Finite Group Databasebeta

← #13285087

Subgroups — #13285087

Order 1792.

2734 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6459C211yesmaximal, characteristic
2C22×C42×D7C211yesmaximal, characteristic
3C2×#89439C211yesmaximal, characteristic
4C2×#89418C212yesmaximal, complemented
5C7×#1436C211yesmaximal, characteristic, complemented
6#11729318C211yesmaximal, characteristic, complemented
7#11729290C211yesmaximal, characteristic, complemented
8(C2×D7)⋊#95C212yesmaximal
9(C2×D7)⋊#95C212yesmaximal
10#11727159C211yesmaximal, characteristic, complemented
11Dic7⋊#95C212yesmaximal, complemented
12C23×C4×D7C2212yes—
13C23×C4×D7C2211yescharacteristic
14C23×Dic14C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C42×C7C2211yescharacteristic
17C22×(C2×D7)⋊C4C2212yes—
18C22×(C2×D7)⋊C4C2211yescharacteristic
19C22×(C2×D7)⋊C4C2211yescharacteristic
20C22×Dic7⋊C4C2212yes—
21C22×Dic7⋊C4C2211yescharacteristic
22C22×Dic7⋊C4C2211yescharacteristic
23C22×C4×Dic7C2211yescharacteristic
24C22×C4×Dic7C2212yes—
25C2×#6459—24no—
26C2×#6459—24no—
27C2×#6459C418yescomplemented
28C2×C42×D7—24no—
29C2×C42×D7—28no—
30C2×C7×#95C2212yes—
31C2×C7×#95C2211yescharacteristic
32C2×C7×#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C2×C7⋊#95C2212yes—
39C2×C7⋊#95C2211yescharacteristic
40#89439—24no—
41#89418—28no—
42#14702—71nomaximal, Sylow
43C24×D7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2312yes—
49C23×Dic7C2312yes—
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2312yes—
53C23×Dic7C2311yescharacteristic
54C22×C4×D7—24no—
55C22×C4×D7—24no—
56C22×C4×D7—24no—
57C22×C4×D7—24no—
58C22×C4×D7—24no—
59C22×C4×D7—24no—
60C22×C4×D7—24no—
61C22×C4×D7—24no—
62C22×C4×D7C2×C414yes—
63C22×C4×D7D414yes—
64C22×C4×D7D414yes—
65C22×C4×D7Q818yescomplemented
66C22×Dic14—24no—
67C22×Dic14—22no—
68C22×Dic14—22no—
69C22×Dic14—24no—
70C22×Dic14C2×C414yescomplemented
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4C2×C414yescomplemented
74C2×C42×C7—22no—
75C2×C42×C7—24no—
76C2×(C2×D7)⋊C4—22no—
77C2×(C2×D7)⋊C4—22no—
78C2×(C2×D7)⋊C4—22no—
79C2×(C2×D7)⋊C4—22no—
80C2×(C2×D7)⋊C4—24no—
81C2×(C2×D7)⋊C4—24no—
82C2×(C2×D7)⋊C4—24no—
83C2×(C2×D7)⋊C4C2×C414yes—
84C2×(C2×D7)⋊C4C2×C414yes—
85C2×Dic7⋊C4—22no—
86C2×Dic7⋊C4—22no—
87C2×Dic7⋊C4—22no—
88C2×Dic7⋊C4—22no—
89C2×Dic7⋊C4—24no—
90C2×Dic7⋊C4—24no—
91C2×Dic7⋊C4—24no—
92C2×Dic7⋊C4C2×C414yescomplemented
93C2×Dic7⋊C4C2×C414yescomplemented
94C2×C4×Dic7—22no—
95C2×C4×Dic7—24no—
96C2×C4×Dic7—24no—
97C2×C4×Dic7—24no—
98C2×C4×Dic7D418yes—
99#6459—232no—
100C42×D7—232no—
101C7×#95—24no—
102C7×#95—22no—
103C7×#95—24no—
104C7⋊#95—24no—
105C7⋊#95—22no—
106C7⋊#95—24no—
107C7⋊#95—24no—
108C7⋊#95—24no—
109C7⋊#95—24no—
110C7⋊#95—24no—
111C22×C22⋊Q8—71no—
112C23×C42—71no—
113C2×#391—72no—
114C2×#380—71no—
115#1436—71no—
116#1436—71no—
117#1436—71no—
118#1436D711yescharacteristic, complemented
119C4⋊#95—72no—
120C22⋊#95—72no—
121C22⋊#95—72no—
122C24×C7C2411yescharacteristic
123C23×D7—22no—
124C23×D7—22no—
125C23×D7—22no—
126C23×D7—22no—
127C23×D7C22×C412yes—
128C23×D7C2×D412yes—
129C23×D7C2×D412yes—
130C23×D7C2×Q814yes—
131C23×D7#4114yes—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
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×C7C22×C412yes—
155C22×C4×C7C22×C412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D414yes—
159C22×C4×C7C2×Q814yes—
160C22×C4×C7#4114yes—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
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×Dic7C22×C412yes—
181C22×Dic7C22×C412yes—
182C22×Dic7C22×C412yes—
183C22×Dic7C2×D414yes—
184C22×Dic7C2×D412yes—
185C22×Dic7C2×D412yes—
186C22×Dic7C2×D414yes—
187C22×Dic7C2×Q814yes—
188C22×Dic7#4114yes—
189C22×Dic7#4114yes—
190C2×C4×D7—216no—
191C2×C4×D7—216no—
192C2×C4×D7—216no—
193C2×C4×D7—216no—
194C2×C4×D7—216no—
195C2×C4×D7—216no—
196C2×C4×D7—216no—
197C2×C4×D7—28no—
198C2×C4×D7—216no—
199C2×C4×D7—216no—
200C2×C4×D7—28no—
201C2×C4×D7C22⋊C4116yes—
202C2×Dic14—28no—
203C2×Dic14—216no—
204C2×Dic14—44no—
205C2×Dic14—48no—
206C2×Dic14—44no—
207C7×C4⋊C4—28no—
208C42×C7—28no—
209(C2×D7)⋊C4—28no—
210(C2×D7)⋊C4—28no—
211(C2×D7)⋊C4—48no—
212Dic7⋊C4—28no—
213Dic7⋊C4—28no—
214Dic7⋊C4—48no—
215C4×Dic7—28no—
216C4×Dic7—216no—
217C23×Q8—71no—
218C24×C4—71no—
219C24×C4—72no—
220C2×C22⋊Q8—78no—
221C2×C22⋊Q8—144no—
222C2×C22⋊Q8—144no—
223C22×C4⋊C4—71no—
224C22×C4⋊C4—71no—
225C22×C4⋊C4—72no—
226C22×C4⋊C4C2×D711yescharacteristic
227C22×C22⋊C4—71no—
228C22×C22⋊C4—71no—
229C22×C22⋊C4—72no—
230C22×C42—72no—
231C22×C42—71no—
232C22×C42—148no—
233C22×C42—144no—
234C22×C42C2×D711yescharacteristic
235#391—148no—
236#380—144no—
237C2×#95—72no—
238C2×#95—72no—
239C2×#95—72no—
240C2×#95—72no—
241C2×#95—72no—
242C2×#95—72no—
243C2×#95—71no—
244C2×#95C2×D712yes—
245C2×#95C2×D712yes—
246C2×#95C2×D711yescharacteristic
247C23×C7C23×C411yescharacteristic
248C23×C7C22×D411yescharacteristic
249C23×C7C22×D411yescharacteristic
250C23×C7C22×D412yes—
251C23×C7C22×Q812yes—
252C23×C7C2×#4112yes—
253C23×C7C2×#4112yes—
254C23×C7C2×#4112yes—
255C23×C7C2×#4112yes—
256C22×D7—24no—
257C22×D7—22no—
258C22×D7—22no—
259C22×D7—22no—
260C22×D7—24no—
261C22×D7—24no—
262C22×D7—24no—
263C22×D7—22no—
264C22×D7—24no—
265C22×D7—24no—
266C22×D7—24no—
267C22×D7—24no—
268C22×D7—24no—
269C22×D7—24no—
270C22×D7—24no—
271C22×D7—24no—
272C22×D7C2×C22⋊C414yes—
273C22×D7C4×Q818yes—
274C22×D7C22⋊Q818yes—
275C22×D7#12414yes—
276C22×D7C4⋊Q814yes—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—22no—
284C2×C4×C7—24no—
285C2×C4×C7—24no—
286C2×C4×C7—24no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—24no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—41no—
293C2×C4×C7—42no—
294C2×C4×C7—41no—
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×C7C2×C22⋊C414yes—
305C2×C4×C7C22⋊Q818yescomplemented
306C2×Dic7—24no—
307C2×Dic7—24no—
308C2×Dic7—24no—
309C2×Dic7—24no—
310C2×Dic7—24no—
311C2×Dic7—24no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—24no—
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—41no—
318C2×Dic7—42no—
319C2×Dic7—41no—
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×Dic7C2×C22⋊C414yes—
330C2×Dic7C4×D418yes—
331C2×Dic7#12014yes—
332C2×Dic7#12118yes—
333C2×Dic7C22⋊Q818yescomplemented
334C2×Dic7#12314yes—
335C4×D7—264no—
336C4×D7—232no—
337Dic14—416no—
338C25—71no—
339C22×Q8—74no—
340C22×Q8—142no—
341C22×Q8—142no—
342C22×Q8—144no—
343C22×Q8—144no—
344C23×C4—71no—
345C23×C4—72no—
346C23×C4—74no—
347C23×C4—74no—
348C23×C4—74no—
349C23×C4—71no—
350C23×C4—71no—
351C23×C4—78no—
352C23×C4—72no—
353C23×C4—144no—
354C23×C4—144no—
355C23×C4—144no—
356C23×C4—144no—
357C23×C4—144no—
358C23×C4—144no—
359C23×C4—144no—
360C23×C4—144no—
361C23×C4C22×D712yes—
362C23×C4C22×D711yescharacteristic
363C23×C4C22×D711yescharacteristic
364C23×C4C22×D711yescharacteristic
365C23×C4C22×D712yes—
366C22⋊Q8—1432no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—74no—
370C2×C4⋊C4—74no—
371C2×C4⋊C4—142no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—142no—
375C2×C4⋊C4—144no—
376C2×C4⋊C4—144no—
377C2×C4⋊C4—144no—
378C2×C4⋊C4C4×D714yescomplemented
379C2×C22⋊C4—74no—
380C2×C22⋊C4—74no—
381C2×C22⋊C4—142no—
382C2×C22⋊C4—142no—
383C2×C22⋊C4—142no—
384C2×C22⋊C4—142no—
385C2×C22⋊C4—144no—
386C2×C22⋊C4—144no—
387C2×C22⋊C4—144no—
388C2×C42—22no—
389C2×C42—24no—
390C2×C42—78no—
391C2×C42—1432no—
392C2×C42—144no—
393C2×C42—144no—
394C2×C42—144no—
395C2×C42—142no—
396#95—24no—
397#95—22no—
398#95—24no—
399#95—144no—
400#95—144no—
401#95—144no—
402#95—144no—
403#95—144no—
404#95—142no—
405#95—144no—
406C22×C7C22×C22⋊C411yescharacteristic
407C22×C7C2×C4×D412yes—
408C22×C7C2×C4×Q812yes—
409C22×C7C4×#4111yescharacteristic
410C22×C7C4×#4111yescharacteristic
411C22×C7C2×#12011yescharacteristic
412C22×C7C2×#12112yes—
413C22×C7C2×C22⋊Q812yes—
414C22×C7C2×C22⋊Q812yes—
415C22×C7C2×C22⋊Q812yes—
416C22×C7C2×#12311yescharacteristic
417C22×C7#52512yes—
418C22×C7#52512yes—
419C22×C7#52512yes—
420C22×C7C2×#12411yescharacteristic
421C22×C7#52912yes—
422C22×C7#52912yes—
423C22×C7C2×C4⋊Q811yescharacteristic
424C22×C7#53212yes—
425C22×C7#53312yes—
426C22×C7#53312yes—
427C2×D7—28no—
428C2×D7—28no—
429C2×D7—28no—
430C2×D7—28no—
431C2×D7—24no—
432C2×D7—24no—
433C2×D7—28no—
434C2×D7—28no—
435C2×D7#39118yescomplemented
436C4×C7—28no—
437C4×C7—24no—
438C4×C7—42no—
439C4×C7—42no—
440C4×C7—82no—
441Dic7—28no—
442Dic7—42no—
443Dic7—42no—
444Dic7—82no—
445Dic7#38618yes—
446C24—72no—
447C24—72no—
448C24—72no—
449C24—74no—
450C24—74no—
451C24—142no—
452C24—142no—
453C24—142no—
454C24—142no—
455C24C23×D711yescharacteristic
456C2×Q8—148no—
457C2×Q8—1416no—
458C2×Q8—284no—
459C2×Q8—288no—
460C2×Q8—284no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—72no—
483C22×C4—74no—
484C22×C4—72no—
485C22×C4—72no—
486C22×C4—74no—
487C22×C4—74no—
488C22×C4—74no—
489C22×C4—74no—
490C22×C4—716no—
491C22×C4—72no—
492C22×C4—72no—
493C22×C4—148no—
494C22×C4—1416no—
495C22×C4—1416no—
496C22×C4—1416no—
497C22×C4—148no—
498C22×C4—1416no—
499C22×C4—1416no—
500C22×C4—1416no—
501C22×C4—1416no—
502C22×C4—1416no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—142no—
506C22×C4—142no—
507C22×C4—1416no—
508C22×C4—142no—
509C22×C4—142no—
510C22×C4—141no—
511C22×C4—141no—
512C22×C4—141no—
513C22×C4—141no—
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×C4C2×C4×D712yes—
524C22×C4C2×C4×D712yes—
525C22×C4C2×C4×D712yes—
526C22×C4#115714yes—
527C22×C4D4×D712yes—
528C22×C4D4×D712yes—
529C22×C4D4×D714yes—
530C22×C4Q8×D714yes—
531C4⋊C4—28no—
532C4⋊C4—148no—
533C4⋊C4—148no—
534C4⋊C4—288no—
535C22⋊C4—148no—
536C22⋊C4—148no—
537C22⋊C4—288no—
538C42—28no—
539C42—1416no—
540C42—148no—
541C2×C7C2×#38611yescharacteristic
542C2×C7C2×#39111yescharacteristic
543C2×C7#229711yescharacteristic
544C2×C7C4×C22⋊Q812yes—
545C2×C7C4×C22⋊Q812yes—
546C2×C7#284812yes—
547C2×C7#285012yes—
548C2×C7#285512yes—
549C2×C7#285712yes—
550D7—28no—
551C23—74no—
552C23—74no—
553C23—78no—
554C23—78no—
555C23—74no—
556C23—144no—
557C23—144no—
558C23—142no—
559C23—144no—
560C23—142no—
561C23—144no—
562C23—144no—
563C23—144no—
564C23—144no—
565C23—144no—
566C23—144no—
567C23—142no—
568C23—144no—
569C23—144no—
570C23—142no—
571C23—144no—
572C23C22×C4×D711yescharacteristic
573C23C2×#115712yes—
574C23C2×D4×D711yescharacteristic
575C23C2×D4×D711yescharacteristic
576C23C2×D4×D712yes—
577C23C2×Q8×D712yes—
578C23D7×#4112yes—
579C23D7×#4112yes—
580C23D7×#4112yes—
581Q8—2816no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—74no—
610C2×C4—78no—
611C2×C4—74no—
612C2×C4—78no—
613C2×C4—74no—
614C2×C4—78no—
615C2×C4—1432no—
616C2×C4—1464no—
617C2×C4—144no—
618C2×C4—144no—
619C2×C4—144no—
620C2×C4—144no—
621C2×C4—144no—
622C2×C4—144no—
623C2×C4—142no—
624C2×C4—142no—
625C2×C4—142no—
626C2×C4—142no—
627C2×C4—144no—
628C2×C4—282no—
629C2×C4—282no—
630C2×C4—281no—
631C2×C4—282no—
632C2×C4—281no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—282no—
637C2×C4—282no—
638C2×C4—282no—
639C2×C4—282no—
640C2×C4D7×C22⋊C414yes—
641C2×C4#645918yescomplemented
642C7#1470211yescharacteristic, complemented, Sylow
643C22—78no—
644C22—148no—
645C22—144no—
646C22—144no—
647C22—148no—
648C22—148no—
649C22—148no—
650C22—148no—
651C22—148no—
652C22C4×#115711yescharacteristic
653C22C2×D7×C22⋊C411yescharacteristic
654C22C2×#645912yes—
655C22#9021311yescharacteristic
656C22#9021912yes—
657C22C4×D4×D712yes—
658C22#9023812yes—
659C22#9023812yes—
660C22#9023912yes—
661C22C4×Q8×D712yes—
662C22#9026912yes—
663C22D7×#12011yescharacteristic
664C22D7×#12112yes—
665C22D7×C22⋊Q812yes—
666C22D7×C22⋊Q812yes—
667C22D7×#12311yescharacteristic
668C22#9034712yes—
669C22D7×#12411yescharacteristic
670C22#9037912yes—
671C22#9039712yes—
672C22D7×C4⋊Q811yescharacteristic
673C4—24no—
674C4—28no—
675C4—42no—
676C4—42no—
677C4—78no—
678C4—82no—
679C4—148no—
680C4—282no—
681C4—282no—
682C4—562no—
683C2—148no—
684C2#1173100811yescharacteristic
685C2C4×#645912yes—
686C2#1173210012yes—
687C2D7×#38611yescharacteristic
688C2#1173238612yes—
689C2#1173289312yes—
690C2#1173340212yes—
691C2#1173373312yes—
692C2D7×#39111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1436C2
socle, cosocleC24×C7C24
prime radicalC24×D7C23
hypercenter#1436D7
hypocenterC7#14702
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
4. C2×#89418C2semidirect
5. C7×#1436C2semidirect
6. #11729318C2semidirect
7. #11729290C2semidirect
10. #11727159C2semidirect
11. Dic7⋊#95C2semidirect
27. C2×#6459C4semidirect
65. C22×C4×D7Q8semidirect
70. C22×Dic14C2×C4semidirect
73. C2×C7×C4⋊C4C2×C4semidirect
92. C2×Dic7⋊C4C2×C4semidirect
93. C2×Dic7⋊C4C2×C4semidirect
118. #1436D7semidirect
305. C2×C4×C7C22⋊Q8semidirect
333. C2×Dic7C22⋊Q8semidirect
378. C2×C4⋊C4C4×D7semidirect
435. C2×D7#391semidirect
641. C2×C4#6459semidirect
642. C7#14702semidirect

Maximal subgroups

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

subgroupindex
C22×#64592
C22×C42×D72
C2×#894392
C2×#894182
C7×#14362
#117293182
#117292902
(C2×D7)⋊#952
#117271592
Dic7⋊#952
#147027