Finite Group Databasebeta

← #13360040

Subgroups — #13360040

Order 1792.

1260 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×#127C211yesmaximal, characteristic, complemented
2C2×#89844C211yesmaximal, characteristic
3C2×#89786C212yesmaximal, complemented
4C42×D28C211yesmaximal, characteristic, complemented
5#11729515C211yesmaximal, characteristic, complemented
6#11729363C212yesmaximal, complemented
7#11727411C212yesmaximal, complemented
8#11727391C212yesmaximal, complemented
9#11727338C211yesmaximal, characteristic, complemented
10#11727322C211yesmaximal, characteristic, complemented
11#11727145C211yesmaximal, characteristic, complemented
12C2×C7×#127C2211yescharacteristic
13C2×C4×C7×D4—21no—
14C2×C4×C7×D4—21no—
15C2×C4×C7×D4C2212yes—
16C22×#1141C2212yes—
17C2×C4×D28C2211yescharacteristic
18C2×C4×D28C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C2×C42×D7—21no—
21C7×#390—21no—
22C7×#390—21no—
23C7×#390C2212yes—
24C7×#384C2211yescharacteristic
25C43×C7C2211yescharacteristic
26C2×#6406C2212yes—
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29#89844C418yescomplemented
30#89786—28no—
31C2×#6383C2212yes—
32C2×#6382C2212yes—
33C4×(C2×D7)⋊C4—21no—
34C4×(C2×D7)⋊C4—21no—
35C4×(C2×D7)⋊C4—21no—
36C4×C4⋊Dic7C2211yescharacteristic
37C4×C4⋊Dic7C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89319C2212yes—
45#21503—71nomaximal, Sylow
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4C2312yes—
49C23×C4×C7—22no—
50C22×D28C2311yescharacteristic
51C22×C4×D7—21no—
52C22×C4×D7—21no—
53C22×C4×D7—21no—
54C7×#127—22no—
55C7×#127C2×C414yescomplemented
56C4×C7×D4—24no—
57C4×C7×D4—24no—
58C4×C7×D4—44no—
59C4×C7×D4—44no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4C2312yes—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C42×C7D411yescharacteristic
67C2×C42×C7D411yescharacteristic
68C2×C42×C7D411yescharacteristic
69C2×C42×C7D411yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×#1141—24no—
74C2×#1141—24no—
75C2×#1141C2×C418yescomplemented
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4×Dic7—21no—
83C2×C4×Dic7—21no—
84C2×C4×Dic7—21no—
85C22×C7⋊C8C2312yes—
86C22×C7⋊C8C2312yes—
87C4×D28—22no—
88C4×D28—22no—
89C4×D28—22no—
90C4×D28—22no—
91C4×D28C2×C414yescomplemented
92C4×D28D414yescomplemented
93C42×D7—44no—
94#6406—24no—
95#6406—24no—
96#6406C2×C418yescomplemented
97C7⋊#95—21no—
98C7⋊#95—21no—
99C7⋊#95—21no—
100#6383—24no—
101#6382—24no—
102C4⋊(C7⋊C8)—22no—
103C4⋊(C7⋊C8)C2×C414yescomplemented
104#6377—24no—
105C2×#459—71no—
106C2×#442—72no—
107C4×#127D711yescharacteristic, complemented
108C42×D4—71no—
109#1982—71no—
110#1948—71no—
111#1926—72no—
112#1879—72no—
113#1802—71no—
114#1767—71no—
115#1762—72no—
116C24×C7—22no—
117C23×D7—21no—
118C2×C7×D4—24no—
119C2×C7×D4—24no—
120C2×C7×D4—22no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—42no—
126C2×C7×D4—42no—
127C2×C7×D4—42no—
128C2×C7×D4—42no—
129C2×C7×D4C22×C414yes—
130C22×C4×C7—22no—
131C22×C4×C7—44no—
132C22×C4×C7—44no—
133C22×C4×C7—44no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2411yescharacteristic
141C22×Dic7—21no—
142C22×Dic7—21no—
143C22×Dic7—21no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28C22×C412yes—
148C2×D28C2×D412yes—
149C2×D28#4112yes—
150C2×C4×D7—24no—
151C2×C4×D7—24no—
152C2×C4×D7—24no—
153C2×C4×D7—42no—
154C2×C4×D7—42no—
155C2×C4×D7—42no—
156C2×C4×D7—42no—
157C2×C4×D7—42no—
158C2×C4×D7—42no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—42no—
162C7×C4⋊C4—42no—
163C7×C22⋊C4—44no—
164C7×C22⋊C4—44no—
165C7×C22⋊C4—44no—
166C42×C7—22no—
167C42×C7—22no—
168C42×C7—22no—
169C42×C7—21no—
170C42×C7—21no—
171C42×C7—41no—
172C42×C7C22×C412yes—
173C42×C7C2×D412yes—
174C42×C7#4112yes—
175C42×C7#4112yes—
176#1141—216no—
177(C2×D7)⋊C4—24no—
178(C2×D7)⋊C4—24no—
179(C2×D7)⋊C4—24no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7C22×C412yes—
184C4⋊Dic7C2×D412yes—
185C4⋊Dic7#4112yes—
186C4×Dic7—42no—
187C4×Dic7—42no—
188C4×Dic7—42no—
189C2×C7⋊C8—22no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8C22×C414yes—
195C22×D8—72no—
196C2×#127C2×D711yescharacteristic
197C2×C4×D4—21no—
198C2×C4×D4—21no—
199C2×C4×D4—71no—
200C2×C4×D4—71no—
201C2×C4×D4—71no—
202C2×C4×D4C2×D712yes—
203C22×C42—141no—
204#459—78no—
205#442—148no—
206C2×#106—72no—
207C2×C4⋊C8—71no—
208C2×C4⋊C8—71no—
209C2×#102—72no—
210C2×#102—72no—
211C2×#102—72no—
212C2×#102—72no—
213C2×#97—72no—
214#390—21no—
215#390—21no—
216#390C2×D712yes—
217#384C2×D711yescharacteristic
218C4×C4⋊C4—71no—
219C4×C4⋊C4—71no—
220C4×C4⋊C4—71no—
221C4×C22⋊C4—141no—
222C4×C22⋊C4—141no—
223C4×C22⋊C4—141no—
224C43C2×D711yescharacteristic
225#340—72no—
226#340—72no—
227#336—72no—
228C23×C7—24no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7C22×D411yescharacteristic
236C22×D7—22no—
237C22×D7—22no—
238C22×D7—22no—
239C22×D7—41no—
240C22×D7—41no—
241C22×D7—41no—
242C22×D7—41no—
243C7×D4—24no—
244C7×D4—24no—
245C7×D4—48no—
246C7×D4—42no—
247C7×D4—42no—
248C7×D4—48no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—88no—
264C2×C4×C7C4×D412yes—
265C2×C4×C7C4×D412yes—
266C2×C4×C7#12112yes—
267C2×C4×C7#12112yes—
268C2×C4×C7AGL(1,ℤ/8)12yes—
269C2×C4×C7AGL(1,ℤ/8)12yes—
270C2×C4×C7AGL(1,ℤ/8)12yes—
271C2×C4×C7C23×C411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286D28—22no—
287D28—22no—
288D28—22no—
289D28C4×D414yescomplemented
290C4×D7—48no—
291C4×D7—48no—
292C4×D7—48no—
293C7⋊C8—24no—
294C7⋊C8—42no—
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—71no—
298C22×D4C22×D712yes—
299C23×C4—22no—
300C23×C4—141no—
301C23×C4—141no—
302C23×C4—141no—
303C2×D8—78no—
304C2×D8—144no—
305C2×D8—144no—
306C22×C8—72no—
307C22×C8—72no—
308#127—22no—
309#127C4×D714yescomplemented
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4—74no—
315C4×D4—74no—
316C4×D4—142no—
317C4×D4—142no—
318C4×D4—142no—
319C4×D4—142no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4C22×D712yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—141no—
330C2×C22⋊C4—141no—
331C2×C22⋊C4—141no—
332C2×C42—141no—
333C2×C42—141no—
334C2×C42—141no—
335C2×C42—284no—
336C2×C42C22⋊D711yescharacteristic
337C2×C42C22⋊D711yescharacteristic
338C2×C42C22⋊D711yescharacteristic
339C2×C42C22⋊D711yescharacteristic
340C2×C42C22×D711yescharacteristic
341C2×C42C22×D711yescharacteristic
342C2×C42C22×D711yescharacteristic
343#106—144no—
344C4⋊C8—74no—
345C4⋊C8—142no—
346#102—78no—
347#102—144no—
348#102—144no—
349#102—144no—
350#97—144no—
351#95—141no—
352#95—141no—
353#95—141no—
354C22×C7—44no—
355C22×C7—44no—
356C22×C7—44no—
357C22×C7—82no—
358C22×C7—82no—
359C22×C7—82no—
360C22×C7—82no—
361C22×C7C2×C4×D411yescharacteristic
362C22×C7C2×#12111yescharacteristic
363C22×C7#52511yescharacteristic
364C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
365C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
366C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
367C22×C7#57511yescharacteristic
368C2×D7—24no—
369C2×D7—42no—
370C2×D7—42no—
371C2×D7—42no—
372C2×D7—42no—
373C2×D7—42no—
374C2×D7—42no—
375C4×C7—22no—
376C4×C7—22no—
377C4×C7—22no—
378C4×C7—22no—
379C4×C7—41no—
380C4×C7—82no—
381C4×C7#44214yescomplemented
382C4×C7C2×C4×D411yescharacteristic
383C4×C7C4×#4111yescharacteristic
384C4×C7#52911yescharacteristic
385C4×C7#53211yescharacteristic
386Dic7—42no—
387Dic7—42no—
388Dic7—42no—
389C24—22no—
390C24—141no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—72no—
403C2×D4—72no—
404C2×D4—72no—
405C2×D4—141no—
406C2×D4—141no—
407C2×D4—141no—
408C2×D4C2×C4×D714yes—
409C22×C4—22no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—144no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—144no—
418C22×C4—144no—
419C22×C4—282no—
420C22×C4—282no—
421C22×C4—282no—
422C22×C4—282no—
423C22×C4—282no—
424C22×C4—282no—
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C2×C22⋊D711yescharacteristic
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C23×D711yescharacteristic
432D8—1416no—
433C2×C8—74no—
434C2×C8—142no—
435C2×C8—142no—
436C2×C8—142no—
437C2×C8—142no—
438C2×C8—142no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—72no—
444C4⋊C4—72no—
445C4⋊C4—72no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—144no—
453C22⋊C4—144no—
454C22⋊C4—144no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—21no—
459C42—21no—
460C42—41no—
461C42—282no—
462C42—282no—
463C42—282no—
464C42C2×C4×D712yes—
465C42#115712yes—
466C42D4×D712yes—
467C42#115912yes—
468C2×C7—84no—
469C2×C7C4×#12111yescharacteristic
470C2×C7C2×#44211yescharacteristic
471C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
472C2×C7#303611yescharacteristic
473C2×C7#303811yescharacteristic
474C2×C7#310911yescharacteristic
475C2×C7#311511yescharacteristic
476D7—44no—
477C23—24no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—142no—
485C23—142no—
486C23—142no—
487C23—281no—
488C23—281no—
489C23—281no—
490C23—281no—
491C23C22×C22⋊D711yescharacteristic
492D4—24no—
493D4—24no—
494D4—48no—
495D4—42no—
496D4—42no—
497D4—48no—
498D4—74no—
499D4—142no—
500D4—142no—
501D4—142no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—88no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—142no—
520C2×C4—288no—
521C2×C4—288no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—288no—
529C2×C4C4×C22⋊D712yes—
530C2×C4C4×C22⋊D712yes—
531C2×C4#649512yes—
532C2×C4#649512yes—
533C2×C4#650112yes—
534C2×C4#650112yes—
535C2×C4#651212yes—
536C2×C4C22×C4×D711yescharacteristic
537C2×C4C2×#115711yescharacteristic
538C2×C4C2×D4×D711yescharacteristic
539C2×C4C2×#115911yescharacteristic
540C2×C4D7×#4111yescharacteristic
541C2×C4D7×#4111yescharacteristic
542C8—144no—
543C8—282no—
544C7#2150311yescharacteristic, complemented, Sylow
545C22—44no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—144no—
553C22—282no—
554C22—282no—
555C22—282no—
556C22—282no—
557C22—282no—
558C22—282no—
559C22C2×C4×C22⋊D711yescharacteristic
560C22C2×#649511yescharacteristic
561C22C2×#649511yescharacteristic
562C22C2×#650111yescharacteristic
563C22C2×#651211yescharacteristic
564C22#9051311yescharacteristic
565C22#9051911yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—82no—
572C4—282no—
573C4—282no—
574C4—282no—
575C4#8978614yescomplemented
576C4C4×#115911yescharacteristic
577C4C4×D4×D711yescharacteristic
578C4#9023911yescharacteristic
579C4#9024811yescharacteristic
580C2—84no—
581C2—284no—
582C2C2×#8978611yescharacteristic
583C2C4×#649511yescharacteristic
584C2C4×#650111yescharacteristic
585C2#1174061011yescharacteristic
586C2#1174074211yescharacteristic
587C2#1174074511yescharacteristic
588C2#1174074611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#127C2
socleC23×C7C22×D4
prime radicalC2×#89844C2
cosocleC22×C4×C7C24
hypercenterC4×#127D7
hypocenterC7#21503
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C4×C7×#127C2semidirect
3. C2×#89786C2semidirect
4. C42×D28C2semidirect
5. #11729515C2semidirect
6. #11729363C2semidirect
7. #11727411C2semidirect
8. #11727391C2semidirect
9. #11727338C2semidirect
10. #11727322C2semidirect
11. #11727145C2semidirect
29. #89844C4semidirect
55. C7×#127C2×C4semidirect
75. C2×#1141C2×C4semidirect
91. C4×D28C2×C4semidirect
92. C4×D28D4semidirect
96. #6406C2×C4semidirect
103. C4⋊(C7⋊C8)C2×C4semidirect
107. C4×#127D7semidirect
289. D28C4×D4semidirect
309. #127C4×D7semidirect
381. C4×C7#442semidirect
544. C7#21503semidirect
575. C4#89786semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1272
C2×#898442
C2×#897862
C42×D282
#117295152
#117293632
#117274112
#117273912
#117273382
#117273222
#117271452
#215037