Finite Group Databasebeta

← #13283064

Subgroups — #13283064

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×#386C211yesmaximal, characteristic
2C2×#89993C211yesmaximal, characteristic
3C2×#89992C211yesmaximal, characteristic
4#11729881C212yesmaximal, complemented
5#11729301C212yesmaximal, complemented
6#11729301C212yesmaximal, complemented
7#11729298C212yesmaximal, complemented
8#11727169C212yesmaximal, complemented
9#11727160C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C24×Dic7C2211yescharacteristic
13C22×C7×C22⋊C4C2212yes—
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C22⋊Dic7C2212yes—
16C22×C22⋊Dic7C2212yes—
17C22×C22⋊Dic7C2212yes—
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×Dic7⋊C4C2212yes—
21C22×C4×Dic7C2211yescharacteristic
22C7×#386—24no—
23C2×C7×#95C2212yes—
24#89993—24no—
25#89992C418yescomplemented
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×C7—22no—
45C23×C4×C7—24no—
46C23×C4×C7—41no—
47C23×C4×C7—41no—
48C23×C4×C7—42no—
49C23×C4×C7C2312yes—
50C23×C4×C7C2312yes—
51C23×C4×C7C2311yescharacteristic
52C23×Dic7—41no—
53C23×Dic7—41no—
54C23×Dic7—42no—
55C23×Dic7—42no—
56C23×Dic7C2×C414yescomplemented
57C23×Dic7C2312yes—
58C23×Dic7C2312yes—
59C23×Dic7C2312yes—
60C23×Dic7C2312yes—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—24no—
64C2×C7×C22⋊C4—42no—
65C2×C7×C22⋊C4—41no—
66C2×C7×C22⋊C4—41no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7—24no—
70C2×C22⋊Dic7—24no—
71C2×C22⋊Dic7—42no—
72C2×C22⋊Dic7—42no—
73C2×C22⋊Dic7—41no—
74C2×C22⋊Dic7—41no—
75C2×C22⋊Dic7—42no—
76C2×C22⋊Dic7—42no—
77C2×C22⋊Dic7C2×C418yes—
78C2×C22⋊Dic7C2×C414yescomplemented
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—41no—
81C2×C4⋊Dic7—41no—
82C2×Dic7⋊C4—24no—
83C2×Dic7⋊C4—42no—
84C2×Dic7⋊C4—42no—
85C2×C4×Dic7—22no—
86C2×C4×Dic7—41no—
87C2×C4×Dic7—41no—
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×#390—71no—
98C2×#386—71no—
99C2×#386D711yescharacteristic
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—22no—
116C24×C7—21no—
117C24×C7—21no—
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×C7—44no—
154C22×C4×C7—48no—
155C22×C4×C7—48no—
156C22×C4×C7—44no—
157C22×C4×C7—44no—
158C22×C4×C7C22×C412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×Q812yes—
161C22×C4×C7#4114yes—
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—22no—
183C22×Dic7—22no—
184C22×Dic7—22no—
185C22×Dic7—22no—
186C22×Dic7—22no—
187C22×Dic7—48no—
188C22×Dic7—44no—
189C22×Dic7—48no—
190C22×Dic7—44no—
191C22×Dic7—44no—
192C22×Dic7C22×C414yes—
193C22×Dic7C22×C414yes—
194C22×Dic7#4114yes—
195C7×C22⋊C4—48no—
196C7×C22⋊C4—44no—
197C22⋊Dic7—48no—
198C22⋊Dic7—44no—
199C22⋊Dic7—48no—
200C22⋊Dic7—48no—
201C4⋊Dic7—44no—
202Dic7⋊C4—48no—
203C4×Dic7—44no—
204C23×D4C2×D711yescharacteristic
205C24×C4—71no—
206C24×C4C2×D711yescharacteristic
207C22×C4⋊C4—72no—
208C22×C4⋊C4—71no—
209C22×C22⋊C4—72no—
210C22×C22⋊C4—71no—
211C22×C22⋊C4—72no—
212C22×C22⋊C4—72no—
213C22×C22⋊C4C2×D712yes—
214C22×C22⋊C4C2×D711yescharacteristic
215C22×C42—71no—
216#390—144no—
217#386—24no—
218#386—78no—
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×#95C2×D712yes—
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—42no—
267C23×C7—41no—
268C23×C7—42no—
269C23×C7—41no—
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—416no—
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×C7—42no—
307C2×C4×C7C2×C4⋊C414yes—
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—24no—
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—22no—
347C23×C4—24no—
348C23×C4—41no—
349C23×C4—41no—
350C23×C4—42no—
351C23×C4—72no—
352C23×C4—72no—
353C23×C4—72no—
354C23×C4—74no—
355C23×C4—72no—
356C23×C4—281no—
357C23×C4—281no—
358C23×C4—282no—
359C23×C4—282no—
360C23×C4C22×D712yes—
361C23×C4C22×D712yes—
362C23×C4C22×D711yescharacteristic
363C2×C4⋊C4—144no—
364C2×C4⋊C4—142no—
365C2×C4⋊C4—282no—
366C2×C4⋊C4—282no—
367C2×C4⋊C4—281no—
368C2×C4⋊C4—281no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—78no—
376C2×C22⋊C4—74no—
377C2×C22⋊C4—144no—
378C2×C22⋊C4—144no—
379C2×C22⋊C4—144no—
380C2×C22⋊C4—144no—
381C2×C22⋊C4—281no—
382C2×C22⋊C4—281no—
383C2×C22⋊C4—282no—
384C2×C22⋊C4—282no—
385C2×C22⋊C4—282no—
386C2×C22⋊C4—282no—
387C2×C42—142no—
388C2×C42—281no—
389C2×C42—281no—
390#95—24no—
391#95—78no—
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#56012yes—
442C22×C7#56011yescharacteristic
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—21no—
455C24—21no—
456C24—22no—
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—44no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—74no—
504C22×C4—74no—
505C22×C4—74no—
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—142no—
527C22×C4—142no—
528C22×C4—142no—
529C22×C4—142no—
530C22×C4—142no—
531C22×C4—284no—
532C22×C4—288no—
533C22×C4—284no—
534C22×C4—288no—
535C22×C4—284no—
536C22×C4C2×Dic1412yes—
537C22×C4C2×C4×D712yes—
538C22×C4#115914yes—
539C22×C4C2×C22⋊D712yes—
540C4⋊C4—284no—
541C4⋊C4—288no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—284no—
545C22⋊C4—288no—
546C22⋊C4—288no—
547C22⋊C4—288no—
548C42—284no—
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—41no—
587C23—41no—
588C23—42no—
589C23—42no—
590C23—41no—
591C23—41no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23C2×Dic7⋊C414yes—
607C23C2×Dic7⋊C414yes—
608C23C22×Dic1411yescharacteristic
609C23C22×C4×D711yescharacteristic
610C23C2×#115912yes—
611C23#654811yescharacteristic
612C23#654811yescharacteristic
613C23#654811yescharacteristic
614C23#654811yescharacteristic
615C23#654812yes—
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—416no—
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—42no—
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—144no—
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—44no—
679C22—44no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22#9017411yescharacteristic
695C22#9017411yescharacteristic
696C22#9017711yescharacteristic
697C22#9017711yescharacteristic
698C22C4×#115912yes—
699C22#9023012yes—
700C22#9023412yes—
701C22#9023612yes—
702C22#9027811yescharacteristic
703C22#9028412yes—
704C22#9028511yescharacteristic
705C22#9030412yes—
706C22#9030512yes—
707C22#9031012yes—
708C22#9031112yes—
709C22#9035412yes—
710C22#9035511yescharacteristic
711C22#9035711yescharacteristic
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#1173211312yes—
727C2#1173212611yescharacteristic
728C2#1173228311yescharacteristic
729C2#1173230212yes—
730C2#1173231111yescharacteristic
731C2#1173232711yescharacteristic
732C2#1173954011yescharacteristic
733C2#1173977912yes—
734C2#1174004311yescharacteristic
735C2#1174013312yes—
736C2#1174032911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#386C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#386D7
hypocenterC7#14513
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
4. #11729881C2semidirect
5. #11729301C2semidirect
6. #11729301C2semidirect
7. #11729298C2semidirect
8. #11727169C2semidirect
9. #11727160C2semidirect
25. #89992C4semidirect
56. C23×Dic7C2×C4semidirect
78. 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×#3862
C2×#899932
C2×#899922
#117298812
#117293012
#117292982
#117271692
#117271602
#145137