Finite Group Databasebeta

← #13363533

Subgroups — #13363533

Order 1792.

1234 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90271C211yesmaximal, characteristic
2C2×#89607C211yesmaximal, characteristic, complemented
3D7×#384C211yesmaximal, characteristic, complemented
4C7×#1803C211yesmaximal, characteristic, complemented
5#11729380C211yesmaximal, characteristic, complemented
6#11728233C211yesmaximal, characteristic, complemented
7#11728022C212yesmaximal, complemented
8#11727981C212yesmaximal, complemented
9#11727338C211yesmaximal, characteristic, complemented
10#11726634C212yesmaximal
11#11726559C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13C2×#6510C2211yescharacteristic
14C2×#6509C2211yescharacteristic
15#90271—24no—
16#90271C418yescomplemented
17C2×#6461—21no—
18C2×#6458C2211yescharacteristic
19C2×#6457—21no—
20C2×D7×C4⋊C4—21no—
21C2×D7×C4⋊C4—21no—
22C2×D7×C4⋊C4C2212yes—
23C2×#6453C2211yescharacteristic
24C2×C4×D28—21no—
25C2×C4×D28C2211yescharacteristic
26C2×#6435—21no—
27C2×C42×D7C2211yescharacteristic
28C2×C7×C4⋊C8C2211yescharacteristic
29C2×C7×#103C2212yes—
30C7×#384C2211yescharacteristic
31C2×#6409C2212yes—
32C2×#6395C2212yes—
33C2×(C2×D7)⋊C8C2212yes—
34C2×#6389C2212yes—
35#89748—21no—
36#89748—21no—
37#89748C2212yes—
38C2×#6383C2212yes—
39#89707C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41#89607—24no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21867—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C22×D28—21no—
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C22×C4×D7—22no—
52C22×C4×D7D411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7C2311yescharacteristic
55C22×C7×C8C2312yes—
56C4×C7×Q8—22no—
57C4×C7×Q8C2×C414yescomplemented
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C4⋊C4C2311yescharacteristic
63C2×C42×C7—21no—
64C2×C42×C7C2311yescharacteristic
65#6510—22no—
66#6510C2×C414yescomplemented
67#6509—22no—
68#6509C2×C414yes—
69C2×(C2×D7)⋊C4—21no—
70C2×(C2×D7)⋊C4—21no—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—22no—
82C2×Dic7⋊C4—22no—
83C2×Dic7⋊C4—22no—
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7C2311yescharacteristic
87C22×C7⋊C8C2312yes—
88#6461—28no—
89#6458—22no—
90#6458C2×C414yes—
91#6457—28no—
92D7×C4⋊C4—42no—
93D7×C4⋊C4—42no—
94D7×C4⋊C4—44no—
95D7×C4⋊C4—44no—
96#6453—22no—
97#6453C2×C414yescomplemented
98C4×D28—28no—
99C4×D28—22no—
100C4×D28C2×C414yescomplemented
101#6435—28no—
102C42×D7C2×C414yes—
103C42×D7D414yes—
104C7×C4⋊C8—22no—
105C7×#103—42no—
106#6409—42no—
107#6395—42no—
108(C2×D7)⋊C8—24no—
109#6389—24no—
110#6383—42no—
111C4⋊(C7⋊C8)—22no—
112C2×#529—71no—
113C2×#432—71no—
114C2×#384—71no—
115#1992—72no—
116#1959—72no—
117#1874—72no—
118#1872—72no—
119#1803—71no—
120#1803D711yescharacteristic, complemented
121#1802—71no—
122#1802—71no—
123C23×D7—21no—
124C23×D7C2×D411yescharacteristic
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8C22×C412yes—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2411yescharacteristic
137C22×Dic7—21no—
138C22×Dic7—22no—
139C22×Dic7C2×D411yescharacteristic
140C22×Dic7C2×D411yescharacteristic
141C22×Dic7C2×D411yescharacteristic
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28—24no—
147C2×D28—21no—
148C2×D28—42no—
149C2×D28—42no—
150C2×D28C22×C412yes—
151C2×C4×D7—22no—
152C2×C4×D7—22no—
153C2×C4×D7—22no—
154C2×C4×D7—22no—
155C2×C4×D7—21no—
156C2×C4×D7—21no—
157C2×C4×D7—21no—
158C2×C4×D7—21no—
159C2×C4×D7—24no—
160C2×C4×D7—42no—
161C2×C4×D7—42no—
162C2×C4×D7—44no—
163C2×C4×D7—44no—
164C2×C4×D7—44no—
165C2×C4×D7C22⋊C414yes—
166C2×C4×D7C22⋊C414yes—
167C2×C4×D7C22×C412yes—
168C2×C4×D7C2×D412yes—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—22no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4C22×C412yes—
179C42×C7—24no—
180C42×C7C22×C412yes—
181C42×C7C2×D412yes—
182(C2×D7)⋊C4—24no—
183(C2×D7)⋊C4—24no—
184(C2×D7)⋊C4—24no—
185(C2×D7)⋊C4—24no—
186(C2×D7)⋊C4—24no—
187C4⋊Dic7—21no—
188C4⋊Dic7—24no—
189C4⋊Dic7—22no—
190C4⋊Dic7—22no—
191C4⋊Dic7—42no—
192C4⋊Dic7—42no—
193C4⋊Dic7C22×C412yes—
194Dic7⋊C4—24no—
195Dic7⋊C4—24no—
196Dic7⋊C4—44no—
197Dic7⋊C4—44no—
198Dic7⋊C4—44no—
199C4×Dic7C22⋊C414yes—
200C4×Dic7C22⋊C414yes—
201C4×Dic7C22×C412yes—
202C4×Dic7C2×D412yes—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206#529—78no—
207#529—144no—
208C2×#126—141no—
209C2×#125—71no—
210C2×#124—71no—
211C2×#123—141no—
212C2×C22⋊Q8—71no—
213C2×#121—71no—
214C2×C4×Q8C2×D711yescharacteristic
215C2×C4×D4—71no—
216C2×C4×D4—141no—
217C2×#117—141no—
218C22×C4⋊C4—72no—
219C22×C4⋊C4—141no—
220C22×C4⋊C4—141no—
221C22×C42—71no—
222#432—144no—
223C2×C4⋊C8—71no—
224C2×C4⋊C8C2×D711yescharacteristic
225C2×#103—72no—
226C2×#103C2×D712yes—
227C2×#102—72no—
228C2×#102—72no—
229C2×C22⋊C8—72no—
230C2×#97—72no—
231#384—72no—
232#384—71no—
233#384—141no—
234#384—141no—
235#384C2×D711yescharacteristic
236#340—72no—
237#340—72no—
238#340—72no—
239#340C2×D712yes—
240C23×C7C22×D411yescharacteristic
241C22×D7—21no—
242C22×D7—21no—
243C22×D7—21no—
244C22×D7—21no—
245C22×D7—22no—
246C22×D7—41no—
247C22×D7—41no—
248C22×D7—41no—
249C22×D7—41no—
250C22×D7—41no—
251C22×D7—41no—
252C22×D7C2×C22⋊C412yes—
253C22×D7AGL(1,ℤ/8)12yes—
254C22×D7#13712yes—
255C7×Q8—22no—
256C7×Q8—22no—
257C7×Q8—41no—
258C7×Q8—41no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7C2×C22⋊C412yes—
271C2×C4×C7C2×C22⋊C412yes—
272C2×C4×C7AGL(1,ℤ/8)12yes—
273C2×C4×C7#13712yes—
274C2×C4×C7C23×C411yescharacteristic
275C2×C4×C7C22×D411yescharacteristic
276C2×C4×C7H(5,𝔽2)11yescharacteristic
277C2×C4×C7H(5,𝔽2)11yescharacteristic
278C2×C4×C72−1+411yescharacteristic
279C2×C4×C72−1+411yescharacteristic
280C2×Dic7—21no—
281C2×Dic7—21no—
282C2×Dic7—21no—
283C2×Dic7—21no—
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7—22no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7C2×C22⋊C412yes—
293C2×Dic7C2×C22⋊C412yes—
294C2×Dic7C2×C22⋊C412yes—
295D28—22no—
296D28—22no—
297D28—48no—
298D28—41no—
299D28—41no—
300C4×D7—28no—
301C4×D7—22no—
302C4×D7—22no—
303C4×D7—22no—
304C4×D7—22no—
305C4×D7—28no—
306C4×D7—48no—
307C4×D7—88no—
308C7×C8—42no—
309C7⋊C8—42no—
310C22×Q8C22×D711yescharacteristic
311C22×D4—71no—
312C22×D4—141no—
313C23×C4—71no—
314C23×C4—71no—
315C23×C4—71no—
316C23×C4—142no—
317C23×C4—141no—
318C22×C8—72no—
319C22×C8C22×D712yes—
320#126—148no—
321#125—74no—
322#125—142no—
323#124—74no—
324#124—142no—
325#123—148no—
326C22⋊Q8—74no—
327C22⋊Q8—142no—
328#121—74no—
329#121—142no—
330C4×Q8—22no—
331C4×Q8C4×D714yescomplemented
332C4×D4—74no—
333C4×D4—148no—
334C4×D4—142no—
335#117—148no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—282no—
350C2×C4⋊C4—282no—
351C2×C4⋊C4—284no—
352C2×C4⋊C4—284no—
353C2×C4⋊C4C22×D712yes—
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C22⋊C4—141no—
356C2×C22⋊C4—141no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C42—21no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42—71no—
364C2×C42—74no—
365C2×C42—74no—
366C2×C42C22×D711yescharacteristic
367C4⋊C8—22no—
368C4⋊C8—142no—
369#103—42no—
370#103—282no—
371#102—282no—
372#102—282no—
373C22⋊C8—144no—
374#97—144no—
375C22×C7C22×C22⋊C411yescharacteristic
376C22×C7#53511yescharacteristic
377C22×C7#53611yescharacteristic
378C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
379C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
380C22×C7C2×#13711yescharacteristic
381C22×C7C2×#13711yescharacteristic
382C2×D7—22no—
383C2×D7—22no—
384C2×D7—22no—
385C2×D7—22no—
386C2×D7—22no—
387C2×D7—22no—
388C2×D7—42no—
389C2×D7—42no—
390C2×D7—42no—
391C2×D7—81no—
392C2×D7—81no—
393C2×D7—81no—
394C2×D7—81no—
395C2×D7#41714yes—
396C4×C7—22no—
397C4×C7—42no—
398C4×C7—42no—
399C4×C7—82no—
400C4×C7#41714yes—
401C4×C7#52011yescharacteristic
402C4×C7#52011yescharacteristic
403C4×C7#53711yescharacteristic
404C4×C7#53711yescharacteristic
405Dic7—22no—
406Dic7—22no—
407Dic7—22no—
408Dic7—42no—
409Dic7—82no—
410C24—71no—
411C24—141no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8C2×C4×D712yes—
418C2×D4—72no—
419C2×D4—144no—
420C2×D4—141no—
421C2×D4—141no—
422C2×D4—141no—
423C2×D4—141no—
424C2×D4—141no—
425C2×D4—282no—
426C2×D4—282no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—71no—
433C22×C4—74no—
434C22×C4—74no—
435C22×C4—72no—
436C22×C4—72no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—141no—
446C22×C4—144no—
447C22×C4—142no—
448C22×C4—284no—
449C22×C4—284no—
450C22×C4—284no—
451C22×C4—282no—
452C22×C4—282no—
453C22×C4D4×D711yescharacteristic
454C22×C4D4×D711yescharacteristic
455C22×C4C23×D711yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—142no—
460C2×C8—142no—
461C2×C8—142no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—72no—
469C4⋊C4—144no—
470C4⋊C4—144no—
471C4⋊C4—144no—
472C4⋊C4—141no—
473C4⋊C4—142no—
474C4⋊C4—142no—
475C4⋊C4—282no—
476C4⋊C4—284no—
477C4⋊C4—284no—
478C4⋊C4—284no—
479C4⋊C4—282no—
480C4⋊C4C2×C4×D712yes—
481C22⋊C4—144no—
482C22⋊C4—144no—
483C22⋊C4—144no—
484C22⋊C4—144no—
485C22⋊C4—144no—
486C42—24no—
487C42—74no—
488C42—74no—
489C42—72no—
490C42—72no—
491C42C2×C4×D712yes—
492C42D4×D712yes—
493C2×C7#231111yescharacteristic
494C2×C7C2×#41711yescharacteristic
495C2×C7C2×#41711yescharacteristic
496C2×C7#321711yescharacteristic
497C2×C7#321811yescharacteristic
498C2×C7#323811yescharacteristic
499C2×C7#323911yescharacteristic
500D7—24no—
501D7—82no—
502C23—72no—
503C23—72no—
504C23—72no—
505C23—141no—
506C23—141no—
507C23—141no—
508C23—141no—
509C23—142no—
510C23—281no—
511C23—281no—
512C23—281no—
513C23—281no—
514C23—281no—
515C23—281no—
516C23C2×D4×D711yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—142no—
522D4—142no—
523D4—288no—
524D4—281no—
525D4—281no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—72no—
538C2×C4—72no—
539C2×C4—72no—
540C2×C4—148no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—148no—
552C2×C4—142no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—288no—
556C2×C4—282no—
557C2×C4—282no—
558C2×C4—282no—
559C2×C4—568no—
560C2×C4D7×C22⋊C412yes—
561C2×C4D7×C22⋊C412yes—
562C2×C4#647712yes—
563C2×C4#648112yes—
564C2×C4C22×C4×D711yescharacteristic
565C2×C4C2×D4×D711yescharacteristic
566C2×C4#654811yescharacteristic
567C2×C4#655111yescharacteristic
568C2×C4#655311yescharacteristic
569C2×C4#655411yescharacteristic
570C8—42no—
571C8—282no—
572C7#2186711yescharacteristic, complemented, Sylow
573C22—74no—
574C22—142no—
575C22—142no—
576C22—142no—
577C22—142no—
578C22—142no—
579C22—142no—
580C22—282no—
581C22—282no—
582C22—282no—
583C22—561no—
584C22—561no—
585C22—561no—
586C22—561no—
587C22C2×D7×C22⋊C411yescharacteristic
588C22#9029811yescharacteristic
589C22#9031911yescharacteristic
590C22C2×#647711yescharacteristic
591C22C2×#648111yescharacteristic
592C22D7×AGL(1,ℤ/8)11yescharacteristic
593C22D7×#13711yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—142no—
599C4—142no—
600C4—142no—
601C4—282no—
602C4—562no—
603C4#8957514yes—
604C4#9019511yescharacteristic
605C4#9021411yescharacteristic
606C4#9029911yescharacteristic
607C4#9032011yescharacteristic
608C2—144no—
609C2—562no—
610C2#1173144811yescharacteristic
611C2C2×#8957511yescharacteristic
612C2D7×#41711yescharacteristic
613C2#1173783511yescharacteristic
614C2#1173792811yescharacteristic
615C2#1173846311yescharacteristic
616C2#1173856011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1803C2
socleC23×C7C22×D4
prime radicalC2×#6458C22
cosocleC22×C4×C7C24
hypercenter#1803D7
hypocenterC7#21867
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C2×#89607C2semidirect
3. D7×#384C2semidirect
4. C7×#1803C2semidirect
5. #11729380C2semidirect
6. #11728233C2semidirect
7. #11728022C2semidirect
8. #11727981C2semidirect
9. #11727338C2semidirect
11. #11726559C2semidirect
16. #90271C4semidirect
57. C4×C7×Q8C2×C4semidirect
66. #6510C2×C4semidirect
97. #6453C2×C4semidirect
100. C4×D28C2×C4semidirect
120. #1803D7semidirect
331. C4×Q8C4×D7semidirect
572. C7#21867semidirect

Maximal subgroups

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

subgroupindex
C2×#902712
C2×#896072
D7×#3842
C7×#18032
#117293802
#117282332
#117280222
#117279812
#117273382
#117266342
#117265592
#218677