Finite Group Databasebeta

← #13275551

Subgroups — #13275551

Order 1792.

1790 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6450C211yesmaximal, characteristic
2C2×C4⋊C4×Dic7C211yesmaximal, characteristic
3C2×#89420C211yesmaximal, characteristic
4C7×#1441C211yesmaximal, characteristic
5#11729320C211yesmaximal, characteristic
6#11729319C211yesmaximal, characteristic
7#11727178C212yesmaximal
8#11727175C212yesmaximal
9#11727164C211yesmaximal, characteristic
10Dic7×#95C212yesmaximal
11Dic7×#95C212yesmaximal
12C23×Dic14C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C4⋊Dic7C2211yescharacteristic
16C22×Dic7⋊C4C2211yescharacteristic
17C22×Dic7⋊C4C2211yescharacteristic
18C22×C4×Dic7C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C22×C4×Dic7C2211yescharacteristic
21C22×C4×Dic7C2212yes—
22C22×C4×Dic7C2211yescharacteristic
23C2×#6450—24no—
24C2×#6450—24no—
25C2×#6450C418yescomplemented
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2211yescharacteristic
38C4⋊C4×Dic7—24no—
39#89420—24no—
40#13850—71nomaximal, Sylow
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2312yes—
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C22×Dic14—22no—
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—22no—
56C22×Dic14—22no—
57C22×Dic14—22no—
58C22×Dic14C2×C414yes—
59C2×C7×C4⋊C4—22no—
60C2×C7×C4⋊C4—22no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4C2×C414yes—
64C2×C4⋊Dic7—22no—
65C2×C4⋊Dic7—24no—
66C2×Dic7⋊C4—22no—
67C2×Dic7⋊C4—22no—
68C2×Dic7⋊C4—22no—
69C2×Dic7⋊C4—22no—
70C2×Dic7⋊C4C2×C414yes—
71C2×Dic7⋊C4C2×C414yes—
72C2×C4×Dic7—22no—
73C2×C4×Dic7—22no—
74C2×C4×Dic7—22no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7—24no—
77C2×C4×Dic7—24no—
78C2×C4×Dic7—24no—
79C2×C4×Dic7—24no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7C2×C414yes—
82C2×C4×Dic7C2×C414yes—
83C2×C4×Dic7C2×C414yes—
84C2×C4×Dic7D414yes—
85C2×C4×Dic7D414yes—
86C2×C4×Dic7Q818yescomplemented
87#6450—232no—
88C7×#95—22no—
89C7×#95—24no—
90C7×#95—24no—
91C7⋊#95—22no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C7⋊#95—24no—
96C7⋊#95—24no—
97C7⋊#95—24no—
98C7⋊#95—24no—
99C7⋊#95—24no—
100C22×C4×Q8—71no—
101C2×C4×C4⋊C4—71no—
102C2×#376—71no—
103#1446—72no—
104#1445—72no—
105#1441—71no—
106#1441—71no—
107#1441—71no—
108#1441D711yescharacteristic
109C4×#95—72no—
110C4×#95—72no—
111C24×C7C2411yescharacteristic
112C22×C4×C7—21no—
113C22×C4×C7—21no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C22×C412yes—
136C22×C4×C7C22×C412yes—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×Q814yes—
140C22×C4×C7#4114yes—
141C22×C4×C7#4112yes—
142C22×C4×C7#4112yes—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7C22×C412yes—
168C22×Dic7C22×C412yes—
169C22×Dic7C22×C412yes—
170C22×Dic7C22×C412yes—
171C22×Dic7C2×D412yes—
172C22×Dic7C2×D412yes—
173C22×Dic7C2×D412yes—
174C22×Dic7C2×D412yes—
175C22×Dic7C2×Q814yes—
176C22×Dic7C2×Q814yes—
177C22×Dic7#4114yes—
178C22×Dic7#4112yes—
179C22×Dic7#4112yes—
180C22×Dic7#4114yes—
181C2×Dic14—28no—
182C2×Dic14—28no—
183C2×Dic14—28no—
184C2×Dic14—28no—
185C2×Dic14—28no—
186C2×Dic14—28no—
187C2×Dic14—28no—
188C7×C4⋊C4—28no—
189C7×C4⋊C4—28no—
190C4⋊Dic7—28no—
191Dic7⋊C4—28no—
192Dic7⋊C4—28no—
193C4×Dic7—28no—
194C4×Dic7—28no—
195C4×Dic7—216no—
196C4×Dic7—48no—
197C4×Dic7C22⋊C4116yes—
198C23×Q8—71no—
199C2×C4×Q8—78no—
200C2×C4×Q8—144no—
201C2×C4×Q8—144no—
202C22×C4⋊C4—71no—
203C22×C4⋊C4—71no—
204C22×C4⋊C4—71no—
205C22×C4⋊C4C2×D711yescharacteristic
206C22×C4⋊C4C2×D711yescharacteristic
207C22×C42—71no—
208C22×C42—71no—
209C22×C42—71no—
210C22×C42—72no—
211C22×C42—72no—
212C4×C4⋊C4—144no—
213#376—144no—
214C2×#95—72no—
215C2×#95—72no—
216C2×#95—72no—
217C2×#95—72no—
218C2×#95—72no—
219C2×#95—72no—
220C2×#95—71no—
221C2×#95—72no—
222C2×#95—72no—
223C2×#95C2×D712yes—
224C2×#95C2×D712yes—
225C2×#95C2×D711yescharacteristic
226C23×C7C23×C411yescharacteristic
227C23×C7C22×D411yescharacteristic
228C23×C7C22×D411yescharacteristic
229C23×C7C22×Q812yes—
230C23×C7C2×#4111yescharacteristic
231C23×C7C2×#4112yes—
232C23×C7C2×#4111yescharacteristic
233C23×C7C2×#4111yescharacteristic
234C23×C7C2×#4111yescharacteristic
235C23×C7C2×#4112yes—
236C23×C7C2×#4112yes—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—24no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7C2×C22⋊C414yes—
267C2×C4×C7#11714yes—
268C2×Dic7—22no—
269C2×Dic7—22no—
270C2×Dic7—22no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—24no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7—41no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—41no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7C2×C22⋊C414yes—
300C2×Dic7C2×C22⋊C414yes—
301C2×Dic7#11714yes—
302C2×Dic7C4×Q818yes—
303C2×Dic7C22⋊Q818yes—
304C2×Dic7#12414yes—
305C2×Dic7C4⋊Q814yes—
306Dic14—232no—
307C22×Q8—74no—
308C22×Q8—142no—
309C22×Q8—142no—
310C22×Q8—142no—
311C22×Q8—142no—
312C22×Q8—142no—
313C22×Q8—142no—
314C23×C4—71no—
315C23×C4—71no—
316C23×C4—71no—
317C23×C4—71no—
318C23×C4—72no—
319C23×C4—72no—
320C23×C4C22×D711yescharacteristic
321C23×C4C22×D712yes—
322C23×C4C22×D712yes—
323C23×C4C22×D711yescharacteristic
324C23×C4C22×D711yescharacteristic
325C4×Q8—1432no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—74no—
331C2×C4⋊C4—74no—
332C2×C4⋊C4—142no—
333C2×C4⋊C4—142no—
334C2×C4⋊C4—144no—
335C2×C4⋊C4—142no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4—142no—
338C2×C4⋊C4C4×D714yes—
339C2×C42—74no—
340C2×C42—74no—
341C2×C42—74no—
342C2×C42—74no—
343C2×C42—74no—
344C2×C42—78no—
345C2×C42—142no—
346C2×C42—142no—
347C2×C42—142no—
348C2×C42—142no—
349C2×C42—144no—
350C2×C42—144no—
351C2×C42—144no—
352C2×C42—144no—
353C2×C42—144no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—144no—
358#95—144no—
359#95—144no—
360#95—144no—
361#95—144no—
362#95—144no—
363#95—142no—
364#95—144no—
365#95—144no—
366C22×C7C22×C22⋊C411yescharacteristic
367C22×C7C2×#11711yescharacteristic
368C22×C7C2×#11711yescharacteristic
369C22×C7C2×C4×Q812yes—
370C22×C7C4×#4112yes—
371C22×C7C2×C22⋊Q812yes—
372C22×C7#52511yescharacteristic
373C22×C7#52511yescharacteristic
374C22×C7#52511yescharacteristic
375C22×C7#52511yescharacteristic
376C22×C7#52511yescharacteristic
377C22×C7#52511yescharacteristic
378C22×C7C2×#12411yescharacteristic
379C22×C7#52912yes—
380C22×C7#52912yes—
381C22×C7#52912yes—
382C22×C7#52912yes—
383C22×C7C2×C4⋊Q811yescharacteristic
384C22×C7#53212yes—
385C22×C7#53312yes—
386C22×C7#53312yes—
387C22×C7#53312yes—
388C22×C7#53312yes—
389C4×C7—24no—
390C4×C7—24no—
391C4×C7—44no—
392C4×C7—44no—
393C4×C7—42no—
394Dic7—24no—
395Dic7—24no—
396Dic7—44no—
397Dic7—44no—
398Dic7—42no—
399Dic7#39118yes—
400C24C23×D711yescharacteristic
401C2×Q8—148no—
402C2×Q8—148no—
403C2×Q8—148no—
404C2×Q8—148no—
405C2×Q8—148no—
406C2×Q8—148no—
407C2×Q8—148no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—72no—
431C22×C4—72no—
432C22×C4—72no—
433C22×C4—74no—
434C22×C4—74no—
435C22×C4—72no—
436C22×C4—72no—
437C22×C4—72no—
438C22×C4—72no—
439C22×C4—72no—
440C22×C4—72no—
441C22×C4—72no—
442C22×C4—74no—
443C22×C4—74no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4—142no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4C2×C4×D712yes—
469C22×C4C2×C4×D712yes—
470C22×C4C2×C4×D712yes—
471C22×C4D4×D712yes—
472C22×C4D4×D712yes—
473C22×C4#115914yes—
474C22×C4#115912yes—
475C22×C4Q8×D714yes—
476C22×C4#116112yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—148no—
480C4⋊C4—148no—
481C4⋊C4—148no—
482C42—716no—
483C42—1416no—
484C42—148no—
485C42—148no—
486C42—288no—
487C2×C7#228011yescharacteristic
488C2×C7C2×#39111yescharacteristic
489C2×C7#229512yes—
490C2×C7#229512yes—
491C2×C7#229711yescharacteristic
492C2×C7#239612yes—
493C2×C7#239612yes—
494C2×C7#239811yescharacteristic
495C2×C7#239811yescharacteristic
496C2×C7#239911yescharacteristic
497C2×C7#239911yescharacteristic
498C23C22×C4×D711yescharacteristic
499C23C2×D4×D711yescharacteristic
500C23C2×D4×D711yescharacteristic
501C23C2×#115912yes—
502C23C2×#115911yescharacteristic
503C23C2×Q8×D712yes—
504C23C2×#116111yescharacteristic
505C23D7×#4111yescharacteristic
506C23D7×#4112yes—
507C23D7×#4112yes—
508C23D7×#4111yescharacteristic
509Q8—1432no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—74no—
540C2×C4—74no—
541C2×C4—78no—
542C2×C4—78no—
543C2×C4—74no—
544C2×C4—74no—
545C2×C4—74no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—142no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—144no—
559C2×C4—144no—
560C2×C4—144no—
561C2×C4—144no—
562C2×C4—142no—
563C2×C4—144no—
564C2×C4—144no—
565C2×C4—144no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—281no—
570C2×C4—282no—
571C2×C4—282no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4—281no—
576C2×C4—282no—
577C2×C4D7×C22⋊C414yes—
578C2×C4#645514yes—
579C7#1385011yescharacteristic, complemented, Sylow
580C22C2×D7×C22⋊C411yescharacteristic
581C22C2×#645511yescharacteristic
582C22D7×#11711yescharacteristic
583C22C4×#115912yes—
584C22C4×Q8×D712yes—
585C22#9027711yescharacteristic
586C22#9028712yes—
587C22#9029112yes—
588C22#9029711yescharacteristic
589C22#9029711yescharacteristic
590C22#9031312yes—
591C22#9031312yes—
592C22D7×C22⋊Q812yes—
593C22#9031811yescharacteristic
594C22#9031811yescharacteristic
595C22#9034012yes—
596C22#9034711yescharacteristic
597C22D7×#12411yescharacteristic
598C22#9037912yes—
599C22#9038212yes—
600C22#9039712yes—
601C22D7×C4⋊Q811yescharacteristic
602C22#9041912yes—
603C4—24no—
604C4—24no—
605C4—44no—
606C4—42no—
607C4—44no—
608C4—78no—
609C4—144no—
610C4—144no—
611C4—284no—
612C4—282no—
613C4—284no—
614C2#1173130711yescharacteristic
615C2#1173210311yescharacteristic
616C2#1173238312yes—
617C2#1173340712yes—
618C2D7×#39111yescharacteristic
619C2#1173433811yescharacteristic
620C2#1173461611yescharacteristic
621C2#1173472712yes—
622C2#1173518412yes—
623C2#1173518511yescharacteristic
624C2#1173557611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1441C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1441D7
hypocenterC7#13850
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
25. C2×#6450C4semidirect
86. C2×C4×Dic7Q8semidirect
579. C7#13850semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
C2×C4⋊C4×Dic72
C2×#894202
C7×#14412
#117293202
#117293192
#117271782
#117271752
#117271642
Dic7×#952
#138507