Finite Group Databasebeta

← #13353293

Subgroups — #13353293

Order 1792.

1510 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1159C211yesmaximal, characteristic
2C2×#89609C211yesmaximal, characteristic, complemented
3C2×#89543C212yesmaximal
4D7×#382C211yesmaximal, characteristic, complemented
5C7×#1804C211yesmaximal, characteristic, complemented
6#11729359C211yesmaximal, characteristic, complemented
7#11728240C211yesmaximal, characteristic, complemented
8#11727908C212yesmaximal, complemented
9#11727313C211yesmaximal, characteristic, complemented
10#11726659C212yesmaximal, complemented
11#11726621C212yesmaximal, complemented
12C22×#1159C2211yescharacteristic
13C2×C4×C7×D4C2211yescharacteristic
14C2×D4×Dic7C2211yescharacteristic
15C2×C4×C22⋊D7—21no—
16C22×C4×Dic7—21no—
17C22×#1131C2212yes—
18C4×#1159—24no—
19C4×#1159C418yescomplemented
20C2×#6455C2211yescharacteristic
21C2×D7×C4⋊C4C2212yes—
22C2×#6450C2211yescharacteristic
23C2×(C22⋊D7)⋊C4—21no—
24C2×#6440—21no—
25C2×C42×D7C2211yescharacteristic
26C2×C4×Dic14C2211yescharacteristic
27C2×C7×C4⋊C8C2211yescharacteristic
28C2×C7×#102C2212yes—
29C7×#382C2211yescharacteristic
30C2×#6406C2212yes—
31C2×#6390C2212yes—
32C2×Dic7⋊C8C2212yes—
33#89749C2212yes—
34C2×#6384C2212yes—
35#89708C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89609—24no—
38#89543—28no—
39D7×#95—21no—
40D7×#95—21no—
41#89422—21no—
42#89422—21no—
43C7×#340C2212yes—
44#89346C2212yes—
45#89324C2212yes—
46#89324C2212yes—
47#20649—71nomaximal, Sylow
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7—21no—
50C22×C22⋊D7—21no—
51C23×Dic7—21no—
52C23×Dic7—21no—
53C2×#1159—22no—
54C2×#1159—22no—
55C2×#1159—22no—
56C2×#1159C2×C414yes—
57C2×#1159D414yes—
58C2×#1159Q814yes—
59C22×C4×D7—22no—
60C22×C4×D7D411yescharacteristic
61C22×C4×D7D411yescharacteristic
62C22×C4×D7C2311yescharacteristic
63C22×Dic14C2311yescharacteristic
64C22×C7×C8C2312yes—
65C4×C7×D4—22no—
66C4×C7×D4C2×C414yescomplemented
67C2×C7×C4⋊C4C2312yes—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C22⋊C4—21no—
70C2×C42×C7C2311yescharacteristic
71C2×C22⋊Dic7—21no—
72D4×Dic7—22no—
73D4×Dic7C2×C414yescomplemented
74C4×C22⋊D7—28no—
75C2×(C2×D7)⋊C4—21no—
76C2×C4⋊Dic7C2312yes—
77C2×C4⋊Dic7C2311yescharacteristic
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—22no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—21no—
84C2×C4×Dic7—28no—
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C2×#1131—24no—
90C2×#1131—24no—
91C2×#1131—24no—
92#6455—22no—
93#6455C2×C414yes—
94D7×C4⋊C4—24no—
95D7×C4⋊C4—24no—
96#6450—22no—
97#6450C2×C414yescomplemented
98(C22⋊D7)⋊C4—28no—
99#6440—28no—
100C42×D7C2×C414yes—
101C42×D7D414yes—
102C4×Dic14—22no—
103C4×Dic14C2×C414yescomplemented
104C7×C4⋊C8—22no—
105C7×#102—24no—
106C7×#95—21no—
107C7×#95—21no—
108#6406—24no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—22no—
112C7⋊#95—22no—
113#6390—24no—
114Dic7⋊C8—24no—
115#6384—24no—
116C4⋊(C7⋊C8)—22no—
117C2×C4×#41—71no—
118C2×#423—71no—
119C2×#417—72no—
120C2×#382—71no—
121#1805—71no—
122#1805—71no—
123#1805—72no—
124#1804—72no—
125#1804—71no—
126#1804D711yescharacteristic, complemented
127#1736—72no—
128C24×C7—21no—
129C23×D7C2×D411yescharacteristic
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4C22×C412yes—
134C2×C7×D4C2×D412yes—
135C2×C7×D4C2×Q812yes—
136C22×C4×C7—21no—
137C22×C4×C7—24no—
138C22×C4×C7—22no—
139C22×C4×C7—44no—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C2×C22⋊D7—24no—
144C2×C22⋊D7—24no—
145C2×C22⋊D7—24no—
146C22×Dic7—21no—
147C22×Dic7—24no—
148C22×Dic7—24no—
149C22×Dic7—24no—
150C22×Dic7—21no—
151C22×Dic7—24no—
152C22×Dic7—22no—
153C22×Dic7—44no—
154C22×Dic7C2×D411yescharacteristic
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157#1159—28no—
158#1159—28no—
159#1159—28no—
160#1159C4⋊C4116yes—
161C2×C4×D7—42no—
162C2×C4×D7—42no—
163C2×C4×D7—44no—
164C2×C4×D7—44no—
165C2×C4×D7—44no—
166C2×C4×D7C22⋊C414yes—
167C2×C4×D7C22⋊C414yes—
168C2×C4×D7C22×C412yes—
169C2×C4×D7C2×D412yes—
170C2×C4×D7C2×D412yes—
171C2×C4×D7C2×Q812yes—
172C2×C4×D7#4112yes—
173C2×C4×D7#4112yes—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14C22×C412yes—
178C2×Dic14C2×D412yes—
179C2×Dic14C2×Q812yes—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C2×C7×C8—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4C22×C412yes—
187C7×C22⋊C4—24no—
188C42×C7C22×C412yes—
189C42×C7C2×D412yes—
190C22⋊Dic7—24no—
191(C2×D7)⋊C4—24no—
192C4⋊Dic7—21no—
193C4⋊Dic7—22no—
194C4⋊Dic7—22no—
195C4⋊Dic7C22×C412yes—
196Dic7⋊C4—24no—
197Dic7⋊C4—24no—
198Dic7⋊C4—44no—
199C4×Dic7—24no—
200C4×Dic7—24no—
201C4×Dic7—24no—
202C4×Dic7C22⋊C414yes—
203C4×Dic7C22⋊C414yes—
204C4×Dic7C22×C412yes—
205C4×Dic7C2×D412yes—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C2×C7⋊C8—22no—
209#1131—48no—
210C22×#41—71no—
211C22×M16—72no—
212C4×#41—78no—
213C4×#41—144no—
214C2×C4×Q8—71no—
215C2×C4×Q8—71no—
216C2×C4×D4—71no—
217C2×C4×D4—141no—
218C2×C4×D4—141no—
219C2×C4×D4C2×D711yescharacteristic
220C2×#117—71no—
221C2×#117—141no—
222C22×C4⋊C4—72no—
223C22×C42—71no—
224C22×C42—141no—
225#423—144no—
226C2×C4⋊C8—72no—
227C2×C4⋊C8—71no—
228C2×C4⋊C8C2×D711yescharacteristic
229#417—148no—
230C2×#103—72no—
231C2×#103—72no—
232C2×#102—72no—
233C2×#102C2×D712yes—
234#382—72no—
235#382—71no—
236#382—141no—
237#382—141no—
238#382C2×D711yescharacteristic
239C2×#95—141no—
240C2×#95—141no—
241#340—72no—
242#340—72no—
243#340—72no—
244#340C2×D712yes—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7—22no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7C22×D411yescharacteristic
251C22×D7—22no—
252C22×D7—22no—
253C22×D7C2×C22⋊C412yes—
254C22×D7#12112yes—
255C22×D7C22⋊Q812yes—
256C7×D4—22no—
257C7×D4—22no—
258C7×D4—22no—
259C7×D4C2×C4⋊C414yes—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—48no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7C2×C22⋊C412yes—
269C2×C4×C7C2×C22⋊C412yes—
270C2×C4×C7C23×C411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×Q811yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C22⋊D7—216no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—216no—
283C2×Dic7—22no—
284C2×Dic7—48no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7—42no—
289C2×Dic7C2×C22⋊C412yes—
290C2×Dic7C2×C22⋊C412yes—
291C2×Dic7C2×C22⋊C412yes—
292C2×Dic7#12112yes—
293C2×Dic7C22⋊Q812yes—
294C4×D7—44no—
295C4×D7—44no—
296C4×D7—88no—
297C4×D7C2×C4⋊C414yes—
298C4×D7#11714yes—
299C4×D7#12114yes—
300C4×D7C22⋊Q814yes—
301Dic14—22no—
302Dic14—22no—
303Dic14—22no—
304Dic14C2×C4⋊C414yes—
305C7×C8—42no—
306C7⋊C8—42no—
307C2×#41—74no—
308C2×#41—74no—
309C2×#41—74no—
310C2×#41—142no—
311C2×#41—142no—
312C2×#41—142no—
313C22×Q8—71no—
314C22×D4—141no—
315C22×D4C22×D711yescharacteristic
316C23×C4—21no—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—71no—
320C23×C4—142no—
321C23×C4—141no—
322C23×C4—141no—
323C2×M16—144no—
324C2×M16—144no—
325C2×M16—144no—
326C22×C8—72no—
327C22×C8C22×D712yes—
328C4×Q8—74no—
329C4×Q8—74no—
330C4×Q8—142no—
331C4×Q8—142no—
332C4×D4—22no—
333C4×D4—74no—
334C4×D4—142no—
335C4×D4—148no—
336C4×D4—148no—
337C4×D4C4×D714yescomplemented
338#117—74no—
339#117—148no—
340#117—142no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—72no—
343C2×C4⋊C4—144no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4C22×D712yes—
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—74no—
357C2×C42—74no—
358C2×C42—148no—
359C2×C42—141no—
360C2×C42—141no—
361C2×C42—141no—
362C2×C42C22×D711yescharacteristic
363C4⋊C8—22no—
364C4⋊C8—144no—
365C4⋊C8—142no—
366#103—144no—
367#103—144no—
368#102—24no—
369#102—144no—
370#95—21no—
371#95—21no—
372#95—142no—
373#95—142no—
374#95—141no—
375#95—141no—
376C22×C7—24no—
377C22×C7—44no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—44no—
381C22×C7C22×C22⋊C411yescharacteristic
382C22×C7C2×#12111yescharacteristic
383C22×C7C2×C22⋊Q811yescharacteristic
384C22×C7#57512yes—
385C22×C7#57512yes—
386C2×D7—24no—
387C2×D7—22no—
388C2×D7—22no—
389C2×D7—24no—
390C2×D7#38014yes—
391C4×C7—22no—
392C4×C7—22no—
393C4×C7—42no—
394C4×C7—82no—
395C4×C7C22×C4⋊C411yescharacteristic
396C4×C7C2×#11711yescharacteristic
397C4×C7C2×#12111yescharacteristic
398C4×C7C2×C22⋊Q811yescharacteristic
399Dic7—22no—
400Dic7—22no—
401Dic7—24no—
402Dic7—42no—
403Dic7—82no—
404Dic7#38014yes—
405C24—21no—
406C24—71no—
407#41—716no—
408#41—148no—
409#41—148no—
410#41—148no—
411C2×Q8—72no—
412C2×Q8—72no—
413C2×Q8—72no—
414C2×Q8—141no—
415C2×Q8—141no—
416C2×Q8—141no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—144no—
421C2×D4—144no—
422C2×D4—144no—
423C2×D4C2×C4×D712yes—
424C2×D4D4×D712yes—
425C2×D4Q8×D712yes—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—44no—
430C22×C4—71no—
431C22×C4—72no—
432C22×C4—72no—
433C22×C4—72no—
434C22×C4—72no—
435C22×C4—72no—
436C22×C4—72no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—74no—
440C22×C4—74no—
441C22×C4—144no—
442C22×C4—144no—
443C22×C4—144no—
444C22×C4—144no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—142no—
448C22×C4—282no—
449C22×C4—282no—
450C22×C4—284no—
451C22×C4—284no—
452C22×C4—284no—
453C22×C4—284no—
454C22×C4D4×D711yescharacteristic
455C22×C4D4×D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457M16—288no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—142no—
462C2×C8—142no—
463C2×C8—142no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—72no—
468C4⋊C4—144no—
469C4⋊C4—144no—
470C4⋊C4—141no—
471C4⋊C4—142no—
472C4⋊C4—142no—
473C4⋊C4—284no—
474C4⋊C4C2×C4×D712yes—
475C22⋊C4—24no—
476C22⋊C4—144no—
477C22⋊C4—144no—
478C42—74no—
479C42—74no—
480C42—72no—
481C42—72no—
482C42—144no—
483C42—144no—
484C42—144no—
485C42C2×C4×D712yes—
486C42D4×D712yes—
487C2×C7—44no—
488C2×C7C2×#38011yescharacteristic
489C2×C7#289611yescharacteristic
490C2×C7#289611yescharacteristic
491C2×C7#304012yes—
492C2×C7#308012yes—
493D7—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23—72no—
500C23—72no—
501C23—72no—
502C23—142no—
503C23—142no—
504C23C2×D4×D711yescharacteristic
505Q8—74no—
506Q8—142no—
507Q8—142no—
508Q8—142no—
509D4—22no—
510D4—22no—
511D4—22no—
512D4—1416no—
513D4D7×C4⋊C414yes—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—72no—
523C2×C4—72no—
524C2×C4—72no—
525C2×C4—74no—
526C2×C4—74no—
527C2×C4—74no—
528C2×C4—74no—
529C2×C4—72no—
530C2×C4—72no—
531C2×C4—1416no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—142no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—284no—
539C2×C4—284no—
540C2×C4—288no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—568no—
546C2×C4D7×C22⋊C412yes—
547C2×C4D7×C22⋊C412yes—
548C2×C4C22×C4×D711yescharacteristic
549C2×C4C2×D4×D711yescharacteristic
550C2×C4C2×D4×D711yescharacteristic
551C2×C4C2×Q8×D711yescharacteristic
552C2×C4D7×#4111yescharacteristic
553C2×C4D7×#4111yescharacteristic
554C8—42no—
555C8—282no—
556C7#2064911yescharacteristic, complemented, Sylow
557C22—24no—
558C22—42no—
559C22—44no—
560C22—42no—
561C22—44no—
562C22—74no—
563C22—142no—
564C22—144no—
565C22—142no—
566C22—144no—
567C22C2×D7×C22⋊C411yescharacteristic
568C22D7×#12111yescharacteristic
569C22D7×C22⋊Q811yescharacteristic
570C22#9045912yes—
571C22#9046412yes—
572C4—22no—
573C4—22no—
574C4—42no—
575C4—74no—
576C4—82no—
577C4—142no—
578C4—142no—
579C4—144no—
580C4—282no—
581C4—562no—
582C4C2×D7×C4⋊C411yescharacteristic
583C4D7×#11711yescharacteristic
584C4D7×#12111yescharacteristic
585C4D7×C22⋊Q811yescharacteristic
586C2—44no—
587C2—144no—
588C2D7×#38011yescharacteristic
589C2#1173627511yescharacteristic
590C2#1173636911yescharacteristic
591C2#1173798012yes—
592C2#1173861212yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1804C2
socleC23×C7C22×D4
prime radicalC22×#1159C22
cosocleC22×C4×C7C24
hypercenter#1804D7
hypocenterC7#20649
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#89609C2semidirect
4. D7×#382C2semidirect
5. C7×#1804C2semidirect
6. #11729359C2semidirect
7. #11728240C2semidirect
8. #11727908C2semidirect
9. #11727313C2semidirect
10. #11726659C2semidirect
11. #11726621C2semidirect
19. C4×#1159C4semidirect
66. C4×C7×D4C2×C4semidirect
73. D4×Dic7C2×C4semidirect
97. #6450C2×C4semidirect
103. C4×Dic14C2×C4semidirect
126. #1804D7semidirect
337. C4×D4C4×D7semidirect
556. C7#20649semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11592
C2×#896092
C2×#895432
D7×#3822
C7×#18042
#117293592
#117282402
#117279082
#117273132
#117266592
#117266212
#206497