Finite Group Databasebeta

← #13284831

Subgroups — #13284831

Order 1792.

1702 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6459C211yesmaximal, characteristic
2C2×#89759C211yesmaximal, characteristic
3C2×#89436C211yesmaximal, characteristic
4C7×#1441C211yesmaximal, characteristic, complemented
5#11729320C211yesmaximal, characteristic, complemented
6#11729319C211yesmaximal, characteristic, complemented
7#11727183C212yesmaximal
8#11727183C212yesmaximal
9#11727178C212yesmaximal, complemented
10#11727175C212yesmaximal, complemented
11#11727164C211yesmaximal, characteristic, complemented
12C23×C4×D7C2211yescharacteristic
13C23×Dic14C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×(C2×D7)⋊C4C2211yescharacteristic
17C22×(C2×D7)⋊C4C2211yescharacteristic
18C22×(C2×D7)⋊C4C2212yes—
19C22×(C2×D7)⋊C4C2212yes—
20C22×C4⋊Dic7C2211yescharacteristic
21C22×Dic7⋊C4C2211yescharacteristic
22C22×Dic7⋊C4C2211yescharacteristic
23C2×#6459—24no—
24C2×#6459—24no—
25C2×#6459C418yescomplemented
26C2×C7×#95C2211yescharacteristic
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38#89759—24no—
39#89436—24no—
40#14660—71nomaximal, Sylow
41C24×D7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C22×C4×D7—42no—
53C22×C4×D7—42no—
54C22×C4×D7C2×C414yes—
55C22×C4×D7D414yes—
56C22×C4×D7D414yes—
57C22×Dic14—22no—
58C22×Dic14—22no—
59C22×Dic14—42no—
60C22×Dic14—42no—
61C22×Dic14C2×C414yes—
62C2×C7×C4⋊C4—22no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—42no—
66C2×C7×C4⋊C4C2×C414yes—
67C2×(C2×D7)⋊C4—22no—
68C2×(C2×D7)⋊C4—22no—
69C2×(C2×D7)⋊C4—22no—
70C2×(C2×D7)⋊C4—22no—
71C2×(C2×D7)⋊C4—24no—
72C2×(C2×D7)⋊C4—24no—
73C2×(C2×D7)⋊C4—42no—
74C2×(C2×D7)⋊C4—42no—
75C2×(C2×D7)⋊C4—42no—
76C2×(C2×D7)⋊C4—42no—
77C2×(C2×D7)⋊C4C2×C414yes—
78C2×(C2×D7)⋊C4C2×C414yes—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—42no—
81C2×Dic7⋊C4—22no—
82C2×Dic7⋊C4—22no—
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4—22no—
85C2×Dic7⋊C4C2×C414yes—
86C2×Dic7⋊C4C2×C414yes—
87#6459—232no—
88C7×#95—22no—
89C7×#95—42no—
90C7×#95—42no—
91C7⋊#95—22no—
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—71no—
106#1441—71no—
107#1441—71no—
108#1441D711yescharacteristic, complemented
109#1440—72no—
110#1440—72no—
111C24×C7C2411yescharacteristic
112C23×D7—21no—
113C23×D7—21no—
114C23×D7—22no—
115C23×D7—22no—
116C23×D7—22no—
117C23×D7—22no—
118C23×D7—21no—
119C23×D7—21no—
120C23×D7C22×C412yes—
121C23×D7C2×D412yes—
122C23×D7C2×D412yes—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
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×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7C22×C412yes—
150C22×C4×C7C22×C412yes—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—22no—
180C22×Dic7C22×C412yes—
181C22×Dic7C22×C412yes—
182C22×Dic7C22×C412yes—
183C22×Dic7C2×D412yes—
184C22×Dic7C2×D412yes—
185C2×C4×D7—48no—
186C2×C4×D7—44no—
187C2×C4×D7—44no—
188C2×C4×D7—48no—
189C2×C4×D7C22⋊C4116yes—
190C2×Dic14—28no—
191C2×Dic14—48no—
192C2×Dic14—44no—
193C2×Dic14—48no—
194C2×Dic14—44no—
195C7×C4⋊C4—28no—
196C7×C4⋊C4—44no—
197(C2×D7)⋊C4—28no—
198(C2×D7)⋊C4—28no—
199(C2×D7)⋊C4—48no—
200(C2×D7)⋊C4—48no—
201C4⋊Dic7—44no—
202Dic7⋊C4—28no—
203Dic7⋊C4—28no—
204C23×Q8—71no—
205C24×C4—71no—
206C2×C22⋊Q8—78no—
207C2×C22⋊Q8—144no—
208C2×C22⋊Q8—144no—
209C22×C4⋊C4—71no—
210C22×C4⋊C4—71no—
211C22×C4⋊C4—71no—
212C22×C4⋊C4C2×D711yescharacteristic
213C22×C4⋊C4C2×D711yescharacteristic
214C22×C22⋊C4—71no—
215C22×C22⋊C4—71no—
216C22×C22⋊C4—72no—
217C22×C22⋊C4—72no—
218#381—144no—
219#380—144no—
220C2×#95—71no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95—72no—
224C2×#95—72no—
225C2×#95—72no—
226C2×#95—72no—
227C2×#95—72no—
228C2×#95—72no—
229C2×#95C2×D711yescharacteristic
230C2×#95C2×D712yes—
231C2×#95C2×D712yes—
232C23×C7C23×C411yescharacteristic
233C23×C7C22×D411yescharacteristic
234C23×C7C22×D411yescharacteristic
235C23×C7H(5,𝔽2)12yes—
236C23×C7H(5,𝔽2)11yescharacteristic
237C23×C7H(5,𝔽2)12yes—
238C23×C7H(5,𝔽2)11yescharacteristic
239C23×C72−1+412yes—
240C23×C72−1+411yescharacteristic
241C23×C72−1+412yes—
242C23×C72−1+411yescharacteristic
243C22×D7—24no—
244C22×D7—24no—
245C22×D7—22no—
246C22×D7—22no—
247C22×D7—24no—
248C22×D7—22no—
249C22×D7—22no—
250C22×D7—22no—
251C22×D7—22no—
252C22×D7—24no—
253C22×D7—22no—
254C22×D7—22no—
255C22×D7—22no—
256C22×D7—22no—
257C22×D7—42no—
258C22×D7—42no—
259C22×D7—42no—
260C22×D7—42no—
261C22×D7—42no—
262C22×D7—42no—
263C22×D7—42no—
264C22×D7—42no—
265C22×D7C2×C22⋊C414yes—
266C2×C4×C7—24no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—24no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—41no—
276C2×C4×C7—42no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7C2×C22⋊C414yes—
297C2×Dic7—24no—
298C2×Dic7—22no—
299C2×Dic7—24no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—24no—
304C2×Dic7—22no—
305C2×Dic7—22no—
306C2×Dic7—41no—
307C2×Dic7—42no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—42no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323C2×Dic7—42no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—42no—
327C2×Dic7C2×C22⋊C414yes—
328C4×D7—416no—
329Dic14—416no—
330C25—71no—
331C22×Q8—74no—
332C22×Q8—142no—
333C22×Q8—142no—
334C22×Q8—282no—
335C22×Q8—282no—
336C23×C4—71no—
337C23×C4—74no—
338C23×C4—74no—
339C23×C4—74no—
340C23×C4—71no—
341C23×C4—71no—
342C23×C4—72no—
343C23×C4—72no—
344C23×C4—282no—
345C23×C4—282no—
346C23×C4C22×D712yes—
347C23×C4C22×D712yes—
348C23×C4C22×D711yescharacteristic
349C23×C4C22×D711yescharacteristic
350C23×C4C22×D711yescharacteristic
351C22⋊Q8—1432no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—74no—
357C2×C4⋊C4—74no—
358C2×C4⋊C4—142no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4—142no—
361C2×C4⋊C4—142no—
362C2×C4⋊C4—142no—
363C2×C4⋊C4—282no—
364C2×C4⋊C4C4×D714yes—
365C2×C22⋊C4—74no—
366C2×C22⋊C4—74no—
367C2×C22⋊C4—142no—
368C2×C22⋊C4—142no—
369C2×C22⋊C4—142no—
370C2×C22⋊C4—142no—
371C2×C22⋊C4—144no—
372C2×C22⋊C4—144no—
373C2×C22⋊C4—282no—
374C2×C22⋊C4—282no—
375C2×C22⋊C4—282no—
376C2×C22⋊C4—282no—
377#95—22no—
378#95—42no—
379#95—42no—
380#95—144no—
381#95—144no—
382#95—142no—
383#95—282no—
384#95—282no—
385#95—282no—
386#95—282no—
387#95—282no—
388#95—282no—
389C22×C7C22×C22⋊C411yescharacteristic
390C22×C7#51812yes—
391C22×C7#51912yes—
392C22×C7#52011yescharacteristic
393C22×C7#52011yescharacteristic
394C22×C7#53411yescharacteristic
395C22×C7#53412yes—
396C22×C7#53411yescharacteristic
397C22×C7#53511yescharacteristic
398C22×C7#53611yescharacteristic
399C22×C7#53611yescharacteristic
400C22×C7#53612yes—
401C22×C7#53611yescharacteristic
402C22×C7#53711yescharacteristic
403C22×C7#53711yescharacteristic
404C22×C7#56212yes—
405C22×C7#56212yes—
406C22×C7#56212yes—
407C22×C7#56212yes—
408C22×C7#56312yes—
409C22×C7#56312yes—
410C22×C7#56312yes—
411C22×C7#56312yes—
412C2×D7—24no—
413C2×D7—24no—
414C2×D7—24no—
415C2×D7—42no—
416C2×D7—42no—
417C2×D7—42no—
418C2×D7—42no—
419C2×D7—44no—
420C2×D7—44no—
421C2×D7—44no—
422C2×D7—44no—
423C4×C7—24no—
424C4×C7—42no—
425C4×C7—42no—
426C4×C7—82no—
427C4×C7—82no—
428Dic7—24no—
429Dic7—42no—
430Dic7—42no—
431Dic7—82no—
432Dic7—82no—
433C24—72no—
434C24—72no—
435C24—72no—
436C24—141no—
437C24—142no—
438C24—142no—
439C24—141no—
440C24—141no—
441C24—142no—
442C24—141no—
443C24—142no—
444C24C23×D711yescharacteristic
445C2×Q8—148no—
446C2×Q8—288no—
447C2×Q8—284no—
448C2×Q8—288no—
449C2×Q8—284no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—72no—
477C22×C4—72no—
478C22×C4—72no—
479C22×C4—716no—
480C22×C4—72no—
481C22×C4—72no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—141no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4—141no—
488C22×C4—141no—
489C22×C4—142no—
490C22×C4—141no—
491C22×C4—141no—
492C22×C4—142no—
493C22×C4—141no—
494C22×C4—142no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—142no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—142no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—288no—
509C22×C4—284no—
510C22×C4—288no—
511C22×C4—284no—
512C22×C4C2×C4×D712yes—
513C22×C4C2×C4×D712yes—
514C22×C4C2×C4×D712yes—
515C22×C4D4×D712yes—
516C22×C4D4×D712yes—
517C4⋊C4—28no—
518C4⋊C4—44no—
519C4⋊C4—148no—
520C4⋊C4—148no—
521C4⋊C4—284no—
522C22⋊C4—148no—
523C22⋊C4—148no—
524C22⋊C4—288no—
525C22⋊C4—288no—
526C2×C7#230811yescharacteristic
527C2×C7#231011yescharacteristic
528C2×C7#231111yescharacteristic
529C2×C7#237812yes—
530C2×C7#237812yes—
531C2×C7#281412yes—
532C2×C7#281412yes—
533C2×C7#281611yescharacteristic
534C2×C7#281611yescharacteristic
535C2×C7#281711yescharacteristic
536C2×C7#281711yescharacteristic
537D7—44no—
538C23—74no—
539C23—142no—
540C23—142no—
541C23—142no—
542C23—144no—
543C23—144no—
544C23—142no—
545C23—144no—
546C23—142no—
547C23—142no—
548C23—144no—
549C23—142no—
550C23—142no—
551C23—142no—
552C23—142no—
553C23—282no—
554C23—282no—
555C23—282no—
556C23—282no—
557C23—282no—
558C23—282no—
559C23—282no—
560C23—282no—
561C23C22×C4×D711yescharacteristic
562C23C2×D4×D711yescharacteristic
563C23C2×D4×D711yescharacteristic
564C23#654812yes—
565C23#654811yescharacteristic
566C23#655111yescharacteristic
567C23#655112yes—
568C23#655312yes—
569C23#655311yescharacteristic
570C23#655412yes—
571C23#655411yescharacteristic
572Q8—2816no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—74no—
604C2×C4—144no—
605C2×C4—142no—
606C2×C4—144no—
607C2×C4—142no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—142no—
612C2×C4—144no—
613C2×C4—2816no—
614C2×C4—281no—
615C2×C4—282no—
616C2×C4—281no—
617C2×C4—281no—
618C2×C4—281no—
619C2×C4—281no—
620C2×C4—281no—
621C2×C4—282no—
622C2×C4—282no—
623C2×C4—282no—
624C2×C4—282no—
625C2×C4—282no—
626C2×C4—282no—
627C2×C4—282no—
628C2×C4—282no—
629C2×C4—282no—
630C2×C4—282no—
631C2×C4—282no—
632C2×C4—282no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4D7×C22⋊C414yes—
636C7#1466011yescharacteristic, complemented, Sylow
637C22—144no—
638C22—144no—
639C22—144no—
640C22—282no—
641C22—284no—
642C22—284no—
643C22—282no—
644C22—282no—
645C22—284no—
646C22—284no—
647C22—282no—
648C22C2×D7×C22⋊C411yescharacteristic
649C22#9019511yescharacteristic
650C22#9021411yescharacteristic
651C22#9023612yes—
652C22#9026312yes—
653C22#9028011yescharacteristic
654C22#9029612yes—
655C22#9029811yescharacteristic
656C22#9029911yescharacteristic
657C22#9031212yes—
658C22#9031912yes—
659C22#9031911yescharacteristic
660C22#9032011yescharacteristic
661C22#9033712yes—
662C22#9033812yes—
663C22#9034411yescharacteristic
664C22#9035812yes—
665C22#9036611yescharacteristic
666C22#9039012yes—
667C22#9039112yes—
668C22#9040312yes—
669C22#9041511yescharacteristic
670C22#9042612yes—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—144no—
677C4—282no—
678C4—282no—
679C4—562no—
680C4—562no—
681C2—284no—
682C2#1173144811yescharacteristic
683C2#1173231911yescharacteristic
684C2#1173242112yes—
685C2#1173343712yes—
686C2#1173405511yescharacteristic
687C2#1173453011yescharacteristic
688C2#1173468611yescharacteristic
689C2#1173489412yes—
690C2#1173545012yes—
691C2#1173545211yescharacteristic
692C2#1173576111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1441C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1441D7
hypocenterC7#14660
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
4. C7×#1441C2semidirect
5. #11729320C2semidirect
6. #11729319C2semidirect
9. #11727178C2semidirect
10. #11727175C2semidirect
11. #11727164C2semidirect
25. C2×#6459C4semidirect
108. #1441D7semidirect
636. C7#14660semidirect

Maximal subgroups

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

subgroupindex
C22×#64592
C2×#897592
C2×#894362
C7×#14412
#117293202
#117293192
#117271832
#117271782
#117271752
#117271642
#146607