Finite Group Databasebeta

← #13284826

Subgroups — #13284826

Order 1792.

1702 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6441C211yesmaximal, characteristic
2C2×#89715C211yesmaximal, characteristic
3C2×#89714C211yesmaximal, characteristic
4C7×#1440C211yesmaximal, characteristic
5#11729303C211yesmaximal, characteristic
6#11729303C211yesmaximal, characteristic
7#11729302C211yesmaximal, characteristic
8#11727178C212yesmaximal, complemented
9#11727175C212yesmaximal, complemented
10#11727165C212yesmaximal, complemented
11#11727163C212yesmaximal, complemented
12C24×Dic7C2211yescharacteristic
13C23×Dic14C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C22⋊Dic7C2211yescharacteristic
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C22⋊Dic7C2211yescharacteristic
20C22×C4⋊Dic7C2211yescharacteristic
21C22×Dic7⋊C4C2211yescharacteristic
22C22×Dic7⋊C4C2211yescharacteristic
23C22×Dic7⋊C4C2212yes—
24C2×#6441—24no—
25C2×#6441—24no—
26C2×#6441C418yescomplemented
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89715—24no—
39#89714—24no—
40#14660—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×Dic7—42no—
47C23×Dic7—42no—
48C23×Dic7C2×C414yes—
49C23×Dic7D414yes—
50C23×Dic7D414yes—
51C23×Dic7C2312yes—
52C23×Dic7C2312yes—
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2311yescharacteristic
55C23×Dic7C2311yescharacteristic
56C23×Dic7C2311yescharacteristic
57C22×Dic14—22no—
58C22×Dic14—22no—
59C22×Dic14—42no—
60C22×Dic14—42no—
61C22×Dic14C2×C414yes—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—42no—
67C2×C7×C22⋊C4—42no—
68C2×C7×C22⋊C4C2×C414yes—
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—22no—
71C2×C22⋊Dic7—22no—
72C2×C22⋊Dic7—22no—
73C2×C22⋊Dic7—42no—
74C2×C22⋊Dic7—42no—
75C2×C22⋊Dic7C2×C414yes—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7C2×C414yes—
79C2×Dic7⋊C4—22no—
80C2×Dic7⋊C4—22no—
81C2×Dic7⋊C4—22no—
82C2×Dic7⋊C4—22no—
83C2×Dic7⋊C4—24no—
84C2×Dic7⋊C4—42no—
85C2×Dic7⋊C4—42no—
86C2×Dic7⋊C4C2×C414yes—
87C2×Dic7⋊C4C2×C414yes—
88#6441—232no—
89C7×#95—42no—
90C7×#95—42no—
91C7⋊#95—24no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—42no—
95C7⋊#95—42no—
96C7⋊#95—42no—
97C7⋊#95—42no—
98C7⋊#95—42no—
99C7⋊#95—42no—
100C22×C22⋊Q8—71no—
101C2×#381—71no—
102C2×#380—71no—
103#1446—72no—
104#1445—72no—
105#1441—72no—
106#1441—72no—
107#1440—71no—
108#1440—71no—
109#1440—71no—
110#1440D711yescharacteristic
111C24×C7—22no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—22no—
117C24×C7C22×C412yes—
118C24×C7C2×D412yes—
119C24×C7C2×D412yes—
120C24×C7C2411yescharacteristic
121C22×C4×C7—21no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7C22×C412yes—
144C22×C4×C7C22×C412yes—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—48no—
172C22×Dic7—48no—
173C22×Dic7—44no—
174C22×Dic7—44no—
175C22×Dic7C22⋊C4116yes—
176C22×Dic7C22×C412yes—
177C22×Dic7C22×C412yes—
178C22×Dic7C22×C412yes—
179C22×Dic7C22×C412yes—
180C22×Dic7C2×D412yes—
181C22×Dic7C2×D412yes—
182C22×Dic7C2×D412yes—
183C22×Dic7C2×D412yes—
184C2×Dic14—28no—
185C2×Dic14—48no—
186C2×Dic14—44no—
187C2×Dic14—48no—
188C2×Dic14—44no—
189C7×C22⋊C4—28no—
190C7×C22⋊C4—44no—
191C7×C22⋊C4—44no—
192C22⋊Dic7—28no—
193C22⋊Dic7—44no—
194C22⋊Dic7—44no—
195C4⋊Dic7—28no—
196Dic7⋊C4—28no—
197Dic7⋊C4—28no—
198Dic7⋊C4—48no—
199C23×Q8—71no—
200C24×C4—71no—
201C2×C22⋊Q8—78no—
202C2×C22⋊Q8—144no—
203C2×C22⋊Q8—144no—
204C22×C4⋊C4—71no—
205C22×C4⋊C4—71no—
206C22×C4⋊C4—71no—
207C22×C4⋊C4—72no—
208C22×C22⋊C4—71no—
209C22×C22⋊C4—71no—
210C22×C22⋊C4—71no—
211C22×C22⋊C4C2×D711yescharacteristic
212C22×C22⋊C4C2×D711yescharacteristic
213C22×C22⋊C4C2×D711yescharacteristic
214#381—144no—
215#380—144no—
216C2×#95—72no—
217C2×#95—72no—
218C2×#95—72no—
219C2×#95—72no—
220C2×#95—72no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95—72no—
224C2×#95—72no—
225C2×#95C2×D712yes—
226C2×#95C2×D712yes—
227C23×C7—24no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—24no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—24no—
234C23×C7—22no—
235C23×C7—24no—
236C23×C7—24no—
237C23×C7—22no—
238C23×C7—24no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7C2×C22⋊C414yes—
248C23×C7C23×C411yescharacteristic
249C23×C7C22×D411yescharacteristic
250C23×C7C22×D411yescharacteristic
251C23×C7H(5,𝔽2)12yes—
252C23×C7H(5,𝔽2)12yes—
253C23×C7H(5,𝔽2)12yes—
254C23×C72−1+412yes—
255C23×C72−1+412yes—
256C23×C72−1+412yes—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—41no—
262C2×C4×C7—42no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—22no—
285C2×Dic7—24no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—416no—
294C2×Dic7—41no—
295C2×Dic7—42no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—42no—
308C2×Dic7—42no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312C2×Dic7—42no—
313C2×Dic7—42no—
314C2×Dic7C2×C22⋊C414yes—
315C2×Dic7C2×C22⋊C414yes—
316Dic14—416no—
317C25C22×D711yescharacteristic
318C22×Q8—74no—
319C22×Q8—142no—
320C22×Q8—142no—
321C22×Q8—282no—
322C22×Q8—282no—
323C23×C4—71no—
324C23×C4—71no—
325C23×C4—74no—
326C23×C4—74no—
327C23×C4—74no—
328C23×C4—71no—
329C23×C4—71no—
330C23×C4—72no—
331C23×C4—72no—
332C23×C4—282no—
333C23×C4—282no—
334C23×C4C22×D712yes—
335C23×C4C22×D712yes—
336C23×C4C22×D711yescharacteristic
337C23×C4C22×D711yescharacteristic
338C22⋊Q8—1432no—
339C2×C4⋊C4—74no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—74no—
342C2×C4⋊C4—142no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4—282no—
350C2×C4⋊C4—282no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—74no—
358C2×C22⋊C4—142no—
359C2×C22⋊C4—142no—
360C2×C22⋊C4—142no—
361C2×C22⋊C4—142no—
362C2×C22⋊C4—282no—
363C2×C22⋊C4—282no—
364C2×C22⋊C4C4×D714yes—
365#95—42no—
366#95—42no—
367#95—144no—
368#95—144no—
369#95—144no—
370#95—282no—
371#95—282no—
372#95—282no—
373#95—282no—
374#95—282no—
375#95—282no—
376C22×C7—24no—
377C22×C7—24no—
378C22×C7—24no—
379C22×C7—44no—
380C22×C7—44no—
381C22×C7—44no—
382C22×C7—44no—
383C22×C7—44no—
384C22×C7—44no—
385C22×C7C22×C22⋊C411yescharacteristic
386C22×C7#51812yes—
387C22×C7#51911yescharacteristic
388C22×C7#51911yescharacteristic
389C22×C7#52012yes—
390C22×C7#53411yescharacteristic
391C22×C7#53412yes—
392C22×C7#53411yescharacteristic
393C22×C7#53511yescharacteristic
394C22×C7#53612yes—
395C22×C7#53611yescharacteristic
396C22×C7#53612yes—
397C22×C7#53712yes—
398C22×C7#56212yes—
399C22×C7#56212yes—
400C22×C7#56212yes—
401C22×C7#56212yes—
402C22×C7#56312yes—
403C22×C7#56312yes—
404C22×C7#56312yes—
405C22×C7#56312yes—
406C4×C7—42no—
407C4×C7—42no—
408C4×C7—82no—
409C4×C7—82no—
410Dic7—24no—
411Dic7—24no—
412Dic7—42no—
413Dic7—42no—
414Dic7—82no—
415Dic7—82no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24C2×C4×D712yes—
423C24D4×D712yes—
424C24D4×D712yes—
425C24C23×D711yescharacteristic
426C2×Q8—148no—
427C2×Q8—288no—
428C2×Q8—284no—
429C2×Q8—288no—
430C2×Q8—284no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—72no—
454C22×C4—72no—
455C22×C4—72no—
456C22×C4—72no—
457C22×C4—72no—
458C22×C4—72no—
459C22×C4—716no—
460C22×C4—72no—
461C22×C4—72no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—142no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—142no—
488C22×C4—288no—
489C22×C4—284no—
490C22×C4—288no—
491C22×C4—284no—
492C22×C4C2×C4×D712yes—
493C22×C4C2×C4×D712yes—
494C4⋊C4—148no—
495C4⋊C4—148no—
496C4⋊C4—148no—
497C4⋊C4—288no—
498C22⋊C4—28no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—148no—
502C22⋊C4—284no—
503C22⋊C4—284no—
504C2×C7—44no—
505C2×C7#230811yescharacteristic
506C2×C7#231011yescharacteristic
507C2×C7#231111yescharacteristic
508C2×C7#237812yes—
509C2×C7#237812yes—
510C2×C7#281412yes—
511C2×C7#281412yes—
512C2×C7#281612yes—
513C2×C7#281712yes—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23D7×C22⋊C414yes—
535C23C22×C4×D711yescharacteristic
536C23C2×D4×D711yescharacteristic
537C23C2×D4×D711yescharacteristic
538C23#654812yes—
539C23#654812yes—
540C23#655312yes—
541C23#655412yes—
542C23#655412yes—
543C23#655412yes—
544Q8—2816no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—74no—
570C2×C4—74no—
571C2×C4—144no—
572C2×C4—144no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—144no—
576C2×C4—144no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—144no—
582C2×C4—144no—
583C2×C4—2816no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—282no—
587C2×C4—282no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—282no—
591C2×C4—282no—
592C2×C4—282no—
593C2×C4—282no—
594C2×C4—282no—
595C2×C4—282no—
596C2×C4—282no—
597C2×C4—282no—
598C2×C4—282no—
599C2×C4—282no—
600C2×C4—282no—
601C2×C4—282no—
602C2×C4—282no—
603C2×C4—282no—
604C7#1466011yescharacteristic, complemented, Sylow
605C22—24no—
606C22—24no—
607C22—24no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22C2×D7×C22⋊C411yescharacteristic
615C22#9020711yescharacteristic
616C22#9020711yescharacteristic
617C22#9023612yes—
618C22#9023712yes—
619C22#9028011yescharacteristic
620C22#9028311yescharacteristic
621C22#9028812yes—
622C22#9029612yes—
623C22#9031212yes—
624C22#9031212yes—
625C22#9033912yes—
626C22#9033912yes—
627C22#9033911yescharacteristic
628C22#9034212yes—
629C22#9034411yescharacteristic
630C22#9035812yes—
631C22#9036312yes—
632C22#9037512yes—
633C22#9040312yes—
634C22#9040312yes—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4—144no—
640C4—144no—
641C4—282no—
642C4—282no—
643C4—562no—
644C4—562no—
645C2—44no—
646C2#1173149811yescharacteristic
647C2#1173210711yescharacteristic
648C2#1173231911yescharacteristic
649C2#1173293612yes—
650C2#1173293712yes—
651C2#1173452912yes—
652C2#1173453112yes—
653C2#1173509112yes—
654C2#1173517212yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1440C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenter#1440D7
hypocenterC7#14660
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
8. #11727178C2semidirect
9. #11727175C2semidirect
10. #11727165C2semidirect
11. #11727163C2semidirect
26. C2×#6441C4semidirect
604. C7#14660semidirect

Maximal subgroups

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

subgroupindex
C22×#64412
C2×#897152
C2×#897142
C7×#14402
#117293032
#117293022
#117271782
#117271752
#117271652
#117271632
#146607