Finite Group Databasebeta

← #13355149

Subgroups — #13355149

Order 1792.

1858 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#135C211yesmaximal, characteristic
2C2×#41×Dic7C211yesmaximal, characteristic, complemented
3#11741531C211yesmaximal, characteristic, complemented
4C2×#90005C211yesmaximal, characteristic, complemented
5C2×#90005C211yesmaximal, characteristic, complemented
6#11729819C212yesmaximal, complemented
7#11729802C212yesmaximal, complemented
8#11729801C212yesmaximal, complemented
9#11729791C212yesmaximal, complemented
10#11729723C211yesmaximal, characteristic, complemented
11C2×#89345C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×#41C2211yescharacteristic
14C2×C7×#135—24no—
15C2×C7×#135—24no—
16C2×C7×#135C418yescomplemented
17C22×C7×Q16C2211yescharacteristic
18C22×C7×SD16C2211yescharacteristic
19C22×C7×SD16C2211yescharacteristic
20C22×C7×D8C2211yescharacteristic
21C23×C7×C8C2211yescharacteristic
22#41×Dic7—24no—
23#41×Dic7—24no—
24C2×Q8×Dic7C2212yes—
25C2×D4×Dic7—22no—
26C2×D4×Dic7C2212yes—
27C2×#6489—21no—
28C2×#6489C2211yescharacteristic
29C2×#6489C2211yescharacteristic
30C22×C4×Dic7—21no—
31C22×C4×Dic7C2211yescharacteristic
32C23×C7⋊C8C2211yescharacteristic
33#90005—24no—
34#90005—24no—
35#90002C2212yes—
36C2×#6409C2212yes—
37C2×#6409C2212yes—
38#89993C2212yes—
39#89992—22no—
40C2×#6406C2212yes—
41C2×#6406C2212yes—
42C4×C22⋊Dic7—21no—
43C4×C22⋊Dic7—21no—
44#89346C2212yes—
45#89346C2212yes—
46#89345C2212yes—
47#89345C2212yes—
48#20924—71nomaximal, Sylow
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41—22no—
55C2×C7×#41—22no—
56C2×C7×#41—22no—
57C2×C7×#41C2×C414yes—
58C2×C7×#41C2×C414yescomplemented
59C2×C7×#41D414yes—
60C2×C7×#41D414yes—
61C22×C7×Q8C2311yescharacteristic
62C22×C7×Q8C2311yescharacteristic
63C22×C7×D4—21no—
64C22×C7×D4—21no—
65C22×C7×D4C2311yescharacteristic
66C22×C7×D4C2311yescharacteristic
67C23×C4×C7—21no—
68C23×C4×C7—21no—
69C23×C4×C7C2311yescharacteristic
70C23×Dic7—22no—
71C23×Dic7D412yes—
72C7×#135—232no—
73C2×C7×Q16—22no—
74C2×C7×Q16—22no—
75C2×C7×Q16C2×C414yescomplemented
76C2×C7×SD16—22no—
77C2×C7×SD16—22no—
78C2×C7×SD16—22no—
79C2×C7×SD16—22no—
80C2×C7×SD16C2×C414yescomplemented
81C2×C7×SD16C2×C414yescomplemented
82C2×C7×D8—22no—
83C2×C7×D8—22no—
84C2×C7×D8C2×C414yescomplemented
85C22×C7×C8—22no—
86C22×C7×C8—22no—
87C22×C7×C8C2×C414yescomplemented
88C22×C7×C8C2311yescharacteristic
89C22×C7×C8C2311yescharacteristic
90C2×C22⋊Dic7—21no—
91C2×C22⋊Dic7—22no—
92C2×C22⋊Dic7—21no—
93C2×C22⋊Dic7—22no—
94C2×C22⋊Dic7—22no—
95Q8×Dic7—24no—
96Q8×Dic7—24no—
97D4×Dic7—216no—
98D4×Dic7—24no—
99D4×Dic7—24no—
100#6489—28no—
101#6489—22no—
102#6489—22no—
103#6489—22no—
104#6489D414yes—
105C2×C4⋊Dic7—22no—
106C2×C4⋊Dic7C2312yes—
107C2×C4⋊Dic7C2312yes—
108C2×C4×Dic7—28no—
109C2×C4×Dic7—21no—
110C2×C4×Dic7—22no—
111C2×C4×Dic7—22no—
112C2×C4×Dic7—22no—
113C2×C4×Dic7—21no—
114C2×C4×Dic7D411yescharacteristic
115C2×C4×Dic7D411yescharacteristic
116C2×C4×Dic7C2312yes—
117C22×C7⋊C8—22no—
118C22×C7⋊C8—22no—
119C22×C7⋊C8—22no—
120C22×C7⋊C8C2312yes—
121#6409—24no—
122#6409—24no—
123#6406—24no—
124#6406—24no—
125C7⋊#95—22no—
126C7⋊#95—22no—
127C22×#135D711yescharacteristic
128C2×C4×#41—71no—
129C2×#416—71no—
130C2×#416—71no—
131#2297—71no—
132#1880—72no—
133#1879—72no—
134#1878—72no—
135#1877—72no—
136#1735—71no—
137C2×#336—71no—
138C24×C7—21no—
139C24×C7—21no—
140C24×C7C2×D411yescharacteristic
141C7×#41—28no—
142C7×#41—28no—
143C7×#41—28no—
144C7×#41—28no—
145C7×#41—28no—
146C7×#41—44no—
147C7×#41—44no—
148C7×#41C22⋊C4116yescomplemented
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8C22×C412yes—
158C2×C7×Q8C22×C412yes—
159C2×C7×Q8C2×D412yes—
160C2×C7×Q8C2×D412yes—
161C2×C7×D4—24no—
162C2×C7×D4—24no—
163C2×C7×D4—24no—
164C2×C7×D4—24no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—21no—
169C2×C7×D4—21no—
170C2×C7×D4—21no—
171C2×C7×D4—21no—
172C2×C7×D4—21no—
173C2×C7×D4—42no—
174C2×C7×D4—42no—
175C2×C7×D4C22×C412yes—
176C2×C7×D4C22×C412yes—
177C2×C7×D4C2×D412yes—
178C2×C7×D4C2×D412yes—
179C22×C4×C7—21no—
180C22×C4×C7—24no—
181C22×C4×C7—24no—
182C22×C4×C7—24no—
183C22×C4×C7—21no—
184C22×C4×C7—24no—
185C22×C4×C7—42no—
186C22×C4×C7—42no—
187C22×C4×C7C22×C412yes—
188C22×C4×C7C2×D412yes—
189C22×C4×C7C2×D412yes—
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D412yes—
192C22×C4×C7#4112yes—
193C22×C4×C7#4112yes—
194C22×C4×C7C2411yescharacteristic
195C22×Dic7—28no—
196C22×Dic7—24no—
197C22×Dic7—22no—
198C22×Dic7—24no—
199C22×Dic7—22no—
200C22×Dic7—42no—
201C22×Dic7—44no—
202C22×Dic7—44no—
203C22×Dic7C2×D412yes—
204C22×Dic7C2×D412yes—
205C7×Q16—28no—
206C7×SD16—28no—
207C7×SD16—28no—
208C7×D8—28no—
209C2×C7×C8—28no—
210C2×C7×C8—21no—
211C2×C7×C8—21no—
212C2×C7×C8—21no—
213C2×C7×C8—21no—
214C2×C7×C8C22×C412yes—
215C2×C7×C8C22×C412yes—
216C22⋊Dic7—28no—
217C22⋊Dic7—24no—
218C22⋊Dic7—44no—
219C22⋊Dic7—42no—
220C22⋊Dic7—44no—
221C4⋊Dic7—28no—
222C4⋊Dic7—22no—
223C4⋊Dic7—22no—
224C4⋊Dic7—22no—
225C4⋊Dic7C2×D414yes—
226C4×Dic7—28no—
227C4×Dic7—22no—
228C4×Dic7—24no—
229C4×Dic7—22no—
230C4×Dic7—22no—
231C4×Dic7—22no—
232C4×Dic7—42no—
233C2×C7⋊C8—22no—
234C2×C7⋊C8—22no—
235C2×C7⋊C8—22no—
236C2×C7⋊C8—44no—
237C22×#41C2×D711yescharacteristic
238C22×#41C2×D711yescharacteristic
239C2×#135—24no—
240C2×#135—24no—
241C2×#135Dic718yescomplemented
242C22×Q16C2×D711yescharacteristic
243C22×SD16C2×D711yescharacteristic
244C22×SD16C2×D711yescharacteristic
245C22×D8C2×D711yescharacteristic
246C23×C8—71no—
247C23×C8C2×D711yescharacteristic
248C4×#41—144no—
249C4×#41—144no—
250C2×C4×Q8—72no—
251C2×C4×D4—72no—
252C2×C4×D4—142no—
253C2×#117—71no—
254C2×#117—71no—
255C2×#117—141no—
256C22×C42—71no—
257C22×C42—141no—
258#416—144no—
259#416—144no—
260C2×#103—72no—
261C2×#103—72no—
262C2×#102—72no—
263C2×#102—72no—
264#391—72no—
265#390—72no—
266#386—142no—
267C4×C22⋊C4—141no—
268C4×C22⋊C4—141no—
269#340—72no—
270#340—72no—
271#336—72no—
272#336—72no—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—22no—
276C23×C7—22no—
277C23×C7—22no—
278C23×C7—22no—
279C23×C7—42no—
280C23×C7—42no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—42no—
284C23×C7—42no—
285C23×C7C4×D412yes—
286C23×C7#12012yes—
287C23×C7#12112yes—
288C23×C7C22×D411yescharacteristic
289C7×Q8—22no—
290C7×Q8—22no—
291C7×Q8—22no—
292C7×Q8—22no—
293C7×Q8—22no—
294C7×Q8—41no—
295C7×Q8—41no—
296C7×Q8C2×C22⋊C414yes—
297C7×D4—216no—
298C7×D4—22no—
299C7×D4—22no—
300C7×D4—22no—
301C7×D4—22no—
302C7×D4—22no—
303C7×D4—48no—
304C7×D4—41no—
305C7×D4—41no—
306C7×D4C2×C22⋊C414yes—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—216no—
311C2×C4×C7—22no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—48no—
315C2×C4×C7C2×C22⋊C414yes—
316C2×C4×C7C4×D414yes—
317C2×C4×C7C4×D412yes—
318C2×C4×C7#12012yes—
319C2×C4×C7#12114yes—
320C2×C4×C7#12112yes—
321C2×C4×C7#12314yes—
322C2×C4×C7C23×C411yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—24no—
332C2×Dic7—22no—
333C2×Dic7—22no—
334C2×Dic7—48no—
335C2×Dic7—416no—
336C2×Dic7—42no—
337C2×Dic7—42no—
338C2×Dic7—42no—
339C2×Dic7—42no—
340C2×Dic7—42no—
341C2×Dic7#13514yescomplemented
342C7×C8—22no—
343C7×C8—22no—
344C7⋊C8—42no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41C2×Dic714yes—
354C2×#41C2×Dic714yescomplemented
355C2×#41C22⋊D714yes—
356C2×#41C22⋊D714yes—
357C22×Q8C22×D711yescharacteristic
358C22×Q8C22×D711yescharacteristic
359C22×D4—21no—
360C22×D4—21no—
361C22×D4C22×D711yescharacteristic
362C22×D4C22×D711yescharacteristic
363C23×C4—21no—
364C23×C4—21no—
365C23×C4—72no—
366C23×C4—142no—
367C23×C4C22×D711yescharacteristic
368#135—232no—
369C2×Q16—22no—
370C2×Q16—22no—
371C2×Q16C2×Dic714yescomplemented
372C2×SD16—22no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16—22no—
376C2×SD16C2×Dic714yescomplemented
377C2×SD16C2×Dic714yescomplemented
378C2×D8—22no—
379C2×D8—22no—
380C2×D8C2×Dic714yescomplemented
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—72no—
384C22×C8—142no—
385C22×C8—142no—
386C22×C8—142no—
387C22×C8C2×Dic714yescomplemented
388C22×C8C22×D711yescharacteristic
389C22×C8C22×D711yescharacteristic
390C4×Q8—144no—
391C4×Q8—144no—
392C4×D4—144no—
393C4×D4—1416no—
394C4×D4—144no—
395#117—74no—
396#117—148no—
397#117—142no—
398#117—142no—
399#117—142no—
400C2×C4⋊C4—72no—
401C2×C4⋊C4—72no—
402C2×C4⋊C4—142no—
403C2×C22⋊C4—142no—
404C2×C22⋊C4—141no—
405C2×C22⋊C4—142no—
406C2×C22⋊C4—142no—
407C2×C22⋊C4—141no—
408C2×C42—71no—
409C2×C42—71no—
410C2×C42—72no—
411C2×C42—142no—
412C2×C42—148no—
413C2×C42—142no—
414C2×C42—141no—
415C2×C42—142no—
416C2×C42—141no—
417#103—144no—
418#103—144no—
419#102—144no—
420#102—144no—
421#95—142no—
422#95—142no—
423C22×C7—22no—
424C22×C7—22no—
425C22×C7—24no—
426C22×C7—24no—
427C22×C7—24no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—44no—
431C22×C7—44no—
432C22×C7—42no—
433C22×C7—42no—
434C22×C7—42no—
435C22×C7—82no—
436C22×C7—82no—
437C22×C7#38614yes—
438C22×C7C2×C4×D411yescharacteristic
439C22×C7C2×#12011yescharacteristic
440C22×C7C2×#12111yescharacteristic
441C22×C7C2×#13512yes—
442C22×C7#57512yes—
443C4×C7—24no—
444C4×C7—42no—
445C4×C7#38614yes—
446C4×C7C22×C22⋊C411yescharacteristic
447C4×C7C2×C4×D411yescharacteristic
448C4×C7C2×#12111yescharacteristic
449C4×C7C2×#12311yescharacteristic
450Dic7—24no—
451Dic7—42no—
452Dic7—44no—
453Dic7—82no—
454C24—21no—
455C24—21no—
456C24D4×D711yescharacteristic
457#41—28no—
458#41—28no—
459#41—28no—
460#41—28no—
461#41—28no—
462#41—44no—
463#41—44no—
464#41C22⋊Dic7116yescomplemented
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8C22×Dic712yes—
474C2×Q8C22×Dic712yes—
475C2×Q8C2×C22⋊D712yes—
476C2×Q8C2×C22⋊D712yes—
477C2×D4—24no—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—42no—
490C2×D4—42no—
491C2×D4C22×Dic712yes—
492C2×D4C22×Dic712yes—
493C2×D4C2×C22⋊D712yes—
494C2×D4C2×C22⋊D712yes—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—72no—
504C22×C4—72no—
505C22×C4—144no—
506C22×C4—148no—
507C22×C4—144no—
508C22×C4—142no—
509C22×C4—142no—
510C22×C4—284no—
511C22×C4—282no—
512C22×C4—284no—
513C22×C4D4×D712yes—
514C22×C4D4×D711yescharacteristic
515C22×C4#115912yes—
516C22×C4#115912yes—
517C22×C4C22×Dic712yes—
518C22×C4C2×C22⋊D712yes—
519C22×C4C2×C22⋊D712yes—
520C22×C4C23×D711yescharacteristic
521Q16—28no—
522SD16—28no—
523SD16—28no—
524D8—28no—
525C2×C8—28no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—142no—
531C2×C8—142no—
532C2×C8—142no—
533C2×C8—284no—
534C2×C8C22×Dic712yes—
535C2×C8C22×Dic712yes—
536C4⋊C4—74no—
537C4⋊C4—148no—
538C4⋊C4—142no—
539C4⋊C4—142no—
540C4⋊C4—142no—
541C22⋊C4—148no—
542C22⋊C4—144no—
543C22⋊C4—284no—
544C22⋊C4—284no—
545C22⋊C4—282no—
546C42—148no—
547C42—142no—
548C42—142no—
549C42—144no—
550C42—142no—
551C42—142no—
552C42—282no—
553C2×C7—24no—
554C2×C7—44no—
555C2×C7—82no—
556C2×C7C2×#38611yescharacteristic
557C2×C7C4×#13511yescharacteristic
558C2×C7#294311yescharacteristic
559C2×C7#300212yes—
560C2×C7#303512yes—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—41no—
573C23D4×Dic712yes—
574C23#650012yes—
575C23#650212yes—
576C23C2×D4×D711yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—41no—
583Q8—41no—
584Q8C2×C22⋊Dic714yes—
585D4—216no—
586D4—22no—
587D4—22no—
588D4—22no—
589D4—22no—
590D4—22no—
591D4—48no—
592D4—41no—
593D4—41no—
594D4C2×C22⋊Dic714yes—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—216no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—74no—
604C2×C4—142no—
605C2×C4—142no—
606C2×C4—142no—
607C2×C4—142no—
608C2×C4—144no—
609C2×C4—142no—
610C2×C4—288no—
611C2×C4—2816no—
612C2×C4—282no—
613C2×C4—282no—
614C2×C4—282no—
615C2×C4—282no—
616C2×C4—282no—
617C2×C4D4×Dic714yes—
618C2×C4D4×Dic712yes—
619C2×C4#649814yes—
620C2×C4#650012yes—
621C2×C4#650214yes—
622C2×C4#650212yes—
623C2×C4C2×C22⋊Dic714yes—
624C2×C4C2×D4×D711yescharacteristic
625C2×C4C2×#115911yescharacteristic
626C2×C4C2×#115911yescharacteristic
627C2×C4C23×Dic711yescharacteristic
628C2×C4C22×C22⋊D711yescharacteristic
629C2×C4C22×C22⋊D711yescharacteristic
630C8—22no—
631C8—22no—
632C8—282no—
633C7#2092411yescharacteristic, complemented, Sylow
634C22—22no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—82no—
647C22—82no—
648C22#8999214yes—
649C22D7×#13512yes—
650C22#9045912yes—
651C22C2×D4×Dic711yescharacteristic
652C22C2×#650011yescharacteristic
653C22C2×#650211yescharacteristic
654C4—24no—
655C4—42no—
656C4—144no—
657C4—282no—
658C4—284no—
659C4—562no—
660C4#8999214yes—
661C4C2×D4×Dic711yescharacteristic
662C4C2×#649811yescharacteristic
663C4C2×#650211yescharacteristic
664C4C22×C22⋊Dic711yescharacteristic
665C2—24no—
666C2—44no—
667C2—82no—
668C2Dic7×#13511yescharacteristic
669C2#1174134211yescharacteristic
670C2#1174134512yes—
671C2#1174135612yes—
672C2C2×#8999211yescharacteristic
subgroupG/H
centerC22×C4D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC22×C7×#135C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC22×#135D7
hypocenterC7#20924
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
2. C2×#41×Dic7C2semidirect
3. #11741531C2semidirect
4. C2×#90005C2semidirect
5. C2×#90005C2semidirect
6. #11729819C2semidirect
7. #11729802C2semidirect
8. #11729801C2semidirect
9. #11729791C2semidirect
10. #11729723C2semidirect
11. C2×#89345C2semidirect
16. C2×C7×#135C4semidirect
58. C2×C7×#41C2×C4semidirect
75. C2×C7×Q16C2×C4semidirect
80. C2×C7×SD16C2×C4semidirect
81. C2×C7×SD16C2×C4semidirect
84. C2×C7×D8C2×C4semidirect
87. C22×C7×C8C2×C4semidirect
148. C7×#41C22⋊C4semidirect
241. C2×#135Dic7semidirect
341. C2×Dic7#135semidirect
354. C2×#41C2×Dic7semidirect
371. C2×Q16C2×Dic7semidirect
376. C2×SD16C2×Dic7semidirect
377. C2×SD16C2×Dic7semidirect
380. C2×D8C2×Dic7semidirect
387. C22×C8C2×Dic7semidirect
464. #41C22⋊Dic7semidirect
633. C7#20924semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1352
C2×#41×Dic72
#117415312
C2×#900052
#117298192
#117298022
#117298012
#117297912
#117297232
C2×#893452
#209247