Finite Group Databasebeta

← #13369259

Subgroups — #13369259

Order 1792.

1542 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90244C211yesmaximal, characteristic
2C7×#2354C211yesmaximal, characteristic, complemented
3C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
4C2×#89774C212yesmaximal, complemented
5#11729340C212yesmaximal, complemented
6#11727342C211yesmaximal, characteristic, complemented
7#11727334C211yesmaximal, characteristic, complemented
8#11727276C212yesmaximal, complemented
9#11727269C212yesmaximal, complemented
10#11727241C211yesmaximal, characteristic, complemented
11C4×#6384C211yesmaximal, characteristic, complemented
12C22×#1157C2212yes—
13C2×C4×C7×D4C2211yescharacteristic
14C2×C7×#117C2212yes—
15C2×#6503—21no—
16C2×#6493C2212yes—
17C23×C7⋊C8C2212yes—
18#90244—24no—
19#90244C418yescomplemented
20C2×#6456—21no—
21C2×#6447—22no—
22C2×#6437C2211yescharacteristic
23C2×C4×Dic14C2211yescharacteristic
24C7×#390—21no—
25C7×#388—22no—
26C7×#384C2211yescharacteristic
27C7×#380C2212yes—
28C4×C7×C4⋊C4—21no—
29C4×C7×C4⋊C4C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2212yes—
31D4×C7⋊C8—24no—
32#89774—28no—
33C2×#6384C2212yes—
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6383C2212yes—
37C2×#6383C2212yes—
38C2×#6383C2212yes—
39C2×#6383C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2211yescharacteristic
43#89324C2211yescharacteristic
44#89324C2212yes—
45#22418—71nomaximal, Sylow
46C22×C7×D4D411yescharacteristic
47C23×C4×C7C2312yes—
48C22×C22⋊D7—22no—
49C22×C22⋊D7—22no—
50C2×#1157—24no—
51C2×#1157—24no—
52C2×#1157—24no—
53C2×#1157—42no—
54C2×#1157—42no—
55C2×#1157C2×C418yes—
56C22×D28—21no—
57C22×D28—21no—
58C22×D28C2312yes—
59C22×C4×D7—22no—
60C22×Dic14C2311yescharacteristic
61C4×C7×D4C2×C414yes—
62C4×C7×D4D414yes—
63C7×#117—24no—
64C7×#117—42no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4C2312yes—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4D412yes—
74C2×C42×C7—22no—
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77#6503—28no—
78#6493—24no—
79#6493C2×C418yes—
80C2×(C2×D7)⋊C4—22no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×C4×Dic7—21no—
84C22×C7⋊C8—24no—
85C22×C7⋊C8—24no—
86C22×C7⋊C8—42no—
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C22×C7⋊C8C2311yescharacteristic
90#6456—28no—
91#6447—216no—
92#6437—22no—
93#6437C2×C414yes—
94C4×Dic14—22no—
95C4×Dic14C2×C414yes—
96C7×#95—22no—
97C7×#95—21no—
98C22⋊(C7⋊C8)—24no—
99#6384—24no—
100#6384—22no—
101#6384D414yescomplemented
102#6383—24no—
103#6383—24no—
104#6383—42no—
105#6383—42no—
106C4⋊(C7⋊C8)—22no—
107C4×C7⋊C8—22no—
108C2×#550—71no—
109C2×C8×D4—71no—
110C2×#416—72no—
111#2354D711yescharacteristic, complemented
112#1982—71no—
113#1957—71no—
114#1893—72no—
115#1875—72no—
116#1803—71no—
117#1788—72no—
118C4×#103—71no—
119C24×C7C2×D412yes—
120C23×D7—22no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4C22⋊C414yes—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7C22×C414yes—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D412yes—
134C22×C4×C7C2×D414yes—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2411yescharacteristic
137C2×C22⋊D7—28no—
138C2×C22⋊D7—28no—
139C2×C22⋊D7—44no—
140C2×C22⋊D7—44no—
141C2×C22⋊D7—82no—
142C2×C22⋊D7—82no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145#1157—216no—
146#1157—48no—
147#1157—48no—
148#1157—48no—
149C2×D28—22no—
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—22no—
153C2×D28—24no—
154C2×D28—22no—
155C2×D28—24no—
156C2×D28—42no—
157C2×D28—42no—
158C2×D28—81no—
159C2×D28—81no—
160C2×D28C22×C414yes—
161C2×C4×D7—28no—
162C2×C4×D7—44no—
163C2×C4×D7—44no—
164C2×Dic14—21no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14C22×C412yes—
168C2×Dic14C2×D412yes—
169C2×Dic14#4112yes—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4—81no—
177C7×C4⋊C4C22⋊C414yes—
178C7×C4⋊C4C2×D412yes—
179C7×C22⋊C4—44no—
180C7×C22⋊C4—82no—
181C7×C22⋊C4C22⋊C418yes—
182C42×C7—44no—
183C42×C7—42no—
184C42×C7C22×C412yes—
185C42×C7C2×D412yes—
186(C2×D7)⋊C4—28no—
187C4⋊Dic7—21no—
188C4⋊Dic7C22×C412yes—
189Dic7⋊C4—24no—
190C4×Dic7—24no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8—21no—
193C2×C7⋊C8—21no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—48no—
200C2×C7⋊C8C2×D412yes—
201C22×#41—72no—
202C23×C8—72no—
203#550—78no—
204#550—144no—
205C2×#127—71no—
206C2×#127—141no—
207C2×#121—72no—
208C2×#121—142no—
209C2×C4×Q8—71no—
210C2×C4×D4—141no—
211C2×C4×D4C2×D711yescharacteristic
212C2×#117C2×D712yes—
213C8×D4—144no—
214C2×C4⋊C8—71no—
215#416—148no—
216C2×#103—71no—
217C2×#103—72no—
218C2×#103—71no—
219C2×#102—72no—
220C2×#102—72no—
221C2×#102—72no—
222C2×#102—72no—
223C2×C22⋊C8—72no—
224C2×C4×C8—71no—
225#390—21no—
226#388—22no—
227#384C2×D711yescharacteristic
228#380C2×D712yes—
229C4×C4⋊C4—21no—
230C4×C4⋊C4C2×D711yescharacteristic
231#340—72no—
232#340—71no—
233#340—71no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7C2×C22⋊C414yes—
241C23×C7C22×D411yescharacteristic
242C22×D7—24no—
243C22×D7—42no—
244C22×D7—42no—
245C22×D7—42no—
246C22×D7—42no—
247C22×D7—42no—
248C22×D7—42no—
249C7×D4—28no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—22no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7C2×C22⋊C412yes—
268C2×C4×C7C2×C22⋊C414yes—
269C2×C4×C7C2×C22⋊C412yes—
270C2×C4×C7#13512yes—
271C2×C4×C7#13512yes—
272C2×C4×C7C23×C411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7H(5,𝔽2)11yescharacteristic
277C2×C4×C7H(5,𝔽2)11yescharacteristic
278C22⋊D7—416no—
279C22⋊D7—88no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—22no—
283C2×Dic7—22no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286D28—24no—
287D28—24no—
288D28—42no—
289D28—42no—
290D28—48no—
291D28—84no—
292C4×D7—416no—
293Dic14—22no—
294Dic14—22no—
295Dic14—22no—
296Dic14C4×D414yes—
297C7⋊C8—22no—
298C7⋊C8—42no—
299C7⋊C8—41no—
300C2×#41—78no—
301C2×#41—144no—
302C2×#41—144no—
303C2×#41—144no—
304C2×#41—282no—
305C2×#41—282no—
306C22×Q8—71no—
307C22×D4—72no—
308C22×D4—142no—
309C22×D4—142no—
310C22×D4—141no—
311C22×D4—141no—
312C22×D4C22⋊D711yescharacteristic
313C23×C4—142no—
314C23×C4C22×D712yes—
315C22×C8—71no—
316C22×C8—72no—
317C22×C8—71no—
318C22×C8—144no—
319C22×C8—144no—
320C22×C8—282no—
321#127—74no—
322#127—148no—
323#127—142no—
324#121—78no—
325#121—144no—
326#121—1416no—
327C4×Q8—74no—
328C4×Q8—142no—
329C4×D4—148no—
330C4×D4C4×D714yes—
331C4×D4D2814yes—
332#117—24no—
333#117—42no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4C22⋊D711yescharacteristic
339C2×C4⋊C4C22×D712yes—
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—142no—
345C2×C22⋊C4C22⋊D712yes—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—141no—
349C2×C42C22×D711yescharacteristic
350C4⋊C8—142no—
351#103—74no—
352#103—144no—
353#103—142no—
354#102—144no—
355#102—144no—
356#102—282no—
357#102—282no—
358C22⋊C8—144no—
359C4×C8—142no—
360#95—22no—
361#95—21no—
362C22×C7—24no—
363C22×C7—24no—
364C22×C7—24no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7C22×C22⋊C411yescharacteristic
370C22×C7D4211yescharacteristic
371C22×C7#54611yescharacteristic
372C22×C7C2×#13511yescharacteristic
373C22×C7C2×#13511yescharacteristic
374C22×C7#57611yescharacteristic
375C22×C7#57711yescharacteristic
376C2×D7—44no—
377C2×D7—44no—
378C2×D7—44no—
379C2×D7—82no—
380C2×D7—82no—
381C2×D7—82no—
382C2×D7—82no—
383C4×C7—24no—
384C4×C7—22no—
385C4×C7—42no—
386C4×C7—82no—
387C4×C7—81no—
388C4×C7#41614yescomplemented
389C4×C7C2×C4×D411yescharacteristic
390C4×C7#51811yescharacteristic
391C4×C7D4211yescharacteristic
392C4×C7#54611yescharacteristic
393Dic7—24no—
394Dic7—42no—
395C24—142no—
396C24C2×C22⋊D712yes—
397#41—1416no—
398#41—288no—
399#41—288no—
400#41—288no—
401C2×Q8—72no—
402C2×Q8—72no—
403C2×Q8—72no—
404C2×Q8—141no—
405C2×Q8—141no—
406C2×Q8—141no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—74no—
410C2×D4—148no—
411C2×D4—148no—
412C2×D4—144no—
413C2×D4—144no—
414C2×D4—142no—
415C2×D4—142no—
416C2×D4—142no—
417C2×D4—142no—
418C2×D4—142no—
419C2×D4—284no—
420C2×D4—284no—
421C2×D4—282no—
422C2×D4—282no—
423C2×D4—561no—
424C2×D4—561no—
425C2×D4—562no—
426C2×D4—562no—
427C2×D4(C2×D7)⋊C414yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—148no—
438C22×C4—284no—
439C22×C4—284no—
440C22×C4C2×C4×D714yes—
441C22×C4C2×D2814yes—
442C22×C4C2×C22⋊D712yes—
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C23×D711yescharacteristic
446C2×C8—72no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—142no—
451C2×C8—142no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—142no—
455C2×C8—288no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—42no—
462C4⋊C4—72no—
463C4⋊C4—81no—
464C4⋊C4—144no—
465C4⋊C4—141no—
466C4⋊C4(C2×D7)⋊C414yes—
467C4⋊C4D4×D712yes—
468C22⋊C4—44no—
469C22⋊C4—82no—
470C22⋊C4—148no—
471C22⋊C4(C2×D7)⋊C418yes—
472C42—44no—
473C42—42no—
474C42—144no—
475C42C2×C4×D712yes—
476C42C2×D2812yes—
477C2×C7—44no—
478C2×C7D4×C22⋊C411yescharacteristic
479C2×C7C2×#41611yescharacteristic
480C2×C7#289711yescharacteristic
481C2×C7#328411yescharacteristic
482C2×C7#328711yescharacteristic
483C2×C7#330411yescharacteristic
484C2×C7#330511yescharacteristic
485D7—84no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—144no—
493C23—282no—
494C23—282no—
495C23—282no—
496C23—282no—
497C23—282no—
498C23—282no—
499C23C2×(C2×D7)⋊C414yes—
500C23C22×C22⋊D711yescharacteristic
501Q8—74no—
502Q8—142no—
503Q8—142no—
504Q8—142no—
505D4—28no—
506D4—144no—
507D4—144no—
508D4—2816no—
509D4—282no—
510D4—282no—
511D4—288no—
512D4—564no—
513D4—568no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—2816no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4C2×(C2×D7)⋊C412yes—
539C2×C4C2×(C2×D7)⋊C414yes—
540C2×C4C2×(C2×D7)⋊C412yes—
541C2×C4#651312yes—
542C2×C4#651312yes—
543C2×C4C22×C4×D711yescharacteristic
544C2×C4C22×D2811yescharacteristic
545C2×C4C2×D4×D711yescharacteristic
546C2×C4#654811yescharacteristic
547C2×C4D7×#4111yescharacteristic
548C2×C4#655311yescharacteristic
549C8—142no—
550C8—282no—
551C8—281no—
552C7#2241811yescharacteristic, complemented, Sylow
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—284no—
561C22—284no—
562C22—284no—
563C22—562no—
564C22—562no—
565C22—562no—
566C22—562no—
567C22C22×(C2×D7)⋊C411yescharacteristic
568C22D4×C22⋊D711yescharacteristic
569C22C2×#651311yescharacteristic
570C22C2×#651311yescharacteristic
571C22#9052011yescharacteristic
572C22#9052611yescharacteristic
573C22#9052811yescharacteristic
574C4—22no—
575C4—24no—
576C4—42no—
577C4—82no—
578C4—81no—
579C4—144no—
580C4—282no—
581C4#8977414yescomplemented
582C4C4×D4×D711yescharacteristic
583C4#9023611yescharacteristic
584C4D4×D2811yescharacteristic
585C4#9024511yescharacteristic
586C2—44no—
587C2—564no—
588C2C2×#8977411yescharacteristic
589C2D4×(C2×D7)⋊C411yescharacteristic
590C2#1174000811yescharacteristic
591C2#1174097511yescharacteristic
592C2#1174103011yescharacteristic
593C2#1174110011yescharacteristic
594C2#1174111811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2354C2
socleC23×C7C22×D4
prime radicalC2×#90244C2
cosocleC22×C4×C7C24
hypercenter#2354D7
hypocenterC7#22418
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. C7×#2354C2semidirect
3. C2×D4×C7⋊C8C2semidirect
4. C2×#89774C2semidirect
5. #11729340C2semidirect
6. #11727342C2semidirect
7. #11727334C2semidirect
8. #11727276C2semidirect
9. #11727269C2semidirect
10. #11727241C2semidirect
11. C4×#6384C2semidirect
19. #90244C4semidirect
101. #6384D4semidirect
111. #2354D7semidirect
388. C4×C7#416semidirect
552. C7#22418semidirect
581. C4#89774semidirect

Maximal subgroups

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

subgroupindex
C2×#902442
C7×#23542
C2×D4×C7⋊C82
C2×#897742
#117293402
#117273422
#117273342
#117272762
#117272692
#117272412
C4×#63842
#224187