋊C4×C73C2211yescharacteristic
33#385×C73C418yescomplemented
34C2×#95×C73C2212yes—
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C23×Dic73C2312yes—
38C23×Dic73C2312yes—
39C2×C4⋊Dic73—24no—
40C2×C4⋊Dic73—24no—
41C2×C4⋊Dic73—24no—
42C2×C4⋊Dic73—24no—
43C2×C4⋊Dic73—24no—
44C2×C4⋊Dic73—24no—
45C2×C4⋊Dic73—24no—
46C2×C4⋊Dic73—42no—
47C2×C4⋊Dic73—42no—
48C2×C4⋊Dic73—42no—
49C2×C4⋊Dic73—41no—
50C2×C4⋊Dic73—41no—
51C2×C4⋊Dic73—42no—
52C2×C22⋊Dic73—24no—
53C2×C22⋊Dic73—24no—
54C2×C22⋊Dic73—24no—
55C2×C22⋊Dic73—24no—
56C2×C22⋊Dic73—42no—
57C2×C22⋊Dic73—42no—
58C2×C4×Dic73—24no—
59C2×C4×Dic73—24no—
60C2×C4×Dic73—42no—
61#14556498—24no—
62#14556498—24no—
63#14556498—24no—
64#14556498—24no—
65#14556498—41no—
66#14556498—41no—
67#14556498—42no—
68#14556498D418yes—
69C25×C73C2311yescharacteristic
70C23×C4×C73—24no—
71C23×C4×C73—24no—
72C23×C4×C73—41no—
73C23×C4×C73—41no—
74C23×C4×C73C2×C414yes—
75C23×C4×C73C2311yescharacteristic
76C23×C4×C73C2311yescharacteristic
77C23×C4×C73C2312yes—
78C23×C4×C73C2312yes—
79C2×C4⋊C4×C73—24no—
80C2×C4⋊C4×C73—24no—
81C2×C4⋊C4×C73C2×C418yes—
82C2×C22⋊C4×C73—41no—
83C2×C22⋊C4×C73—41no—
84C2×C22⋊C4×C73C2×C414yes—
85C2×C22⋊C4×C73C2×C414yes—
86C2×C22⋊C4×C73D414yes—
87C2×C22⋊C4×C73Q814yes—
88#95×C73C2×C418yes—
89C22×Dic73—22no—
90C22×Dic73—22no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—22no—
95C22×Dic73—22no—
96C22×Dic73—22no—
97C22×Dic73—22no—
98C22×Dic73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73C2×D414yes—
116C22×Dic73C2×D414yes—
117C4⋊Dic73—44no—
118C4⋊Dic73—48no—
119C4⋊Dic73—48no—
120C4⋊Dic73—48no—
121C4⋊Dic73—44no—
122C22⋊Dic73—48no—
123C22⋊Dic73—48no—
124C4×Dic73—48no—
125C24×C73—22no—
126C24×C73—22no—
127C24×C73—21no—
128C24×C73—21no—
129C24×C73—22no—
130C24×C73—22no—
131C24×C73—21no—
132C24×C73—21no—
133C24×C73C22×C412yes—
134C24×C73C2×D412yes—
135C24×C73C2×Q812yes—
136C24×C73C2411yescharacteristic
137C22×C4×C73—22no—
138C22×C4×C73—22no—
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—216no—
154C22×C4×C73—44no—
155C22×C4×C73—44no—
156C22×C4×C73—48no—
157C22×C4×C73—44no—
158C22×C4×C73C22×C412yes—
159C22×C4×C73C22×C412yes—
160C22×C4×C73C22×C414yes—
161C22×C4×C73C22×C414yes—
162C22×C4×C73C2×D414yes—
163C22×C4×C73C2×D414yes—
164C22×C4×C73C2×Q814yes—
165C22×C4×C73#4114yes—
166C4⋊C4×C73—216no—
167C22⋊C4×C73—44no—
168C22⋊C4×C73C4⋊C4116yescomplemented
169C23×C73—24no—
170C23×C73—22no—
171C23×C73—22no—
172C23×C73—24no—
173C23×C73—22no—
174C23×C73—22no—
175C23×C73—22no—
176C23×C73—22no—
177C23×C73—24no—
178C23×C73—24no—
179C23×C73—24no—
180C23×C73—24no—
181C23×C73—42no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73—41no—
186C23×C73—42no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73—42no—
190C23×C73—41no—
191C23×C73C2×C4⋊C414yes—
192C23×C73C23×C411yescharacteristic
193C23×C73C22×D412yes—
194C23×C73C22×D411yescharacteristic
195C23×C73C22×Q811yescharacteristic
196C23×C73C2×#4112yes—
197C23×C73H(5,𝔽2)11yescharacteristic
198C23×C73H(5,𝔽2)11yescharacteristic
199C23×C73H(5,𝔽2)12yes—
200C23×C732−1+412yes—
201C23×C732−1+411yescharacteristic
202C23×C732−1+411yescharacteristic
203C2×C4×C73—24no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—24no—
207C2×C4×C73—24no—
208C2×C4×C73—24no—
209C2×C4×C73—24no—
210C2×C4×C73—24no—
211C2×C4×C73—24no—
212C2×C4×C73—24no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—24no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—24no—
219C2×C4×C73—416no—
220C2×C4×C73—41no—
221C2×C4×C73—42no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—42no—
225C2×C4×C73—41no—
226C2×C4×C73C2×C4⋊C418yes—
227C2×C4×C73C4×D418yes—
228C2×Dic73—24no—
229C2×Dic73—24no—
230C2×Dic73—24no—
231C2×Dic73—24no—
232C2×Dic73—24no—
233C2×Dic73—24no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C22×C73—24no—
257C22×C73—28no—
258C22×C73—44no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—44no—
262C22×C73—42no—
263C22×C73—42no—
264C22×C73—44no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73C22×C4⋊C411yescharacteristic
268C22×C73C2×C4×D412yes—
269C22×C73#51811yescharacteristic
270C22×C73#51911yescharacteristic
271C22×C73#52012yes—
272C22×C73#53612yes—
273C22×C73#53712yes—
274C22×C73#53711yescharacteristic
275C22×C73#53711yescharacteristic
276C22×C73#54112yes—
277C22×C73#54212yes—
278C22×C73#54411yescharacteristic
279C22×C73#54411yescharacteristic
280C22×C73D4211yescharacteristic
281C22×C73D4212yes—
282C22×C73#54612yes—
283C22×C73#54712yes—
284C22×C73D4×Q812yes—
285C22×C73#55111yescharacteristic
286C22×C73#55412yes—
287C22×C73#56011yescharacteristic
288C22×C73#56212yes—
289C22×C73#56311yescharacteristic
290C4×C73—28no—
291C4×C73—28no—
292C4×C73—42no—
293C4×C73—42no—
294Dic73—44no—
295Dic73—44no—
296Dic73—82no—
297Dic73—82no—
298#14402—731nomaximal, Sylow
299C2×C73—44no—
300C2×C73#231611yescharacteristic
301C2×C73#232312yes—
302C2×C73D4×C4⋊C412yes—
303C2×C73#235811yescharacteristic
304C2×C73#237611yescharacteristic
305C2×C73#267212yes—
306C2×C73#267511yescharacteristic
307C2×C73#267711yescharacteristic
308C2×C73#270412yes—
309C2×C73#272211yescharacteristic
310C2×C73#274211yescharacteristic
311C2×#389—732no—
312C2×#388—732no—
313C2×#385D7311yescharacteristic
314C2×#380—732no—
315#1444—732no—
316#1441—732no—
317#1439—731no—
318#1439—731no—
319#1436—732no—
320C73#1440211yescharacteristic, complemented, Sylow
321C24×C4C2×D7311yescharacteristic
322C22×C4⋊C4—731no—
323C22×C4⋊C4—732no—
324C22×C4⋊C4—731no—
325C22×C4⋊C4—732no—
326C22×C4⋊C4—732no—
327C22×C4⋊C4C2×D7312yes—
328C22×C22⋊C4—732no—
329C22×C22⋊C4—732no—
330C22×C22⋊C4C2×D7311yescharacteristic
331C22×C22⋊C4C2×D7311yescharacteristic
332C22×C42—732no—
333#389—1468no—
334#388—1468no—
335#385Dic7318yescomplemented
336#380—1468no—
337C2×#95—731no—
338C2×#95—732no—
339C2×#95—731no—
340C2×#95—732no—
341C2×#95—732no—
342C2×#95—732no—
343C2×#95—732no—
344C2×#95—732no—
345C2×#95C2×D7312yes—
346C25C22×D7311yescharacteristic
347C23×C4—24no—
348C23×C4—24no—
349C23×C4—41no—
350C23×C4—41no—
351C23×C4—732no—
352C23×C4—732no—
353C23×C4—732no—
354C23×C4—732no—
355C23×C4C2×Dic7314yes—
356C23×C4C22×D7311yescharacteristic
357C23×C4C22×D7311yescharacteristic
358C23×C4C22×D7312yes—
359C23×C4C22×D7312yes—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—2922no—
363C2×C4⋊C4—2922no—
364C2×C4⋊C4—2922no—
365C2×C4⋊C4—2921no—
366C2×C4⋊C4—2922no—
367C2×C4⋊C4—2921no—
368C2×C4⋊C4—1464no—
369C2×C4⋊C4—1464no—
370C2×C4⋊C4—1464no—
371C2×C4⋊C4—1464no—
372C2×C4⋊C4—1464no—
373C2×C4⋊C4—1464no—
374C2×C4⋊C4—1464no—
375C2×C4⋊C4C2×Dic7318yes—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—2922no—
379C2×C22⋊C4—2922no—
380C2×C22⋊C4—1464no—
381C2×C22⋊C4—1464no—
382C2×C22⋊C4—1464no—
383C2×C22⋊C4—1464no—
384C2×C22⋊C4Dic14614yes—
385C2×C22⋊C4D29214yes—
386C2×C22⋊C4C2×Dic7314yes—
387C2×C22⋊C4C2×Dic7314yes—
388C2×C42—2922no—
389C2×C42—1464no—
390C2×C42—1464no—
391#95—2921no—
392#95—2921no—
393#95—2922no—
394#95—738no—
395#95—1464no—
396#95—1464no—
397#95—1464no—
398#95—1464no—
399#95C2×Dic7318yes—
400C24—22no—
401C24—22no—
402C24—21no—
403C24—21no—
404C24—22no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24C22×Dic7312yes—
409C24C2×D29212yes—
410C24C2×Dic14612yes—
411C24C23×D7311yescharacteristic
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—734no—
434C22×C4—734no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1462no—
459C22×C4—1462no—
460C22×C4—1462no—
461C22×C4C22×Dic7312yes—
462C22×C4C22×Dic7312yes—
463C22×C4C22×Dic7314yes—
464C22×C4C22×Dic7314yes—
465C22×C4D4×D7314yes—
466C22×C4C2×D29214yes—
467C22×C4C2×Dic14614yes—
468C22×C4#1192402714yes—
469C4⋊C4—216no—
470C4⋊C4—2928no—
471C4⋊C4—2928no—
472C4⋊C4—2928no—
473C4⋊C4—2924no—
474C4⋊C4—2924no—
475C22⋊C4—44no—
476C22⋊C4—2928no—
477C22⋊C4—2928no—
478C22⋊C4C4⋊Dic73116yescomplemented
479C42—2928no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23C2×C4⋊Dic7314yes—
503C23C23×Dic7311yescharacteristic
504C23C2×D4×D7312yes—
505C23C22×D29211yescharacteristic
506C23C22×Dic14611yescharacteristic
507C23C2×#1192402712yes—
508C23#1455663611yescharacteristic
509C23#1455663611yescharacteristic
510C23#1455663712yes—
511C23#1455663812yes—
512C23#1455663911yescharacteristic
513C23#1455663911yescharacteristic
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—416no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—2922no—
540C2×C4—2922no—
541C2×C4—2922no—
542C2×C4—2922no—
543C2×C4—2922no—
544C2×C4—2922no—
545C2×C4—2922no—
546C2×C4—2922no—
547C2×C4—2922no—
548C2×C4—2922no—
549C2×C4—2922no—
550C2×C4—2922no—
551C2×C4—2922no—
552C2×C4—2922no—
553C2×C4—2922no—
554C2×C4—2922no—
555C2×C4—2922no—
556C2×C4—2922no—
557C2×C4—2922no—
558C2×C4—2922no—
559C2×C4—1464no—
560C2×C4—1464no—
561C2×C4—1464no—
562C2×C4—1464no—
563C2×C4—1464no—
564C2×C4—1464no—
565C2×C4C2×C4⋊Dic7318yes—
566C2×C4D4×Dic7318yes—
567C22—24no—
568C22—28no—
569C22—42no—
570C22—44no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22C22×C4⋊Dic7311yescharacteristic
579C22C2×D4×Dic7312yes—
580C22#1798906211yescharacteristic
581C22#1798906511yescharacteristic
582C22#1798907012yes—
583C22#1798916012yes—
584C22#1798916712yes—
585C22#1798916811yescharacteristic
586C22#1798916811yescharacteristic
587C22#1798919712yes—
588C22#1798921012yes—
589C22#1798921811yescharacteristic
590C22#1798921811yescharacteristic
591C22#1798921911yescharacteristic
592C22D4×D29212yes—
593C22#1798922412yes—
594C22#1798924112yes—
595C22D4×Dic14612yes—
596C22#1798926811yescharacteristic
597C22#1798929412yes—
598C22#1798932711yescharacteristic
599C22#1798933412yes—
600C22#1798934011yescharacteristic
601C4—28no—
602C4—28no—
603C4—42no—
604C4—42no—
605C4—2924no—
606C4—2924no—
607C4—5842no—
608C4—5842no—
609C2—44no—
610C2#2261254911yescharacteristic
611C2#2261261512yes—
612C2D4×C4⋊Dic7312yes—
613C2#2261291511yescharacteristic
614C2#2261307911yescharacteristic
615C2#2261618512yes—
616C2#2261622211yescharacteristic
617C2#2261624611yescharacteristic
618C2#2261659312yes—
619C2#2261683711yescharacteristic
620C2#2261706011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#385×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#385D73
hypocenterC73#14402
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. C2×#17988486C2semidirect
4. #22608196C2semidirect
5. #22608184C2semidirect
8. #22608165C2semidirect
33. #385×C73C4semidirect
168. C22⋊C4×C73C4⋊C4semidirect
320. C73#14402semidirect
335. #385Dic73semidirect
478. C22⋊C4C4⋊Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884862
C2×#179884802
C2×#179884472
#226081962
#226081842
#226081762
#226081652
C2×#385×C732
#1440273