Finite Group Databasebeta

← #13354530

Subgroups — #13354530

Order 1792.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D7×C4⋊C4C211yesmaximal, characteristic
2C2×#89633C212yesmaximal, complemented
3C2×#89498C211yesmaximal, characteristic
4#11731307C211yesmaximal, characteristic
5C7×#1853C211yesmaximal, characteristic, complemented
6#11729309C211yesmaximal, characteristic, complemented
7#11728333C212yesmaximal, complemented
8#11727681C211yesmaximal, characteristic, complemented
9#11727259C211yesmaximal, characteristic, complemented
10#11726628C212yesmaximal, complemented
11#11726621C212yesmaximal, complemented
12C23×C4×D7—21no—
13C23×C4×D7C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic7C2211yescharacteristic
16C22×Dic7⋊C4—21no—
17C22×#1131C2212yes—
18C2×#6455C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4—21no—
21C2×D7×C4⋊C4—21no—
22C2×D7×C4⋊C4—28no—
23C2×D7×C4⋊C4C2212yes—
24C2×D7×C4⋊C4C2212yes—
25C2×C7×#106C2212yescomplemented
26C2×C7×C22⋊C8C2211yescharacteristic
27C7×#380C2211yescharacteristic
28#89984C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×(C2×D7)⋊C8C2212yes—
31C2×#6392C2212yescomplemented
32#89759C2211yescharacteristic
33#89759C2211yescharacteristic
34C2×#6382C2212yescomplemented
35C2×#6381C2212yescomplemented
36#89727—21no—
37C22⋊C4×Dic7—21no—
38#89633—28no—
39#89498C418yescomplemented
40#89438—21no—
41#89436—21no—
42C7×#340C2212yescomplemented
43#89346C2212yescomplemented
44#89324C2212yescomplemented
45#89324C2212yescomplemented
46#20844—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—28no—
55C22×C4×D7—22no—
56C22×C4×D7—22no—
57C22×C4×D7—44no—
58C22×C4×D7—44no—
59C22×C4×D7C2×C414yes—
60C22×C4×D7D411yescharacteristic
61C22×C4×D7D411yescharacteristic
62C22×C4×D7Q814yes—
63C22×C4×D7C2311yescharacteristic
64C22×C4×D7C2311yescharacteristic
65C22×C7×C8C2312yes—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C22⋊C4—21no—
72C2×C22⋊Dic7—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—24no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2312yes—
82C2×Dic7⋊C4—28no—
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4—22no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C22×C7⋊C8C2312yes—
88C2×#1131—24no—
89C2×#1131—24no—
90C2×#1131C2×C418yescomplemented
91#6455—24no—
92#6455—24no—
93D7×C4⋊C4—28no—
94D7×C4⋊C4—28no—
95D7×C4⋊C4—24no—
96D7×C4⋊C4—24no—
97D7×C4⋊C4—24no—
98D7×C4⋊C4D418yes—
99C7×#106—24no—
100C7×C22⋊C8C2×C414yescomplemented
101C7×#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C22⋊(C7⋊C8)C2×C414yescomplemented
106(C2×D7)⋊C8C2×C418yescomplemented
107#6392—24no—
108#6382—24no—
109#6381—24no—
110C23×C4⋊C4—71no—
111C2×#428—72no—
112C2×#407—71no—
113#2280—71no—
114#1853—72no—
115#1853—71no—
116#1853D711yescharacteristic, complemented
117#1852—71no—
118#1852—72no—
119#1852—71no—
120#1736—72no—
121C24×C7C2×D411yescharacteristic
122C23×D7—22no—
123C23×D7—22no—
124C23×D7—22no—
125C23×D7—22no—
126C23×D7—22no—
127C23×D7—22no—
128C23×D7C4⋊C414yes—
129C23×D7C2×D411yescharacteristic
130C23×D7C2×D411yescharacteristic
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—24no—
134C22×C4×C7—22no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×Q812yes—
142C22×C4×C7C2411yescharacteristic
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—24no—
146C22×Dic7—22no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—42no—
150C22×Dic7—42no—
151C22×Dic7C4⋊C414yes—
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C2×C4×D7—22no—
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—22no—
159C2×C4×D7—22no—
160C2×C4×D7—216no—
161C2×C4×D7—216no—
162C2×C4×D7—28no—
163C2×C4×D7—28no—
164C2×C4×D7—432no—
165C2×C4×D7—44no—
166C2×C4×D7—44no—
167C2×C4×D7—44no—
168C2×C4×D7—44no—
169C2×C4×D7C4⋊C414yes—
170C2×C4×D7C4⋊C414yes—
171C2×C4×D7C22×C412yes—
172C2×C4×D7C22×C412yes—
173C2×C4×D7C2×D414yes—
174C2×C4×D7C2×Q812yes—
175C2×C4×D7C2×Q812yes—
176C2×C4×D7#4114yes—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8C22×C414yescomplemented
180C7×C4⋊C4—24no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4C2×D414yes—
185C7×C22⋊C4—42no—
186C22⋊Dic7—42no—
187(C2×D7)⋊C4—42no—
188(C2×D7)⋊C4—42no—
189(C2×D7)⋊C4—42no—
190(C2×D7)⋊C4—42no—
191C4⋊Dic7—24no—
192C4⋊Dic7—22no—
193C4⋊Dic7—22no—
194C4⋊Dic7—22no—
195C4⋊Dic7C2×D414yes—
196Dic7⋊C4—28no—
197Dic7⋊C4—28no—
198C4×Dic7—42no—
199C4×Dic7—42no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8C22×C414yescomplemented
203#1131—216no—
204C24×C4—71no—
205C24×C4—141no—
206C22×M16—72no—
207C2×#117—71no—
208C2×#117—71no—
209C22×C4⋊C4—72no—
210C22×C4⋊C4—71no—
211C22×C4⋊C4—72no—
212C22×C4⋊C4—148no—
213C22×C4⋊C4—141no—
214C22×C4⋊C4—141no—
215C22×C4⋊C4—141no—
216C22×C4⋊C4C2×D711yescharacteristic
217#428—148no—
218C2×#107—72no—
219C2×#107—72no—
220C2×#106—72no—
221C2×#106C2×D712yescomplemented
222#407—78no—
223C2×C22⋊C8—71no—
224C2×C22⋊C8—72no—
225C2×C22⋊C8C2×D711yescharacteristic
226#381—141no—
227#380—71no—
228#380—71no—
229#380—71no—
230#380C2×D711yescharacteristic
231#379—141no—
232C4×C22⋊C4—141no—
233C4×C22⋊C4—141no—
234#340—72no—
235#340—72no—
236#340—72no—
237#340C2×D712yescomplemented
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7C2×C4⋊C412yes—
243C23×C7AGL(1,ℤ/8)12yes—
244C23×C7#13712yes—
245C23×C7C22×D411yescharacteristic
246C22×D7—28no—
247C22×D7—28no—
248C22×D7—28no—
249C22×D7—28no—
250C22×D7—21no—
251C22×D7—21no—
252C22×D7—21no—
253C22×D7—21no—
254C22×D7—21no—
255C22×D7—21no—
256C22×D7—21no—
257C22×D7—21no—
258C22×D7—28no—
259C22×D7—28no—
260C22×D7—28no—
261C22×D7C2×C4⋊C412yes—
262C22×D7C2×C4⋊C412yes—
263C22×D7#12012yes—
264C22×D7#12312yes—
265C22×D7AGL(1,ℤ/8)12yes—
266C22×D7#13712yes—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—48no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7C2×C4⋊C412yes—
282C2×C4×C7C23×C411yescharacteristic
283C2×C4×C7C22×D412yes—
284C2×C4×C7C22×Q811yescharacteristic
285C2×C4×C7C2×#4112yes—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—28no—
289C2×Dic7—24no—
290C2×Dic7—48no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7C2×C4⋊C412yes—
300C2×Dic7C2×C4⋊C412yes—
301C2×Dic7#12012yes—
302C2×Dic7#12312yes—
303C4×D7—28no—
304C4×D7—24no—
305C4×D7—24no—
306C4×D7—28no—
307C4×D7—416no—
308C4×D7—416no—
309C4×D7C4×D418yes—
310C4×D7C22⋊Q818yes—
311C7×C8—24no—
312C7⋊C8—24no—
313C25—71no—
314C23×C4—21no—
315C23×C4—71no—
316C23×C4—71no—
317C23×C4—74no—
318C23×C4—71no—
319C23×C4—71no—
320C23×C4—74no—
321C23×C4—71no—
322C23×C4—144no—
323C23×C4—144no—
324C23×C4—142no—
325C23×C4—142no—
326C23×C4—148no—
327C23×C4—141no—
328C23×C4—284no—
329C23×C4—284no—
330C23×C4C22×D711yescharacteristic
331C2×M16—78no—
332C2×M16—144no—
333C2×M16—144no—
334C22×C8—72no—
335C22×C8C22×D712yes—
336#117—144no—
337#117—144no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—78no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—144no—
346C2×C4⋊C4—148no—
347C2×C4⋊C4—148no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4—142no—
350C2×C4⋊C4—142no—
351C2×C4⋊C4—148no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—144no—
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C4⋊C4C22×D712yes—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C42—141no—
364C2×C42—141no—
365#107—144no—
366#107—144no—
367#106—24no—
368#106—144no—
369C22⋊C8—74no—
370C22⋊C8—78no—
371C22⋊C8C4×D714yescomplemented
372#95—21no—
373#95—141no—
374#95—141no—
375#95—141no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7#42814yescomplemented
383C22×C7C22×C4⋊C411yescharacteristic
384C22×C7C2×#12011yescharacteristic
385C22×C7C2×#12311yescharacteristic
386C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
388C22×C7C2×#13711yescharacteristic
389C22×C7C2×#13711yescharacteristic
390C2×D7—232no—
391C2×D7—22no—
392C2×D7—22no—
393C2×D7—22no—
394C2×D7—22no—
395C2×D7—22no—
396C2×D7—22no—
397C2×D7—22no—
398C2×D7—22no—
399C2×D7—22no—
400C2×D7—22no—
401C2×D7—22no—
402C2×D7—22no—
403C2×D7#38514yes—
404C2×D7#42814yescomplemented
405C4×C7—22no—
406C4×C7—44no—
407C4×C7—81no—
408C4×C7—81no—
409C4×C7C2×C4×D412yes—
410C4×C7C2×C22⋊Q812yes—
411Dic7—22no—
412Dic7—44no—
413Dic7—81no—
414Dic7—81no—
415Dic7#38514yes—
416C24—71no—
417C24—71no—
418C24—74no—
419C24—142no—
420C24—142no—
421C24—142no—
422C24—142no—
423C24—142no—
424C24—142no—
425C24D4×D711yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—71no—
435C22×C4—72no—
436C22×C4—74no—
437C22×C4—72no—
438C22×C4—74no—
439C22×C4—74no—
440C22×C4—71no—
441C22×C4—74no—
442C22×C4—74no—
443C22×C4—72no—
444C22×C4—72no—
445C22×C4—142no—
446C22×C4—142no—
447C22×C4—1416no—
448C22×C4—1416no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—148no—
452C22×C4—148no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—144no—
458C22×C4—142no—
459C22×C4—141no—
460C22×C4—141no—
461C22×C4—284no—
462C22×C4—2832no—
463C22×C4—284no—
464C22×C4—284no—
465C22×C4—284no—
466C22×C4—282no—
467C22×C4—282no—
468C22×C4C2×C4×D712yes—
469C22×C4D4×D711yescharacteristic
470C22×C4Q8×D712yes—
471C22×C4C23×D711yescharacteristic
472M16—1416no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—74no—
476C2×C8—142no—
477C2×C8—142no—
478C2×C8C2×C4×D714yescomplemented
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—74no—
484C4⋊C4—148no—
485C4⋊C4—148no—
486C4⋊C4—144no—
487C4⋊C4—142no—
488C4⋊C4—142no—
489C4⋊C4—142no—
490C4⋊C4D4×D714yes—
491C22⋊C4—42no—
492C22⋊C4—282no—
493C22⋊C4—282no—
494C22⋊C4—282no—
495C22⋊C4—282no—
496C22⋊C4—282no—
497C42—282no—
498C42—282no—
499C2×C7—24no—
500C2×C7C2×#38511yescharacteristic
501C2×C7C2×#42811yescharacteristic
502C2×C7C2×#42811yescharacteristic
503C2×C7#300311yescharacteristic
504C2×C7#300411yescharacteristic
505C2×C7#309211yescharacteristic
506C2×C7#309211yescharacteristic
507D7—24no—
508D7—24no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—72no—
514C23—72no—
515C23—72no—
516C23—72no—
517C23—72no—
518C23—72no—
519C23—148no—
520C23—148no—
521C23—148no—
522C23—148no—
523C23—148no—
524C23—148no—
525C23—148no—
526C23—141no—
527C23—141no—
528C23—141no—
529C23—141no—
530C23—141no—
531C23—141no—
532C23—141no—
533C23—141no—
534C23D7×C4⋊C412yes—
535C23#647712yes—
536C23#647812yes—
537C23C2×D4×D711yescharacteristic
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—72no—
555C2×C4—78no—
556C2×C4—78no—
557C2×C4—72no—
558C2×C4—148no—
559C2×C4—148no—
560C2×C4—148no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—144no—
564C2×C4—144no—
565C2×C4—144no—
566C2×C4—2816no—
567C2×C4—2816no—
568C2×C4—288no—
569C2×C4—282no—
570C2×C4—282no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4D7×C4⋊C412yes—
578C2×C4C22×C4×D711yescharacteristic
579C2×C4C2×D4×D712yes—
580C2×C4C2×Q8×D711yescharacteristic
581C2×C4D7×#4112yes—
582C8—24no—
583C8—144no—
584C7#2084411yescharacteristic, complemented, Sylow
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—74no—
592C22—74no—
593C22—1432no—
594C22—142no—
595C22—142no—
596C22—142no—
597C22—142no—
598C22—142no—
599C22—142no—
600C22—142no—
601C22—142no—
602C22—142no—
603C22—142no—
604C22—142no—
605C22—142no—
606C22#8963314yescomplemented
607C22C2×D7×C4⋊C411yescharacteristic
608C22D7×#12011yescharacteristic
609C22D7×#12311yescharacteristic
610C22C2×#647711yescharacteristic
611C22C2×#647811yescharacteristic
612C22D7×AGL(1,ℤ/8)11yescharacteristic
613C22D7×#13711yescharacteristic
614C4—22no—
615C4—44no—
616C4—74no—
617C4—81no—
618C4—81no—
619C4—142no—
620C4—284no—
621C4—561no—
622C4—561no—
623C4C4×D4×D712yes—
624C4D7×C22⋊Q812yes—
625C2—24no—
626C2—144no—
627C2—144no—
628C2D7×#38511yescharacteristic
629C2C2×#8963311yescharacteristic
630C2D7×#42811yescharacteristic
631C2#1173748411yescharacteristic
632C2#1173759111yescharacteristic
633C2#1173877011yescharacteristic
634C2#1173887211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1853C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1853D7
hypocenterC7#20844
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#89633C2semidirect
5. C7×#1853C2semidirect
6. #11729309C2semidirect
7. #11728333C2semidirect
8. #11727681C2semidirect
9. #11727259C2semidirect
10. #11726628C2semidirect
11. #11726621C2semidirect
25. C2×C7×#106C22semidirect
31. C2×#6392C22semidirect
34. C2×#6382C22semidirect
35. C2×#6381C22semidirect
39. #89498C4semidirect
42. C7×#340C22semidirect
43. #89346C22semidirect
44. #89324C22semidirect
45. #89324C22semidirect
90. C2×#1131C2×C4semidirect
100. C7×C22⋊C8C2×C4semidirect
105. C22⋊(C7⋊C8)C2×C4semidirect
106. (C2×D7)⋊C8C2×C4semidirect
116. #1853D7semidirect
179. C2×C7×C8C22×C4semidirect
202. C2×C7⋊C8C22×C4semidirect
221. C2×#106C2×D7semidirect
237. #340C2×D7semidirect
371. C22⋊C8C4×D7semidirect
382. C22×C7#428semidirect
404. C2×D7#428semidirect
478. C2×C8C2×C4×D7semidirect
584. C7#20844semidirect
606. C22#89633semidirect

Maximal subgroups

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

subgroupindex
C22×D7×C4⋊C42
C2×#896332
C2×#894982
#117313072
C7×#18532
#117293092
#117283332
#117276812
#117272592
#117266282
#117266212
#208447