Finite Group Databasebeta

← #13283059

Subgroups — #13283059

Order 1792.

1918 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#390C211yesmaximal, characteristic
2C2×#89992C211yesmaximal, characteristic
3C2×#89992C211yesmaximal, characteristic
4#11729321C212yesmaximal, complemented
5#11729301C212yesmaximal, complemented
6#11729301C212yesmaximal, complemented
7#11729298C212yesmaximal, complemented
8#11729298C212yesmaximal, complemented
9#11729290C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C24×Dic7C2211yescharacteristic
12C24×Dic7C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2212yes—
15C22×C7×C22⋊C4C2212yes—
16C22×C42×C7C2211yescharacteristic
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C22⋊Dic7C2212yes—
20C22×C22⋊Dic7C2211yescharacteristic
21C22×Dic7⋊C4C2212yes—
22C7×#390—24no—
23#89992—24no—
24#89992C418yescomplemented
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34#14513—71nomaximal, Sylow
35C25×C7C2311yescharacteristic
36C25×C7C2311yescharacteristic
37C22×C7×D4—41no—
38C22×C7×D4—41no—
39C22×C7×D4—41no—
40C22×C7×D4—41no—
41C22×C7×D4C2×C414yes—
42C22×C7×D4D414yes—
43C22×C7×D4Q814yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—22no—
48C23×Dic7—24no—
49C23×Dic7—41no—
50C23×Dic7—41no—
51C23×Dic7—41no—
52C23×Dic7—41no—
53C23×Dic7—42no—
54C23×Dic7—42no—
55C23×Dic7—42no—
56C23×Dic7C2×C414yescomplemented
57C23×Dic7C2312yes—
58C23×Dic7C2312yes—
59C23×Dic7C2312yes—
60C23×Dic7C2312yes—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—41no—
63C2×C7×C4⋊C4—41no—
64C2×C7×C22⋊C4—24no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4—24no—
68C2×C7×C22⋊C4—42no—
69C2×C7×C22⋊C4—42no—
70C2×C42×C7—22no—
71C2×C42×C7—41no—
72C2×C42×C7—41no—
73C2×C22⋊Dic7—24no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—24no—
76C2×C22⋊Dic7—42no—
77C2×C22⋊Dic7—42no—
78C2×C22⋊Dic7—41no—
79C2×C22⋊Dic7—41no—
80C2×C22⋊Dic7—42no—
81C2×C22⋊Dic7—41no—
82C2×C22⋊Dic7—41no—
83C2×C22⋊Dic7C2×C418yes—
84C2×C22⋊Dic7C2×C414yescomplemented
85C2×Dic7⋊C4—24no—
86C2×Dic7⋊C4—42no—
87C2×Dic7⋊C4—42no—
88C7⋊#95—24no—
89C7⋊#95—24no—
90C7⋊#95—42no—
91C7⋊#95—42no—
92C7⋊#95—42no—
93C7⋊#95—42no—
94C7⋊#95—42no—
95C7⋊#95—42no—
96C7⋊#95C2×C418yescomplemented
97C2×#390D711yescharacteristic
98C2×#386—71no—
99C2×#386—71no—
100#1442—72no—
101#1439—72no—
102#1439—72no—
103#1436—72no—
104C22⋊#95—72no—
105C22⋊#95—72no—
106C24×C7—22no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7—22no—
112C24×C7—22no—
113C24×C7—21no—
114C24×C7—21no—
115C24×C7—21no—
116C24×C7—21no—
117C24×C7—22no—
118C24×C7—21no—
119C24×C7—21no—
120C24×C7—21no—
121C24×C7—21no—
122C24×C7C22×C412yes—
123C24×C7C22×C412yes—
124C24×C7C2×D412yes—
125C24×C7C2×D412yes—
126C24×C7C2×Q812yes—
127C24×C7C2×Q812yes—
128C24×C7C2411yescharacteristic
129C2×C7×D4—44no—
130C2×C7×D4—44no—
131C2×C7×D4—44no—
132C2×C7×D4—44no—
133C2×C7×D4—44no—
134C2×C7×D4—44no—
135C2×C7×D4C4⋊C4116yescomplemented
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
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×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7C22×C412yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×Q812yes—
156C22×C4×C7#4114yes—
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—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×Dic7—22no—
181C22×Dic7—22no—
182C22×Dic7—48no—
183C22×Dic7—44no—
184C22×Dic7—44no—
185C22×Dic7—48no—
186C22×Dic7—48no—
187C22×Dic7—44no—
188C22×Dic7—44no—
189C22×Dic7—48no—
190C22×Dic7—44no—
191C22×Dic7—44no—
192C22×Dic7C22×C414yes—
193C22×Dic7C22×C414yes—
194C22×Dic7#4114yes—
195C7×C4⋊C4—44no—
196C7×C22⋊C4—48no—
197C7×C22⋊C4—48no—
198C42×C7—44no—
199C22⋊Dic7—48no—
200C22⋊Dic7—44no—
201C22⋊Dic7—48no—
202C22⋊Dic7—44no—
203Dic7⋊C4—48no—
204C23×D4C2×D711yescharacteristic
205C24×C4—71no—
206C24×C4—71no—
207C22×C4⋊C4—72no—
208C22×C4⋊C4C2×D711yescharacteristic
209C22×C22⋊C4—72no—
210C22×C22⋊C4—71no—
211C22×C22⋊C4—72no—
212C22×C22⋊C4—71no—
213C22×C22⋊C4C2×D712yes—
214C22×C22⋊C4C2×D712yes—
215C22×C42C2×D711yescharacteristic
216#390—24no—
217#386—78no—
218#386—144no—
219C2×#95—72no—
220C2×#95—72no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95—72no—
224C2×#95—72no—
225C2×#95—72no—
226C2×#95—72no—
227C2×#95—72no—
228C23×C7—24no—
229C23×C7—24no—
230C23×C7—22no—
231C23×C7—24no—
232C23×C7—22no—
233C23×C7—24no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—22no—
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—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—42no—
256C23×C7—42no—
257C23×C7—42no—
258C23×C7—42no—
259C23×C7—42no—
260C23×C7—41no—
261C23×C7—41no—
262C23×C7—42no—
263C23×C7—42no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7—41no—
267C23×C7—42no—
268C23×C7—41no—
269C23×C7—42no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7C2×C4⋊C414yes—
273C23×C7C2×C4⋊C414yes—
274C23×C7C23×C411yescharacteristic
275C23×C7C22×D411yescharacteristic
276C23×C7C22×Q811yescharacteristic
277C23×C7C2×#4112yes—
278C23×C7C2×#4112yes—
279C23×C7H(5,𝔽2)11yescharacteristic
280C23×C7H(5,𝔽2)11yescharacteristic
281C23×C7H(5,𝔽2)11yescharacteristic
282C23×C7H(5,𝔽2)11yescharacteristic
283C23×C7H(5,𝔽2)12yes—
284C23×C7H(5,𝔽2)12yes—
285C7×D4—416no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—24no—
293C2×C4×C7—24no—
294C2×C4×C7—24no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7C2×C4⋊C414yes—
307C2×Dic7—24no—
308C2×Dic7—24no—
309C2×Dic7—24no—
310C2×Dic7—24no—
311C2×Dic7—24no—
312C2×Dic7—24no—
313C2×Dic7—24no—
314C2×Dic7—24no—
315C2×Dic7—24no—
316C2×Dic7—416no—
317C2×Dic7—416no—
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×Dic7—42no—
328C2×Dic7—42no—
329C2×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7—42no—
332C2×Dic7—42no—
333C2×Dic7—42no—
334C2×Dic7—42no—
335C2×Dic7—42no—
336C2×Dic7—42no—
337C25C22×D711yescharacteristic
338C25C22×D711yescharacteristic
339C22×D4—41no—
340C22×D4—41no—
341C22×D4—41no—
342C22×D4—41no—
343C22×D4Dic1414yes—
344C22×D4C4×D714yes—
345C22×D4C22⋊D714yes—
346C23×C4—72no—
347C23×C4—72no—
348C23×C4—72no—
349C23×C4—74no—
350C23×C4—72no—
351C23×C4—144no—
352C23×C4—142no—
353C23×C4—281no—
354C23×C4—281no—
355C23×C4—281no—
356C23×C4—281no—
357C23×C4—282no—
358C23×C4—282no—
359C23×C4—282no—
360C23×C4C22×D712yes—
361C23×C4C22×D712yes—
362C23×C4C22×D711yescharacteristic
363C2×C4⋊C4—22no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—144no—
367C2×C4⋊C4—282no—
368C2×C4⋊C4—282no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4—78no—
376C2×C22⋊C4—74no—
377C2×C22⋊C4—144no—
378C2×C22⋊C4—144no—
379C2×C22⋊C4—142no—
380C2×C22⋊C4—281no—
381C2×C22⋊C4—282no—
382C2×C22⋊C4—281no—
383C2×C22⋊C4—282no—
384C2×C22⋊C4—282no—
385C2×C22⋊C4—281no—
386C2×C22⋊C4—281no—
387C2×C42—22no—
388C2×C42—41no—
389C2×C42—41no—
390#95—78no—
391#95—144no—
392#95—144no—
393#95—282no—
394#95—282no—
395#95—282no—
396#95—282no—
397#95—282no—
398#95—282no—
399C22×C7—24no—
400C22×C7—24no—
401C22×C7—28no—
402C22×C7—28no—
403C22×C7—44no—
404C22×C7—44no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—42no—
411C22×C7—44no—
412C22×C7—44no—
413C22×C7—42no—
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—44no—
418C22×C7—44no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7C22×C4⋊C411yescharacteristic
422C22×C7C4×#4112yes—
423C22×C7#51811yescharacteristic
424C22×C7#51811yescharacteristic
425C22×C7#51812yes—
426C22×C7#53411yescharacteristic
427C22×C7#53411yescharacteristic
428C22×C7#53812yes—
429C22×C7#53812yes—
430C22×C7#54012yes—
431C22×C7#54012yes—
432C22×C7#54311yescharacteristic
433C22×C7#54311yescharacteristic
434C22×C7#54612yes—
435C22×C7#54612yes—
436C22×C7#55111yescharacteristic
437C22×C7#55312yes—
438C22×C7#55412yes—
439C22×C7#55412yes—
440C22×C7#55911yescharacteristic
441C22×C7#56011yescharacteristic
442C22×C7#56012yes—
443C22×C7H(3,𝔽4)11yescharacteristic
444C4×C7—24no—
445C4×C7—44no—
446C4×C7—82no—
447Dic7—44no—
448Dic7—44no—
449Dic7—44no—
450Dic7—82no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—21no—
456C24—21no—
457C24—21no—
458C24—21no—
459C24—21no—
460C24—21no—
461C24—21no—
462C24—21no—
463C24—22no—
464C24—22no—
465C24—22no—
466C24—22no—
467C24C2×Dic1412yes—
468C24C2×Dic1412yes—
469C24C2×C4×D712yes—
470C24C2×C4×D712yes—
471C24C2×C22⋊D712yes—
472C24C2×C22⋊D712yes—
473C24C23×D711yescharacteristic
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—44no—
479C2×D4—44no—
480C2×D4Dic7⋊C4116yescomplemented
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—74no—
499C22×C4—74no—
500C22×C4—74no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—142no—
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—142no—
515C22×C4—142no—
516C22×C4—142no—
517C22×C4—142no—
518C22×C4—142no—
519C22×C4—142no—
520C22×C4—142no—
521C22×C4—142no—
522C22×C4—142no—
523C22×C4—142no—
524C22×C4—142no—
525C22×C4—142no—
526C22×C4—284no—
527C22×C4—288no—
528C22×C4—284no—
529C22×C4—288no—
530C22×C4—288no—
531C22×C4—284no—
532C22×C4—284no—
533C22×C4—284no—
534C22×C4—284no—
535C22×C4—288no—
536C22×C4C2×Dic1412yes—
537C22×C4C2×C4×D712yes—
538C22×C4#115914yes—
539C22×C4C2×C22⋊D712yes—
540C4⋊C4—44no—
541C4⋊C4—288no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—284no—
545C22⋊C4—288no—
546C22⋊C4—288no—
547C22⋊C4—284no—
548C42—44no—
549C2×C7—44no—
550C2×C7—44no—
551C2×C7#231511yescharacteristic
552C2×C7#233312yes—
553C2×C7#234912yes—
554C2×C7#237411yescharacteristic
555C2×C7#237511yescharacteristic
556C2×C7#268411yescharacteristic
557C2×C7#268812yes—
558C2×C7#270911yescharacteristic
559C2×C7#274811yescharacteristic
560C2×C7#278012yes—
561C2×C7#280411yescharacteristic
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—24no—
584C23—24no—
585C23—24no—
586C23—42no—
587C23—42no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—42no—
595C23—42no—
596C23—41no—
597C23—41no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23C2×Dic7⋊C414yes—
607C23C2×Dic7⋊C414yes—
608C23C22×Dic1411yescharacteristic
609C23C22×C4×D711yescharacteristic
610C23C2×#115912yes—
611C23#654811yescharacteristic
612C23#654811yescharacteristic
613C23#654812yes—
614C23#654811yescharacteristic
615C23#654811yescharacteristic
616C23D7×#4112yes—
617C23#655312yes—
618C23C22×C22⋊D711yescharacteristic
619D4—416no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—144no—
641C2×C4—144no—
642C2×C4—144no—
643C2×C4—144no—
644C2×C4—144no—
645C2×C4—144no—
646C2×C4—144no—
647C2×C4—144no—
648C2×C4—144no—
649C2×C4—2816no—
650C2×C4—2816no—
651C2×C4—282no—
652C2×C4—282no—
653C2×C4—282no—
654C2×C4—282no—
655C2×C4—282no—
656C2×C4—282no—
657C2×C4—282no—
658C2×C4—282no—
659C2×C4—282no—
660C2×C4—282no—
661C2×C4—282no—
662C2×C4—282no—
663C2×C4—282no—
664C2×C4—282no—
665C2×C4—282no—
666C2×C4—282no—
667C2×C4—282no—
668C2×C4—282no—
669C2×C4—282no—
670C2×C4C2×Dic7⋊C414yes—
671C7#1451311yescharacteristic, complemented, Sylow
672C22—28no—
673C22—28no—
674C22—24no—
675C22—24no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—42no—
693C22—42no—
694C22#9017411yescharacteristic
695C22#9017411yescharacteristic
696C22#9017711yescharacteristic
697C22#9017711yescharacteristic
698C22C4×#115912yes—
699C22#9023012yes—
700C22#9023412yes—
701C22#9023612yes—
702C22#9029212yes—
703C22#9030612yes—
704C22#9030812yes—
705C22#9031112yes—
706C22#9036912yes—
707C22#9037011yescharacteristic
708C22#9037411yescharacteristic
709C22#9040411yescharacteristic
710C22#9041012yes—
711C22#9041111yescharacteristic
712C22C22×Dic7⋊C411yescharacteristic
713C22#9049511yescharacteristic
714C22#9049511yescharacteristic
715C22#9049712yes—
716C22#9052012yes—
717C4—24no—
718C4—44no—
719C4—82no—
720C4—284no—
721C4—284no—
722C4—284no—
723C4—562no—
724C2—44no—
725C2—44no—
726C2#1173374412yes—
727C2#1173375311yescharacteristic
728C2#1173385011yescharacteristic
729C2#1173385812yes—
730C2#1173386611yescharacteristic
731C2#1173387911yescharacteristic
732C2#1173954011yescharacteristic
733C2#1173977912yes—
734C2#1174013412yes—
735C2#1174045011yescharacteristic
736C2#1174062811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#390C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#390D7
hypocenterC7#14513
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. #11729321C2semidirect
5. #11729301C2semidirect
6. #11729301C2semidirect
7. #11729298C2semidirect
8. #11729298C2semidirect
9. #11729290C2semidirect
24. #89992C4semidirect
56. C23×Dic7C2×C4semidirect
84. C2×C22⋊Dic7C2×C4semidirect
96. C7⋊#95C2×C4semidirect
135. C2×C7×D4C4⋊C4semidirect
480. C2×D4Dic7⋊C4semidirect
671. C7#14513semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3902
C2×#899922
#117293212
#117293012
#117292982
#117292902
#145137