Finite Group Databasebeta

← #13280174

Subgroups — #13280174

Order 1792.

2142 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6460C211yesmaximal, characteristic
2C22×#6435C211yesmaximal, characteristic
3C2×#89759C211yesmaximal, characteristic
4C2×#89746C211yesmaximal, characteristic, complemented
5C2×#89701C211yesmaximal, characteristic, complemented
6C7×C4⋊#95C211yesmaximal, characteristic, complemented
7#11729317C211yesmaximal, characteristic, complemented
8#11727183C214yesmaximal
9#11727165C214yesmaximal, complemented
10C23×C4×D7C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×(C2×D7)⋊C4C2212yes—
16C22×(C2×D7)⋊C4C2212yes—
17C22×(C2×D7)⋊C4C2212yes—
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2212yes—
20C22×Dic7⋊C4C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C2×#6460—24no—
23C2×#6460—24no—
24C2×#6460C418yescomplemented
25C2×#6435—24no—
26C2×#6435—24no—
27C2×#6435—24no—
28C2×C7×#95C2214yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2214yes—
31C2×C7⋊#95C2214yes—
32C2×C7⋊#95C2214yes—
33#89759—24no—
34#89746—24no—
35#89701—24no—
36#14258—71nomaximal, Sylow
37C24×D7C2311yescharacteristic
38C23×C4×C7C2312yes—
39C23×C4×C7C2312yes—
40C23×C4×C7C2312yes—
41C23×C4×C7C2311yescharacteristic
42C23×Dic7C2312yes—
43C23×Dic7C2312yes—
44C23×Dic7C2312yes—
45C23×Dic7C2311yescharacteristic
46C22×C4×D7—22no—
47C22×C4×D7—22no—
48C22×C4×D7—22no—
49C22×C4×D7—22no—
50C22×C4×D7C2×C414yes—
51C22×C4×D7D414yes—
52C22×C4×D7D414yes—
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—22no—
56C22×Dic14—22no—
57C22×Dic14—22no—
58C22×Dic14—22no—
59C22×Dic14C2×C414yes—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4C2×C414yes—
66C2×C42×C7—22no—
67C2×C42×C7—22no—
68C2×C42×C7D414yes—
69C2×(C2×D7)⋊C4—24no—
70C2×(C2×D7)⋊C4—24no—
71C2×(C2×D7)⋊C4—24no—
72C2×(C2×D7)⋊C4—24no—
73C2×(C2×D7)⋊C4—24no—
74C2×(C2×D7)⋊C4—24no—
75C2×(C2×D7)⋊C4—44no—
76C2×(C2×D7)⋊C4C2×C418yes—
77C2×C4⋊Dic7—24no—
78C2×C4⋊Dic7—24no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7C2×C418yes—
83C2×Dic7⋊C4—24no—
84C2×Dic7⋊C4—24no—
85C2×Dic7⋊C4—24no—
86C2×C4×Dic7—22no—
87C2×C4×Dic7—22no—
88C2×C4×Dic7D414yes—
89#6460—232no—
90#6435—232no—
91C7×#95—44no—
92C7⋊#95—28no—
93C7⋊#95—28no—
94C7⋊#95—24no—
95C7⋊#95—44no—
96C22×C22⋊Q8—71no—
97C22×#117—71no—
98C2×#391—71no—
99C2×#391—71no—
100C2×#380—71no—
101#1441—74no—
102#1440—74no—
103C4⋊#95—71no—
104C4⋊#95D711yescharacteristic, complemented
105C24×C7C2411yescharacteristic
106C23×D7—21no—
107C23×D7—21no—
108C23×D7—21no—
109C23×D7—21no—
110C23×D7—22no—
111C23×D7—22no—
112C23×D7—22no—
113C23×D7—22no—
114C23×D7C22×C412yes—
115C23×D7C2×D412yes—
116C23×D7C2×D412yes—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—24no—
122C22×C4×C7—22no—
123C22×C4×C7—24no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—24no—
131C22×C4×C7—24no—
132C22×C4×C7C22×C414yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C2×D414yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7#4112yes—
140C22×C4×C7#4112yes—
141C22×Dic7—21no—
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—24no—
161C22×Dic7—24no—
162C22×Dic7C22×C414yes—
163C22×Dic7C22×C412yes—
164C22×Dic7C2×D414yes—
165C22×Dic7C2×D412yes—
166C22×Dic7C2×D412yes—
167C2×C4×D7—28no—
168C2×C4×D7—28no—
169C2×C4×D7—28no—
170C2×C4×D7—28no—
171C2×C4×D7—28no—
172C2×C4×D7—28no—
173C2×C4×D7C22⋊C4116yes—
174C2×Dic14—28no—
175C2×Dic14—28no—
176C2×Dic14—28no—
177C2×Dic14—44no—
178C2×Dic14—44no—
179C2×Dic14—44no—
180C2×Dic14—44no—
181C7×C4⋊C4—28no—
182C7×C4⋊C4—44no—
183C42×C7—28no—
184(C2×D7)⋊C4—216no—
185(C2×D7)⋊C4—216no—
186(C2×D7)⋊C4—48no—
187C4⋊Dic7—216no—
188C4⋊Dic7—44no—
189Dic7⋊C4—216no—
190C4×Dic7—28no—
191C23×Q8—71no—
192C24×C4—71no—
193C2×C22⋊Q8—78no—
194C2×C22⋊Q8—144no—
195C2×C22⋊Q8—144no—
196C2×#117—144no—
197C2×#117—144no—
198C2×#117—144no—
199C22×C4⋊C4—72no—
200C22×C4⋊C4—72no—
201C22×C4⋊C4—71no—
202C22×C4⋊C4C2×D711yescharacteristic
203C22×C4⋊C4C2×D711yescharacteristic
204C22×C22⋊C4—72no—
205C22×C22⋊C4—72no—
206C22×C22⋊C4—72no—
207C22×C42—71no—
208C22×C42C2×D711yescharacteristic
209#391—144no—
210#391—144no—
211#380—144no—
212C2×#95—74no—
213C2×#95—74no—
214C2×#95—72no—
215C2×#95—74no—
216C2×#95C2×D714yes—
217C23×C7C23×C411yescharacteristic
218C23×C7C22×D411yescharacteristic
219C23×C7C22×D411yescharacteristic
220C23×C7C22×D411yescharacteristic
221C23×C7C22×D411yescharacteristic
222C23×C7C2×#4111yescharacteristic
223C23×C7C2×#4111yescharacteristic
224C23×C7H(5,𝔽2)12yes—
225C23×C72−1+412yes—
226C23×C72−1+412yes—
227C23×C72−1+412yes—
228C22×D7—22no—
229C22×D7—22no—
230C22×D7—24no—
231C22×D7—24no—
232C22×D7—22no—
233C22×D7—22no—
234C22×D7—22no—
235C22×D7—22no—
236C22×D7—22no—
237C22×D7—22no—
238C22×D7—24no—
239C22×D7—22no—
240C22×D7—22no—
241C22×D7—24no—
242C22×D7—42no—
243C22×D7—42no—
244C22×D7—42no—
245C22×D7—42no—
246C22×D7—42no—
247C22×D7—42no—
248C22×D7—42no—
249C22×D7—42no—
250C22×D7C2×C22⋊C414yes—
251C2×C4×C7—24no—
252C2×C4×C7—28no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—44no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—44no—
267C2×C4×C7—44no—
268C2×C4×C7C2×C22⋊C414yes—
269C2×C4×C7C4×D414yes—
270C2×C4×C7C4×D414yes—
271C2×C4×C7#12014yes—
272C2×C4×C7#12114yes—
273C2×C4×C7#12114yes—
274C2×C4×C7#12314yes—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—44no—
299C2×Dic7—44no—
300C2×Dic7C2×C22⋊C414yes—
301C4×D7—232no—
302Dic14—416no—
303C25—71no—
304C22×Q8—74no—
305C22×Q8—142no—
306C22×Q8—142no—
307C22×Q8—142no—
308C22×Q8—142no—
309C22×Q8—142no—
310C22×Q8—142no—
311C23×C4—71no—
312C23×C4—74no—
313C23×C4—74no—
314C23×C4—74no—
315C23×C4—72no—
316C23×C4—72no—
317C23×C4—72no—
318C23×C4—142no—
319C23×C4—142no—
320C23×C4—142no—
321C23×C4—142no—
322C23×C4C22×D712yes—
323C23×C4C22×D712yes—
324C23×C4C22×D712yes—
325C23×C4C22×D711yescharacteristic
326C22⋊Q8—1432no—
327#117—1432no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—78no—
334C2×C4⋊C4—144no—
335C2×C4⋊C4—144no—
336C2×C4⋊C4—144no—
337C2×C4⋊C4—144no—
338C2×C4⋊C4—144no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—142no—
341C2×C4⋊C4—142no—
342C2×C4⋊C4C4×D714yes—
343C2×C22⋊C4—78no—
344C2×C22⋊C4—144no—
345C2×C22⋊C4—144no—
346C2×C22⋊C4—144no—
347C2×C22⋊C4—144no—
348C2×C22⋊C4—144no—
349C2×C22⋊C4—144no—
350C2×C22⋊C4—284no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—74no—
354C2×C42—142no—
355C2×C42—142no—
356C2×C42D2814yes—
357#95—44no—
358#95—144no—
359#95—148no—
360#95—148no—
361#95—284no—
362C22×C7C22×C22⋊C411yescharacteristic
363C22×C7C2×C4×D411yescharacteristic
364C22×C7C2×C4×D411yescharacteristic
365C22×C7#51912yes—
366C22×C7#52012yes—
367C22×C7C2×#12011yescharacteristic
368C22×C7C2×#12111yescharacteristic
369C22×C7C2×#12111yescharacteristic
370C22×C7C2×#12311yescharacteristic
371C22×C7#53511yescharacteristic
372C22×C7#53611yescharacteristic
373C22×C7#53612yes—
374C22×C7#53612yes—
375C22×C7#53612yes—
376C22×C7#53611yescharacteristic
377C22×C7#53611yescharacteristic
378C22×C7#53712yes—
379C22×C7#53712yes—
380C22×C7#53712yes—
381C22×C7#54112yes—
382C22×C7#54112yes—
383C22×C7#54212yes—
384C22×C7#54212yes—
385C2×D7—24no—
386C2×D7—24no—
387C2×D7—24no—
388C2×D7—44no—
389C2×D7—44no—
390C2×D7—44no—
391C2×D7—42no—
392C2×D7—42no—
393C2×D7—42no—
394C2×D7—42no—
395C2×D7—44no—
396C4×C7—44no—
397C4×C7—44no—
398C4×C7—82no—
399C4×C7#38618yes—
400Dic7—24no—
401Dic7—44no—
402Dic7—44no—
403Dic7—82no—
404C24—72no—
405C24—72no—
406C24—72no—
407C24—141no—
408C24—141no—
409C24—141no—
410C24—142no—
411C24—141no—
412C24—142no—
413C24—142no—
414C24—142no—
415C24C23×D711yescharacteristic
416C2×Q8—148no—
417C2×Q8—148no—
418C2×Q8—148no—
419C2×Q8—284no—
420C2×Q8—284no—
421C2×Q8—284no—
422C2×Q8—284no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—72no—
439C22×C4—72no—
440C22×C4—72no—
441C22×C4—716no—
442C22×C4—74no—
443C22×C4—74no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—148no—
449C22×C4—148no—
450C22×C4—148no—
451C22×C4—148no—
452C22×C4—148no—
453C22×C4—148no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—144no—
469C22×C4—142no—
470C22×C4—144no—
471C22×C4C2×C4×D714yes—
472C22×C4C2×C4×D712yes—
473C22×C4C2×D2814yes—
474C22×C4C2×D2812yes—
475C22×C4#115712yes—
476C22×C4D4×D712yes—
477C22×C4D4×D712yes—
478C22×C4D4×D712yes—
479C22×C4#116112yes—
480C4⋊C4—28no—
481C4⋊C4—44no—
482C4⋊C4—1416no—
483C4⋊C4—1416no—
484C4⋊C4—284no—
485C22⋊C4—1416no—
486C22⋊C4—1416no—
487C22⋊C4—288no—
488C42—28no—
489C42—148no—
490C2×C7C2×#38611yescharacteristic
491C2×C7#231011yescharacteristic
492C2×C7#231111yescharacteristic
493C2×C7#232312yes—
494C2×C7#232312yes—
495C2×C7#261612yes—
496C2×C7#262011yescharacteristic
497C2×C7#262011yescharacteristic
498C2×C7#262111yescharacteristic
499C2×C7#262111yescharacteristic
500C2×C7#262512yes—
501D7—44no—
502C23—74no—
503C23—142no—
504C23—142no—
505C23—142no—
506C23—142no—
507C23—142no—
508C23—142no—
509C23—142no—
510C23—142no—
511C23—144no—
512C23—144no—
513C23—144no—
514C23—144no—
515C23—142no—
516C23—142no—
517C23—282no—
518C23—282no—
519C23—282no—
520C23—282no—
521C23—282no—
522C23—282no—
523C23—282no—
524C23—282no—
525C23C22×C4×D711yescharacteristic
526C23C22×D2811yescharacteristic
527C23C2×#115711yescharacteristic
528C23C2×D4×D711yescharacteristic
529C23C2×D4×D711yescharacteristic
530C23C2×D4×D711yescharacteristic
531C23#654812yes—
532C23C2×#116111yescharacteristic
533C23#655112yes—
534C23#655412yes—
535C23#655412yes—
536Q8—2816no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—74no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—1432no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—144no—
565C2×C4—144no—
566C2×C4—144no—
567C2×C4—144no—
568C2×C4—282no—
569C2×C4—282no—
570C2×C4—282no—
571C2×C4—282no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4—282no—
576C2×C4—282no—
577C2×C4—282no—
578C2×C4—284no—
579C2×C4—282no—
580C2×C4—284no—
581C2×C4C4×D2814yes—
582C2×C4D7×C22⋊C414yes—
583C2×C4#644514yes—
584C2×C4#644714yes—
585C2×C4#645614yes—
586C2×C4#645714yes—
587C2×C4#645814yes—
588C7#1425811yescharacteristic, complemented, Sylow
589C22—144no—
590C22—144no—
591C22—144no—
592C22—284no—
593C22—284no—
594C22—282no—
595C22—282no—
596C22—284no—
597C22—282no—
598C22—282no—
599C22—284no—
600C22C2×C4×D2811yescharacteristic
601C22C2×D7×C22⋊C411yescharacteristic
602C22C2×#644511yescharacteristic
603C22C2×#644711yescharacteristic
604C22C2×#645611yescharacteristic
605C22C2×#645711yescharacteristic
606C22C2×#645811yescharacteristic
607C22#9020112yes—
608C22#9021712yes—
609C22#9023712yes—
610C22#9025212yes—
611C22#9026312yes—
612C22#9027412yes—
613C22#9029912yes—
614C22#9029912yes—
615C22#9031912yes—
616C22#9031912yes—
617C22#9036611yescharacteristic
618C22#9036611yescharacteristic
619C22#9038812yes—
620C22#9040611yescharacteristic
621C22#9041511yescharacteristic
622C22#9042412yes—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—144no—
627C4—284no—
628C4—284no—
629C4—562no—
630C4#8943718yes—
631C2—284no—
632C2C2×#8943711yescharacteristic
633C2#1173120712yes—
634C2#1173139512yes—
635C2#1173163811yescharacteristic
636C2#1173181511yescharacteristic
637C2#1173354712yes—
638C2#1173364212yes—
639C2#1173373911yescharacteristic
640C2#1173390211yescharacteristic
641C2#1173405511yescharacteristic
642C2#1173408611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×C4⋊#95C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenterC4⋊#95D7
hypocenterC7#14258
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. C2×#89746C2semidirect
5. C2×#89701C2semidirect
6. C7×C4⋊#95C2semidirect
7. #11729317C2semidirect
9. #11727165C2semidirect
24. C2×#6460C4semidirect
104. C4⋊#95D7semidirect
588. C7#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#64602
C22×#64352
C2×#897592
C2×#897462
C2×#897012
C7×C4⋊#952
#117293172
#117271832
#117271652
#142587