Finite Group Databasebeta

← #33015164

Subgroups — #33015164

Order 18688.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988480C212yesmaximal, complemented
2#22608199C212yesmaximal, complemented
3#22608176C212yesmaximal, complemented
4#22608172C211yesmaximal, characteristic
5#22608172C211yesmaximal, characteristic
6#22608168C212yesmaximal, complemented
7#22608160C212yesmaximal, complemented
8#22608154C212yesmaximal, complemented
9C22×#123×C73C211yesmaximal, characteristic
10C22×C4⋊Dic73C2212yes—
11C22×C22⋊Dic73C2212yes—
12C22×C22⋊Dic73C2212yes—
13C22×C22⋊Dic73C2211yescharacteristic
14C22×C22⋊Dic73C2212yes—
15C22×C22⋊Dic73C2211yescharacteristic
16C22×C4×Dic73C2212yes—
17#17988480—28no—
18C2×#14556498C2212yes—
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C2×#14556498C2212yes—
23C2×#14556498C2212yes—
24C2×#14556498C2211yescharacteristic
25C2×#14556498C2212yes—
26C2×#14556498C2211yescharacteristic
27C23×D4×C73C2211yescharacteristic
28C24×C4×C73C2211yescharacteristic
29C2×#123×C73—28no—
30C2×#123×C73C418yescomplemented
31C22×C4⋊C4×C73C2212yes—
32C22×C22⋊C4×C73C2212yes—
33C22×C22⋊C4×C73C2211yescharacteristic
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C23×Dic73C2312yes—
38C2×C4⋊Dic73—24no—
39C2×C4⋊Dic73—24no—
40C2×C4⋊Dic73—42no—
41C2×C22⋊Dic73—24no—
42C2×C22⋊Dic73—24no—
43C2×C22⋊Dic73—24no—
44C2×C22⋊Dic73—24no—
45C2×C22⋊Dic73—24no—
46C2×C22⋊Dic73—24no—
47C2×C22⋊Dic73—42no—
48C2×C22⋊Dic73—42no—
49C2×C22⋊Dic73—41no—
50C2×C22⋊Dic73—42no—
51C2×C22⋊Dic73—41no—
52C2×C22⋊Dic73—42no—
53C2×C22⋊Dic73—42no—
54C2×C4×Dic73—24no—
55C2×C4×Dic73—24no—
56C2×C4×Dic73—42no—
57#14556498—24no—
58#14556498—24no—
59#14556498—42no—
60#14556498—42no—
61#14556498—42no—
62#14556498—42no—
63#14556498—41no—
64#14556498—42no—
65#14556498—41no—
66C25×C73C2311yescharacteristic
67C25×C73C2311yescharacteristic
68C22×D4×C73—24no—
69C22×D4×C73—42no—
70C22×D4×C73—41no—
71C22×D4×C73—41no—
72C22×D4×C73C2×C414yes—
73C23×C4×C73—24no—
74C23×C4×C73—42no—
75C23×C4×C73—41no—
76C23×C4×C73—41no—
77C23×C4×C73C2×C414yes—
78C23×C4×C73C2312yes—
79C23×C4×C73C2311yescharacteristic
80C23×C4×C73C2312yes—
81#123×C73—232no—
82C2×C4⋊C4×C73—24no—
83C2×C4⋊C4×C73—24no—
84C2×C4⋊C4×C73C2×C418yes—
85C2×C22⋊C4×C73—24no—
86C2×C22⋊C4×C73—24no—
87C2×C22⋊C4×C73C2×C418yes—
88C2×C22⋊C4×C73C2×C414yes—
89C2×C22⋊C4×C73D418yes—
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×Dic73—22no—
116C22×Dic73#4114yes—
117C22×Dic73#4114yes—
118C4⋊Dic73—48no—
119C22⋊Dic73—44no—
120C22⋊Dic73—48no—
121C22⋊Dic73—48no—
122C22⋊Dic73—48no—
123C22⋊Dic73—44no—
124C4×Dic73—48no—
125C24×C73—22no—
126C24×C73—22no—
127C24×C73—22no—
128C24×C73—21no—
129C24×C73—21no—
130C24×C73—22no—
131C24×C73—22no—
132C24×C73—22no—
133C24×C73—22no—
134C24×C73—22no—
135C24×C73—22no—
136C24×C73—21no—
137C24×C73—21no—
138C24×C73—22no—
139C24×C73—22no—
140C24×C73C22×C412yes—
141C24×C73C22×C412yes—
142C24×C73C2×D414yes—
143C24×C73C2411yescharacteristic
144C2×D4×C73—28no—
145C2×D4×C73—44no—
146C2×D4×C73—44no—
147C2×D4×C73—48no—
148C2×D4×C73—48no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—28no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—44no—
165C22×C4×C73—44no—
166C22×C4×C73—48no—
167C22×C4×C73—48no—
168C22×C4×C73C22×C414yes—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C22×C414yes—
171C22×C4×C73C2×D414yes—
172C22×C4×C73C2×D414yes—
173C4⋊C4×C73—216no—
174C22⋊C4×C73—216no—
175C22⋊C4×C73C22⋊C4116yescomplemented
176C23×C73—24no—
177C23×C73—22no—
178C23×C73—22no—
179C23×C73—24no—
180C23×C73—24no—
181C23×C73—24no—
182C23×C73—24no—
183C23×C73—24no—
184C23×C73—24no—
185C23×C73—24no—
186C23×C73—22no—
187C23×C73—24no—
188C23×C73—22no—
189C23×C73—22no—
190C23×C73—24no—
191C23×C73—42no—
192C23×C73—42no—
193C23×C73—42no—
194C23×C73—42no—
195C23×C73—41no—
196C23×C73—42no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—42no—
200C23×C73—41no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—42no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73—42no—
208C23×C73—42no—
209C23×C73—42no—
210C23×C73—42no—
211C23×C73—41no—
212C23×C73—42no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—42no—
216C23×C73—41no—
217C23×C73C2×C22⋊C414yes—
218C23×C73C23×C411yescharacteristic
219C23×C73C22×D412yes—
220C23×C73C2×#4112yes—
221C23×C73C2×#4112yes—
222C23×C73H(5,𝔽2)12yes—
223C23×C73H(5,𝔽2)11yescharacteristic
224C23×C73H(5,𝔽2)11yescharacteristic
225C23×C73H(5,𝔽2)12yes—
226C23×C732−1+412yes—
227D4×C73—416no—
228C2×C4×C73—24no—
229C2×C4×C73—24no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—24no—
233C2×C4×C73—24no—
234C2×C4×C73—24no—
235C2×C4×C73—22no—
236C2×C4×C73—24no—
237C2×C4×C73—24no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—416no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×C4×C73C2×C22⋊C418yes—
248C2×Dic73—24no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—24no—
254C2×Dic73—42no—
255C2×Dic73—42no—
256C2×Dic73—42no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C22×C73—24no—
277C22×C73—28no—
278C22×C73—44no—
279C22×C73—42no—
280C22×C73—42no—
281C22×C73—44no—
282C22×C73—44no—
283C22×C73—44no—
284C22×C73—44no—
285C22×C73—42no—
286C22×C73—44no—
287C22×C73—44no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73—82no—
291C22×C73—82no—
292C22×C73—82no—
293C22×C73—82no—
294C22×C73C22×C22⋊C411yescharacteristic
295C22×C73C4×#4112yes—
296C22×C73#51811yescharacteristic
297C22×C73#51811yescharacteristic
298C22×C73#52012yes—
299C22×C73#53512yes—
300C22×C73#53512yes—
301C22×C73#53812yes—
302C22×C73#53912yes—
303C22×C73#54012yes—
304C22×C73#54212yes—
305C22×C73#54612yes—
306C22×C73#54612yes—
307C22×C73#54712yes—
308C22×C73#54712yes—
309C22×C73#55111yescharacteristic
310C22×C73#55312yes—
311C22×C73#55911yescharacteristic
312C22×C73#56011yescharacteristic
313C22×C73#56212yes—
314C22×C73#56311yescharacteristic
315C4×C73—28no—
316C4×C73—44no—
317C4×C73—42no—
318Dic73—44no—
319Dic73—44no—
320Dic73—82no—
321Dic73—82no—
322#14506—731nomaximal, Sylow
323C2×C73—44no—
324C2×C73—82no—
325C2×C73#230911yescharacteristic
326C2×C73#233512yes—
327C2×C73#233912yes—
328C2×C73#237411yescharacteristic
329C2×C73#237611yescharacteristic
330C2×C73#270312yes—
331C2×C73#270612yes—
332C2×C73#271012yes—
333C2×C73#271812yes—
334C22×#123D7311yescharacteristic
335C2×#388—732no—
336#1445—732no—
337#1439—732no—
338#1438—731no—
339#1438—731no—
340#1437—732no—
341C4⋊#95—732no—
342C22⋊#95—732no—
343C73#1450611yescharacteristic, complemented, Sylow
344C23×D4C2×D7311yescharacteristic
345C24×C4C2×D7311yescharacteristic
346C2×#123—28no—
347C2×#123Dic7318yescomplemented
348C22×C4⋊C4—732no—
349C22×C4⋊C4C2×D7312yes—
350C22×C22⋊C4—731no—
351C22×C22⋊C4—732no—
352C22×C22⋊C4—732no—
353C22×C22⋊C4—732no—
354C22×C22⋊C4—731no—
355C22×C22⋊C4C2×D7312yes—
356C22×C22⋊C4C2×D7311yescharacteristic
357C22×C42—732no—
358#388—1468no—
359C2×#95—732no—
360C2×#95—732no—
361C2×#95—732no—
362C2×#95—732no—
363C2×#95—731no—
364C2×#95—732no—
365C2×#95—731no—
366C2×#95—732no—
367C2×#95—732no—
368C25C22×D7311yescharacteristic
369C25C22×D7311yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×Dic7314yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4—732no—
380C23×C4—732no—
381C23×C4—732no—
382C23×C4—732no—
383C23×C4C2×Dic7314yes—
384C23×C4C22×D7312yes—
385C23×C4C22×D7311yescharacteristic
386C23×C4C22×D7312yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—2922no—
391C2×C4⋊C4—1464no—
392C2×C4⋊C4—1464no—
393C2×C4⋊C4C2×Dic7318yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—2921no—
397C2×C22⋊C4—2922no—
398C2×C22⋊C4—2922no—
399C2×C22⋊C4—2922no—
400C2×C22⋊C4—2922no—
401C2×C22⋊C4—2922no—
402C2×C22⋊C4—2921no—
403C2×C22⋊C4—1464no—
404C2×C22⋊C4—1464no—
405C2×C22⋊C4—1464no—
406C2×C22⋊C4—1464no—
407C2×C22⋊C4—1464no—
408C2×C22⋊C4—1464no—
409C2×C22⋊C4C2×Dic7318yes—
410C2×C22⋊C4C2×Dic7314yes—
411C2×C22⋊C4C22⋊D7318yes—
412C2×C42—2922no—
413C2×C42—1464no—
414C2×C42—1464no—
415#95—2922no—
416#95—2922no—
417#95—2922no—
418#95—2922no—
419#95—2921no—
420#95—2922no—
421#95—2921no—
422#95—1464no—
423#95—1464no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24—22no—
438C24—22no—
439C24C22×Dic7312yes—
440C24C22×Dic7312yes—
441C24C2×C22⋊D7314yes—
442C24C23×D7311yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—734no—
468C22×C4—734no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4C22×Dic7314yes—
496C22×C4C22×Dic7312yes—
497C22×C4C22×Dic7314yes—
498C22×C4C2×C22⋊D7314yes—
499C22×C4C2×C22⋊D7314yes—
500C4⋊C4—216no—
501C4⋊C4—2928no—
502C22⋊C4—216no—
503C22⋊C4—2924no—
504C22⋊C4—2928no—
505C22⋊C4—2928no—
506C22⋊C4—2928no—
507C22⋊C4—2924no—
508C22⋊C4C22⋊Dic73116yescomplemented
509C42—2928no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊Dic7314yes—
552C23C23×Dic7311yescharacteristic
553C23C22×C22⋊D7312yes—
554C23#41×D7312yes—
555C23#41×D7312yes—
556C23#1455663612yes—
557C23#1455663611yescharacteristic
558C23#1455663611yescharacteristic
559C23#1455663712yes—
560C23#1455663812yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—2922no—
592C2×C4—2922no—
593C2×C4—2922no—
594C2×C4—2922no—
595C2×C4—2922no—
596C2×C4—2922no—
597C2×C4—2922no—
598C2×C4—2922no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—2922no—
602C2×C4—2922no—
603C2×C4—1464no—
604C2×C4—1464no—
605C2×C4—1464no—
606C2×C4—1464no—
607C2×C4—1464no—
608C2×C4—1464no—
609C2×C4C2×C22⋊Dic7318yes—
610C22—28no—
611C22—24no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊Dic7311yescharacteristic
629C22#41×Dic7312yes—
630C22#1798906211yescharacteristic
631C22#1798906211yescharacteristic
632C22#1798907012yes—
633C22#1798915412yes—
634C22#1798915412yes—
635C22#1798917312yes—
636C22#1798918012yes—
637C22#1798918912yes—
638C22#1798920512yes—
639C22#1798922512yes—
640C22#1798922512yes—
641C22#1798923612yes—
642C22#1798923612yes—
643C22#1798926511yescharacteristic
644C22#1798928212yes—
645C22#1798932111yescharacteristic
646C22#1798932711yescharacteristic
647C22#1798933412yes—
648C22#1798934011yescharacteristic
649C4—28no—
650C4—44no—
651C4—42no—
652C4—2924no—
653C4—2924no—
654C4—5842no—
655C4—5842no—
656C2—44no—
657C2—82no—
658C2#2261251311yescharacteristic
659C2#2261272512yes—
660C2#2261276112yes—
661C2#2261306711yescharacteristic
662C2#2261307911yescharacteristic
663C2#2261656812yes—
664C2#2261661312yes—
665C2#2261666912yes—
666C2#2261677712yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC22×#123×C73C2
socle, cosocleC24×C73C24
prime radicalC23×D4×C73C22
hypercenterC22×#123D73
hypocenterC73#14506
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C2×#17988480C2semidirect
2. #22608199C2semidirect
3. #22608176C2semidirect
6. #22608168C2semidirect
7. #22608160C2semidirect
8. #22608154C2semidirect
30. C2×#123×C73C4semidirect
175. C22⋊C4×C73C22⋊C4semidirect
343. C73#14506semidirect
347. C2×#123Dic73semidirect
508. C22⋊C4C22⋊Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884802
#226081992
#226081762
#226081722
#226081682
#226081602
#226081542
C22×#123×C732
#1450673