Finite Group Databasebeta

← #13354776

Subgroups — #13354776

Order 1792.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1157C211yesmaximal, characteristic
2C2×#89905C211yesmaximal, characteristic
3C2×#89905C211yesmaximal, characteristic
4C2×#89893C211yesmaximal, characteristic, complemented
5C2×#89700C211yesmaximal, characteristic, complemented
6C7×#1728C211yesmaximal, characteristic, complemented
7#11729739C211yesmaximal, characteristic, complemented
8#11726438C212yesmaximal, complemented
9#11726437C212yesmaximal, complemented
10#11726425C212yesmaximal, complemented
11#11726424C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C22×C42×C7C2211yescharacteristic
16C2×C4×C22⋊D7—21no—
17C2×C4×C22⋊D7—21no—
18C2×#6489C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×C4⋊Dic7C2211yescharacteristic
21C22×Dic7⋊C4—21no—
22C4×#1157—24no—
23C4×#1157—24no—
24C2×C4×D28C2211yescharacteristic
25C2×C4×D28C2211yescharacteristic
26C2×#6435—21no—
27C2×C42×D7—21no—
28C2×C4×Dic14C2211yescharacteristic
29C2×C4×Dic14C2211yescharacteristic
30C2×C7⋊#95—21no—
31#89905—24no—
32#89905C418yescomplemented
33#89893—24no—
34C2×#6395C2212yes—
35C2×#6395C2212yes—
36C2×#6392C2212yes—
37C2×#6391C2212yes—
38C2×#6390C2212yes—
39C2×#6390C2212yes—
40#89700—21no—
41#89700—21no—
42#89700C2212yes—
43#89700C2212yes—
44C7×#336C2212yes—
45C7×#336C2212yes—
46#89346C2212yes—
47#89346C2212yes—
48#20875—71nomaximal, Sylow
49C23×C4×C7D411yescharacteristic
50C23×C4×C7D411yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C22×C22⋊D7—21no—
53C23×Dic7—21no—
54C23×Dic7—21no—
55C2×#1157—22no—
56C2×#1157—22no—
57C2×#1157—22no—
58C2×#1157C2×C414yes—
59C2×#1157D414yes—
60C2×#1157Q814yes—
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×Dic14C2311yescharacteristic
66C2×C7×M16—22no—
67C2×C7×M16—22no—
68C2×C7×M16—22no—
69C2×C7×M16—22no—
70C2×C7×M16—22no—
71C2×C7×M16C2×C414yescomplemented
72C22×C7×C8C2312yes—
73C22×C7×C8C2312yes—
74C2×C42×C7—22no—
75C2×C42×C7—22no—
76C2×C42×C7D411yescharacteristic
77C2×C42×C7D411yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—21no—
81C4×C22⋊D7—28no—
82C4×C22⋊D7—28no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85#6489—22no—
86#6489D414yes—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—22no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2×C414yescomplemented
93C2×C4⋊Dic7C2312yes—
94C2×C4⋊Dic7C2312yes—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—22no—
100C2×Dic7⋊C4—22no—
101C2×Dic7⋊C4—44no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7—21no—
104C2×C4×Dic7—21no—
105C4×D28—22no—
106C4×D28—22no—
107C4×D28—22no—
108C4×D28—22no—
109#6435—28no—
110C42×D7—28no—
111C4×Dic14—22no—
112C4×Dic14—22no—
113C4×Dic14—22no—
114C4×Dic14—22no—
115C7⋊#95—22no—
116C7⋊#95—22no—
117#6395—24no—
118#6395C2×C418yescomplemented
119#6392—24no—
120#6391—24no—
121#6390—24no—
122#6390C2×C418yescomplemented
123C2×C4×#41—71no—
124C2×#428—71no—
125C2×#417—71no—
126C2×#417—71no—
127C2×#389—71no—
128#1865—72no—
129#1864—72no—
130#1863—72no—
131#1862—72no—
132#1736—71no—
133#1728D711yescharacteristic, complemented
134C24×C7C2×D411yescharacteristic
135C23×D7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—41no—
141C22×C4×C7—41no—
142C22×C4×C7C22×C412yes—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×Q812yes—
151C22×C4×C7#4112yes—
152C22×C4×C7#4112yes—
153C22×C4×C7C2411yescharacteristic
154C2×C22⋊D7—24no—
155C2×C22⋊D7—24no—
156C2×C22⋊D7—24no—
157C22×Dic7—21no—
158C22×Dic7—24no—
159C22×Dic7—22no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—22no—
163C22×Dic7—42no—
164C22×Dic7—42no—
165C22×Dic7—42no—
166C22×Dic7—42no—
167C22×Dic7—42no—
168#1157—28no—
169#1157—28no—
170#1157—28no—
171#1157C4⋊C4116yescomplemented
172C2×D28—21no—
173C2×D28—21no—
174C2×D28—21no—
175C2×D28C22×C412yes—
176C2×D28C2×D412yes—
177C2×D28C2×Q812yes—
178C2×C4×D7—24no—
179C2×C4×D7—24no—
180C2×C4×D7—24no—
181C2×C4×D7—24no—
182C2×C4×D7—24no—
183C2×C4×D7—42no—
184C2×C4×D7—42no—
185C2×C4×D7—42no—
186C2×C4×D7—42no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14—21no—
190C2×Dic14C22×C412yes—
191C2×Dic14C2×D412yes—
192C2×Dic14C2×Q812yes—
193C7×M16—28no—
194C7×M16—44no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C2×C7×C8—22no—
198C2×C7×C8—22no—
199C2×C7×C8—22no—
200C2×C7×C8C22×C414yes—
201C42×C7—22no—
202C42×C7—22no—
203C42×C7—21no—
204C42×C7—21no—
205C42×C7—21no—
206C42×C7—21no—
207C22⋊Dic7—24no—
208(C2×D7)⋊C4—24no—
209(C2×D7)⋊C4—24no—
210C4⋊Dic7—24no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—22no—
214C4⋊Dic7—22no—
215C4⋊Dic7—22no—
216C4⋊Dic7—42no—
217C4⋊Dic7C22×C414yes—
218C4⋊Dic7C2×D412yes—
219C4⋊Dic7C2×D412yes—
220Dic7⋊C4—24no—
221Dic7⋊C4—24no—
222Dic7⋊C4—44no—
223Dic7⋊C4—44no—
224C4×Dic7—24no—
225C4×Dic7—24no—
226C4×Dic7—24no—
227C22×#41—71no—
228C22×M16C2×D711yescharacteristic
229C22×M16C2×D711yescharacteristic
230C4×#41—144no—
231C4×#41—144no—
232C2×C4×Q8—71no—
233C2×C4×Q8—71no—
234C2×C4×D4—71no—
235C2×C4×D4—71no—
236C2×C4×D4—141no—
237C2×C4×D4—141no—
238C2×#117—71no—
239C2×#117—141no—
240C22×C4⋊C4—71no—
241C22×C4⋊C4—71no—
242C22×C4⋊C4—141no—
243C22×C42—141no—
244C22×C42C2×D711yescharacteristic
245#428—144no—
246C2×#107—72no—
247C2×#106—72no—
248#417—78no—
249#417—144no—
250C2×#103—72no—
251C2×#103—72no—
252C2×#102—72no—
253C2×#102—72no—
254#389—72no—
255#389—72no—
256#389—141no—
257#389—141no—
258C2×#95—141no—
259#340—72no—
260#340—72no—
261#336C2×D712yes—
262#336C2×D712yes—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7C4×D412yes—
268C23×C7#12012yes—
269C23×C7C22⋊Q812yes—
270C23×C7C22×D411yescharacteristic
271C22×D7—22no—
272C22×D7—22no—
273C22×D7—22no—
274C22×D7—41no—
275C22×D7—41no—
276C22×D7—41no—
277C22×D7—41no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—44no—
289C2×C4×C7—44no—
290C2×C4×C7C2×C4⋊C414yes—
291C2×C4×C7C4×D412yes—
292C2×C4×C7C4×D414yes—
293C2×C4×C7#12012yes—
294C2×C4×C7C22⋊Q814yes—
295C2×C4×C7C22⋊Q812yes—
296C2×C4×C7#12314yes—
297C2×C4×C7AGL(1,ℤ/8)12yes—
298C2×C4×C7#13712yes—
299C2×C4×C7C23×C411yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×Q811yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C22⋊D7—216no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—24no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—48no—
313C2×Dic7—42no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—84no—
323D28—22no—
324D28—22no—
325D28—22no—
326D28C2×C4⋊C414yescomplemented
327C4×D7—216no—
328C4×D7—48no—
329C4×D7—48no—
330Dic14—22no—
331Dic14—22no—
332Dic14—22no—
333Dic14C2×C4⋊C414yescomplemented
334C7×C8—24no—
335C7×C8—42no—
336C2×#41—74no—
337C2×#41—74no—
338C2×#41—74no—
339C2×#41—142no—
340C2×#41—142no—
341C2×#41—142no—
342C22×Q8—71no—
343C22×D4—71no—
344C22×D4—141no—
345C23×C4—141no—
346C23×C4—141no—
347C23×C4—141no—
348C23×C4—141no—
349C23×C4—141no—
350C23×C4D2811yescharacteristic
351C23×C4D2811yescharacteristic
352C23×C4C22×D711yescharacteristic
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C4×D714yescomplemented
359C22×C8C22×D712yes—
360C22×C8C22×D712yes—
361C4×Q8—142no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×D4—142no—
366C4×D4—142no—
367C4×D4—142no—
368C4×D4—142no—
369C4×D4—148no—
370C4×D4—148no—
371#117—74no—
372#117—148no—
373#117—142no—
374C2×C4⋊C4—74no—
375C2×C4⋊C4—72no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—72no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—142no—
383C2×C4⋊C4—142no—
384C2×C4⋊C4—142no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—142no—
387C2×C4⋊C4—142no—
388C2×C4⋊C4—284no—
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C22⋊C4—141no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—148no—
395C2×C42—141no—
396C2×C42—141no—
397C2×C42—141no—
398C2×C42D2811yescharacteristic
399C2×C42D2811yescharacteristic
400C2×C42C22×D711yescharacteristic
401C2×C42C22×D711yescharacteristic
402#107—144no—
403#106—144no—
404#103—78no—
405#103—144no—
406#102—78no—
407#102—144no—
408#95—142no—
409#95—142no—
410C22×C7—22no—
411C22×C7—22no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—22no—
415C22×C7—22no—
416C22×C7#38514yes—
417C22×C7C2×C4×D411yescharacteristic
418C22×C7C2×#12011yescharacteristic
419C22×C7C2×C22⋊Q811yescharacteristic
420C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
421C22×C7C2×#13711yescharacteristic
422C22×C7#57511yescharacteristic
423C22×C7#57511yescharacteristic
424C2×D7—24no—
425C2×D7—42no—
426C2×D7—42no—
427C2×D7—42no—
428C2×D7—42no—
429C2×D7—42no—
430C2×D7—42no—
431C4×C7—22no—
432C4×C7—22no—
433C4×C7—41no—
434C4×C7—41no—
435C4×C7#38514yes—
436C4×C7C22×C4⋊C411yescharacteristic
437C4×C7C2×C4×D411yescharacteristic
438C4×C7C2×C22⋊Q811yescharacteristic
439C4×C7C2×#12311yescharacteristic
440Dic7—24no—
441Dic7—44no—
442Dic7—42no—
443Dic7—42no—
444Dic7—82no—
445C24—141no—
446C24C2×D2811yescharacteristic
447#41—716no—
448#41—148no—
449#41—148no—
450#41—148no—
451C2×Q8—72no—
452C2×Q8—72no—
453C2×Q8—72no—
454C2×Q8—141no—
455C2×Q8—141no—
456C2×Q8—141no—
457C2×D4—72no—
458C2×D4—72no—
459C2×D4—72no—
460C2×D4—144no—
461C2×D4—144no—
462C2×D4—144no—
463C2×D4—141no—
464C2×D4—141no—
465C2×D4—141no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—144no—
473C22×C4—144no—
474C22×C4—141no—
475C22×C4—144no—
476C22×C4—144no—
477C22×C4—144no—
478C22×C4—144no—
479C22×C4—142no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—142no—
483C22×C4—282no—
484C22×C4—282no—
485C22×C4—282no—
486C22×C4—282no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4—282no—
490C22×C4—282no—
491C22×C4—282no—
492C22×C4C2×C4×D712yes—
493C22×C4C2×D2811yescharacteristic
494C22×C4C2×D2811yescharacteristic
495C22×C4C2×D2811yescharacteristic
496C22×C4C2×D2811yescharacteristic
497C22×C4C2×D2811yescharacteristic
498C22×C4#115712yes—
499C22×C4D4×D712yes—
500C22×C4D4×D712yes—
501C22×C4#115912yes—
502C22×C4Q8×D712yes—
503C22×C4C23×D711yescharacteristic
504M16—28no—
505M16—44no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8C2×C4×D714yes—
512C4⋊C4—74no—
513C4⋊C4—72no—
514C4⋊C4—72no—
515C4⋊C4—144no—
516C4⋊C4—144no—
517C4⋊C4—144no—
518C4⋊C4—142no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4—142no—
522C4⋊C4—142no—
523C4⋊C4—284no—
524C4⋊C4—282no—
525C4⋊C4—284no—
526C22⋊C4—144no—
527C22⋊C4—144no—
528C22⋊C4—144no—
529C42—22no—
530C42—22no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—144no—
536C42—144no—
537C42—144no—
538C2×C7—24no—
539C2×C7C2×#38511yescharacteristic
540C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
541C2×C7C4×#13711yescharacteristic
542C2×C7#300611yescharacteristic
543C2×C7#300711yescharacteristic
544C2×C7#307911yescharacteristic
545C2×C7#308011yescharacteristic
546D7—44no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—142no—
552C23—142no—
553C23—142no—
554C23—281no—
555C23—281no—
556C23—281no—
557C23—281no—
558C23C4×D2812yes—
559C23#644512yes—
560C23#646012yes—
561C23C22×D2811yescharacteristic
562Q8—74no—
563Q8—142no—
564Q8—142no—
565Q8—142no—
566D4—74no—
567D4—1416no—
568D4—142no—
569D4—142no—
570D4—142no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—142no—
584C2×C4—142no—
585C2×C4—142no—
586C2×C4—1416no—
587C2×C4—144no—
588C2×C4—142no—
589C2×C4—142no—
590C2×C4—288no—
591C2×C4—288no—
592C2×C4—288no—
593C2×C4—282no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—282no—
599C2×C4—282no—
600C2×C4—282no—
601C2×C4—282no—
602C2×C4—564no—
603C2×C4C4×D2812yes—
604C2×C4(C22⋊D7)⋊C414yes—
605C2×C4#644512yes—
606C2×C4#644614yes—
607C2×C4D7×C4⋊C414yes—
608C2×C4#645914yes—
609C2×C4#646012yes—
610C2×C4#647112yes—
611C2×C4#647212yes—
612C2×C4C22×C4×D711yescharacteristic
613C2×C4C2×#115711yescharacteristic
614C2×C4C2×D4×D711yescharacteristic
615C2×C4C2×D4×D711yescharacteristic
616C2×C4C2×#115911yescharacteristic
617C2×C4C2×Q8×D711yescharacteristic
618C8—24no—
619C8—42no—
620C7#2087511yescharacteristic, complemented, Sylow
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—144no—
628C22—282no—
629C22—282no—
630C22—282no—
631C22—282no—
632C22—282no—
633C22—282no—
634C22#8944014yes—
635C22C2×C4×D2811yescharacteristic
636C22C2×#644511yescharacteristic
637C22C2×#646011yescharacteristic
638C22C2×#647111yescharacteristic
639C22C2×#647211yescharacteristic
640C22#9043911yescharacteristic
641C22#9043911yescharacteristic
642C4—22no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—144no—
647C4—282no—
648C4—282no—
649C4—284no—
650C4—562no—
651C4#8944014yes—
652C4C2×(C22⋊D7)⋊C411yescharacteristic
653C4C2×#644611yescharacteristic
654C4C2×D7×C4⋊C411yescharacteristic
655C4C2×#645911yescharacteristic
656C2—24no—
657C2—284no—
658C2C2×#8944011yescharacteristic
659C2C4×#647111yescharacteristic
660C2C4×#647211yescharacteristic
661C2#1173608411yescharacteristic
662C2#1173608511yescharacteristic
663C2#1173650511yescharacteristic
664C2#1173650611yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1728C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#1728D7
hypocenterC7#20875
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
4. C2×#89893C2semidirect
5. C2×#89700C2semidirect
6. C7×#1728C2semidirect
7. #11729739C2semidirect
8. #11726438C2semidirect
9. #11726437C2semidirect
10. #11726425C2semidirect
11. #11726424C2semidirect
32. #89905C4semidirect
71. C2×C7×M16C2×C4semidirect
92. C2×C4⋊Dic7C2×C4semidirect
118. #6395C2×C4semidirect
122. #6390C2×C4semidirect
133. #1728D7semidirect
171. #1157C4⋊C4semidirect
326. D28C2×C4⋊C4semidirect
333. Dic14C2×C4⋊C4semidirect
358. C2×M16C4×D7semidirect
620. C7#20875semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11572
C2×#899052
C2×#898932
C2×#897002
C7×#17282
#117297392
#117264382
#117264372
#117264252
#117264242
#208757