Finite Group Databasebeta

← #13285122

Subgroups — #13285122

Order 1792.

2842 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6498C211yesmaximal, characteristic
2C23×C4×Dic7C211yesmaximal, characteristic, complemented
3C2×C7×#386C211yesmaximal, characteristic
4C2×#89992C211yesmaximal, characteristic
5C2×#89715C211yesmaximal, characteristic, complemented
6#11729881C211yesmaximal, characteristic, complemented
7#11729299C214yesmaximal, complemented
8#11729298C211yesmaximal, characteristic, complemented
9#11727160C214yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C24×Dic7C2211yescharacteristic
13C24×Dic7C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C22⋊Dic7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C22⋊Dic7C2212yes—
20C2×#6498—24no—
21C2×#6498—24no—
22C2×#6498C418yescomplemented
23C22×Dic7⋊C4C2212yes—
24C22×C4×Dic7—24no—
25C22×C4×Dic7—24no—
26C22×C4×Dic7—24no—
27C22×C4×Dic7C2214yes—
28C7×#386—24no—
29C2×C7×#95C2212yes—
30#89992—24no—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2214yes—
33C2×C7⋊#95C2214yes—
34C2×C7⋊#95C2214yes—
35#89715—24no—
36#14706—71nomaximal, Sylow
37C25×C7C2311yescharacteristic
38C25×C7C2311yescharacteristic
39C22×C7×D4—22no—
40C22×C7×D4—22no—
41C22×C7×D4—22no—
42C22×C7×D4—22no—
43C22×C7×D4—22no—
44C22×C7×D4—22no—
45C22×C7×D4C2×C414yes—
46C23×C4×C7—22no—
47C23×C4×C7—22no—
48C23×C4×C7—28no—
49C23×C4×C7D414yes—
50C23×C4×C7C2312yes—
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2312yes—
53C23×Dic7—22no—
54C23×Dic7—22no—
55C23×Dic7—22no—
56C23×Dic7—22no—
57C23×Dic7—22no—
58C23×Dic7—22no—
59C23×Dic7—28no—
60C23×Dic7C2×C414yes—
61C23×Dic7D414yes—
62C23×Dic7D414yes—
63C23×Dic7D414yes—
64C23×Dic7C2312yes—
65C23×Dic7C2312yes—
66C23×Dic7C2312yes—
67C23×Dic7C2312yes—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—24no—
72C2×C7×C22⋊C4—28no—
73C2×C22⋊Dic7—22no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—24no—
76C2×C22⋊Dic7—24no—
77C2×C22⋊Dic7—22no—
78C2×C22⋊Dic7—22no—
79C2×C22⋊Dic7—22no—
80C2×C22⋊Dic7—24no—
81C2×C22⋊Dic7—28no—
82C2×C22⋊Dic7C2×C414yes—
83C2×C22⋊Dic7C2×C418yescomplemented
84#6498—232no—
85C2×Dic7⋊C4—24no—
86C2×Dic7⋊C4—24no—
87C2×Dic7⋊C4C2×C418yescomplemented
88C2×C4×Dic7—232no—
89C2×C4×Dic7—28no—
90C2×C4×Dic7—28no—
91C2×C4×Dic7—28no—
92C7×#95—24no—
93C7⋊#95—28no—
94C7⋊#95—28no—
95C7⋊#95—24no—
96C7⋊#95—28no—
97C22×#123—71no—
98C23×C42—71no—
99C2×#386—71no—
100C2×#386D711yescharacteristic
101C2×#381—71no—
102#1437—74no—
103#1436—74no—
104C22⋊#95—71no—
105C22⋊#95—71no—
106C24×C7—24no—
107C24×C7—24no—
108C24×C7—21no—
109C24×C7—21no—
110C24×C7—21no—
111C24×C7—21no—
112C24×C7—24no—
113C24×C7—24no—
114C24×C7—21no—
115C24×C7—21no—
116C24×C7C22×C412yes—
117C24×C7C22×C412yes—
118C24×C7C2×D412yes—
119C24×C7C2×D412yes—
120C24×C7C2×D412yes—
121C24×C7C2×D412yes—
122C24×C7#4112yes—
123C24×C7#4112yes—
124C24×C7C2411yescharacteristic
125C2×C7×D4—28no—
126C2×C7×D4—28no—
127C2×C7×D4—28no—
128C2×C7×D4—44no—
129C2×C7×D4—44no—
130C2×C7×D4—44no—
131C2×C7×D4—44no—
132C22×C4×C7—216no—
133C22×C4×C7—216no—
134C22×C4×C7—28no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—24no—
138C22×C4×C7—216no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—24no—
150C22×C4×C7C22×C412yes—
151C22×C4×C7C2×D414yes—
152C22×C4×C7#4118yes—
153C22×Dic7—216no—
154C22×Dic7—216no—
155C22×Dic7—28no—
156C22×Dic7—28no—
157C22×Dic7—28no—
158C22×Dic7—28no—
159C22×Dic7—28no—
160C22×Dic7—28no—
161C22×Dic7—28no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—24no—
169C22×Dic7—216no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—24no—
173C22×Dic7—24no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—24no—
177C22×Dic7—22no—
178C22×Dic7—24no—
179C22×Dic7C22⋊C4116yescomplemented
180C22×Dic7C22×C414yes—
181C22×Dic7C22×C414yes—
182C22×Dic7C2×D414yes—
183C22×Dic7C2×D414yes—
184C22×Dic7C2×D414yes—
185C22×Dic7#4118yes—
186C7×C22⋊C4—44no—
187C7×C22⋊C4—48no—
188C22⋊Dic7—28no—
189C22⋊Dic7—216no—
190C22⋊Dic7—44no—
191C22⋊Dic7—48no—
192Dic7⋊C4—216no—
193C4×Dic7—232no—
194C23×D4C2×D711yescharacteristic
195C24×C4—71no—
196C24×C4—71no—
197C24×C4C2×D711yescharacteristic
198C2×#123—78no—
199C2×#123—144no—
200C2×#123—144no—
201C22×C4⋊C4—72no—
202C22×C22⋊C4—71no—
203C22×C22⋊C4—72no—
204C22×C22⋊C4—71no—
205C22×C22⋊C4—72no—
206C22×C22⋊C4C2×D711yescharacteristic
207C22×C22⋊C4C2×D712yes—
208C22×C42—74no—
209C22×C42—144no—
210C22×C42—144no—
211C22×C42—144no—
212#386—24no—
213#386—144no—
214#381—144no—
215C2×#95—74no—
216C2×#95—74no—
217C2×#95—72no—
218C2×#95—74no—
219C2×#95C2×D712yes—
220C23×C7—28no—
221C23×C7—24no—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
227C23×C7—28no—
228C23×C7—28no—
229C23×C7—28no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—28no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—41no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—44no—
245C23×C7—44no—
246C23×C7—44no—
247C23×C7C2×C22⋊C414yes—
248C23×C7C4×D414yes—
249C23×C7C4×D414yes—
250C23×C7#12014yes—
251C23×C7#12114yes—
252C23×C7#12114yes—
253C23×C7#12314yes—
254C23×C7C23×C411yescharacteristic
255C23×C7C22×D411yescharacteristic
256C23×C7C22×D411yescharacteristic
257C23×C7C22×D411yescharacteristic
258C23×C7C22×D411yescharacteristic
259C23×C7C2×#4111yescharacteristic
260C23×C7C2×#4111yescharacteristic
261C23×C7C2×#4114yes—
262C23×C7C2×#4114yes—
263C7×D4—416no—
264C2×C4×C7—232no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—28no—
268C2×C4×C7—28no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—44no—
280C2×C4×C7—44no—
281C2×C4×C7#12318yescomplemented
282C2×Dic7—232no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7—24no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—28no—
292C2×Dic7—28no—
293C2×Dic7—232no—
294C2×Dic7—24no—
295C2×Dic7—28no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—44no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—44no—
303C2×Dic7—44no—
304C2×Dic7C2×C22⋊C418yes—
305C2×Dic7#12318yescomplemented
306C25C22×D711yescharacteristic
307C25C22×D711yescharacteristic
308C22×D4—22no—
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—22no—
314C22×D4C4×D714yes—
315C23×C4—22no—
316C23×C4—22no—
317C23×C4—28no—
318C23×C4—74no—
319C23×C4—74no—
320C23×C4—74no—
321C23×C4—72no—
322C23×C4—72no—
323C23×C4—74no—
324C23×C4—72no—
325C23×C4—72no—
326C23×C4—148no—
327C23×C4—142no—
328C23×C4—142no—
329C23×C4—142no—
330C23×C4—142no—
331C23×C4—142no—
332C23×C4—142no—
333C23×C4C22⋊D714yes—
334C23×C4C22×D712yes—
335C23×C4C22×D711yescharacteristic
336C23×C4C22×D712yes—
337#123—1432no—
338C2×C4⋊C4—78no—
339C2×C4⋊C4—144no—
340C2×C4⋊C4—144no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—28no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—74no—
347C2×C22⋊C4—78no—
348C2×C22⋊C4—144no—
349C2×C22⋊C4—144no—
350C2×C22⋊C4—142no—
351C2×C22⋊C4—142no—
352C2×C22⋊C4—148no—
353C2×C22⋊C4—144no—
354C2×C22⋊C4—142no—
355C2×C22⋊C4—142no—
356C2×C22⋊C4—142no—
357C2×C42—148no—
358C2×C42—148no—
359C2×C42—1432no—
360C2×C42—148no—
361#95—24no—
362#95—148no—
363#95—148no—
364#95—144no—
365#95—148no—
366C22×C7—24no—
367C22×C7—24no—
368C22×C7—28no—
369C22×C7—28no—
370C22×C7—28no—
371C22×C7—28no—
372C22×C7—28no—
373C22×C7—28no—
374C22×C7—42no—
375C22×C7—44no—
376C22×C7—44no—
377C22×C7—44no—
378C22×C7—81no—
379C22×C7—81no—
380C22×C7—82no—
381C22×C7—82no—
382C22×C7—82no—
383C22×C7#38618yes—
384C22×C7C22×C22⋊C411yescharacteristic
385C22×C7C2×C4×D411yescharacteristic
386C22×C7C2×C4×D411yescharacteristic
387C22×C7C4×#4114yes—
388C22×C7C2×#12011yescharacteristic
389C22×C7C2×#12111yescharacteristic
390C22×C7C2×#12111yescharacteristic
391C22×C7C2×#12311yescharacteristic
392C22×C7C2×#12312yes—
393C22×C7C2×#12312yes—
394C22×C7#52512yes—
395C22×C7#52512yes—
396C22×C7#52512yes—
397C22×C7#52512yes—
398C22×C7#52914yes—
399C22×C7#52914yes—
400C22×C7#53214yes—
401C4×C7—28no—
402C4×C7—42no—
403C4×C7—82no—
404Dic7—28no—
405Dic7—28no—
406Dic7—44no—
407Dic7—82no—
408C24—24no—
409C24—24no—
410C24—21no—
411C24—21no—
412C24—21no—
413C24—21no—
414C24—24no—
415C24—21no—
416C24—24no—
417C24—21no—
418C24C2×C4×D712yes—
419C24C2×C4×D712yes—
420C24#115712yes—
421C24D4×D712yes—
422C24D4×D712yes—
423C24D4×D712yes—
424C24#115912yes—
425C24C2×C22⋊D712yes—
426C24C23×D711yescharacteristic
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—28no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—216no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—716no—
453C22×C4—74no—
454C22×C4—74no—
455C22×C4—74no—
456C22×C4—74no—
457C22×C4—78no—
458C22×C4—74no—
459C22×C4—1416no—
460C22×C4—1416no—
461C22×C4—1416no—
462C22×C4—148no—
463C22×C4—148no—
464C22×C4—148no—
465C22×C4—148no—
466C22×C4—148no—
467C22×C4—148no—
468C22×C4—148no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—144no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4—142no—
478C22×C4—144no—
479C22×C4—144no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—144no—
484C22×C4—144no—
485C22×C4C2×C4×D712yes—
486C22×C4#115918yes—
487C22×C4C2×C22⋊D714yes—
488C4⋊C4—1416no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—1416no—
492C22⋊C4—148no—
493C22⋊C4—288no—
494C22⋊C4—284no—
495C42—1432no—
496C2×C7—28no—
497C2×C7—82no—
498C2×C7C2×#38611yescharacteristic
499C2×C7#229712yes—
500C2×C7C4×#12312yes—
501C2×C7C4×#12312yes—
502C2×C7#284911yescharacteristic
503C2×C7#284911yescharacteristic
504C2×C7#285312yes—
505C2×C7#285412yes—
506C2×C7#285412yes—
507C23—28no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—41no—
522C23—41no—
523C23—44no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23D7×C22⋊C414yes—
535C23(C22⋊D7)⋊C414yes—
536C23#644614yes—
537C23#644714yes—
538C23C4×C22⋊D714yes—
539C23#650014yes—
540C23#650114yes—
541C23C22×C4×D711yescharacteristic
542C23C2×#115711yescharacteristic
543C23C2×D4×D711yescharacteristic
544C23C2×D4×D711yescharacteristic
545C23C2×D4×D711yescharacteristic
546C23C2×#115911yescharacteristic
547C23C2×#115914yes—
548C23D7×#4114yes—
549C23C22×C22⋊D711yescharacteristic
550D4—416no—
551C2×C4—232no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—78no—
569C2×C4—78no—
570C2×C4—1432no—
571C2×C4—144no—
572C2×C4—144no—
573C2×C4—144no—
574C2×C4—144no—
575C2×C4—144no—
576C2×C4—144no—
577C2×C4—144no—
578C2×C4—1432no—
579C2×C4—148no—
580C2×C4—144no—
581C2×C4—148no—
582C2×C4—144no—
583C2×C4—148no—
584C2×C4—282no—
585C2×C4—284no—
586C2×C4—282no—
587C2×C4—282no—
588C2×C4—284no—
589C2×C4—282no—
590C2×C4—284no—
591C2×C4—282no—
592C2×C4#649818yescomplemented
593C7#1470611yescharacteristic, complemented, Sylow
594C22—28no—
595C22—28no—
596C22—28no—
597C22—28no—
598C22—28no—
599C22—24no—
600C22—24no—
601C22—28no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—82no—
607C22—82no—
608C22—81no—
609C22—81no—
610C22—82no—
611C22#8972818yes—
612C22C2×D7×C22⋊C411yescharacteristic
613C22C2×(C22⋊D7)⋊C411yescharacteristic
614C22C2×#644611yescharacteristic
615C22C2×#644711yescharacteristic
616C22C4×#115914yes—
617C22#9022914yes—
618C22#9027712yes—
619C22#9027712yes—
620C22#9028714yes—
621C22#9029114yes—
622C22D7×#12312yes—
623C22#9034712yes—
624C22C2×C4×C22⋊D711yescharacteristic
625C22C2×#649812yes—
626C22C2×#650011yescharacteristic
627C22C2×#650111yescharacteristic
628C22#9051912yes—
629C4—28no—
630C4—42no—
631C4—82no—
632C4—148no—
633C4—148no—
634C4—284no—
635C4—562no—
636C2—28no—
637C2—82no—
638C2#1173210312yes—
639C2#1173210612yes—
640C2#1173231612yes—
641C2#1173231712yes—
642C2C2×#8972811yescharacteristic
643C2C4×#649812yes—
644C2#1174004711yescharacteristic
645C2#1174014412yes—
646C2#1174035711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#386C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#386D7
hypocenterC7#14706
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C23×C4×Dic7C2semidirect
5. C2×#89715C2semidirect
6. #11729881C2semidirect
7. #11729299C2semidirect
8. #11729298C2semidirect
9. #11727160C2semidirect
22. C2×#6498C4semidirect
83. C2×C22⋊Dic7C2×C4semidirect
87. C2×Dic7⋊C4C2×C4semidirect
179. C22×Dic7C22⋊C4semidirect
281. C2×C4×C7#123semidirect
305. C2×Dic7#123semidirect
592. C2×C4#6498semidirect
593. C7#14706semidirect

Maximal subgroups

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

subgroupindex
C22×#64982
C23×C4×Dic72
C2×C7×#3862
C2×#899922
C2×#897152
#117298812
#117292992
#117292982
#117271602
#147067