Finite Group Databasebeta

← #13355506

Subgroups — #13355506

Order 1792.

1894 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D7×M16C211yesmaximal, characteristic
2C22×#6460C211yesmaximal, characteristic
3C2×#89892C211yesmaximal, characteristic, complemented
4#11739712C211yesmaximal, characteristic
5C2×#89776C211yesmaximal, characteristic, complemented
6C7×#1736C211yesmaximal, characteristic, complemented
7#11729312C211yesmaximal, characteristic, complemented
8#11726638C214yesmaximal, complemented
9#11726636C214yesmaximal, complemented
10C23×C4×D7C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×(C2×D7)⋊C4—21no—
16C2×#6489C2211yescharacteristic
17C22×C4⋊Dic7—21no—
18C22×#1136C2211yescharacteristic
19C2×D7×M16—24no—
20C2×D7×M16—24no—
21C2×D7×M16—24no—
22C22×#1131C2212yes—
23C22×C8×D7C2212yes—
24C2×#6460—21no—
25C2×#6460—21no—
26C2×#6460—24no—
27C2×#6460C418yescomplemented
28C2×#6460C2212yes—
29C2×#6460C2212yes—
30#89892—24no—
31C2×#6390C2214yescomplemented
32#89776—24no—
33#89759C2212yes—
34#89759C2212yes—
35C2×#6384C2214yescomplemented
36#89727—21no—
37#89715—21no—
38C4×(C2×D7)⋊C4—21no—
39C4×(C2×D7)⋊C4—21no—
40C7×#340C2214yescomplemented
41#89324C2214yescomplemented
42#20972—71nomaximal, Sylow
43C24×D7D411yescharacteristic
44C23×C4×C7—21no—
45C23×C4×C7C2311yescharacteristic
46C23×Dic7—21no—
47C23×Dic7D411yescharacteristic
48C22×C4×D7—22no—
49C22×C4×D7—22no—
50C22×C4×D7—22no—
51C22×C4×D7—22no—
52C22×C4×D7C2×C414yes—
53C22×C4×D7D411yescharacteristic
54C22×C4×D7D411yescharacteristic
55C22×C4×D7D414yes—
56C22×C4×D7C2311yescharacteristic
57C22×C4×D7C2311yescharacteristic
58C22×Dic14—22no—
59C22×Dic14—22no—
60C22×Dic14—22no—
61C22×Dic14—22no—
62C22×Dic14—21no—
63C22×Dic14—22no—
64C22×Dic14C2×C414yes—
65C22×Dic14C2312yes—
66C2×C7×M16—22no—
67C2×C7×M16—22no—
68C2×C7×M16D414yes—
69C22×C7×C8C2312yes—
70C7×#117—22no—
71C7×#117—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4C2×C414yes—
75C2×C7×C4⋊C4C2312yes—
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C22⋊Dic7—21no—
80C2×(C2×D7)⋊C4—28no—
81C2×(C2×D7)⋊C4—22no—
82C2×(C2×D7)⋊C4—22no—
83C2×(C2×D7)⋊C4—22no—
84C2×(C2×D7)⋊C4—22no—
85#6489—22no—
86#6489—22no—
87C2×C4⋊Dic7—28no—
88C2×C4⋊Dic7—22no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7C2312yes—
91C2×C4×Dic7—21no—
92C2×#1136—22no—
93C2×#1136—22no—
94C2×#1136D414yes—
95C22×C7⋊C8C2312yes—
96D7×M16—232no—
97C2×#1131—24no—
98C2×#1131—24no—
99C2×#1131—24no—
100C2×C8×D7—24no—
101C2×C8×D7—24no—
102C2×C8×D7—24no—
103#6460—28no—
104#6460—28no—
105#6460—24no—
106#6460—24no—
107#6460C2×C418yes—
108#6460C2×C418yes—
109C7⋊#95—22no—
110C7⋊#95—22no—
111#6390—28no—
112#6384—28no—
113C23×M16—71no—
114C22×C22⋊Q8—71no—
115C2×#419—71no—
116C2×#419—71no—
117#2280—71no—
118#1894—74no—
119#1892—74no—
120#1736—71no—
121#1736D711yescharacteristic, complemented
122C24×C7C2×D411yescharacteristic
123C23×D7—22no—
124C23×D7—22no—
125C23×D7—24no—
126C23×D7—42no—
127C23×D7C22⋊C414yes—
128C23×D7C2×D411yescharacteristic
129C23×D7C2×D411yescharacteristic
130C22×C4×C7—24no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—44no—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7#4112yes—
140C22×C4×C7#4112yes—
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—24no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—42no—
148C22×Dic7—42no—
149C22×Dic7C22⋊C414yes—
150C22×Dic7C2×D411yescharacteristic
151C22×Dic7C2×D411yescharacteristic
152C2×C4×D7—28no—
153C2×C4×D7—28no—
154C2×C4×D7—28no—
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—22no—
159C2×C4×D7—28no—
160C2×C4×D7—21no—
161C2×C4×D7—21no—
162C2×C4×D7—21no—
163C2×C4×D7—21no—
164C2×C4×D7—21no—
165C2×C4×D7—21no—
166C2×C4×D7—21no—
167C2×C4×D7—21no—
168C2×C4×D7C22⋊C414yes—
169C2×C4×D7C22⋊C414yes—
170C2×C4×D7C22×C412yes—
171C2×C4×D7C22×C412yes—
172C2×C4×D7C2×D412yes—
173C2×C4×D7C2×D412yes—
174C2×Dic14—28no—
175C2×Dic14—28no—
176C2×Dic14—22no—
177C2×Dic14—22no—
178C2×Dic14—22no—
179C2×Dic14—22no—
180C2×Dic14—24no—
181C2×Dic14—22no—
182C2×Dic14—44no—
183C2×Dic14—42no—
184C2×Dic14—42no—
185C2×Dic14—44no—
186C2×Dic14C22×C414yes—
187C7×M16—28no—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C2×C7×C8C2×D414yes—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4C22×C414yes—
196C7×C22⋊C4—42no—
197C42×C7—42no—
198C22⋊Dic7—42no—
199(C2×D7)⋊C4—28no—
200(C2×D7)⋊C4—28no—
201(C2×D7)⋊C4—44no—
202(C2×D7)⋊C4—44no—
203C4⋊Dic7—28no—
204C4⋊Dic7—28no—
205C4⋊Dic7—22no—
206C4⋊Dic7—22no—
207C4×Dic7—42no—
208#1136—28no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8C2×D414yes—
212#1131—216no—
213C8×D7—216no—
214C23×Q8—71no—
215C24×C4—71no—
216C22×M16—71no—
217C22×M16—72no—
218C22×M16—144no—
219C22×M16—144no—
220C22×M16—144no—
221C22×M16C2×D711yescharacteristic
222C23×C8—72no—
223C2×C22⋊Q8—78no—
224C2×C22⋊Q8—72no—
225C2×C22⋊Q8—72no—
226C2×C22⋊Q8—144no—
227C2×C22⋊Q8—141no—
228C2×C22⋊Q8—141no—
229C2×#117—71no—
230C2×#117C2×D711yescharacteristic
231C22×C4⋊C4—141no—
232C22×C4⋊C4C2×D711yescharacteristic
233C22×C22⋊C4—141no—
234#419—144no—
235#419—144no—
236C2×#103—74no—
237C2×#103—74no—
238#381—141no—
239#380—72no—
240#380—72no—
241#379—141no—
242C4×C22⋊C4—141no—
243C4×C22⋊C4—141no—
244#340—74no—
245#340C2×D714yescomplemented
246C23×C7—22no—
247C23×C7—22no—
248C23×C7C2×C22⋊C412yes—
249C23×C7#12112yes—
250C23×C7#12112yes—
251C23×C7C22×D411yescharacteristic
252C22×D7—21no—
253C22×D7—21no—
254C22×D7—22no—
255C22×D7—21no—
256C22×D7—21no—
257C22×D7—22no—
258C22×D7—22no—
259C22×D7—28no—
260C22×D7—28no—
261C22×D7—44no—
262C22×D7—48no—
263C22×D7—48no—
264C22×D7C2×C22⋊C412yes—
265C22×D7C2×C22⋊C412yes—
266C22×D7#13714yes—
267C2×C4×C7—24no—
268C2×C4×C7—48no—
269C2×C4×C7—44no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7C2×C22⋊C412yes—
274C2×C4×C7C4×D414yes—
275C2×C4×C7C4×D414yes—
276C2×C4×C7#12014yes—
277C2×C4×C7#12112yes—
278C2×C4×C7#12112yes—
279C2×C4×C7#12314yes—
280C2×C4×C7C23×C411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—28no—
291C2×Dic7—24no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—48no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7C2×C22⋊C412yes—
299C2×Dic7C2×C22⋊C412yes—
300C4×D7—22no—
301C4×D7—22no—
302C4×D7—22no—
303C4×D7—22no—
304C4×D7—22no—
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—22no—
308C4×D7—28no—
309C4×D7—28no—
310Dic14—24no—
311Dic14—24no—
312Dic14—42no—
313Dic14—42no—
314Dic14—48no—
315C7×C8—24no—
316C7⋊C8—24no—
317C25—71no—
318C22×Q8—74no—
319C22×Q8—72no—
320C22×Q8—142no—
321C22×Q8—141no—
322C22×Q8—142no—
323C22×Q8—142no—
324C22×Q8—142no—
325C22×Q8—142no—
326C23×C4—21no—
327C23×C4—71no—
328C23×C4—74no—
329C23×C4—74no—
330C23×C4—71no—
331C23×C4—71no—
332C23×C4—71no—
333C23×C4—71no—
334C23×C4—142no—
335C23×C4—142no—
336C23×C4—142no—
337C23×C4—142no—
338C23×C4—141no—
339C23×C4C22×D711yescharacteristic
340C2×M16—22no—
341C2×M16—22no—
342C2×M16—74no—
343C2×M16—144no—
344C2×M16—144no—
345C2×M16—1432no—
346C2×M16—142no—
347C2×M16—142no—
348C2×M16—144no—
349C2×M16D2814yes—
350C22×C8—72no—
351C22×C8—144no—
352C22×C8—144no—
353C22×C8—144no—
354C22×C8C22×D712yes—
355C22⋊Q8—78no—
356C22⋊Q8—78no—
357C22⋊Q8—148no—
358C22⋊Q8—148no—
359C22⋊Q8—144no—
360C22⋊Q8—144no—
361#117—22no—
362#117—22no—
363#117—142no—
364#117—142no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—72no—
368C2×C4⋊C4—148no—
369C2×C4⋊C4—142no—
370C2×C4⋊C4—142no—
371C2×C4⋊C4C4×D714yes—
372C2×C4⋊C4C22×D712yes—
373C2×C4⋊C4C22×D712yes—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—148no—
376C2×C22⋊C4—142no—
377C2×C22⋊C4—142no—
378C2×C22⋊C4—142no—
379C2×C22⋊C4—142no—
380C2×C22⋊C4—141no—
381C2×C42—21no—
382C2×C42—141no—
383#103—148no—
384#103—148no—
385#95—142no—
386#95—142no—
387C22×C7—22no—
388C22×C7—22no—
389C22×C7—24no—
390C22×C7—24no—
391C22×C7#38614yes—
392C22×C7C22×C22⋊C411yescharacteristic
393C22×C7C2×#12111yescharacteristic
394C22×C7C2×#12111yescharacteristic
395C22×C7C2×#13712yes—
396C22×C7#57512yes—
397C2×D7—22no—
398C2×D7—22no—
399C2×D7—24no—
400C2×D7—24no—
401C2×D7—22no—
402C2×D7—22no—
403C2×D7—22no—
404C2×D7—416no—
405C2×D7—42no—
406C2×D7—42no—
407C2×D7#41914yescomplemented
408C4×C7—44no—
409C4×C7—82no—
410C4×C7#38614yes—
411C4×C7C2×C4×D411yescharacteristic
412C4×C7C2×C4×D411yescharacteristic
413C4×C7C2×#12011yescharacteristic
414C4×C7C2×#12311yescharacteristic
415Dic7—22no—
416Dic7—22no—
417Dic7—44no—
418Dic7—82no—
419C24—71no—
420C24—71no—
421C24—74no—
422C24—144no—
423C24—142no—
424C24—142no—
425C24—282no—
426C24D4×D711yescharacteristic
427C2×Q8—74no—
428C2×Q8—144no—
429C2×Q8—142no—
430C2×Q8—148no—
431C2×Q8—148no—
432C2×Q8—142no—
433C2×Q8—142no—
434C2×Q8—142no—
435C2×Q8—142no—
436C2×Q8—284no—
437C2×Q8—282no—
438C2×Q8—284no—
439C2×Q8—282no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—44no—
444C22×C4—74no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—74no—
448C22×C4—74no—
449C22×C4—72no—
450C22×C4—72no—
451C22×C4—72no—
452C22×C4—72no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—148no—
456C22×C4—148no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—148no—
460C22×C4—148no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—141no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—141no—
471C22×C4—144no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—282no—
475C22×C4—282no—
476C22×C4C2×C4×D712yes—
477C22×C4C2×D2812yes—
478C22×C4#115712yes—
479C22×C4D4×D711yescharacteristic
480C22×C4D4×D712yes—
481C22×C4D4×D712yes—
482C22×C4#116112yes—
483C22×C4C23×D711yescharacteristic
484M16—28no—
485M16—1416no—
486M16—148no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—74no—
490C2×C8—1416no—
491C2×C8—142no—
492C2×C8—142no—
493C2×C8C2×D2814yes—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—148no—
499C4⋊C4—148no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4C2×C4×D714yes—
503C22⋊C4—42no—
504C22⋊C4—148no—
505C22⋊C4—148no—
506C22⋊C4—284no—
507C22⋊C4—284no—
508C22⋊C4—282no—
509C42—42no—
510C42—282no—
511C2×C7—24no—
512C2×C7C2×#38611yescharacteristic
513C2×C7C2×#41911yescharacteristic
514C2×C7#289611yescharacteristic
515C2×C7#305612yes—
516C2×C7#305612yes—
517D7—24no—
518D7—42no—
519C23—22no—
520C23—22no—
521C23—72no—
522C23—74no—
523C23—72no—
524C23—148no—
525C23—141no—
526C23—141no—
527C23—142no—
528C23—141no—
529C23—141no—
530C23—148no—
531C23—142no—
532C23—142no—
533C23—284no—
534C23—288no—
535C23—288no—
536C23D7×C22⋊C412yes—
537C23#644712yes—
538C23#645812yes—
539C23C2×D4×D711yescharacteristic
540Q8—144no—
541Q8—144no—
542Q8—282no—
543Q8—282no—
544Q8—288no—
545C2×C4—24no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—72no—
552C2×C4—72no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—148no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—148no—
567C2×C4—148no—
568C2×C4—144no—
569C2×C4—282no—
570C2×C4—282no—
571C2×C4—288no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4C4×D2814yes—
576C2×C4D7×C22⋊C412yes—
577C2×C4#644514yes—
578C2×C4#644712yes—
579C2×C4#645614yes—
580C2×C4#645714yes—
581C2×C4#645812yes—
582C2×C4C22×C4×D711yescharacteristic
583C2×C4C22×D2811yescharacteristic
584C2×C4C2×#115711yescharacteristic
585C2×C4C2×D4×D711yescharacteristic
586C2×C4C2×D4×D711yescharacteristic
587C2×C4C2×#116111yescharacteristic
588C8—24no—
589C8—144no—
590C7#2097211yescharacteristic, complemented, Sylow
591C22—24no—
592C22—22no—
593C22—22no—
594C22—24no—
595C22—74no—
596C22—144no—
597C22—142no—
598C22—144no—
599C22—142no—
600C22—142no—
601C22—142no—
602C22—142no—
603C22—2816no—
604C22—282no—
605C22—282no—
606C22#8943714yes—
607C22C2×D7×C22⋊C411yescharacteristic
608C22C2×#644711yescharacteristic
609C22C2×#645811yescharacteristic
610C22#9046412yes—
611C22D7×#13712yes—
612C4—44no—
613C4—82no—
614C4—142no—
615C4—142no—
616C4—284no—
617C4—562no—
618C4#8943714yes—
619C4C2×C4×D2811yescharacteristic
620C4C2×#644511yescharacteristic
621C4C2×#645611yescharacteristic
622C4C2×#645711yescharacteristic
623C2—24no—
624C2—144no—
625C2—282no—
626C2C2×#8943711yescharacteristic
627C2#1173633412yes—
628C2#1173636911yescharacteristic
629C2D7×#41911yescharacteristic
630C2#1173665712yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1736C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1736D7
hypocenterC7#20972
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. C2×#89892C2semidirect
5. C2×#89776C2semidirect
6. C7×#1736C2semidirect
7. #11729312C2semidirect
8. #11726638C2semidirect
9. #11726636C2semidirect
27. C2×#6460C4semidirect
31. C2×#6390C22semidirect
35. C2×#6384C22semidirect
40. C7×#340C22semidirect
41. #89324C22semidirect
121. #1736D7semidirect
245. #340C2×D7semidirect
407. C2×D7#419semidirect
590. C7#20972semidirect

Maximal subgroups

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

subgroupindex
C22×D7×M162
C22×#64602
C2×#898922
#117397122
C2×#897762
C7×#17362
#117293122
#117266382
#117266362
#209727