Finite Group Databasebeta

← #13355811

Subgroups — #13355811

Order 1792.

1748 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11744336C211yesmaximal, characteristic
2#11744328C211yesmaximal, characteristic, complemented
3C2×C7×#410C211yesmaximal, characteristic, complemented
4C2×#89778C212yesmaximal, complemented
5#11729346C212yesmaximal, complemented
6#11727778C218yesmaximal, complemented
7C22×C7×#41C2211yescharacteristic
8C22×#1157C2211yescharacteristic
9C2×C7×#117C2212yes—
10#90524—22no—
11#90523—22no—
12#90521C412yescomplemented
13#90519C414yescomplemented
14#90518—24no—
15#41×Dic7—24no—
16#90516C412yescomplemented
17#90507—21no—
18#90503—21no—
19#90496—21no—
20#90495C2214yes—
21#90489—21no—
22#90489C2214yes—
23#90482—22no—
24C2×#6489—22no—
25C2×#6489C2211yescharacteristic
26C2×#6489C2212yes—
27C7×#410—22no—
28C7×#410—22no—
29C2×C7×#99C2214yes—
30#89778—24no—
31#89778—24no—
32C2×#6380C2218yes—
33#89321C2218yes—
34#21014—71nomaximal, Sylow
35C2×C7×#41—22no—
36C2×C7×#41—22no—
37C2×C7×#41—22no—
38C2×C7×#41—22no—
39C2×C7×#41C2×C412yes—
40C2×C7×#41C2×C412yes—
41C2×C7×#41D412yes—
42C2×C7×#41D412yes—
43C22×C7×Q8D411yescharacteristic
44C22×C7×D4D411yescharacteristic
45C22×C7×D4C2312yes—
46C23×C4×C7D412yes—
47C23×C4×C7C2311yescharacteristic
48C22×C22⋊D7C2312yes—
49C2×#1157—21no—
50C2×#1157—21no—
51C2×#1157—21no—
52C2×#1157—21no—
53C2×#1157—22no—
54C2×#1157—22no—
55C2×#1157—22no—
56C2×#1157—22no—
57C22×D28C2×C411yescharacteristic, complemented
58C22×C4×D7C2×C412yescomplemented
59C22×Dic14C2×C411yescharacteristic, complemented
60C7×#117—22no—
61C7×#117—22no—
62C7×#117—22no—
63C7×#117—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C22⋊C4C2314yes—
66C2×C42×C7—22no—
67#6516—24no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7C2312yes—
70C2×C22⋊Dic7C2314yes—
71#6510—22no—
72#6509—24no—
73Q8×Dic7—28no—
74#6507—22no—
75#6503—22no—
76#6502—28no—
77#6501—24no—
78#6500C2×C418yescomplemented
79#6499—22no—
80#6498C2×C418yescomplemented
81D4×Dic7—28no—
82D4×Dic7—216no—
83#6493—24no—
84#6492—24no—
85C4×C22⋊D7—28no—
86#6489—24no—
87#6489—24no—
88#6489—22no—
89#6489—24no—
90#6489—21no—
91#6489—22no—
92#6489—21no—
93#6489C2×C412yescomplemented
94#6489C2×C412yescomplemented
95#6489D414yescomplemented
96#6489D414yescomplemented
97C2×C4⋊Dic7—24no—
98C2×C4⋊Dic7—22no—
99C2×C4⋊Dic7—21no—
100#6487—24no—
101C2×C4×Dic7—24no—
102C2×C4×Dic7—22no—
103C2×C4×Dic7—21no—
104C7×#99—28no—
105#6380—216no—
106#3449—71no—
107#3428—71no—
108C2×#410—72no—
109C2×#410D711yescharacteristic, complemented
110#1913—78no—
111#1740—72no—
112C24×C7C2×D412yes—
113C24×C7C2411yescharacteristic
114C23×D7C22×C412yes—
115C7×#41—24no—
116C7×#41—24no—
117C7×#41—24no—
118C7×#41—24no—
119C7×#41—24no—
120C7×#41—24no—
121C7×#41—24no—
122C7×#41—24no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8C22⋊C412yes—
126C2×C7×Q8C22⋊C412yes—
127C2×C7×D4—24no—
128C2×C7×D4—24no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4C22⋊C412yes—
132C2×C7×D4C22⋊C412yes—
133C2×C7×D4C22×C414yes—
134C2×C7×D4C2×D414yes—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—24no—
140C22×C4×C7C22⋊C414yes—
141C22×C4×C7C22⋊C414yes—
142C22×C4×C7C22×C411yescharacteristic
143C22×C4×C7C22×C411yescharacteristic
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D412yes—
151C22×C4×C7#4112yes—
152C22×C4×C7#4112yes—
153C2×C22⋊D7—22no—
154C2×C22⋊D7—22no—
155C2×C22⋊D7—24no—
156C2×C22⋊D7—24no—
157C2×C22⋊D7—28no—
158C22×Dic7—28no—
159C22×Dic7—24no—
160C22×Dic7—22no—
161C22×Dic7C22×C412yes—
162#1157—41no—
163#1157—42no—
164#1157—41no—
165#1157—42no—
166#1157—42no—
167#1157—42no—
168#1157—42no—
169#1157—41no—
170#1157—41no—
171#1157—42no—
172C2×D28—21no—
173C2×D28—21no—
174C2×D28—22no—
175C2×D28—22no—
176C2×C4×D7—22no—
177C2×C4×D7—22no—
178C2×C4×D7—24no—
179C2×C4×D7—24no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—22no—
183C2×Dic14—22no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4—42no—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—24no—
188C42×C7—42no—
189C42×C7—42no—
190C22⋊Dic7—28no—
191C22⋊Dic7—28no—
192C22⋊Dic7—24no—
193C22⋊Dic7—22no—
194C22⋊Dic7C22×C414yes—
195C22⋊Dic7C2×D418yes—
196(C2×D7)⋊C4—28no—
197C4⋊Dic7—28no—
198C4⋊Dic7—22no—
199C4⋊Dic7—22no—
200C4⋊Dic7—24no—
201C4⋊Dic7—24no—
202Dic7⋊C4—28no—
203C4×Dic7—28no—
204C4×Dic7—22no—
205C4×Dic7—22no—
206C4×Dic7—24no—
207C4×Dic7—24no—
208C22×#41—71no—
209C22×#41C2×D711yescharacteristic
210#537—142no—
211#536—72no—
212#536—141no—
213#535—72no—
214#535—141no—
215#534—74no—
216#532—142no—
217#525—74no—
218#525—142no—
219#520—144no—
220#519—141no—
221#518—74no—
222#518—141no—
223C4×#41—144no—
224C2×#117—72no—
225C2×#117—71no—
226C2×#117—142no—
227C2×#117C2×D712yes—
228#410—22no—
229#410—22no—
230#410—144no—
231#410—144no—
232C2×#99—78no—
233C2×#99C2×D714yes—
234#342—78no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—81no—
238C23×C7—82no—
239C23×C7C2×C22⋊C414yes—
240C23×C7C23×C411yescharacteristic
241C23×C7C22×D411yescharacteristic
242C23×C7C22×D412yes—
243C23×C7C22×D411yescharacteristic
244C23×C7C2×#4112yes—
245C22×D7—22no—
246C22×D7—24no—
247C22×D7—82no—
248C7×Q8—24no—
249C7×Q8—24no—
250C7×D4—28no—
251C7×D4—28no—
252C7×D4—24no—
253C7×D4—24no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—44no—
261C2×C4×C7—84no—
262C2×C4×C7C2×C22⋊C414yes—
263C2×C4×C7C2×C22⋊C411yescharacteristic
264C2×C4×C7C2×C22⋊C411yescharacteristic
265C2×C4×C7C2×C22⋊C411yescharacteristic
266C2×C4×C7C2×C22⋊C411yescharacteristic
267C2×C4×C7C4×D412yes—
268C2×C4×C7C4×D412yes—
269C2×C4×C7C4×D412yes—
270C2×C4×C7C4×D412yes—
271C2×C4×C7#12011yescharacteristic
272C2×C4×C7#12012yes—
273C2×C4×C7#12011yescharacteristic
274C2×C4×C7#12112yes—
275C2×C4×C7#12112yes—
276C2×C4×C7#12112yes—
277C2×C4×C7#12112yes—
278C2×C4×C7#12311yescharacteristic
279C2×C4×C7#12311yescharacteristic
280C2×C4×C7#12312yes—
281C22⋊D7—42no—
282C22⋊D7—44no—
283C22⋊D7—42no—
284C22⋊D7—88no—
285C2×Dic7—24no—
286C2×Dic7—22no—
287C2×Dic7—24no—
288C2×Dic7—28no—
289C2×Dic7—84no—
290C2×Dic7—88no—
291C2×Dic7—82no—
292D28—41no—
293D28—41no—
294D28—42no—
295C4×D7—42no—
296C4×D7—44no—
297C4×D7—42no—
298Dic14—41no—
299Dic14—41no—
300Dic14—42no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—141no—
306C2×#41—141no—
307C2×#41—142no—
308C2×#41—142no—
309C2×#41—142no—
310C2×#41—142no—
311C2×#41—141no—
312C2×#41—141no—
313C2×#41C4×D712yes—
314C2×#41C4×D712yes—
315C2×#41C22⋊D712yes—
316C2×#41C22⋊D712yes—
317C22×Q8—71no—
318C22×Q8D2811yescharacteristic
319C22×D4—72no—
320C22×D4—71no—
321C22×D4D2811yescharacteristic
322C22×D4C22×D712yes—
323C23×C4—72no—
324C23×C4D2812yes—
325C23×C4C22×D711yescharacteristic
326C4⋊Q8—142no—
327#127—142no—
328#124—142no—
329#124—142no—
330#123—78no—
331#123—144no—
332C22⋊Q8—144no—
333C22⋊Q8—144no—
334#121—144no—
335#121—148no—
336#121—144no—
337#120—78no—
338#120—144no—
339C4×Q8—148no—
340C4×D4—1416no—
341C4×D4—148no—
342C4×D4—148no—
343#117—22no—
344#117—22no—
345#117—22no—
346#117—22no—
347#117—72no—
348#117—72no—
349#117—74no—
350#117—74no—
351#117—144no—
352#117—144no—
353#117—142no—
354#117—142no—
355#117—141no—
356#117—141no—
357#117—144no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—144no—
360C2×C4⋊C4—142no—
361C2×C4⋊C4—141no—
362C2×C22⋊C4—74no—
363C2×C22⋊C4—72no—
364C2×C22⋊C4—144no—
365C2×C22⋊C4C22×D714yes—
366C2×C42—22no—
367C2×C42—144no—
368C2×C42—142no—
369C2×C42—141no—
370#99—28no—
371#99—1416no—
372C22×C7—24no—
373C22×C7—42no—
374C22×C7—41no—
375C22×C7—42no—
376C22×C7—41no—
377C22×C7—41no—
378C22×C7—44no—
379C22×C7—44no—
380C22×C7—82no—
381C22×C7C22×C22⋊C411yescharacteristic
382C22×C7C2×C4×D412yes—
383C22×C7C2×#12011yescharacteristic
384C22×C7C2×#12112yes—
385C22×C7C2×#12311yescharacteristic
386C2×D7—44no—
387C2×D7—42no—
388C2×D7—84no—
389C4×C7—24no—
390C4×C7—84no—
391C4×C7#38611yescharacteristic
392C4×C7#38611yescharacteristic
393C4×C7#38612yes—
394C4×C7#38611yescharacteristic
395C4×C7#38612yes—
396C4×C7#38611yescharacteristic
397Dic7—48no—
398Dic7—44no—
399Dic7—42no—
400C24—72no—
401C24C2×D2812yes—
402C24C23×D711yescharacteristic
403#41—24no—
404#41—24no—
405#41—24no—
406#41—24no—
407#41—24no—
408#41—24no—
409#41—24no—
410#41—24no—
411#41—282no—
412#41—281no—
413#41—281no—
414#41—281no—
415#41—282no—
416#41—281no—
417#41—282no—
418#41—282no—
419#41—282no—
420#41—282no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—142no—
424C2×Q8—142no—
425C2×Q8—141no—
426C2×Q8—141no—
427C2×Q8(C2×D7)⋊C412yes—
428C2×Q8(C2×D7)⋊C412yes—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—142no—
434C2×D4—144no—
435C2×D4—144no—
436C2×D4—142no—
437C2×D4—148no—
438C2×D4—142no—
439C2×D4—142no—
440C2×D4—141no—
441C2×D4—141no—
442C2×D4(C2×D7)⋊C412yes—
443C2×D4(C2×D7)⋊C412yes—
444C2×D4C2×C4×D714yes—
445C2×D4C2×C22⋊D714yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—72no—
452C22×C4—148no—
453C22×C4—144no—
454C22×C4—142no—
455C22×C4—144no—
456C22×C4—144no—
457C22×C4—142no—
458C22×C4—142no—
459C22×C4(C2×D7)⋊C414yes—
460C22×C4(C2×D7)⋊C414yes—
461C22×C4C2×C4×D711yescharacteristic
462C22×C4C2×C4×D711yescharacteristic
463C22×C4C2×D2812yes—
464C22×C4C2×D2811yescharacteristic
465C22×C4C2×D2811yescharacteristic
466C22×C4D4×D712yes—
467C22×C4D4×D712yes—
468C22×C4#115912yes—
469C22×C4#115912yes—
470C22×C4C2×C22⋊D711yescharacteristic
471C22×C4C2×C22⋊D711yescharacteristic
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—148no—
475C4⋊C4—144no—
476C4⋊C4—144no—
477C4⋊C4—148no—
478C4⋊C4—142no—
479C4⋊C4—142no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—74no—
483C22⋊C4—78no—
484C22⋊C4—148no—
485C22⋊C4—148no—
486C22⋊C4—144no—
487C22⋊C4—142no—
488C22⋊C4—148no—
489C42—42no—
490C42—42no—
491C42—148no—
492C42—144no—
493C42—144no—
494C42—142no—
495C42—142no—
496C2×C7—21no—
497C2×C7—21no—
498C2×C7—22no—
499C2×C7—22no—
500C2×C7—21no—
501C2×C7—44no—
502C2×C7C2×#38611yescharacteristic
503D7—82no—
504C23—22no—
505C23—22no—
506C23—81no—
507C23—82no—
508C23—142no—
509C23—144no—
510C23—562no—
511C23C2×(C2×D7)⋊C414yes—
512C23C22×C4×D711yescharacteristic
513C23C22×D2811yescharacteristic
514C23C2×D4×D712yes—
515C23C2×#115912yes—
516C23C22×C22⋊D711yescharacteristic
517Q8—24no—
518Q8—24no—
519Q8—281no—
520Q8—281no—
521Q8—282no—
522D4—28no—
523D4—28no—
524D4—24no—
525D4—24no—
526D4—282no—
527D4—282no—
528D4—284no—
529D4—281no—
530D4—281no—
531D4—282no—
532D4—568no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—44no—
540C2×C4—84no—
541C2×C4—144no—
542C2×C4—148no—
543C2×C4—144no—
544C2×C4—142no—
545C2×C4—282no—
546C2×C4—284no—
547C2×C4—282no—
548C2×C4—564no—
549C2×C4—568no—
550C2×C4—562no—
551C2×C4(C22⋊D7)⋊C412yes—
552C2×C4(C22⋊D7)⋊C412yes—
553C2×C4(C22⋊D7)⋊C412yes—
554C2×C4(C22⋊D7)⋊C412yes—
555C2×C4#644512yes—
556C2×C4#644511yescharacteristic
557C2×C4#644511yescharacteristic
558C2×C4#644911yescharacteristic
559C2×C4#644912yes—
560C2×C4#644911yescharacteristic
561C2×C4C2×(C2×D7)⋊C414yes—
562C2×C4C2×(C2×D7)⋊C411yescharacteristic
563C2×C4C2×(C2×D7)⋊C411yescharacteristic
564C2×C4C2×(C2×D7)⋊C411yescharacteristic
565C2×C4C2×(C2×D7)⋊C411yescharacteristic
566C2×C4#650212yes—
567C2×C4#650212yes—
568C2×C4#650212yes—
569C2×C4#650212yes—
570C7#2101411yescharacteristic, complemented, Sylow
571C22—24no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—82no—
580C22—284no—
581C22—282no—
582C22—564no—
583C22C2×(C22⋊D7)⋊C412yes—
584C22C2×#644511yescharacteristic
585C22C2×#644911yescharacteristic
586C22C22×(C2×D7)⋊C411yescharacteristic
587C22C2×#650212yes—
588C4—24no—
589C4—84no—
590C4—288no—
591C4—284no—
592C4—282no—
593C4#8973011yescharacteristic
594C4#8973011yescharacteristic
595C4#8973011yescharacteristic
596C4#8973012yes—
597C4#8973012yes—
598C4#8973011yescharacteristic
599C2—22no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—22no—
604C2—44no—
605C2—562no—
606C2C2×#8973011yescharacteristic
subgroupG/H
centerC4#89730
commutatorC23×C7C23×C4
FrattiniC24C23×D7
FittingC2×C7×#410C2
socleC2×C7C2×#386
prime radical#11744336C2
cosocleC24×C7C24
hypercenterC2×#410D7
hypocenterC7#21014
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. #11744328C2semidirect
3. C2×C7×#410C2semidirect
4. C2×#89778C2semidirect
5. #11729346C2semidirect
6. #11727778C2semidirect
12. #90521C4semidirect
13. #90519C4semidirect
16. #90516C4semidirect
57. C22×D28C2×C4semidirect
58. C22×C4×D7C2×C4semidirect
59. C22×Dic14C2×C4semidirect
78. #6500C2×C4semidirect
80. #6498C2×C4semidirect
93. #6489C2×C4semidirect
94. #6489C2×C4semidirect
95. #6489D4semidirect
96. #6489D4semidirect
109. C2×#410D7semidirect
570. C7#21014semidirect

Maximal subgroups

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

subgroupindex
#117443362
#117443282
C2×C7×#4102
C2×#897782
#117293462
#117277782
#210147