Finite Group Databasebeta

← #13280781

Subgroups — #13280781

Order 1792.

1622 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89701C211yesmaximal, characteristic, complemented
2C2×C4×(C2×D7)⋊C4C211yesmaximal, characteristic
3C2×#89445C211yesmaximal, characteristic
4C7×#1434C211yesmaximal, characteristic, complemented
5#11727181C212yesmaximal
6#11727181C211yesmaximal, characteristic
7(C2×D7)⋊#95C212yesmaximal
8#11727178C211yesmaximal, characteristic, complemented
9#11727173C211yesmaximal, characteristic, complemented
10#11727172C212yesmaximal, complemented
11#11727161C212yesmaximal, complemented
12C23×C4×D7C2211yescharacteristic
13C23×Dic14C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×(C2×D7)⋊C4C2211yescharacteristic
16C22×(C2×D7)⋊C4C2211yescharacteristic
17C22×(C2×D7)⋊C4C2212yes—
18C22×(C2×D7)⋊C4C2212yes—
19C22×C4⋊Dic7C2212yes—
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4×Dic7C2211yescharacteristic
22C2×C7×#95C2212yes—
23C2×C7×#95C2211yescharacteristic
24C2×C7×#95C2212yes—
25C2×C7×#95C2211yescharacteristic
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C4×(C2×D7)⋊C4—24no—
36#89701—24no—
37#89445C418yescomplemented
38#14319—71nomaximal, Sylow
39C24×D7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2311yescharacteristic
45C23×Dic7C2312yes—
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C22×C4×D7—22no—
51C22×C4×D7—22no—
52C22×C4×D7—42no—
53C22×C4×D7—42no—
54C22×C4×D7Q814yescomplemented
55C22×Dic14—22no—
56C22×Dic14—22no—
57C22×Dic14—42no—
58C22×Dic14—42no—
59C22×Dic14C2×C414yescomplemented
60C2×C42×C7—42no—
61C2×C42×C7D414yes—
62C2×(C2×D7)⋊C4—22no—
63C2×(C2×D7)⋊C4—22no—
64C2×(C2×D7)⋊C4—22no—
65C2×(C2×D7)⋊C4—22no—
66C2×(C2×D7)⋊C4—22no—
67C2×(C2×D7)⋊C4—24no—
68C2×(C2×D7)⋊C4—42no—
69C2×(C2×D7)⋊C4—42no—
70C2×(C2×D7)⋊C4—42no—
71C2×(C2×D7)⋊C4—42no—
72C2×(C2×D7)⋊C4C2×C414yes—
73C2×(C2×D7)⋊C4C2×C418yes—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—24no—
76C2×C4⋊Dic7—42no—
77C2×C4⋊Dic7—42no—
78C2×C4⋊Dic7—42no—
79C2×C4×Dic7—22no—
80C2×C4×Dic7—42no—
81C7×#95—22no—
82C7×#95—24no—
83C7×#95—42no—
84C7×#95C2×C414yescomplemented
85C7⋊#95—24no—
86C7⋊#95—24no—
87C7⋊#95—22no—
88C7⋊#95—22no—
89C7⋊#95—42no—
90C7⋊#95—42no—
91C7⋊#95—42no—
92C7⋊#95—42no—
93C7⋊#95C2×C418yescomplemented
94C2×#393—71no—
95C2×#391—71no—
96C2×C4×C22⋊C4—71no—
97#1446—71no—
98#1444—71no—
99#1443—72no—
100#1439—71no—
101#1439—72no—
102#1436—72no—
103#1434D711yescharacteristic, complemented
104C22⋊#95—72no—
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×Q812yes—
116C23×D7#4112yes—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
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×C7C22×C414yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×Q812yes—
144C22×C4×C7#4112yes—
145C22×C4×C7#4112yes—
146C22×Dic7—22no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—22no—
153C22×Dic7—21no—
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×Dic7C22×C412yes—
172C22×Dic7C22×C414yes—
173C22×Dic7C2×Q812yes—
174C22×Dic7#4112yes—
175C2×C4×D7—28no—
176C2×C4×D7—48no—
177C2×C4×D7—44no—
178C2×C4×D7—44no—
179C2×C4×D7—48no—
180C2×Dic14—28no—
181C2×Dic14—48no—
182C2×Dic14—44no—
183C2×Dic14—44no—
184C2×Dic14—48no—
185C42×C7—44no—
186(C2×D7)⋊C4—28no—
187(C2×D7)⋊C4—28no—
188(C2×D7)⋊C4—48no—
189(C2×D7)⋊C4—48no—
190C4⋊Dic7—48no—
191C4⋊Dic7—44no—
192C4×Dic7—44no—
193C23×Q8—71no—
194C24×C4—71no—
195C22×C4⋊C4—72no—
196C22×C4⋊C4—71no—
197C22×C22⋊C4—71no—
198C22×C22⋊C4—72no—
199C22×C22⋊C4—71no—
200C22×C22⋊C4—72no—
201C22×C42—71no—
202C22×C42C2×D711yescharacteristic
203#393—78no—
204#391—144no—
205C4×C22⋊C4—144no—
206C2×#95—72no—
207C2×#95—72no—
208C2×#95—72no—
209C2×#95—71no—
210C2×#95—71no—
211C2×#95—72no—
212C2×#95—72no—
213C2×#95—72no—
214C2×#95—72no—
215C2×#95C2×D712yes—
216C2×#95C2×D711yescharacteristic
217C2×#95C2×D712yes—
218C2×#95C2×D711yescharacteristic
219C23×C7C23×C411yescharacteristic
220C23×C7C22×D411yescharacteristic
221C23×C7C22×Q811yescharacteristic
222C23×C7C2×#4111yescharacteristic
223C23×C7C2×#4111yescharacteristic
224C23×C7C2×#4111yescharacteristic
225C23×C7C2×#4111yescharacteristic
226C23×C7H(5,𝔽2)12yes—
227C23×C7H(5,𝔽2)12yes—
228C23×C7H(5,𝔽2)12yes—
229C23×C72−1+412yes—
230C22×D7—22no—
231C22×D7—22no—
232C22×D7—24no—
233C22×D7—24no—
234C22×D7—22no—
235C22×D7—22no—
236C22×D7—22no—
237C22×D7—22no—
238C22×D7—24no—
239C22×D7—24no—
240C22×D7—24no—
241C22×D7—24no—
242C22×D7—41no—
243C22×D7—41no—
244C22×D7—41no—
245C22×D7—41no—
246C22×D7—42no—
247C22×D7—42no—
248C22×D7—42no—
249C22×D7—42no—
250C22×D7—42no—
251C22×D7—42no—
252C22×D7C4×Q814yes—
253C2×C4×C7—24no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
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×C7C4×D414yes—
281C2×C4×C7C22⋊Q814yes—
282C2×Dic7—22no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—42no—
292C2×Dic7—41no—
293C2×Dic7—41no—
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—
311C4×D7—416no—
312Dic14—416no—
313C25—71no—
314C22×Q8—74no—
315C22×Q8—142no—
316C22×Q8—142no—
317C22×Q8—282no—
318C22×Q8—282no—
319C23×C4—71no—
320C23×C4—72no—
321C23×C4—71no—
322C23×C4—71no—
323C23×C4—74no—
324C23×C4—72no—
325C23×C4—142no—
326C23×C4—142no—
327C23×C4—282no—
328C23×C4—282no—
329C23×C4C22×D712yes—
330C23×C4C22×D712yes—
331C23×C4C22×D711yescharacteristic
332C23×C4C22×D711yescharacteristic
333C23×C4C22×D711yescharacteristic
334C2×C4⋊C4—144no—
335C2×C4⋊C4—142no—
336C2×C4⋊C4—282no—
337C2×C4⋊C4—282no—
338C2×C4⋊C4—282no—
339C2×C22⋊C4—74no—
340C2×C22⋊C4—78no—
341C2×C22⋊C4—142no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4—142no—
344C2×C22⋊C4—142no—
345C2×C22⋊C4—142no—
346C2×C22⋊C4—144no—
347C2×C22⋊C4—282no—
348C2×C22⋊C4—282no—
349C2×C22⋊C4—282no—
350C2×C22⋊C4—282no—
351C2×C42—42no—
352C2×C42—142no—
353C2×C42—282no—
354C2×C42D2814yes—
355#95—22no—
356#95—24no—
357#95—42no—
358#95—78no—
359#95—142no—
360#95—142no—
361#95—144no—
362#95—144no—
363#95—282no—
364#95—282no—
365#95—282no—
366#95—282no—
367#95C4×D714yescomplemented
368C22×C7C2×C4×D411yescharacteristic
369C22×C7C2×C4×Q811yescharacteristic
370C22×C7C4×#4111yescharacteristic
371C22×C7#51812yes—
372C22×C7#52012yes—
373C22×C7C2×C22⋊Q811yescharacteristic
374C22×C7#52511yescharacteristic
375C22×C7#52911yescharacteristic
376C22×C7#53311yescharacteristic
377C22×C7#53411yescharacteristic
378C22×C7#53511yescharacteristic
379C22×C7#53511yescharacteristic
380C22×C7#53611yescharacteristic
381C22×C7#53812yes—
382C22×C7#53812yes—
383C22×C7#53812yes—
384C22×C7#53812yes—
385C22×C7#53912yes—
386C22×C7#54212yes—
387C22×C7#54212yes—
388C22×C7#54212yes—
389C22×C7#54312yes—
390C22×C7#54412yes—
391C2×D7—28no—
392C2×D7—24no—
393C2×D7—42no—
394C2×D7—42no—
395C2×D7—44no—
396C2×D7—42no—
397C2×D7—42no—
398C2×D7—42no—
399C2×D7—42no—
400C2×D7—44no—
401C2×D7—44no—
402C4×C7—24no—
403C4×C7—24no—
404C4×C7—44no—
405C4×C7—42no—
406C4×C7—82no—
407Dic7—44no—
408Dic7—42no—
409Dic7—42no—
410Dic7—42no—
411Dic7—82no—
412C24—72no—
413C24—72no—
414C24—72no—
415C24—141no—
416C24—141no—
417C24—141no—
418C24—142no—
419C24—141no—
420C24—142no—
421C24—142no—
422C24—142no—
423C24C23×D711yescharacteristic
424C2×Q8—148no—
425C2×Q8—288no—
426C2×Q8—284no—
427C2×Q8—284no—
428C2×Q8—288no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
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—72no—
451C22×C4—74no—
452C22×C4—72no—
453C22×C4—72no—
454C22×C4—141no—
455C22×C4—142no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—141no—
459C22×C4—142no—
460C22×C4—141no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—141no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—142no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—148no—
473C22×C4—142no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—288no—
481C22×C4—284no—
482C22×C4—284no—
483C22×C4—288no—
484C22×C4C2×C4×D714yes—
485C22×C4C2×C4×D712yes—
486C22×C4C2×D2812yes—
487C22×C4C2×D2812yes—
488C22×C4C2×D2812yes—
489C22×C4#115712yes—
490C22×C4#115912yes—
491C22×C4Q8×D712yes—
492C4⋊C4—284no—
493C4⋊C4—288no—
494C22⋊C4—148no—
495C22⋊C4—148no—
496C22⋊C4—288no—
497C22⋊C4—288no—
498C42—44no—
499C42—284no—
500C2×C7C4×C22⋊Q811yescharacteristic
501C2×C7#232111yescharacteristic
502C2×C7#232611yescharacteristic
503C2×C7#232912yes—
504C2×C7#233712yes—
505C2×C7#262611yescharacteristic
506C2×C7#262811yescharacteristic
507C2×C7#263111yescharacteristic
508C2×C7#263811yescharacteristic
509C2×C7#265312yes—
510C2×C7#266512yes—
511D7—44no—
512C23—74no—
513C23—144no—
514C23—144no—
515C23—142no—
516C23—142no—
517C23—144no—
518C23—142no—
519C23—142no—
520C23—144no—
521C23—142no—
522C23—142no—
523C23—144no—
524C23—144no—
525C23—282no—
526C23—282no—
527C23—281no—
528C23—281no—
529C23—281no—
530C23—281no—
531C23—282no—
532C23—282no—
533C23—282no—
534C23—282no—
535C23C22×C4×D711yescharacteristic
536C23C22×D2811yescharacteristic
537C23C2×#115711yescharacteristic
538C23C2×#115911yescharacteristic
539C23#654812yes—
540C23C2×Q8×D711yescharacteristic
541C23D7×#4111yescharacteristic
542C23D7×#4111yescharacteristic
543C23#655312yes—
544C23#655312yes—
545C23#655412yes—
546Q8—2816no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
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—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—144no—
575C2×C4—142no—
576C2×C4—144no—
577C2×C4—142no—
578C2×C4—144no—
579C2×C4—142no—
580C2×C4—144no—
581C2×C4—144no—
582C2×C4—144no—
583C2×C4—2816no—
584C2×C4—282no—
585C2×C4—281no—
586C2×C4—282no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—282no—
592C2×C4—281no—
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—
604C2×C4C4×D2814yes—
605C2×C4#646014yes—
606C7#1431911yescharacteristic, complemented, Sylow
607C22—144no—
608C22—148no—
609C22—282no—
610C22—282no—
611C22—282no—
612C22—284no—
613C22—284no—
614C22—282no—
615C22—284no—
616C22—282no—
617C22—282no—
618C22C2×C4×D2811yescharacteristic
619C22#9018411yescharacteristic
620C22C2×#646011yescharacteristic
621C22#9021412yes—
622C22#9021511yescharacteristic
623C22#9021611yescharacteristic
624C22#9021611yescharacteristic
625C22#9021711yescharacteristic
626C22#9022512yes—
627C22C4×#115911yescharacteristic
628C22#9023612yes—
629C22#9024811yescharacteristic
630C22#9025112yes—
631C22C4×Q8×D711yescharacteristic
632C22#9026911yescharacteristic
633C22#9030812yes—
634C22#9032512yes—
635C22#9035112yes—
636C22#9035312yes—
637C22#9037312yes—
638C22#9038012yes—
639C22#9039512yes—
640C22#9039512yes—
641C4—24no—
642C4—24no—
643C4—42no—
644C4—44no—
645C4—82no—
646C4—282no—
647C4—284no—
648C4—282no—
649C4—282no—
650C4—562no—
651C2—284no—
652C2#1173083211yescharacteristic
653C2#1173101411yescharacteristic
654C2C4×#646011yescharacteristic
655C2#1173139711yescharacteristic
656C2#1173153611yescharacteristic
657C2#1173163911yescharacteristic
658C2#1173175411yescharacteristic
659C2#1173259012yes—
660C2#1173274412yes—
661C2#1173291712yes—
662C2#1173335412yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1434D7
hypocenterC7#14319
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C2×#89701C2semidirect
4. C7×#1434C2semidirect
8. #11727178C2semidirect
9. #11727173C2semidirect
10. #11727172C2semidirect
11. #11727161C2semidirect
37. #89445C4semidirect
54. C22×C4×D7Q8semidirect
59. C22×Dic14C2×C4semidirect
84. C7×#95C2×C4semidirect
93. C7⋊#95C2×C4semidirect
103. #1434D7semidirect
367. #95C4×D7semidirect
606. C7#14319semidirect

Maximal subgroups

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

subgroupindex
C2×#897012
C2×C4×(C2×D7)⋊C42
C2×#894452
C7×#14342
#117271812
(C2×D7)⋊#952
#117271782
#117271732
#117271722
#117271612
#143197