Finite Group Databasebeta

← #13502590

Subgroups — #13502590

Order 1792.

2008 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
1C22×C4×C7×D4C211yesmaximal, characteristic, complemented
2C2×#89989C212yesmaximal, complemented
3C2×#89787C212yesmaximal, complemented
4#11739526C211yesmaximal, characteristic, complemented
5#11729367C214yesmaximal, complemented
6#11729358C212yesmaximal, complemented
7#11729314C212yesmaximal, complemented
8#11729284C211yesmaximal, characteristic, complemented
9C23×C7×D4C2211yescharacteristic
10C24×C4×C7—21no—
11C2×C4×C7×D4—28no—
12C2×C4×C7×D4—21no—
13C2×C4×C7×D4—21no—
14C2×C4×C7×D4C2212yes—
15C2×C4×C7×D4C2212yes—
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4—21no—
18C22×C42×C7C2211yescharacteristic
19C22×#1141C2214yes—
20C2×#6493—21no—
21C2×#6493—21no—
22C2×#6493C2212yes—
23C2×#6437C2211yescharacteristic, complemented
24#89989—28no—
25C2×#6406C2214yes—
26C2×C22⋊(C7⋊C8)C2212yes—
27#89787—28no—
28C2×#6383C2214yescomplemented
29#89711—21no—
30#89711—21no—
31#89711C2212yescomplemented
32#89707C2211yescharacteristic, complemented
33C2×C4⋊(C7⋊C8)C2212yescomplemented
34#89324C2214yescomplemented
35#89319C2212yescomplemented
36#32902—71nomaximal, Sylow
37C25×C7—21no—
38C22×C7×D4—24no—
39C22×C7×D4—24no—
40C22×C7×D4—21no—
41C22×C7×D4—22no—
42C22×C7×D4D414yescomplemented
43C22×C7×D4C2312yes—
44C23×C4×C7—28no—
45C23×C4×C7—21no—
46C23×C4×C7—22no—
47C23×C4×C7—22no—
48C23×C4×C7—44no—
49C23×C4×C7—44no—
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—22no—
54C22×D28—21no—
55C22×D28—21no—
56C22×D28C2312yes—
57C4×C7×D4—28no—
58C4×C7×D4—28no—
59C4×C7×D4—24no—
60C4×C7×D4—24no—
61C4×C7×D4—24no—
62C4×C7×D4D418yescomplemented
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4C2312yes—
66C2×C7×C22⋊C4—28no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—22no—
69C2×C42×C7—24no—
70C2×C42×C7D411yescharacteristic
71C2×C42×C7D411yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×#1141—28no—
75C2×#1141—28no—
76C2×#1141—28no—
77#6493—24no—
78#6493—24no—
79#6493—44no—
80#6493—44no—
81C2×(C2×D7)⋊C4—22no—
82C2×(C2×D7)⋊C4—22no—
83C2×(C2×D7)⋊C4—22no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2312yes—
87C22×C7⋊C8C2314yes—
88#6437—24no—
89#6406—28no—
90C22⋊(C7⋊C8)—24no—
91#6383—28no—
92C4⋊(C7⋊C8)—24no—
93C22×C4×D4D711yescharacteristic, complemented
94C2×#459—72no—
95C2×#447—72no—
96#2856—71no—
97#1879—74no—
98#1874—72no—
99#1802—72no—
100#1797—71no—
101C24×C7—24no—
102C24×C7—24no—
103C24×C7—24no—
104C24×C7—22no—
105C24×C7—42no—
106C24×C7—42no—
107C24×C7—42no—
108C24×C7—42no—
109C24×C7C2×D411yescharacteristic
110C23×D7—22no—
111C2×C7×D4—24no—
112C2×C7×D4—24no—
113C2×C7×D4—216no—
114C2×C7×D4—216no—
115C2×C7×D4—24no—
116C2×C7×D4—22no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4C2×D414yes—
120C2×C7×D4C2×D414yes—
121C2×C7×D4#4114yes—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—24no—
127C22×C4×C7—28no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—28no—
132C22×C4×C7—41no—
133C22×C4×C7—44no—
134C22×C4×C7—432no—
135C22×C4×C7—42no—
136C22×C4×C7—42no—
137C22×C4×C7—44no—
138C22×C4×C7—44no—
139C22×C4×C7—44no—
140C22×C4×C7D814yescomplemented
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7#4112yes—
148C22×C4×C7C2411yescharacteristic
149C2×C22⋊D7—44no—
150C2×C22⋊D7—44no—
151C2×C22⋊D7—44no—
152C22×Dic7—22no—
153C2×D28—24no—
154C2×D28—22no—
155C2×D28—22no—
156C2×D28—24no—
157C2×D28—42no—
158C2×D28—42no—
159C2×D28—42no—
160C2×D28—42no—
161C2×D28—42no—
162C2×D28—42no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4C2×D414yes—
166C7×C22⋊C4—28no—
167C7×C22⋊C4—28no—
168C42×C7—22no—
169C42×C7—22no—
170C42×C7—22no—
171C42×C7C2×D414yes—
172#1141—416no—
173(C2×D7)⋊C4—44no—
174(C2×D7)⋊C4—44no—
175(C2×D7)⋊C4—44no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—42no—
179C4⋊Dic7—42no—
180C2×C7⋊C8—24no—
181C2×C7⋊C8—24no—
182C2×C7⋊C8—24no—
183C23×D4C2×D711yescharacteristic
184C24×C4—21no—
185C22×D8—74no—
186C2×#127—71no—
187C2×#121—72no—
188C2×#121—141no—
189C2×#121—141no—
190C2×C4×D4—28no—
191C2×C4×D4—21no—
192C2×C4×D4—21no—
193C2×C4×D4C2×D712yes—
194C2×C4×D4C2×D712yes—
195C22×C4⋊C4C2×D711yescharacteristic
196C22×C22⋊C4—21no—
197C22×C42C2×D711yescharacteristic
198#459—148no—
199#447—148no—
200C2×C4⋊C8—72no—
201C2×#102—74no—
202C2×#102—74no—
203C2×C22⋊C8—72no—
204#390—72no—
205#390—141no—
206#390—141no—
207#384—71no—
208#340—74no—
209#336—72no—
210C23×C7—21no—
211C23×C7—21no—
212C23×C7—21no—
213C23×C7—21no—
214C23×C7—216no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—24no—
218C23×C7—48no—
219C23×C7—48no—
220C23×C7—48no—
221C23×C7—48no—
222C23×C7—48no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—48no—
228C23×C7#12112yes—
229C23×C7C2×D812yes—
230C23×C7AGL(1,ℤ/8)12yes—
231C23×C7C22×D411yescharacteristic
232C22×D7—42no—
233C22×D7—42no—
234C22×D7—42no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7—42no—
238C22×D7—42no—
239C7×D4—216no—
240C7×D4—24no—
241C7×D4—24no—
242C7×D4—24no—
243C7×D4#12118yescomplemented
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—28no—
253C2×C4×C7—21no—
254C2×C4×C7—24no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—44no—
258C2×C4×C7—416no—
259C2×C4×C7—48no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—416no—
263C2×C4×C7#12012yes—
264C2×C4×C7#12012yes—
265C2×C4×C7#12112yes—
266C2×C4×C7C2×D812yes—
267C2×C4×C7C2×D812yes—
268C2×C4×C7AGL(1,ℤ/8)12yes—
269C2×C4×C7C22×D411yescharacteristic
270C2×C4×C7C22×D412yes—
271C2×C4×C7C2×#4112yes—
272C2×C4×C7C2×#4111yescharacteristic
273C22⋊D7—88no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277D28—44no—
278D28—44no—
279D28—84no—
280D28—84no—
281C7⋊C8—44no—
282C25—21no—
283C22×D4—24no—
284C22×D4—24no—
285C22×D4—22no—
286C22×D4—21no—
287C22×D4—72no—
288C22×D4—142no—
289C22×D4—141no—
290C22×D4—141no—
291C22×D4D2814yescomplemented
292C22×D4C22×D712yes—
293C23×C4—28no—
294C23×C4—21no—
295C23×C4—22no—
296C23×C4—22no—
297C23×C4—44no—
298C23×C4—44no—
299C23×C4C22⋊D711yescharacteristic
300C23×C4C22⋊D711yescharacteristic
301C23×C4C22×D711yescharacteristic
302C2×D8—148no—
303C2×D8—148no—
304C2×D8—148no—
305C22×C8—74no—
306#127—144no—
307#121—144no—
308#121—144no—
309#121—284no—
310#121—284no—
311C4×D4—28no—
312C4×D4—28no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4C22⋊D718yescomplemented
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—72no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4C22×D712yes—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—142no—
327C2×C22⋊C4—142no—
328C2×C22⋊C4—142no—
329C2×C42—24no—
330C2×C42C22⋊D711yescharacteristic
331C2×C42C22⋊D711yescharacteristic
332C2×C42C22×D711yescharacteristic
333C2×C42C22×D711yescharacteristic
334C4⋊C8—144no—
335#102—148no—
336#102—148no—
337C22⋊C8—144no—
338C22×C7—22no—
339C22×C7—22no—
340C22×C7—22no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—28no—
345C22×C7—432no—
346C22×C7—44no—
347C22×C7—44no—
348C22×C7—44no—
349C22×C7—44no—
350C22×C7—44no—
351C22×C7—44no—
352C22×C7#45914yescomplemented
353C22×C7C2×#12011yescharacteristic
354C22×C7C2×#12111yescharacteristic
355C22×C7#52511yescharacteristic
356C22×C7C22×D811yescharacteristic
357C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
358C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
359C22×C7#57511yescharacteristic
360C2×D7—82no—
361C2×D7—82no—
362C2×D7—82no—
363C2×D7—82no—
364C2×D7—82no—
365C2×D7—82no—
366C2×D7—82no—
367C4×C7—22no—
368C4×C7—22no—
369C4×C7—22no—
370C4×C7—44no—
371C4×C7—41no—
372C4×C7#44714yescomplemented
373C4×C7C2×#12112yes—
374C4×C7#52512yes—
375Dic7—82no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—142no—
385C24C2×C22⋊D711yescharacteristic
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—216no—
389C2×D4—216no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—142no—
395C2×D4—142no—
396C2×D4—144no—
397C2×D4—144no—
398C2×D4—282no—
399C2×D4—282no—
400C2×D4—284no—
401C2×D4—284no—
402C2×D4—284no—
403C2×D4—282no—
404C2×D4—282no—
405C2×D4—282no—
406C2×D4—282no—
407C2×D4C2×D2814yes—
408C2×D4#115714yes—
409C2×D4C2×C22⋊D714yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—41no—
421C22×C4—44no—
422C22×C4—432no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—142no—
429C22×C4#114114yescomplemented
430C22×C4C2×D2812yes—
431C22×C4#115712yes—
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C2×C22⋊D711yescharacteristic
435C22×C4C2×C22⋊D711yescharacteristic
436C22×C4C2×C22⋊D711yescharacteristic
437C22×C4C23×D711yescharacteristic
438D8—2816no—
439C2×C8—144no—
440C2×C8—144no—
441C2×C8—144no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—142no—
445C4⋊C4—142no—
446C4⋊C4—282no—
447C4⋊C4—282no—
448C4⋊C4C2×C22⋊D714yes—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—284no—
452C22⋊C4—284no—
453C22⋊C4—284no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42C2×C22⋊D714yes—
458C2×C7—24no—
459C2×C7—48no—
460C2×C7#284711yescharacteristic
461C2×C7C2×#44711yescharacteristic
462C2×C7#300611yescharacteristic
463C2×C7C2×#45911yescharacteristic
464C2×C7#303611yescharacteristic
465C2×C7#310011yescharacteristic
466C2×C7#310911yescharacteristic
467D7—162no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—216no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—48no—
477C23—48no—
478C23—48no—
479C23—48no—
480C23—48no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—48no—
486C23—282no—
487C23—282no—
488C23—282no—
489C23—282no—
490C23—282no—
491C23—282no—
492C23—282no—
493C23#649312yes—
494C23C2×#114112yes—
495C23#651212yes—
496C23C22×C22⋊D711yescharacteristic
497D4—216no—
498D4—24no—
499D4—24no—
500D4—24no—
501D4—284no—
502D4—284no—
503D4—564no—
504D4—568no—
505D4—564no—
506D4#649318yescomplemented
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—416no—
522C2×C4—48no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—416no—
526C2×C4—282no—
527C2×C4—282no—
528C2×C4—282no—
529C2×C4#649312yes—
530C2×C4C2×#114112yes—
531C2×C4C2×#114112yes—
532C2×C4#649512yes—
533C2×C4#651612yes—
534C2×C4#651612yes—
535C2×C4C22×D2811yescharacteristic
536C2×C4C2×#115712yes—
537C2×C4C2×#115711yescharacteristic
538C2×C4C22×C22⋊D712yes—
539C8—284no—
540C7#3290211yescharacteristic, complemented, Sylow
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—28no—
548C22—432no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—562no—
556C22—562no—
557C22—562no—
558C22—562no—
559C22—562no—
560C22—562no—
561C22—562no—
562C22#8978714yescomplemented
563C22C2×#649311yescharacteristic
564C22#9048211yescharacteristic
565C22C22×#114111yescharacteristic
566C22C2×#649511yescharacteristic
567C22C2×#651211yescharacteristic
568C22#9051311yescharacteristic
569C22C2×#651611yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—44no—
574C4—41no—
575C4—562no—
576C4#8998914yescomplemented
577C4C2×#649312yes—
578C4#9048212yes—
579C2—24no—
580C2—48no—
581C2—1122no—
582C2C2×#8978711yescharacteristic
583C2#1173975211yescharacteristic
584C2#1173975311yescharacteristic
585C2#1173975511yescharacteristic
586C2#1174147711yescharacteristic
587C2C2×#8998911yescharacteristic
588C2#1174152711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC22×C4×C7×D4C2
socleC23×C7C22×D4
hypercenterC22×C4×D4D7
hypocenterC7#32902
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C7×D4C2semidirect
2. C2×#89989C2semidirect
3. C2×#89787C2semidirect
4. #11739526C2semidirect
5. #11729367C2semidirect
6. #11729358C2semidirect
7. #11729314C2semidirect
8. #11729284C2semidirect
23. C2×#6437C22semidirect
28. C2×#6383C22semidirect
31. #89711C22semidirect
32. #89707C22semidirect
33. C2×C4⋊(C7⋊C8)C22semidirect
34. #89324C22semidirect
35. #89319C22semidirect
42. C22×C7×D4D4semidirect
62. C4×C7×D4D4semidirect
93. C22×C4×D4D7semidirect
140. C22×C4×C7D8semidirect
243. C7×D4#121semidirect
291. C22×D4D28semidirect
316. C4×D4C22⋊D7semidirect
352. C22×C7#459semidirect
372. C4×C7#447semidirect
429. C22×C4#1141semidirect
506. D4#6493semidirect
540. C7#32902semidirect
562. C22#89787semidirect
576. C4#89989semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C7×D42
C2×#899892
C2×#897872
#117395262
#117293672
#117293582
#117293142
#117292842
#329027