Finite Group Databasebeta

← #13354539

Subgroups — #13354539

Order 1792.

1630 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×#6455C211yesmaximal, characteristic
2C2×#89653C212yesmaximal, complemented
3C2×#89498C211yesmaximal, characteristic
4D7×#380C211yesmaximal, characteristic
5C7×#1852C211yesmaximal, characteristic, complemented
6#11729309C211yesmaximal, characteristic, complemented
7#11728387C212yesmaximal, complemented
8#11727682C211yesmaximal, characteristic, complemented
9#11727259C211yesmaximal, characteristic, complemented
10#11726630C212yesmaximal, complemented
11#11726621C212yesmaximal, complemented
12C23×C4×D7C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×(C2×D7)⋊C4—21no—
15C22×C4⋊Dic7C2211yescharacteristic
16C22×C4×Dic7—21no—
17C22×#1131C2212yes—
18C2×#6455—21no—
19C2×#6455—21no—
20C2×#6455—28no—
21C2×#6455C2212yes—
22C2×#6455C2212yes—
23C2×D7×C4⋊C4C2211yescharacteristic
24C2×D7×C4⋊C4C2211yescharacteristic
25C2×D7×C22⋊C4—21no—
26C2×C7×#107C2212yescomplemented
27C2×C7×C22⋊C8C2211yescharacteristic
28C7×#380C2211yescharacteristic
29#89984C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×(C2×D7)⋊C8C2212yes—
32C2×#6391C2212yescomplemented
33#89759C2211yescharacteristic
34#89759C2211yescharacteristic
35C2×#6382C2212yescomplemented
36C2×#6381C2212yescomplemented
37#89714—21no—
38#89653—28no—
39#89498C418yescomplemented
40#89439—21no—
41D7×#95—21no—
42C7×#340C2212yescomplemented
43#89346C2212yescomplemented
44#89324C2212yescomplemented
45#89324C2212yescomplemented
46#20846—71nomaximal, Sylow
47C24×D7D411yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7C2311yescharacteristic
50C23×Dic7—21no—
51C23×Dic7D411yescharacteristic
52C22×C4×D7—24no—
53C22×C4×D7—24no—
54C22×C4×D7—21no—
55C22×C4×D7—21no—
56C22×C4×D7C2×C414yes—
57C22×C4×D7D411yescharacteristic
58C22×C4×D7D411yescharacteristic
59C22×C4×D7Q814yes—
60C22×C4×D7C2311yescharacteristic
61C22×C4×D7C2311yescharacteristic
62C22×C7×C8C2312yes—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2312yes—
68C2×C7×C22⋊C4—21no—
69C2×C22⋊Dic7—21no—
70C2×(C2×D7)⋊C4—28no—
71C2×(C2×D7)⋊C4—22no—
72C2×(C2×D7)⋊C4—22no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—24no—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2312yes—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×C4×Dic7—28no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7—22no—
85C22×C7⋊C8C2312yes—
86C2×#1131—24no—
87C2×#1131—24no—
88C2×#1131C2×C418yescomplemented
89#6455—28no—
90#6455—28no—
91#6455—24no—
92#6455—24no—
93#6455—24no—
94#6455D418yes—
95D7×C4⋊C4—24no—
96D7×C4⋊C4—24no—
97D7×C22⋊C4—44no—
98C7×#107—24no—
99C7×C22⋊C8C2×C414yescomplemented
100C7×#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C22⋊(C7⋊C8)C2×C414yescomplemented
105(C2×D7)⋊C8C2×C418yescomplemented
106#6391—24no—
107#6382—24no—
108#6381—24no—
109C22×#117—71no—
110C2×#428—72no—
111C2×#407—71no—
112C2×#380—71no—
113#1854—72no—
114#1854—72no—
115#1853—71no—
116#1853—71no—
117#1852—71no—
118#1852D711yescharacteristic, complemented
119#1736—72no—
120C24×C7C2×D411yescharacteristic
121C23×D7—22no—
122C23×D7—22no—
123C23×D7—22no—
124C23×D7—22no—
125C23×D7—41no—
126C23×D7—41no—
127C23×D7C4⋊C414yes—
128C23×D7C2×D411yescharacteristic
129C23×D7C2×D411yescharacteristic
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—24no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7C22×C412yes—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×Q812yes—
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—24no—
145C22×Dic7—22no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—42no—
149C22×Dic7—42no—
150C22×Dic7C4⋊C414yes—
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C2×C4×D7—22no—
154C2×C4×D7—22no—
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—22no—
159C2×C4×D7—216no—
160C2×C4×D7—216no—
161C2×C4×D7—42no—
162C2×C4×D7—42no—
163C2×C4×D7—42no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—42no—
167C2×C4×D7C4⋊C414yes—
168C2×C4×D7C4⋊C414yes—
169C2×C4×D7C22×C412yes—
170C2×C4×D7C22×C412yes—
171C2×C4×D7C2×D414yes—
172C2×C4×D7C2×Q812yes—
173C2×C4×D7C2×Q812yes—
174C2×C4×D7#4114yes—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C2×C7×C8C22×C414yes—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4C2×D414yes—
183C7×C22⋊C4—42no—
184C22⋊Dic7—42no—
185(C2×D7)⋊C4—28no—
186(C2×D7)⋊C4—28no—
187(C2×D7)⋊C4—42no—
188(C2×D7)⋊C4—42no—
189C4⋊Dic7—24no—
190C4⋊Dic7—22no—
191C4⋊Dic7—22no—
192C4⋊Dic7—22no—
193C4⋊Dic7C2×D414yes—
194Dic7⋊C4—42no—
195Dic7⋊C4—42no—
196C4×Dic7—28no—
197C4×Dic7—28no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8C22×C414yes—
201#1131—216no—
202C24×C4—71no—
203C22×M16—72no—
204C2×#117—72no—
205C2×#117—72no—
206C2×#117—148no—
207C2×#117—141no—
208C2×#117—141no—
209C22×C4⋊C4—71no—
210C22×C4⋊C4—71no—
211C22×C4⋊C4—71no—
212C22×C4⋊C4C2×D711yescharacteristic
213C22×C22⋊C4—141no—
214C22×C22⋊C4—141no—
215C22×C42—141no—
216#428—148no—
217C2×#107—72no—
218C2×#107C2×D712yescomplemented
219C2×#106—72no—
220C2×#106—72no—
221#407—78no—
222C2×C22⋊C8—71no—
223C2×C22⋊C8—72no—
224C2×C22⋊C8C2×D711yescharacteristic
225#380—71no—
226#380—71no—
227#380—71no—
228#380—141no—
229#380—141no—
230#380C2×D711yescharacteristic
231C2×#95—141no—
232#340—72no—
233#340—72no—
234#340—72no—
235#340C2×D712yescomplemented
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7C2×C4⋊C412yes—
241C23×C7AGL(1,ℤ/8)12yes—
242C23×C7#13712yes—
243C23×C7C22×D411yescharacteristic
244C22×D7—21no—
245C22×D7—21no—
246C22×D7—21no—
247C22×D7—21no—
248C22×D7—21no—
249C22×D7—21no—
250C22×D7—21no—
251C22×D7—21no—
252C22×D7—21no—
253C22×D7—21no—
254C22×D7—28no—
255C22×D7—28no—
256C22×D7—44no—
257C22×D7—44no—
258C22×D7—44no—
259C22×D7—44no—
260C22×D7—44no—
261C22×D7C2×C4⋊C412yes—
262C22×D7C2×C4⋊C412yes—
263C22×D7#12012yes—
264C22×D7#12312yes—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—48no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7C2×C4⋊C412yes—
280C2×C4×C7C23×C411yescharacteristic
281C2×C4×C7C22×D412yes—
282C2×C4×C7C22×Q811yescharacteristic
283C2×C4×C7C2×#4112yes—
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—28no—
287C2×Dic7—24no—
288C2×Dic7—48no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7C2×C4⋊C412yes—
298C2×Dic7C2×C4⋊C412yes—
299C2×Dic7#12012yes—
300C2×Dic7#12312yes—
301C4×D7—28no—
302C4×D7—24no—
303C4×D7—24no—
304C4×D7—28no—
305C4×D7—84no—
306C4×D7—84no—
307C4×D7C4×D418yes—
308C4×D7C22⋊Q818yes—
309C7×C8—24no—
310C7⋊C8—24no—
311C25—71no—
312C23×C4—21no—
313C23×C4—71no—
314C23×C4—71no—
315C23×C4—74no—
316C23×C4—71no—
317C23×C4—71no—
318C23×C4—74no—
319C23×C4—71no—
320C23×C4—144no—
321C23×C4—144no—
322C23×C4—141no—
323C23×C4—141no—
324C23×C4—141no—
325C23×C4C22×D711yescharacteristic
326C2×M16—78no—
327C2×M16—144no—
328C2×M16—144no—
329C22×C8—72no—
330C22×C8C22×D712yes—
331#117—78no—
332#117—144no—
333#117—144no—
334#117—148no—
335#117—148no—
336#117—144no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—72no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—144no—
343C2×C4⋊C4—144no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C4⋊C4C22×D712yes—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—142no—
353C2×C22⋊C4—142no—
354C2×C22⋊C4—148no—
355C2×C22⋊C4—141no—
356C2×C22⋊C4—141no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—284no—
359C2×C42—142no—
360C2×C42—142no—
361C2×C42—148no—
362#107—24no—
363#107—144no—
364#106—144no—
365#106—144no—
366C22⋊C8—74no—
367C22⋊C8—78no—
368C22⋊C8C4×D714yescomplemented
369#95—21no—
370#95—141no—
371#95—141no—
372#95—141no—
373C22×C7—22no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7#42814yescomplemented
380C22×C7C22×C4⋊C411yescharacteristic
381C22×C7C2×#12011yescharacteristic
382C22×C7C2×#12311yescharacteristic
383C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
384C22×C7C2×#13711yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57511yescharacteristic
387C2×D7—22no—
388C2×D7—22no—
389C2×D7—22no—
390C2×D7—22no—
391C2×D7—22no—
392C2×D7—22no—
393C2×D7—22no—
394C2×D7—22no—
395C2×D7—22no—
396C2×D7—416no—
397C2×D7—41no—
398C2×D7—41no—
399C2×D7—41no—
400C2×D7—41no—
401C2×D7#38514yes—
402C4×C7—22no—
403C4×C7—44no—
404C4×C7—81no—
405C4×C7—81no—
406C4×C7C2×C4×D412yes—
407C4×C7C2×C22⋊Q812yes—
408Dic7—22no—
409Dic7—44no—
410Dic7—81no—
411Dic7—81no—
412Dic7#38514yes—
413C24—71no—
414C24—71no—
415C24—74no—
416C24—142no—
417C24—142no—
418C24—142no—
419C24—142no—
420C24—281no—
421C24—281no—
422C24D4×D711yescharacteristic
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—71no—
432C22×C4—72no—
433C22×C4—74no—
434C22×C4—72no—
435C22×C4—74no—
436C22×C4—74no—
437C22×C4—71no—
438C22×C4—74no—
439C22×C4—74no—
440C22×C4—72no—
441C22×C4—72no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—1416no—
445C22×C4—1416no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—142no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—142no—
452C22×C4—144no—
453C22×C4—142no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—282no—
457C22×C4—282no—
458C22×C4—282no—
459C22×C4—282no—
460C22×C4—282no—
461C22×C4—282no—
462C22×C4—282no—
463C22×C4—282no—
464C22×C4C2×C4×D712yes—
465C22×C4D4×D711yescharacteristic
466C22×C4Q8×D712yes—
467C22×C4C23×D711yescharacteristic
468M16—1416no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—74no—
472C2×C8—142no—
473C2×C8—142no—
474C2×C8C2×C4×D714yes—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—74no—
480C4⋊C4—144no—
481C4⋊C4—142no—
482C4⋊C4—142no—
483C4⋊C4—142no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4D4×D714yes—
487C22⋊C4—42no—
488C22⋊C4—148no—
489C22⋊C4—148no—
490C22⋊C4—282no—
491C22⋊C4—282no—
492C22⋊C4—282no—
493C42—148no—
494C42—148no—
495C2×C7—24no—
496C2×C7C2×#38511yescharacteristic
497C2×C7C2×#42811yescharacteristic
498C2×C7#291111yescharacteristic
499C2×C7#300611yescharacteristic
500C2×C7#300711yescharacteristic
501C2×C7#309411yescharacteristic
502C2×C7#309511yescharacteristic
503D7—24no—
504D7—42no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—72no—
510C23—72no—
511C23—72no—
512C23—72no—
513C23—148no—
514C23—141no—
515C23—141no—
516C23—141no—
517C23—141no—
518C23—148no—
519C23—141no—
520C23—141no—
521C23—141no—
522C23—141no—
523C23—141no—
524C23—141no—
525C23—284no—
526C23—284no—
527C23—284no—
528C23—284no—
529C23—284no—
530C23D7×C4⋊C412yes—
531C23#647412yes—
532C23#648112yes—
533C23C2×D4×D711yescharacteristic
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—72no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—78no—
552C2×C4—78no—
553C2×C4—72no—
554C2×C4—148no—
555C2×C4—148no—
556C2×C4—148no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—144no—
560C2×C4—144no—
561C2×C4—144no—
562C2×C4—288no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—564no—
572C2×C4—564no—
573C2×C4D7×C4⋊C412yes—
574C2×C4C22×C4×D711yescharacteristic
575C2×C4C2×D4×D712yes—
576C2×C4C2×Q8×D711yescharacteristic
577C2×C4D7×#4112yes—
578C8—24no—
579C8—144no—
580C7#2084611yescharacteristic, complemented, Sylow
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—74no—
588C22—142no—
589C22—142no—
590C22—142no—
591C22—142no—
592C22—142no—
593C22—142no—
594C22—142no—
595C22—142no—
596C22—142no—
597C22—2816no—
598C22—281no—
599C22—281no—
600C22—281no—
601C22—281no—
602C22#8965314yescomplemented
603C22C2×D7×C4⋊C411yescharacteristic
604C22D7×#12011yescharacteristic
605C22D7×#12311yescharacteristic
606C22C2×#647411yescharacteristic
607C22C2×#648111yescharacteristic
608C22#9046411yescharacteristic
609C22#9046811yescharacteristic
610C4—22no—
611C4—44no—
612C4—74no—
613C4—81no—
614C4—81no—
615C4—142no—
616C4—284no—
617C4—561no—
618C4—561no—
619C4C4×D4×D712yes—
620C4D7×C22⋊Q812yes—
621C2—24no—
622C2—144no—
623C2—282no—
624C2D7×#38511yescharacteristic
625C2C2×#8965311yescharacteristic
626C2#1173665311yescharacteristic
627C2#1173744811yescharacteristic
628C2#1173762911yescharacteristic
629C2#1173872811yescharacteristic
630C2#1173891411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1852C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1852D7
hypocenterC7#20846
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#89653C2semidirect
5. C7×#1852C2semidirect
6. #11729309C2semidirect
7. #11728387C2semidirect
8. #11727682C2semidirect
9. #11727259C2semidirect
10. #11726630C2semidirect
11. #11726621C2semidirect
26. C2×C7×#107C22semidirect
32. C2×#6391C22semidirect
35. C2×#6382C22semidirect
36. C2×#6381C22semidirect
39. #89498C4semidirect
42. C7×#340C22semidirect
43. #89346C22semidirect
44. #89324C22semidirect
45. #89324C22semidirect
88. C2×#1131C2×C4semidirect
99. C7×C22⋊C8C2×C4semidirect
104. C22⋊(C7⋊C8)C2×C4semidirect
105. (C2×D7)⋊C8C2×C4semidirect
118. #1852D7semidirect
218. C2×#107C2×D7semidirect
235. #340C2×D7semidirect
368. C22⋊C8C4×D7semidirect
379. C22×C7#428semidirect
580. C7#20846semidirect
602. C22#89653semidirect

Maximal subgroups

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

subgroupindex
C22×#64552
C2×#896532
C2×#894982
D7×#3802
C7×#18522
#117293092
#117283872
#117276822
#117272592
#117266302
#117266212
#208467