Finite Group Databasebeta

← #13354798

Subgroups — #13354798

Order 1792.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90518C211yesmaximal, characteristic
2C2×C7×#416C211yesmaximal, characteristic
3C2×#90004C211yesmaximal, characteristic
4#11739677C211yesmaximal, characteristic, complemented
5C2×#89767C211yesmaximal, characteristic, complemented
6C2×#89346C211yesmaximal, characteristic, complemented
7#11729719C211yesmaximal, characteristic, complemented
8#11727991C212yesmaximal, complemented
9#11727989C212yesmaximal, complemented
10#11727923C212yesmaximal, complemented
11#11727921C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C2×C7×#117C2211yescharacteristic
15#90518—24no—
16#90518—24no—
17C2×Q8×Dic7C2212yes—
18C2×D4×Dic7—22no—
19C2×D4×Dic7C2212yes—
20C2×#6489—21no—
21C2×#6489C2211yescharacteristic
22C22×C4⋊Dic7—21no—
23C22×C4⋊Dic7C2211yescharacteristic
24C22×C4⋊Dic7C2211yescharacteristic
25C22×#1136C2211yescharacteristic
26C7×#416—24no—
27C2×C7×#103C2211yescharacteristic
28C2×C7×#103C2211yescharacteristic
29C2×C7×#102C2211yescharacteristic
30C2×C7×#102C2211yescharacteristic
31#90004C418yescomplemented
32C2×#6409C2212yes—
33C2×#6406C2212yes—
34#89767—24no—
35#89745C2212yes—
36#89745C2212yes—
37C2×#6382C2212yes—
38C2×#6381C2212yes—
39#89714—22no—
40#89700—21no—
41#89700—21no—
42#89346C2212yes—
43#89346C2212yes—
44#89345C2212yes—
45#89345C2212yes—
46#20878—71nomaximal, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41C2×C414yes—
51C2×C7×#41D414yes—
52C2×C7×#41Q814yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7C2311yescharacteristic
58C23×Dic7—21no—
59C23×Dic7—21no—
60C23×Dic7—22no—
61C22×C7×C8—22no—
62C22×C7×C8—22no—
63C22×C7×C8—22no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C7×#117—22no—
67C7×#117—22no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C42×C7—21no—
72C2×C22⋊Dic7—22no—
73C2×C22⋊Dic7D412yes—
74Q8×Dic7—24no—
75Q8×Dic7—24no—
76D4×Dic7—216no—
77D4×Dic7—24no—
78D4×Dic7—24no—
79#6489—28no—
80#6489—22no—
81#6489—22no—
82C2×C4⋊Dic7—28no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7—22no—
87C2×C4⋊Dic7—22no—
88C2×C4⋊Dic7C2×C414yescomplemented
89C2×C4⋊Dic7D411yescharacteristic
90C2×C4⋊Dic7D411yescharacteristic
91C2×C4⋊Dic7D414yes—
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2312yes—
94C2×Dic7⋊C4—22no—
95C2×Dic7⋊C4—22no—
96C2×C4×Dic7—22no—
97C2×C4×Dic7C2312yes—
98C2×#1136—22no—
99C2×#1136—22no—
100C2×#1136C2×C414yescomplemented
101C22×C7⋊C8C2312yes—
102C7×#103—22no—
103C7×#103—22no—
104C7×#102—22no—
105C7×#102—22no—
106#6409C2×C418yescomplemented
107#6406C2×C418yescomplemented
108C7⋊#95—22no—
109C7⋊#95—22no—
110#6382—24no—
111#6381—24no—
112C2×#520—71no—
113C2×#428—71no—
114C2×#417—71no—
115C2×#416D711yescharacteristic
116#2316—71no—
117#1865—72no—
118#1864—72no—
119#1863—72no—
120#1862—72no—
121C2×#340—71no—
122#1729—71no—
123C24×C7—21no—
124C24×C7C2×D411yescharacteristic
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C7×#41C4⋊C4116yescomplemented
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8C22×C412yes—
133C2×C7×Q8C2×D412yes—
134C2×C7×Q8C2×Q812yes—
135C2×C7×D4—24no—
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C22×C412yes—
142C2×C7×D4C2×D412yes—
143C2×C7×D4C2×Q812yes—
144C22×C4×C7—21no—
145C22×C4×C7—24no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C22×C412yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×Q812yes—
155C22×C4×C7#4112yes—
156C22×C4×C7#4112yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—28no—
161C22×Dic7—24no—
162C22×Dic7—24no—
163C22×Dic7—82no—
164C22×Dic7—82no—
165C22×Dic7—81no—
166C22×Dic7—81no—
167C22×Dic7—81no—
168C22×Dic7—81no—
169C22×Dic7C2×D412yes—
170C22×Dic7C2×D412yes—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—44no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C22⋊C4—42no—
183C42×C7—42no—
184C22⋊Dic7—28no—
185C22⋊Dic7—24no—
186C4⋊Dic7—28no—
187C4⋊Dic7—24no—
188C4⋊Dic7—22no—
189C4⋊Dic7—24no—
190C4⋊Dic7—22no—
191C4⋊Dic7—22no—
192C4⋊Dic7—22no—
193C4⋊Dic7C22×C414yes—
194C4⋊Dic7C2×D414yes—
195Dic7⋊C4—82no—
196Dic7⋊C4—82no—
197C4×Dic7—28no—
198C4×Dic7—22no—
199C4×Dic7—22no—
200#1136—28no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8C22×C414yes—
204C22×#41C2×D711yescharacteristic
205C22×M16—71no—
206C23×C8C2×D711yescharacteristic
207#520—144no—
208#520—144no—
209C2×C4×Q8—72no—
210C2×C4×D4—72no—
211C2×C4×D4—142no—
212C2×#117—71no—
213C2×#117—141no—
214C2×#117C2×D711yescharacteristic
215C22×C4⋊C4—71no—
216C22×C4⋊C4—71no—
217C22×C4⋊C4—141no—
218#428—144no—
219C2×#107—72no—
220C2×#106—72no—
221#417—78no—
222#416—24no—
223C2×#103—72no—
224C2×#103C2×D711yescharacteristic
225C2×#103C2×D711yescharacteristic
226C2×#102—72no—
227C2×#102C2×D711yescharacteristic
228C2×#102C2×D711yescharacteristic
229#389—72no—
230#389—72no—
231#389—141no—
232#389—141no—
233#380—142no—
234#340—72no—
235#340—72no—
236#336—72no—
237#336—72no—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7C4×D412yes—
250C23×C7#12012yes—
251C23×C7C22⋊Q812yes—
252C23×C7C22×D411yescharacteristic
253C7×Q8—22no—
254C7×Q8—22no—
255C7×Q8—22no—
256C7×Q8C2×C4⋊C414yes—
257C7×D4—216no—
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—22no—
261C7×D4C2×C4⋊C414yes—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—216no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7C2×C4⋊C414yes—
273C2×C4×C7C4×D414yes—
274C2×C4×C7C4×D412yes—
275C2×C4×C7#12012yes—
276C2×C4×C7C22⋊Q814yes—
277C2×C4×C7C22⋊Q812yes—
278C2×C4×C7#12314yes—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C22×Q811yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×Dic7—24no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—24no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—22no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—84no—
298C2×Dic7—84no—
299C2×Dic7—88no—
300C7×C8—41no—
301C7×C8—41no—
302C7⋊C8—24no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41Dic1414yes—
307C2×#41C4×D714yes—
308C2×#41C22⋊D714yes—
309C22×Q8C22×D711yescharacteristic
310C22×D4—21no—
311C22×D4C22×D711yescharacteristic
312C23×C4—21no—
313C23×C4—142no—
314C23×C4—141no—
315C23×C4—141no—
316C23×C4C22×D711yescharacteristic
317C2×M16—74no—
318C2×M16—142no—
319C2×M16—142no—
320C22×C8—22no—
321C22×C8—22no—
322C22×C8—22no—
323C22×C8—72no—
324C22×C8C22×D711yescharacteristic
325C22×C8C22×D711yescharacteristic
326C4×Q8—144no—
327C4×Q8—144no—
328C4×D4—144no—
329C4×D4—1416no—
330C4×D4—144no—
331#117—22no—
332#117—22no—
333#117—148no—
334#117—142no—
335#117—142no—
336C2×C4⋊C4—74no—
337C2×C4⋊C4—74no—
338C2×C4⋊C4—72no—
339C2×C4⋊C4—72no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—142no—
343C2×C4⋊C4—148no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—142no—
349C2×C4⋊C4—142no—
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—72no—
354C2×C22⋊C4—142no—
355C2×C42—21no—
356C2×C42—72no—
357C2×C42—142no—
358#107—144no—
359#106—144no—
360#103—22no—
361#103—22no—
362#103—78no—
363#102—22no—
364#102—22no—
365#102—78no—
366#95—142no—
367#95—142no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—24no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7#38514yes—
382C22×C7C2×C4×D411yescharacteristic
383C22×C7C2×#12011yescharacteristic
384C22×C7C2×C22⋊Q811yescharacteristic
385C22×C7#57611yescharacteristic
386C22×C7#57711yescharacteristic
387C22×C7#57711yescharacteristic
388C22×C7#57811yescharacteristic
389C4×C7—24no—
390C4×C7—81no—
391C4×C7—81no—
392C4×C7#38514yes—
393C4×C7C22×C4⋊C411yescharacteristic
394C4×C7C2×C4×D411yescharacteristic
395C4×C7C2×C22⋊Q811yescharacteristic
396C4×C7C2×#12311yescharacteristic
397Dic7—44no—
398Dic7—42no—
399Dic7—44no—
400Dic7—42no—
401C24—21no—
402C24D4×D711yescharacteristic
403#41—28no—
404#41—28no—
405#41—28no—
406#41Dic7⋊C4116yescomplemented
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8C2×Dic1412yes—
411C2×Q8C2×C4×D712yes—
412C2×Q8C2×C22⋊D712yes—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4C2×Dic1412yes—
420C2×D4C2×C4×D712yes—
421C2×D4C2×C22⋊D712yes—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—72no—
429C22×C4—72no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—148no—
433C22×C4—144no—
434C22×C4—144no—
435C22×C4—562no—
436C22×C4—562no—
437C22×C4—561no—
438C22×C4—561no—
439C22×C4—561no—
440C22×C4—561no—
441C22×C4C2×Dic1412yes—
442C22×C4C2×C4×D712yes—
443C22×C4D4×D711yescharacteristic
444C22×C4D4×D712yes—
445C22×C4#115912yes—
446C22×C4#115912yes—
447C22×C4C2×C22⋊D712yes—
448C22×C4C23×D711yescharacteristic
449M16—148no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—44no—
457C2×C8—74no—
458C2×C8—142no—
459C2×C8—142no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—74no—
465C4⋊C4—74no—
466C4⋊C4—148no—
467C4⋊C4—144no—
468C4⋊C4—144no—
469C4⋊C4—142no—
470C4⋊C4—142no—
471C4⋊C4—142no—
472C4⋊C4—142no—
473C4⋊C4—562no—
474C4⋊C4—562no—
475C22⋊C4—42no—
476C22⋊C4—148no—
477C22⋊C4—144no—
478C42—42no—
479C42—148no—
480C42—142no—
481C42—142no—
482C2×C7—24no—
483C2×C7—44no—
484C2×C7C2×#38511yescharacteristic
485C2×C7#294611yescharacteristic
486C2×C7#294711yescharacteristic
487C2×C7#301211yescharacteristic
488C2×C7#301311yescharacteristic
489C2×C7#308311yescharacteristic
490C2×C7#308411yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23#644112yes—
503C23(C22⋊D7)⋊C412yes—
504C23#650012yes—
505C23C2×D4×D711yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8C2×Dic7⋊C414yes—
510D4—216no—
511D4—22no—
512D4—22no—
513D4—22no—
514D4C2×Dic7⋊C414yes—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—216no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—144no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—142no—
529C2×C4—142no—
530C2×C4—142no—
531C2×C4—144no—
532C2×C4—142no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4—282no—
537C2×C4—564no—
538C2×C4—564no—
539C2×C4—568no—
540C2×C4#644114yes—
541C2×C4#644112yes—
542C2×C4(C22⋊D7)⋊C414yes—
543C2×C4(C22⋊D7)⋊C412yes—
544C2×C4C2×Dic7⋊C414yes—
545C2×C4#649814yes—
546C2×C4#650012yes—
547C2×C4C22×Dic1411yescharacteristic
548C2×C4C22×C4×D711yescharacteristic
549C2×C4C2×D4×D711yescharacteristic
550C2×C4C2×#115911yescharacteristic
551C2×C4C2×#115911yescharacteristic
552C2×C4C22×C22⋊D711yescharacteristic
553C8—41no—
554C8—41no—
555C8—144no—
556C7#2087811yescharacteristic, complemented, Sylow
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#8971814yes—
571C22C2×#644111yescharacteristic
572C22C2×(C22⋊D7)⋊C411yescharacteristic
573C22#9046011yescharacteristic
574C22#9046111yescharacteristic
575C22#9046111yescharacteristic
576C22#9046211yescharacteristic
577C22C2×#650011yescharacteristic
578C4—24no—
579C4—81no—
580C4—81no—
581C4—282no—
582C4—284no—
583C4—282no—
584C4—284no—
585C4#8971814yes—
586C4C2×#644111yescharacteristic
587C4C2×(C22⋊D7)⋊C411yescharacteristic
588C4C22×Dic7⋊C411yescharacteristic
589C4C2×#649811yescharacteristic
590C2—24no—
591C2—44no—
592C2#1173626511yescharacteristic
593C2#1173626611yescharacteristic
594C2#1173626811yescharacteristic
595C2#1173626911yescharacteristic
596C2C2×#8971811yescharacteristic
597C2#1174135711yescharacteristic
598C2#1174135811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#416C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#416D7
hypocenterC7#20878
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
4. #11739677C2semidirect
5. C2×#89767C2semidirect
6. C2×#89346C2semidirect
7. #11729719C2semidirect
8. #11727991C2semidirect
9. #11727989C2semidirect
10. #11727923C2semidirect
11. #11727921C2semidirect
31. #90004C4semidirect
88. C2×C4⋊Dic7C2×C4semidirect
100. C2×#1136C2×C4semidirect
106. #6409C2×C4semidirect
107. #6406C2×C4semidirect
128. C7×#41C4⋊C4semidirect
406. #41Dic7⋊C4semidirect
556. C7#20878semidirect

Maximal subgroups

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

subgroupindex
C2×#905182
C2×C7×#4162
C2×#900042
#117396772
C2×#897672
C2×#893462
#117297192
#117279912
#117279892
#117279232
#117279212
#208787