Finite Group Databasebeta

← #13275727

Subgroups — #13275727

Order 1792.

1918 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#381C211yesmaximal, characteristic
2C2×#89983C211yesmaximal, characteristic
3C22×C7⋊#95C211yesmaximal, characteristic
4C2×#89714C211yesmaximal, characteristic
5C2×C22⋊C4×Dic7C211yesmaximal, characteristic
6#11729298C211yesmaximal, characteristic
7#11729298C211yesmaximal, characteristic
8#11727172C212yesmaximal, complemented
9#11727168C212yesmaximal, complemented
10#11727159C212yesmaximal, complemented
11#11727157C212yesmaximal, complemented
12C24×C4×C7C2211yescharacteristic
13C24×Dic7C2211yescharacteristic
14C24×Dic7C2211yescharacteristic
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C22⋊Dic7C2211yescharacteristic
18C22×C22⋊Dic7C2211yescharacteristic
19C22×Dic7⋊C4C2212yes—
20C22×C4×Dic7C2212yes—
21C7×#381—24no—
22C2×C7×#95C2212yes—
23C2×C7×#95C2212yes—
24#89983—24no—
25C2×C7⋊#95C418yescomplemented
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35C2×C7⋊#95C2211yescharacteristic
36C2×C7⋊#95C2211yescharacteristic
37C2×C7⋊#95C2211yescharacteristic
38#89714—24no—
39C22⋊C4×Dic7—24no—
40#13874—71nomaximal, Sylow
41C25×C7C2311yescharacteristic
42C23×C4×C7—22no—
43C23×C4×C7—22no—
44C23×C4×C7—22no—
45C23×C4×C7—22no—
46C23×C4×C7—22no—
47C23×C4×C7—22no—
48C23×C4×C7C2×C414yes—
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2312yes—
53C23×Dic7—22no—
54C23×Dic7—22no—
55C23×Dic7—22no—
56C23×Dic7—22no—
57C23×Dic7—24no—
58C23×Dic7—24no—
59C23×Dic7—42no—
60C23×Dic7—42no—
61C23×Dic7C2×C414yes—
62C23×Dic7C2×C414yes—
63C23×Dic7C2312yes—
64C23×Dic7C2311yescharacteristic
65C23×Dic7C2311yescharacteristic
66C23×Dic7C2312yes—
67C23×Dic7C2311yescharacteristic
68C23×Dic7C2311yescharacteristic
69C2×C7×C22⋊C4—22no—
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4D414yes—
75C2×C22⋊Dic7—22no—
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7—22no—
78C2×C22⋊Dic7—22no—
79C2×C22⋊Dic7—22no—
80C2×C22⋊Dic7D414yes—
81C2×Dic7⋊C4—24no—
82C2×Dic7⋊C4—24no—
83C2×Dic7⋊C4—42no—
84C2×C4×Dic7—24no—
85C2×C4×Dic7—24no—
86C2×C4×Dic7—42no—
87C7×#95—24no—
88C7×#95—24no—
89C7⋊#95—24no—
90C7⋊#95—24no—
91C7⋊#95—24no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C7⋊#95—42no—
96C7⋊#95—42no—
97C7⋊#95C2×C414yes—
98C7⋊#95C2×C414yes—
99C7⋊#95C2×C414yes—
100C7⋊#95C2×C414yes—
101C2×#381—71no—
102C2×#381D711yescharacteristic
103C2×#380—71no—
104C2×C4×C22⋊C4—71no—
105C22×#95—71no—
106#1443—72no—
107#1442—72no—
108#1436—72no—
109#1434—72no—
110C22⋊#95—71no—
111C22⋊#95—71no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7C22×C412yes—
117C24×C7C2×D412yes—
118C24×C7C2×D412yes—
119C24×C7#4112yes—
120C24×C7#4112yes—
121C24×C7#4112yes—
122C24×C7#4112yes—
123C24×C7C2411yescharacteristic
124C22×C4×C7—28no—
125C22×C4×C7—28no—
126C22×C4×C7—28no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
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×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—44no—
153C22×C4×C7—44no—
154C22×C4×C7—44no—
155C22×C4×C7—44no—
156C22×C4×C7C22×C412yes—
157C22×C4×C7C22×C412yes—
158C22×C4×C7C2×D414yes—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—28no—
166C22×Dic7—28no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—22no—
180C22×Dic7—22no—
181C22×Dic7—28no—
182C22×Dic7—28no—
183C22×Dic7—22no—
184C22×Dic7—22no—
185C22×Dic7—22no—
186C22×Dic7—22no—
187C22×Dic7—22no—
188C22×Dic7—22no—
189C22×Dic7—22no—
190C22×Dic7—44no—
191C22×Dic7—44no—
192C22×Dic7—48no—
193C22×Dic7—48no—
194C22×Dic7—48no—
195C22×Dic7—48no—
196C22×Dic7C22×C412yes—
197C22×Dic7C22×C412yes—
198C22×Dic7C22×C412yes—
199C22×Dic7C22×C412yes—
200C22×Dic7C2×D414yes—
201C22×Dic7#4114yes—
202C22×Dic7#4114yes—
203C7×C22⋊C4—28no—
204C7×C22⋊C4—44no—
205C22⋊Dic7—28no—
206C22⋊Dic7—44no—
207Dic7⋊C4—48no—
208C4×Dic7—48no—
209C24×C4—71no—
210C24×C4—71no—
211C24×C4C2×D711yescharacteristic
212C22×C4⋊C4—72no—
213C22×C22⋊C4—71no—
214C22×C22⋊C4—71no—
215C22×C22⋊C4C2×D711yescharacteristic
216C22×C22⋊C4C2×D711yescharacteristic
217C22×C42—72no—
218#381—24no—
219#381—144no—
220#380—144no—
221C4×C22⋊C4—144no—
222C2×#95—78no—
223C2×#95—71no—
224C2×#95—71no—
225C2×#95—71no—
226C2×#95—71no—
227C2×#95—72no—
228C2×#95—72no—
229C2×#95—72no—
230C2×#95—72no—
231C2×#95—72no—
232C2×#95—72no—
233C2×#95—72no—
234C2×#95—72no—
235C2×#95C2×D712yes—
236C2×#95C2×D712yes—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—24no—
242C23×C7—24no—
243C23×C7—24no—
244C23×C7—24no—
245C23×C7—24no—
246C23×C7—24no—
247C23×C7—24no—
248C23×C7—24no—
249C23×C7—24no—
250C23×C7—24no—
251C23×C7—24no—
252C23×C7—24no—
253C23×C7#11714yes—
254C23×C7C4×D414yes—
255C23×C7C4×D414yes—
256C23×C7#12114yes—
257C23×C7#12314yes—
258C23×C7#12414yes—
259C23×C7#12614yes—
260C23×C7C23×C411yescharacteristic
261C23×C7C22×D411yescharacteristic
262C23×C7C22×D411yescharacteristic
263C23×C7C2×#4111yescharacteristic
264C23×C7C2×#4111yescharacteristic
265C23×C7C2×#4111yescharacteristic
266C23×C7C2×#4111yescharacteristic
267C23×C7C2×#4112yes—
268C23×C7C2×#4112yes—
269C23×C7H(5,𝔽2)12yes—
270C23×C72−1+412yes—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—24no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—416no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—24no—
307C2×Dic7—24no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—24no—
315C2×Dic7—24no—
316C2×Dic7—24no—
317C2×Dic7—416no—
318C2×Dic7—416no—
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×Dic7—42no—
328C2×Dic7—42no—
329C2×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7—42no—
332C2×Dic7—42no—
333C2×Dic7—42no—
334C2×Dic7#11714yes—
335C2×Dic7#11714yes—
336C25C22×D711yescharacteristic
337C23×C4—22no—
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—22no—
343C23×C4—71no—
344C23×C4—71no—
345C23×C4—72no—
346C23×C4—71no—
347C23×C4—71no—
348C23×C4—74no—
349C23×C4—74no—
350C23×C4—72no—
351C23×C4—144no—
352C23×C4—144no—
353C23×C4—142no—
354C23×C4—142no—
355C23×C4—142no—
356C23×C4—142no—
357C23×C4—282no—
358C23×C4—282no—
359C23×C4C4×D714yes—
360C23×C4C22×D712yes—
361C23×C4C22×D711yescharacteristic
362C23×C4C22×D711yescharacteristic
363C23×C4C22×D712yes—
364C2×C4⋊C4—144no—
365C2×C4⋊C4—144no—
366C2×C4⋊C4—282no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—74no—
373C2×C22⋊C4—142no—
374C2×C22⋊C4—142no—
375C2×C22⋊C4—142no—
376C2×C22⋊C4—142no—
377C2×C22⋊C4—142no—
378C2×C22⋊C4C22⋊D714yes—
379C2×C42—144no—
380C2×C42—144no—
381C2×C42—282no—
382#95—24no—
383#95—24no—
384#95—74no—
385#95—74no—
386#95—74no—
387#95—74no—
388#95—144no—
389#95—144no—
390#95—144no—
391#95—144no—
392#95—144no—
393#95—144no—
394#95—282no—
395#95—282no—
396C22×C7—24no—
397C22×C7—24no—
398C22×C7—24no—
399C22×C7—24no—
400C22×C7—24no—
401C22×C7—28no—
402C22×C7—28no—
403C22×C7—28no—
404C22×C7—28no—
405C22×C7—24no—
406C22×C7#38818yes—
407C22×C7C2×#11711yescharacteristic
408C22×C7C2×#11711yescharacteristic
409C22×C7C2×#11711yescharacteristic
410C22×C7C2×C4×D411yescharacteristic
411C22×C7C2×C4×D411yescharacteristic
412C22×C7#51811yescharacteristic
413C22×C7#51911yescharacteristic
414C22×C7C2×#12111yescharacteristic
415C22×C7C2×#12311yescharacteristic
416C22×C7C2×#12411yescharacteristic
417C22×C7C2×#12611yescharacteristic
418C22×C7#52912yes—
419C22×C7#52912yes—
420C22×C7#53912yes—
421C22×C7#53912yes—
422C22×C7#54612yes—
423C22×C7#54612yes—
424C22×C7#54712yes—
425C22×C7#54712yes—
426C22×C7#55112yes—
427C22×C7#55112yes—
428C22×C7#55212yes—
429C22×C7#55212yes—
430C4×C7—44no—
431C4×C7—42no—
432C4×C7—42no—
433C4×C7—82no—
434Dic7—24no—
435Dic7—24no—
436Dic7—42no—
437Dic7—42no—
438Dic7—44no—
439Dic7—82no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24C2×C4×D712yes—
445C24#115712yes—
446C24D4×D712yes—
447C24#115912yes—
448C24#115912yes—
449C24#115912yes—
450C24C2×C22⋊D712yes—
451C24C23×D711yescharacteristic
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—72no—
485C22×C4—72no—
486C22×C4—74no—
487C22×C4—74no—
488C22×C4—72no—
489C22×C4—72no—
490C22×C4—74no—
491C22×C4—148no—
492C22×C4—148no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—141no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—141no—
499C22×C4—148no—
500C22×C4—148no—
501C22×C4—142no—
502C22×C4—141no—
503C22×C4—141no—
504C22×C4—142no—
505C22×C4—141no—
506C22×C4—142no—
507C22×C4—142no—
508C22×C4—142no—
509C22×C4—142no—
510C22×C4—142no—
511C22×C4—142no—
512C22×C4—142no—
513C22×C4—142no—
514C22×C4—141no—
515C22×C4—142no—
516C22×C4—142no—
517C22×C4—142no—
518C22×C4—142no—
519C22×C4—142no—
520C22×C4—142no—
521C22×C4—142no—
522C22×C4—284no—
523C22×C4—284no—
524C22×C4—288no—
525C22×C4—288no—
526C22×C4—288no—
527C22×C4—288no—
528C22×C4C2×C4×D712yes—
529C22×C4C2×C4×D712yes—
530C22×C4C2×C22⋊D714yes—
531C4⋊C4—288no—
532C22⋊C4—28no—
533C22⋊C4—44no—
534C22⋊C4—148no—
535C22⋊C4—284no—
536C42—288no—
537C2×C7—28no—
538C2×C7#228311yescharacteristic
539C2×C7C2×#38811yescharacteristic
540C2×C7#231311yescharacteristic
541C2×C7#234211yescharacteristic
542C2×C7#234211yescharacteristic
543C2×C7#234411yescharacteristic
544C2×C7#234411yescharacteristic
545C2×C7#241912yes—
546C2×C7#242512yes—
547C2×C7#248812yes—
548C2×C7#249312yes—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23#644014yes—
566C23#644214yes—
567C23(C22⋊D7)⋊C414yes—
568C23#644614yes—
569C23C4×C22⋊D714yes—
570C23#649914yes—
571C23#650214yes—
572C23C22×C4×D711yescharacteristic
573C23C2×#115711yescharacteristic
574C23C2×D4×D711yescharacteristic
575C23C2×#115911yescharacteristic
576C23C2×#115911yescharacteristic
577C23C2×#115911yescharacteristic
578C23D7×#4112yes—
579C23D7×#4112yes—
580C23#655312yes—
581C23#655412yes—
582C23C22×C22⋊D711yescharacteristic
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—416no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—74no—
613C2×C4—74no—
614C2×C4—142no—
615C2×C4—142no—
616C2×C4—142no—
617C2×C4—144no—
618C2×C4—142no—
619C2×C4—144no—
620C2×C4—144no—
621C2×C4—144no—
622C2×C4—142no—
623C2×C4—144no—
624C2×C4—142no—
625C2×C4—142no—
626C2×C4—142no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—144no—
630C2×C4—144no—
631C2×C4—2816no—
632C2×C4—2816no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—282no—
637C2×C4—282no—
638C2×C4—282no—
639C2×C4—282no—
640C2×C4—282no—
641C2×C4—282no—
642C2×C4—282no—
643C2×C4—282no—
644C2×C4—282no—
645C2×C4—282no—
646C2×C4—282no—
647C2×C4—282no—
648C7#1387411yescharacteristic, complemented, Sylow
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—28no—
654C22—28no—
655C22—24no—
656C22—24no—
657C22—28no—
658C22—28no—
659C22#8971718yes—
660C22C2×#644011yescharacteristic
661C22C2×#644211yescharacteristic
662C22C2×(C22⋊D7)⋊C411yescharacteristic
663C22C2×#644611yescharacteristic
664C22#9020711yescharacteristic
665C22D7×#11711yescharacteristic
666C22D7×#11711yescharacteristic
667C22#9021211yescharacteristic
668C22#9021812yes—
669C22#9022212yes—
670C22#9034012yes—
671C22#9034012yes—
672C22#9034112yes—
673C22#9034512yes—
674C22#9034612yes—
675C22#9034912yes—
676C22#9035612yes—
677C22#9035612yes—
678C22C2×C4×C22⋊D711yescharacteristic
679C22C2×#649911yescharacteristic
680C22C2×#650211yescharacteristic
681C22#9051512yes—
682C22#9052012yes—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—144no—
688C4—144no—
689C4—284no—
690C4—282no—
691C4—282no—
692C4—562no—
693C2—28no—
694C2#1173149611yescharacteristic
695C2#1173149911yescharacteristic
696C2#1173150511yescharacteristic
697C2#1173151012yes—
698C2#1173152312yes—
699C2#1173156711yescharacteristic
700C2C2×#8971711yescharacteristic
701C2#1173968311yescharacteristic
702C2#1173971611yescharacteristic
703C2#1174029212yes—
704C2#1174030212yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#381C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#381D7
hypocenterC7#13874
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
8. #11727172C2semidirect
9. #11727168C2semidirect
10. #11727159C2semidirect
11. #11727157C2semidirect
25. C2×C7⋊#95C4semidirect
648. C7#13874semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3812
C2×#899832
C22×C7⋊#952
C2×#897142
C2×C22⋊C4×Dic72
#117292982
#117271722
#117271682
#117271592
#117271572
#138747