Finite Group Databasebeta

← #13355503

Subgroups — #13355503

Order 1792.

1894 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×#6509C211yesmaximal, characteristic
2C22×D7×M16C211yesmaximal, characteristic
3C2×C7×#419C211yesmaximal, characteristic, complemented
4C2×#89997C211yesmaximal, characteristic, complemented
5#11739712C211yesmaximal, characteristic
6#11729739C211yesmaximal, characteristic, complemented
7#11729335C211yesmaximal, characteristic, complemented
8#11728025C214yesmaximal, complemented
9#11728024C214yesmaximal, complemented
10C23×C7×Q8C2211yescharacteristic
11C23×C4×D7C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C2×#6509—24no—
15C2×#6509—21no—
16C2×#6509—21no—
17C2×#6509C418yescomplemented
18C2×#6509C2212yes—
19C2×#6509C2212yes—
20C22×(C2×D7)⋊C4—21no—
21C2×#6489C2211yescharacteristic
22C22×C4⋊Dic7C2211yescharacteristic
23C22×Dic7⋊C4—21no—
24C22×#1136C2211yescharacteristic
25C2×D7×M16—24no—
26C2×D7×M16—24no—
27C2×D7×M16—24no—
28C22×#1131C2212yes—
29C22×C8×D7C2212yes—
30C7×#419—24no—
31C2×C7×#103C2214yescomplemented
32#89997—24no—
33C2×#6409C2214yescomplemented
34#89759C2212yes—
35#89759C2212yes—
36#89727—21no—
37#89715—21no—
38C4×(C2×D7)⋊C4—21no—
39C4×(C2×D7)⋊C4—21no—
40#89346C2214yescomplemented
41#89324C2214yescomplemented
42#20972—71nomaximal, Sylow
43C24×D7D411yescharacteristic
44C22×C7×Q8—22no—
45C22×C7×Q8—21no—
46C22×C7×Q8—22no—
47C22×C7×Q8—22no—
48C22×C7×Q8—22no—
49C22×C7×Q8—22no—
50C22×C7×Q8C2×C414yes—
51C22×C7×Q8C2312yes—
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C23×Dic7—21no—
55C23×Dic7D411yescharacteristic
56C22×C4×D7—22no—
57C22×C4×D7—22no—
58C22×C4×D7—22no—
59C22×C4×D7—22no—
60C22×C4×D7C2×C414yes—
61C22×C4×D7D411yescharacteristic
62C22×C4×D7D411yescharacteristic
63C22×C4×D7D414yes—
64C22×C4×D7C2311yescharacteristic
65C22×C4×D7C2311yescharacteristic
66C2×C7×M16—22no—
67C2×C7×M16—22no—
68C2×C7×M16D414yes—
69C22×C7×C8C2312yes—
70C7×#117—22no—
71C7×#117—22no—
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C22⋊C4—21no—
74C2×C42×C7—21no—
75C2×C22⋊Dic7—21no—
76#6509—28no—
77#6509—28no—
78#6509—24no—
79#6509—24no—
80#6509C2×C418yes—
81#6509C2×C418yes—
82C2×(C2×D7)⋊C4—28no—
83C2×(C2×D7)⋊C4—22no—
84C2×(C2×D7)⋊C4—22no—
85C2×(C2×D7)⋊C4—22no—
86C2×(C2×D7)⋊C4—22no—
87#6489—22no—
88#6489—22no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7C2×C414yes—
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2312yes—
94C2×Dic7⋊C4—28no—
95C2×Dic7⋊C4—22no—
96C2×Dic7⋊C4—22no—
97C2×C4×Dic7—21no—
98C2×#1136—22no—
99C2×#1136—22no—
100C2×#1136D414yes—
101C22×C7⋊C8C2312yes—
102D7×M16—232no—
103C2×#1131—24no—
104C2×#1131—24no—
105C2×#1131—24no—
106C2×C8×D7—24no—
107C2×C8×D7—24no—
108C2×C8×D7—24no—
109C7×#103—28no—
110#6409—28no—
111C7⋊#95—22no—
112C7⋊#95—22no—
113C23×M16—71no—
114C22×C22⋊Q8—71no—
115C2×#419—71no—
116C2×#419D711yescharacteristic, complemented
117#2280—71no—
118#1894—74no—
119#1892—74no—
120#1736—71no—
121#1736—71no—
122C24×C7C2×D411yescharacteristic
123C23×D7—22no—
124C23×D7—22no—
125C23×D7—24no—
126C23×D7—42no—
127C23×D7C22⋊C414yes—
128C23×D7C2×D411yescharacteristic
129C23×D7C2×D411yescharacteristic
130C2×C7×Q8—24no—
131C2×C7×Q8—22no—
132C2×C7×Q8—28no—
133C2×C7×Q8—28no—
134C2×C7×Q8—22no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—22no—
138C2×C7×Q8—44no—
139C2×C7×Q8—42no—
140C2×C7×Q8—42no—
141C2×C7×Q8—44no—
142C2×C7×Q8C22×C414yes—
143C22×C4×C7—24no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7C22×C412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7#4112yes—
154C22×C4×C7#4112yes—
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—24no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—44no—
162C22×Dic7C22⋊C414yes—
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C2×C4×D7—28no—
166C2×C4×D7—28no—
167C2×C4×D7—28no—
168C2×C4×D7—22no—
169C2×C4×D7—22no—
170C2×C4×D7—22no—
171C2×C4×D7—22no—
172C2×C4×D7—28no—
173C2×C4×D7—21no—
174C2×C4×D7—21no—
175C2×C4×D7—21no—
176C2×C4×D7—21no—
177C2×C4×D7—21no—
178C2×C4×D7—21no—
179C2×C4×D7—21no—
180C2×C4×D7—21no—
181C2×C4×D7C22⋊C414yes—
182C2×C4×D7C22⋊C414yes—
183C2×C4×D7C22×C412yes—
184C2×C4×D7C22×C412yes—
185C2×C4×D7C2×D412yes—
186C2×C4×D7C2×D412yes—
187C7×M16—28no—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C2×C7×C8C2×D414yes—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C22⋊C4—42no—
194C42×C7—42no—
195C22⋊Dic7—42no—
196(C2×D7)⋊C4—28no—
197(C2×D7)⋊C4—28no—
198(C2×D7)⋊C4—44no—
199(C2×D7)⋊C4—44no—
200C4⋊Dic7—24no—
201C4⋊Dic7—22no—
202C4⋊Dic7—22no—
203C4⋊Dic7—22no—
204C4⋊Dic7C22×C414yes—
205Dic7⋊C4—28no—
206Dic7⋊C4—28no—
207C4×Dic7—42no—
208#1136—28no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8C2×D414yes—
212#1131—216no—
213C8×D7—216no—
214C23×Q8C2×D711yescharacteristic
215C24×C4—71no—
216C22×M16—71no—
217C22×M16—72no—
218C22×M16—144no—
219C22×M16—144no—
220C22×M16—144no—
221C22×M16C2×D711yescharacteristic
222C23×C8—72no—
223C2×C22⋊Q8—78no—
224C2×C22⋊Q8—72no—
225C2×C22⋊Q8—72no—
226C2×C22⋊Q8—144no—
227C2×C22⋊Q8—141no—
228C2×C22⋊Q8—141no—
229C2×#117—71no—
230C2×#117C2×D711yescharacteristic
231C22×C4⋊C4—71no—
232C22×C4⋊C4—141no—
233C22×C22⋊C4—141no—
234#419—24no—
235#419—144no—
236C2×#103—74no—
237C2×#103C2×D714yescomplemented
238#381—141no—
239#380—72no—
240#380—72no—
241#379—141no—
242C4×C22⋊C4—141no—
243C4×C22⋊C4—141no—
244#340—74no—
245#340—74no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7C2×C22⋊C412yes—
249C23×C7#12112yes—
250C23×C7#12112yes—
251C23×C7C22×D411yescharacteristic
252C22×D7—21no—
253C22×D7—21no—
254C22×D7—22no—
255C22×D7—21no—
256C22×D7—21no—
257C22×D7—22no—
258C22×D7—22no—
259C22×D7—28no—
260C22×D7—28no—
261C22×D7—44no—
262C22×D7—48no—
263C22×D7—48no—
264C22×D7C2×C22⋊C412yes—
265C22×D7C2×C22⋊C412yes—
266C22×D7#13714yes—
267C7×Q8—24no—
268C7×Q8—24no—
269C7×Q8—42no—
270C7×Q8—42no—
271C7×Q8—48no—
272C2×C4×C7—24no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—48no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7C2×C22⋊C412yes—
280C2×C4×C7C4×D414yes—
281C2×C4×C7C4×D414yes—
282C2×C4×C7#12014yes—
283C2×C4×C7#12112yes—
284C2×C4×C7#12112yes—
285C2×C4×C7#12314yes—
286C2×C4×C7C23×C411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—28no—
297C2×Dic7—24no—
298C2×Dic7—48no—
299C2×Dic7—44no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7C2×C22⋊C412yes—
304C2×Dic7C2×C22⋊C412yes—
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—22no—
311C4×D7—22no—
312C4×D7—22no—
313C4×D7—28no—
314C4×D7—28no—
315C7×C8—24no—
316C7⋊C8—24no—
317C25—71no—
318C22×Q8—22no—
319C22×Q8—21no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8—22no—
324C22×Q8C4×D714yes—
325C22×Q8C22×D712yes—
326C23×C4—21no—
327C23×C4—71no—
328C23×C4—74no—
329C23×C4—74no—
330C23×C4—71no—
331C23×C4—71no—
332C23×C4—71no—
333C23×C4—71no—
334C23×C4—142no—
335C23×C4—142no—
336C23×C4—142no—
337C23×C4—142no—
338C23×C4—141no—
339C23×C4C22×D711yescharacteristic
340C2×M16—22no—
341C2×M16—22no—
342C2×M16—74no—
343C2×M16—144no—
344C2×M16—144no—
345C2×M16—1432no—
346C2×M16—142no—
347C2×M16—142no—
348C2×M16—144no—
349C2×M16C22⋊D714yes—
350C22×C8—72no—
351C22×C8—144no—
352C22×C8—144no—
353C22×C8—144no—
354C22×C8C22×D712yes—
355C22⋊Q8—78no—
356C22⋊Q8—78no—
357C22⋊Q8—148no—
358C22⋊Q8—148no—
359C22⋊Q8—144no—
360C22⋊Q8—144no—
361#117—22no—
362#117—22no—
363#117—142no—
364#117—142no—
365C2×C4⋊C4—74no—
366C2×C4⋊C4—72no—
367C2×C4⋊C4—72no—
368C2×C4⋊C4—148no—
369C2×C4⋊C4—142no—
370C2×C4⋊C4—142no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4C22×D712yes—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—148no—
376C2×C22⋊C4—142no—
377C2×C22⋊C4—142no—
378C2×C22⋊C4—142no—
379C2×C22⋊C4—142no—
380C2×C22⋊C4—141no—
381C2×C42—21no—
382C2×C42—141no—
383#103—28no—
384#103—148no—
385#95—142no—
386#95—142no—
387C22×C7—22no—
388C22×C7—22no—
389C22×C7—24no—
390C22×C7—24no—
391C22×C7#38614yes—
392C22×C7C22×C22⋊C411yescharacteristic
393C22×C7C2×#12111yescharacteristic
394C22×C7C2×#12111yescharacteristic
395C22×C7C2×#13712yes—
396C22×C7#57512yes—
397C2×D7—22no—
398C2×D7—22no—
399C2×D7—24no—
400C2×D7—24no—
401C2×D7—22no—
402C2×D7—22no—
403C2×D7—22no—
404C2×D7—416no—
405C2×D7—42no—
406C2×D7—42no—
407C2×D7#41914yescomplemented
408C4×C7—44no—
409C4×C7—82no—
410C4×C7#38614yes—
411C4×C7C2×C4×D411yescharacteristic
412C4×C7C2×C4×D411yescharacteristic
413C4×C7C2×#12011yescharacteristic
414C4×C7C2×#12311yescharacteristic
415Dic7—22no—
416Dic7—22no—
417Dic7—44no—
418Dic7—82no—
419C24—71no—
420C24—71no—
421C24—74no—
422C24—144no—
423C24—142no—
424C24—142no—
425C24—282no—
426C24D4×D711yescharacteristic
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8C2×C4×D714yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—74no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—74no—
449C22×C4—74no—
450C22×C4—72no—
451C22×C4—72no—
452C22×C4—72no—
453C22×C4—72no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—148no—
457C22×C4—148no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—148no—
461C22×C4—148no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—141no—
471C22×C4—141no—
472C22×C4—144no—
473C22×C4—142no—
474C22×C4—142no—
475C22×C4—284no—
476C22×C4C2×C4×D712yes—
477C22×C4#115712yes—
478C22×C4D4×D711yescharacteristic
479C22×C4D4×D712yes—
480C22×C4D4×D712yes—
481C22×C4#115912yes—
482C22×C4C2×C22⋊D712yes—
483C22×C4C23×D711yescharacteristic
484M16—28no—
485M16—1416no—
486M16—148no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—74no—
490C2×C8—1416no—
491C2×C8—142no—
492C2×C8—142no—
493C2×C8C2×C22⋊D714yes—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—74no—
497C4⋊C4—148no—
498C4⋊C4—148no—
499C4⋊C4—144no—
500C4⋊C4—142no—
501C4⋊C4—142no—
502C4⋊C4—142no—
503C22⋊C4—42no—
504C22⋊C4—148no—
505C22⋊C4—148no—
506C22⋊C4—284no—
507C22⋊C4—284no—
508C22⋊C4—282no—
509C42—42no—
510C42—282no—
511C2×C7—24no—
512C2×C7C2×#38611yescharacteristic
513C2×C7C2×#41911yescharacteristic
514C2×C7#289611yescharacteristic
515C2×C7#305612yes—
516C2×C7#305612yes—
517D7—24no—
518D7—42no—
519C23—22no—
520C23—22no—
521C23—72no—
522C23—74no—
523C23—72no—
524C23—148no—
525C23—141no—
526C23—142no—
527C23—141no—
528C23—141no—
529C23—142no—
530C23—142no—
531C23—141no—
532C23—148no—
533C23—284no—
534C23—288no—
535C23—288no—
536C23D7×C22⋊C412yes—
537C23#644712yes—
538C23#650112yes—
539C23C2×D4×D711yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—42no—
544Q8—48no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—148no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—148no—
568C2×C4—148no—
569C2×C4—144no—
570C2×C4—288no—
571C2×C4—284no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4D7×C22⋊C412yes—
576C2×C4(C22⋊D7)⋊C414yes—
577C2×C4#644614yes—
578C2×C4#644712yes—
579C2×C4C4×C22⋊D714yes—
580C2×C4#650014yes—
581C2×C4#650112yes—
582C2×C4C22×C4×D711yescharacteristic
583C2×C4C2×#115711yescharacteristic
584C2×C4C2×D4×D711yescharacteristic
585C2×C4C2×D4×D711yescharacteristic
586C2×C4C2×#115911yescharacteristic
587C2×C4C22×C22⋊D711yescharacteristic
588C8—24no—
589C8—144no—
590C7#2097211yescharacteristic, complemented, Sylow
591C22—22no—
592C22—22no—
593C22—24no—
594C22—24no—
595C22—74no—
596C22—142no—
597C22—144no—
598C22—142no—
599C22—144no—
600C22—142no—
601C22—142no—
602C22—142no—
603C22—2816no—
604C22—282no—
605C22—282no—
606C22#8972814yes—
607C22C2×D7×C22⋊C411yescharacteristic
608C22C2×#644711yescharacteristic
609C22D7×#13712yes—
610C22#9046812yes—
611C22C2×#650111yescharacteristic
612C4—44no—
613C4—82no—
614C4—142no—
615C4—142no—
616C4—284no—
617C4—562no—
618C4#8972814yes—
619C4C2×(C22⋊D7)⋊C411yescharacteristic
620C4C2×#644611yescharacteristic
621C4C2×C4×C22⋊D711yescharacteristic
622C4C2×#650011yescharacteristic
623C2—24no—
624C2—144no—
625C2—282no—
626C2D7×#41911yescharacteristic
627C2#1173640111yescharacteristic
628C2#1173640712yes—
629C2C2×#8972811yescharacteristic
630C2#1174143012yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#419C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenterC2×#419D7
hypocenterC7#20972
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×C7×#419C2semidirect
4. C2×#89997C2semidirect
6. #11729739C2semidirect
7. #11729335C2semidirect
8. #11728025C2semidirect
9. #11728024C2semidirect
17. C2×#6509C4semidirect
31. C2×C7×#103C22semidirect
33. C2×#6409C22semidirect
40. #89346C22semidirect
41. #89324C22semidirect
116. C2×#419D7semidirect
237. C2×#103C2×D7semidirect
407. C2×D7#419semidirect
590. C7#20972semidirect

Maximal subgroups

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

subgroupindex
C22×#65092
C22×D7×M162
C2×C7×#4192
C2×#899972
#117397122
#117297392
#117293352
#117280252
#117280242
#209727