Finite Group Databasebeta

← #13355265

Subgroups — #13355265

Order 1792.

1814 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6513C211yesmaximal, characteristic
2C2×#90214C211yesmaximal, characteristic, complemented
3C7×#2311C211yesmaximal, characteristic, complemented
4C2×#90004C211yesmaximal, characteristic, complemented
5C2×#89738C211yesmaximal, characteristic, complemented
6#11729333C211yesmaximal, characteristic, complemented
7C2×#89324C211yesmaximal, characteristic, complemented
8#11727444C212yesmaximal, complemented
9#11727441C212yesmaximal, complemented
10#11727414C212yesmaximal, complemented
11#11727411C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C22×#1157C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C2×#6513—24no—
17C2×#6513—24no—
18C2×#6513C418yescomplemented
19C22×#1144C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C22×#1142C2211yescharacteristic
22C22×#1141C2211yescharacteristic
23C22×C4⋊Dic7C2211yescharacteristic
24C22×#1136C2211yescharacteristic
25C23×C7⋊C8C2211yescharacteristic
26#90214—24no—
27#90214—24no—
28C2×#6455—21no—
29C2×D7×C4⋊C4—21no—
30C2×(C22⋊D7)⋊C4—22no—
31C2×C4×D28C2212yes—
32C2×C4×Dic14C2212yes—
33C7×#391C2212yes—
34C7×#390C2212yes—
35C7×#386—22no—
36C7×#380—21no—
37C7×#380—21no—
38#90004—24no—
39C2×#6409C2212yes—
40C2×#6406C2212yes—
41C2×#6384C2212yes—
42#89738—24no—
43C2×#6383C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#89319C2212yes—
48#20937—71nomaximal, Sylow
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41C2×C414yescomplemented
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C22×C22⋊D7—21no—
62C23×Dic7—21no—
63C2×#1157—22no—
64C2×#1157—22no—
65C2×#1157—22no—
66C2×#1157C2×C414yes—
67C2×#1157D414yes—
68C2×#1157D414yes—
69C22×D28C2311yescharacteristic
70C22×C4×D7—21no—
71C22×C4×D7—22no—
72C22×Dic14C2311yescharacteristic
73C7×#117—22no—
74C7×#117—22no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4C2312yes—
81C2×C7×C22⋊C4—22no—
82C2×C7×C22⋊C4—22no—
83C2×C7×C22⋊C4D412yes—
84C2×C42×C7C2312yes—
85#6513—232no—
86C2×#1144—22no—
87C2×#1144—22no—
88C2×#1144C2×C414yescomplemented
89C2×#1143—22no—
90C2×#1143—22no—
91C2×#1143C2×C414yescomplemented
92C2×#1142—22no—
93C2×#1142—22no—
94C2×#1142C2×C414yescomplemented
95C2×#1141—22no—
96C2×#1141—22no—
97C2×#1141C2×C414yescomplemented
98C2×(C2×D7)⋊C4—22no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7—22no—
101C2×C4⋊Dic7D414yes—
102C2×C4⋊Dic7C2312yes—
103C2×Dic7⋊C4—22no—
104C2×C4×Dic7—22no—
105C2×#1136—22no—
106C2×#1136—22no—
107C2×#1136—22no—
108C22×C7⋊C8—22no—
109C22×C7⋊C8—22no—
110C22×C7⋊C8C2×C414yescomplemented
111C22×C7⋊C8C2312yes—
112C22×C7⋊C8C2311yescharacteristic
113C22×C7⋊C8C2311yescharacteristic
114#6455—28no—
115D7×C4⋊C4—28no—
116(C22⋊D7)⋊C4—216no—
117C4×D28—24no—
118C4×D28—24no—
119C4×Dic14—24no—
120C4×Dic14—24no—
121C7×#95—22no—
122C7×#95—22no—
123#6409—24no—
124#6406—24no—
125#6384—24no—
126#6383—24no—
127C22×#135—71no—
128C2×#520—71no—
129C2×#417—71no—
130C2×#417—71no—
131#2311D711yescharacteristic, complemented
132#1880—72no—
133#1879—72no—
134#1878—72no—
135#1877—72no—
136C2×#340—71no—
137#1729—71no—
138C24×C7—21no—
139C24×C7C2×D411yescharacteristic
140C23×D7—21no—
141C7×#41—28no—
142C7×#41—28no—
143C7×#41—44no—
144C7×#41—44no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8C22×C412yes—
151C2×C7×D4—24no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4—21no—
156C2×C7×D4—21no—
157C2×C7×D4—42no—
158C2×C7×D4—42no—
159C2×C7×D4C22×C412yes—
160C22×C4×C7—21no—
161C22×C4×C7—24no—
162C22×C4×C7—22no—
163C22×C4×C7—42no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7—81no—
167C22×C4×C7—81no—
168C22×C4×C7C22×C412yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D412yes—
175C22×C4×C7#4112yes—
176C22×C4×C7#4112yes—
177C22×C4×C7C2411yescharacteristic
178C2×C22⋊D7—24no—
179C2×C22⋊D7—24no—
180C2×C22⋊D7—24no—
181C22×Dic7—21no—
182C22×Dic7—22no—
183C22×Dic7—24no—
184C22×Dic7—81no—
185C22×Dic7—81no—
186#1157—28no—
187#1157—28no—
188#1157—28no—
189#1157C22⋊C4116yescomplemented
190C2×D28—21no—
191C2×D28—21no—
192C2×D28—21no—
193C2×D28C22×C412yes—
194C2×D28C2×D412yes—
195C2×D28C2×D412yes—
196C2×C4×D7—24no—
197C2×C4×D7—24no—
198C2×C4×D7—24no—
199C2×C4×D7—28no—
200C2×C4×D7—82no—
201C2×C4×D7—82no—
202C2×Dic14—21no—
203C2×Dic14—21no—
204C2×Dic14—21no—
205C2×Dic14C22×C412yes—
206C2×Dic14C2×D412yes—
207C2×Dic14C2×D412yes—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—22no—
212C7×C4⋊C4—42no—
213C7×C22⋊C4—24no—
214C7×C22⋊C4—82no—
215C7×C22⋊C4—82no—
216C42×C7—22no—
217C42×C7—22no—
218#1144—28no—
219#1143—28no—
220#1142—28no—
221#1141—28no—
222(C2×D7)⋊C4—28no—
223C4⋊Dic7—24no—
224C4⋊Dic7—22no—
225C4⋊Dic7C2×D414yes—
226Dic7⋊C4—28no—
227C4×Dic7—28no—
228#1136—44no—
229C2×C7⋊C8—28no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—22no—
235C2×C7⋊C8—22no—
236C2×C7⋊C8—22no—
237C2×C7⋊C8C22×C412yes—
238C2×C7⋊C8C22×C412yes—
239C22×#41—71no—
240C22×#41C2×D711yescharacteristic
241C2×#135—78no—
242C2×#135—144no—
243C2×#135—144no—
244C22×Q16—71no—
245C22×SD16—71no—
246C22×SD16—71no—
247C22×D8—71no—
248C22×M16—71no—
249C23×C8—71no—
250#520—144no—
251#520—144no—
252C2×C4×Q8—72no—
253C2×C4×D4—72no—
254C2×C4×D4—142no—
255C2×#117—141no—
256C2×#117C2×D711yescharacteristic
257C22×C4⋊C4—71no—
258C22×C4⋊C4—141no—
259C22×C4⋊C4C2×D711yescharacteristic
260#417—144no—
261#417—144no—
262C2×#103—72no—
263C2×#103—72no—
264C2×#102—72no—
265C2×#102—72no—
266#391C2×D712yes—
267#390C2×D712yes—
268#386—22no—
269#380—21no—
270#380—21no—
271#340—72no—
272#340—72no—
273#336—72no—
274#336—72no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—22no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7C4×D412yes—
287C23×C7#12012yes—
288C23×C7#12112yes—
289C23×C7C22×D411yescharacteristic
290C22×D7—22no—
291C22×D7—22no—
292C22×D7—22no—
293C22×D7—41no—
294C22×D7—41no—
295C22×D7—41no—
296C22×D7—41no—
297C7×Q8—22no—
298C7×Q8—22no—
299C7×Q8—41no—
300C7×Q8—41no—
301C7×D4—22no—
302C7×D4—22no—
303C7×D4—48no—
304C7×D4—41no—
305C7×D4—41no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—48no—
316C2×C4×C7—42no—
317C2×C4×C7—42no—
318C2×C4×C7—42no—
319C2×C4×C7—84no—
320C2×C4×C7C2×C22⋊C414yes—
321C2×C4×C7C4×D414yes—
322C2×C4×C7C4×D412yes—
323C2×C4×C7#12012yes—
324C2×C4×C7#12114yes—
325C2×C4×C7#12112yes—
326C2×C4×C7#12314yes—
327C2×C4×C7C23×C411yescharacteristic
328C2×C4×C7C22×D411yescharacteristic
329C2×C4×C7C22×D411yescharacteristic
330C2×C4×C7C22×D411yescharacteristic
331C2×C4×C7C2×#4111yescharacteristic
332C2×C4×C7C2×#4111yescharacteristic
333C22⋊D7—216no—
334C2×Dic7—22no—
335C2×Dic7—22no—
336C2×Dic7—22no—
337C2×Dic7—24no—
338C2×Dic7—42no—
339C2×Dic7—42no—
340C2×Dic7—84no—
341D28—22no—
342D28—22no—
343D28—22no—
344D28C2×C22⋊C414yes—
345C4×D7—216no—
346C4×D7—88no—
347Dic14—22no—
348Dic14—22no—
349Dic14—22no—
350Dic14C2×C22⋊C414yes—
351C7⋊C8—22no—
352C7⋊C8—22no—
353C7⋊C8—42no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41—74no—
360C2×#41—74no—
361C2×#41—74no—
362C2×#41—142no—
363C2×#41—142no—
364C2×#41—142no—
365C2×#41C4×D714yescomplemented
366C22×Q8—71no—
367C22×Q8C22×D711yescharacteristic
368C22×D4—21no—
369C22×D4—71no—
370C22×D4—141no—
371C22×D4C22×D711yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—141no—
375C23×C4—141no—
376C23×C4—142no—
377C23×C4C22×D711yescharacteristic
378#135—1432no—
379C2×Q16—74no—
380C2×Q16—142no—
381C2×Q16—142no—
382C2×SD16—74no—
383C2×SD16—74no—
384C2×SD16—142no—
385C2×SD16—142no—
386C2×SD16—142no—
387C2×SD16—142no—
388C2×D8—74no—
389C2×D8—142no—
390C2×D8—142no—
391C2×M16—142no—
392C2×M16—142no—
393C2×M16—142no—
394C22×C8—74no—
395C22×C8—71no—
396C22×C8—71no—
397C22×C8—72no—
398C22×C8—142no—
399C22×C8—142no—
400C4×Q8—144no—
401C4×Q8—144no—
402C4×D4—144no—
403C4×D4—144no—
404C4×D4—1416no—
405#117—22no—
406#117—22no—
407#117—148no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—74no—
412C2×C4⋊C4—72no—
413C2×C4⋊C4—148no—
414C2×C4⋊C4—142no—
415C2×C4⋊C4—141no—
416C2×C4⋊C4—142no—
417C2×C4⋊C4C22⋊D711yescharacteristic
418C2×C4⋊C4C22⋊D711yescharacteristic
419C2×C4⋊C4C22×D712yes—
420C2×C22⋊C4—22no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—142no—
423C2×C22⋊C4C22⋊D712yes—
424C2×C42—142no—
425C2×C42C22×D712yes—
426#103—144no—
427#103—144no—
428#102—144no—
429#102—144no—
430#95—22no—
431#95—22no—
432C22×C7—22no—
433C22×C7—22no—
434C22×C7—22no—
435C22×C7—22no—
436C22×C7—22no—
437C22×C7—22no—
438C22×C7—42no—
439C22×C7—42no—
440C22×C7—42no—
441C22×C7—81no—
442C22×C7—81no—
443C22×C7—81no—
444C22×C7—81no—
445C22×C7#38614yes—
446C22×C7C2×C4×D411yescharacteristic
447C22×C7C2×#12011yescharacteristic
448C22×C7C2×#12111yescharacteristic
449C22×C7#57611yescharacteristic
450C22×C7#57711yescharacteristic
451C22×C7#57711yescharacteristic
452C22×C7#57811yescharacteristic
453C2×D7—24no—
454C2×D7—42no—
455C2×D7—42no—
456C2×D7—42no—
457C2×D7—42no—
458C2×D7—42no—
459C2×D7—42no—
460C4×C7—42no—
461C4×C7—42no—
462C4×C7—42no—
463C4×C7—82no—
464C4×C7#38614yes—
465C4×C7C22×C22⋊C411yescharacteristic
466C4×C7C2×C4×D411yescharacteristic
467C4×C7C2×#12111yescharacteristic
468C4×C7C2×#12311yescharacteristic
469Dic7—24no—
470Dic7—44no—
471C24—21no—
472C24—141no—
473C24C2×C22⋊D711yescharacteristic
474#41—28no—
475#41—28no—
476#41—44no—
477#41—44no—
478#41—716no—
479#41—148no—
480#41—148no—
481#41—148no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—72no—
488C2×Q8—72no—
489C2×Q8—72no—
490C2×Q8—141no—
491C2×Q8—141no—
492C2×Q8—141no—
493C2×Q8C2×C4×D712yes—
494C2×D4—24no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—72no—
503C2×D4—72no—
504C2×D4—72no—
505C2×D4—144no—
506C2×D4—144no—
507C2×D4—144no—
508C2×D4—141no—
509C2×D4—141no—
510C2×D4—141no—
511C2×D4C2×C4×D712yes—
512C22×C4—21no—
513C22×C4—24no—
514C22×C4—22no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—81no—
519C22×C4—81no—
520C22×C4—148no—
521C22×C4—141no—
522C22×C4—144no—
523C22×C4—144no—
524C22×C4—144no—
525C22×C4—142no—
526C22×C4—144no—
527C22×C4—562no—
528C22×C4—562no—
529C22×C4—561no—
530C22×C4—561no—
531C22×C4C2×C4×D712yes—
532C22×C4#115712yes—
533C22×C4D4×D712yes—
534C22×C4D4×D712yes—
535C22×C4D4×D712yes—
536C22×C4#115912yes—
537C22×C4C2×C22⋊D712yes—
538C22×C4C2×C22⋊D712yes—
539C22×C4C2×C22⋊D711yescharacteristic
540C22×C4C23×D711yescharacteristic
541Q16—148no—
542SD16—148no—
543SD16—148no—
544D8—148no—
545M16—284no—
546C2×C8—72no—
547C2×C8—72no—
548C2×C8—148no—
549C2×C8—141no—
550C2×C8—141no—
551C2×C8—141no—
552C2×C8—141no—
553C2×C8—142no—
554C2×C8—142no—
555C2×C8—142no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—42no—
561C4⋊C4—74no—
562C4⋊C4—148no—
563C4⋊C4—144no—
564C4⋊C4—142no—
565C22⋊C4—24no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—148no—
569C42—22no—
570C42—22no—
571C42—148no—
572C2×C7—24no—
573C2×C7—82no—
574C2×C7C2×#38611yescharacteristic
575C2×C7#295111yescharacteristic
576C2×C7#295211yescharacteristic
577C2×C7#301211yescharacteristic
578C2×C7#301311yescharacteristic
579C2×C7#304511yescharacteristic
580C2×C7#304611yescharacteristic
581D7—44no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—22no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—142no—
594C23—142no—
595C23—142no—
596C23—281no—
597C23—281no—
598C23—281no—
599C23—281no—
600C23C4×C22⋊D712yes—
601C23#650012yes—
602C23#650112yes—
603C23C22×C22⋊D711yescharacteristic
604Q8—22no—
605Q8—22no—
606Q8—41no—
607Q8—41no—
608Q8—74no—
609Q8—142no—
610Q8—142no—
611Q8—142no—
612D4—22no—
613D4—22no—
614D4—48no—
615D4—41no—
616D4—41no—
617D4—74no—
618D4—1416no—
619D4—142no—
620D4—142no—
621D4—142no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—142no—
637C2×C4—142no—
638C2×C4—142no—
639C2×C4—1416no—
640C2×C4—144no—
641C2×C4—282no—
642C2×C4—282no—
643C2×C4—568no—
644C2×C4—564no—
645C2×C4D7×C22⋊C414yes—
646C2×C4(C22⋊D7)⋊C414yes—
647C2×C4#644614yes—
648C2×C4#644714yes—
649C2×C4C4×C22⋊D712yes—
650C2×C4#650012yes—
651C2×C4#650112yes—
652C2×C4C22×C4×D711yescharacteristic
653C2×C4C2×#115711yescharacteristic
654C2×C4C2×D4×D711yescharacteristic
655C2×C4C2×D4×D711yescharacteristic
656C2×C4C2×D4×D711yescharacteristic
657C2×C4C2×#115911yescharacteristic
658C8—142no—
659C8—142no—
660C8—282no—
661C7#2093711yescharacteristic, complemented, Sylow
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—144no—
676C22—282no—
677C22—282no—
678C22—282no—
679C22—282no—
680C22—282no—
681C22—282no—
682C22#8972814yes—
683C22C2×C4×C22⋊D711yescharacteristic
684C22C2×#650011yescharacteristic
685C22C2×#650111yescharacteristic
686C22#9052611yescharacteristic
687C22#9052711yescharacteristic
688C22#9052811yescharacteristic
689C22#9052911yescharacteristic
690C4—42no—
691C4—42no—
692C4—42no—
693C4—82no—
694C4—144no—
695C4—284no—
696C4#8972814yes—
697C4C2×D7×C22⋊C411yescharacteristic
698C4C2×(C22⋊D7)⋊C411yescharacteristic
699C4C2×#644611yescharacteristic
700C4C2×#644711yescharacteristic
701C2—24no—
702C2—82no—
703C2—284no—
704C2C2×#8972811yescharacteristic
705C2#1174000211yescharacteristic
706C2#1174000311yescharacteristic
707C2#1174083011yescharacteristic
708C2#1174085111yescharacteristic
709C2#1174086911yescharacteristic
710C2#1174087211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2311C2
socleC23×C7C22×D4
prime radicalC22×#6513C2
cosocleC22×C4×C7C24
hypercenter#2311D7
hypocenterC7#20937
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#90214C2semidirect
3. C7×#2311C2semidirect
4. C2×#90004C2semidirect
5. C2×#89738C2semidirect
6. #11729333C2semidirect
7. C2×#89324C2semidirect
8. #11727444C2semidirect
9. #11727441C2semidirect
10. #11727414C2semidirect
11. #11727411C2semidirect
18. C2×#6513C4semidirect
54. C2×C7×#41C2×C4semidirect
88. C2×#1144C2×C4semidirect
91. C2×#1143C2×C4semidirect
94. C2×#1142C2×C4semidirect
97. C2×#1141C2×C4semidirect
110. C22×C7⋊C8C2×C4semidirect
131. #2311D7semidirect
189. #1157C22⋊C4semidirect
365. C2×#41C4×D7semidirect
661. C7#20937semidirect

Maximal subgroups

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

subgroupindex
C22×#65132
C2×#902142
C7×#23112
C2×#900042
C2×#897382
#117293332
C2×#893242
#117274442
#117274412
#117274142
#117274112
#209377