Finite Group Databasebeta

← C2×#11726756 #13354653

Subgroups — C2×#11726756 #13354653

Order 1792.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90195C211yesmaximal, characteristic
2C22×#6399C211yesmaximal, characteristic
3C2×#89905C211yesmaximal, characteristic
4C2×#89893C211yesmaximal, characteristic, complemented
5C2×#89748C211yesmaximal, characteristic, complemented
6C2×C7×#343C211yesmaximal, characteristic, complemented
7C2×#89350C211yesmaximal, characteristic, complemented
8#11726756C218yesmaximal, direct product factor
9C22×#1157C2211yescharacteristic
10C22×C7×M16C2211yescharacteristic
11C22×C7×M16C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C2×C4×C22⋊D7—22no—
14C2×#6489C2211yescharacteristic
15C22×C4⋊Dic7C2211yescharacteristic
16C22×Dic7⋊C4—22no—
17C22×C4×Dic7C2211yescharacteristic
18#90195—22no—
19#90195—22no—
20#90195—22no—
21#90195C2214yes—
22C2×#6456—21no—
23C2×#6455—21no—
24C2×D7×C4⋊C4—21no—
25C2×#6450—21no—
26C2×#6399C418yescomplemented
27C2×#6399C2214yes—
28#89905—22no—
29#89905C2214yes—
30#89893—22no—
31#89893C2214yescomplemented
32C2×#6395—21no—
33C2×#6392—21no—
34C2×#6391—21no—
35C2×#6390—21no—
36#89748—22no—
37#89748C2214yescomplemented
38C7×#343C2214yescomplemented
39#89350C2214yescomplemented
40C2×#1859—71nomaximal, Sylow
41C23×C4×C7D412yes—
42C23×C4×C7C2311yescharacteristic
43C22×C22⋊D7—21no—
44C23×Dic7—21no—
45C23×Dic7—21no—
46C2×#1157—21no—
47C2×#1157—21no—
48C2×#1157—22no—
49C2×#1157—22no—
50C2×#1157C2×C414yes—
51C2×#1157D412yes—
52C2×#1157Q814yes—
53C2×#1157C2312yes—
54C22×D28D411yescharacteristic
55C22×C4×D7—21no—
56C22×C4×D7—22no—
57C22×Dic14D411yescharacteristic
58C2×C7×M16—21no—
59C2×C7×M16—22no—
60C2×C7×M16—21no—
61C2×C7×M16—22no—
62C2×C7×M16—22no—
63C2×C7×M16C2×C414yescomplemented
64C2×C7×M16C2312yes—
65C2×C7×M16C2312yes—
66C22×C7×C8—21no—
67C22×C7×C8—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4D412yes—
73C2×C7×C4⋊C4C2312yes—
74C2×C22⋊Dic7—21no—
75C4×C22⋊D7—28no—
76C4×C22⋊D7—28no—
77C2×(C2×D7)⋊C4—22no—
78#6489—21no—
79#6489—21no—
80#6489D412yes—
81#6489C2312yes—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7D411yescharacteristic
88C2×C4⋊Dic7D411yescharacteristic
89C2×C4⋊Dic7C2312yes—
90C2×Dic7⋊C4—22no—
91C2×Dic7⋊C4—28no—
92C2×Dic7⋊C4—48no—
93C2×C4×Dic7—22no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—22no—
97C2×C4×Dic7C2×C414yescomplemented
98C2×C4×Dic7C2312yes—
99#6456—24no—
100#6456—24no—
101#6455—24no—
102#6455—24no—
103D7×C4⋊C4—24no—
104D7×C4⋊C4—24no—
105#6450—24no—
106#6450—24no—
107#6399C2×C4116yescomplemented
108#6395—24no—
109#6392—24no—
110#6391—24no—
111#6390—24no—
112C2×#520—71no—
113C2×#428—71no—
114C22×#104—71no—
115C2×#417—71no—
116C2×#384—71no—
117#1859—78no—
118C2×#343D711yescharacteristic, complemented
119C2×#339—71no—
120C24×C7C2×D411yescharacteristic
121C23×D7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—44no—
125C22×C4×C7C22×C412yes—
126C22×C4×C7C2×D414yes—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×Q812yes—
133C22×C4×C7#4112yes—
134C22×C4×C7#4112yes—
135C22×C4×C7C2411yescharacteristic
136C2×C22⋊D7—24no—
137C2×C22⋊D7—22no—
138C2×C22⋊D7—24no—
139C2×C22⋊D7—22no—
140C22×Dic7—24no—
141C22×Dic7—24no—
142C22×Dic7—24no—
143C22×Dic7—21no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—22no—
148C22×Dic7—44no—
149C22×Dic7—41no—
150C22×Dic7—41no—
151#1157—24no—
152#1157—24no—
153#1157—24no—
154#1157—24no—
155#1157—24no—
156#1157—22no—
157#1157—22no—
158#1157C4⋊C418yescomplemented
159#1157C22×C414yes—
160#1157C2×Q814yes—
161C2×D28—21no—
162C2×D28—22no—
163C2×D28C4⋊C414yescomplemented
164C2×D28C2×D412yes—
165C2×C4×D7—24no—
166C2×C4×D7—22no—
167C2×C4×D7—24no—
168C2×C4×D7—22no—
169C2×C4×D7—28no—
170C2×C4×D7—84no—
171C2×Dic14—21no—
172C2×Dic14—22no—
173C2×Dic14C4⋊C414yescomplemented
174C2×Dic14C2×D412yes—
175C7×M16—24no—
176C7×M16—22no—
177C7×M16—22no—
178C7×M16—22no—
179C7×M16—42no—
180C7×M16C22×C414yescomplemented
181C2×C7×C8—24no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—42no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C22⋊Dic7—22no—
190C22⋊Dic7—22no—
191(C2×D7)⋊C4—28no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—22no—
195C4⋊Dic7—22no—
196C4⋊Dic7—41no—
197C4⋊Dic7—41no—
198C4⋊Dic7C2×D412yes—
199C4⋊Dic7C2×D412yes—
200Dic7⋊C4—28no—
201Dic7⋊C4—416no—
202C4×Dic7—28no—
203C4×Dic7—24no—
204C4×Dic7—22no—
205C4×Dic7—22no—
206C4×Dic7—22no—
207C4×Dic7C22×C414yescomplemented
208C22×#41—71no—
209C22×M16C2×D711yescharacteristic
210C22×M16C2×D711yescharacteristic
211#520—74no—
212#520—142no—
213#520—142no—
214#520—142no—
215C2×C4×Q8—141no—
216C2×C4×D4—141no—
217C2×C4×D4—142no—
218C2×#117—71no—
219C2×#117—141no—
220C22×C4⋊C4—71no—
221C22×C4⋊C4—141no—
222C22×C4⋊C4—142no—
223C22×C4⋊C4C2×D711yescharacteristic
224C22×C42—71no—
225#428—74no—
226#428—142no—
227C2×#107—141no—
228C2×#106—141no—
229C2×#104—78no—
230C2×#104—74no—
231#417—74no—
232#417—142no—
233C2×#103—141no—
234C2×#102—141no—
235#384—74no—
236#384—142no—
237#343C2×D714yescomplemented
238#339—74no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7C4×D412yes—
242C23×C7#12011yescharacteristic
243C23×C7#12011yescharacteristic
244C23×C7C22⋊Q812yes—
245C23×C7C22×D411yescharacteristic
246C22×D7—22no—
247C22×D7—24no—
248C22×D7—42no—
249C22×D7—42no—
250C2×C4×C7—24no—
251C2×C4×C7—44no—
252C2×C4×C7—44no—
253C2×C4×C7—82no—
254C2×C4×C7C2×C4⋊C412yes—
255C2×C4×C7C2×C4⋊C412yes—
256C2×C4×C7C4×D412yes—
257C2×C4×C7C4×D412yes—
258C2×C4×C7C4×D412yes—
259C2×C4×C7#12011yescharacteristic
260C2×C4×C7#12011yescharacteristic
261C2×C4×C7C22⋊Q812yes—
262C2×C4×C7C22⋊Q812yes—
263C2×C4×C7C22⋊Q812yes—
264C2×C4×C7#12312yes—
265C2×C4×C7#12312yes—
266C2×C4×C7C23×C411yescharacteristic
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C22×Q811yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C22⋊D7—24no—
273C22⋊D7—24no—
274C22⋊D7—28no—
275C2×Dic7—24no—
276C2×Dic7—216no—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—22no—
280C2×Dic7—24no—
281C2×Dic7—22no—
282C2×Dic7—22no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—44no—
286C2×Dic7—42no—
287C2×Dic7—82no—
288C2×Dic7—82no—
289D28—22no—
290D28—24no—
291D28C2×C4⋊C414yescomplemented
292C4×D7—24no—
293C4×D7—24no—
294C4×D7—28no—
295C4×D7—88no—
296Dic14—22no—
297Dic14—24no—
298Dic14C2×C4⋊C414yescomplemented
299C7×C8—24no—
300C7×C8—42no—
301C2×#41—74no—
302C2×#41—72no—
303C2×#41—74no—
304C2×#41—72no—
305C2×#41—141no—
306C2×#41—142no—
307C2×#41—141no—
308C2×#41—142no—
309C22×Q8—71no—
310C22×D4—71no—
311C22×D4—141no—
312C23×C4—142no—
313C23×C4—141no—
314C23×C4—141no—
315C23×C4—141no—
316C23×C4D2812yes—
317C23×C4C22×D711yescharacteristic
318C2×M16—21no—
319C2×M16—22no—
320C2×M16—21no—
321C2×M16—22no—
322C2×M16—22no—
323C2×M16C4×D714yescomplemented
324C2×M16C22×D712yes—
325C2×M16C22×D712yes—
326C22×C8—21no—
327C22×C8—21no—
328C4×Q8—144no—
329C4×Q8—144no—
330C4×D4—144no—
331C4×D4—144no—
332C4×D4—148no—
333C4×D4—148no—
334#117—72no—
335#117—72no—
336#117—144no—
337#117—144no—
338#117—141no—
339#117—141no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—72no—
347C2×C4⋊C4—144no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4—142no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—148no—
356C2×C4⋊C4—288no—
357C2×C4⋊C4D2812yes—
358C2×C4⋊C4C22×D712yes—
359C2×C22⋊C4—142no—
360C2×C22⋊C4—141no—
361C2×C42—74no—
362C2×C42—72no—
363C2×C42—142no—
364C2×C42—141no—
365C2×C42—142no—
366C2×C42—141no—
367#107—144no—
368#106—144no—
369#104—716no—
370#103—144no—
371#102—144no—
372C22×C7—24no—
373C22×C7—24no—
374C22×C7—21no—
375C22×C7—21no—
376C22×C7—21no—
377C22×C7—21no—
378C22×C7#38512yes—
379C22×C7#38512yes—
380C22×C7#45312yes—
381C22×C7#45612yes—
382C22×C7C2×C4×D411yescharacteristic
383C22×C7C2×#12011yescharacteristic
384C22×C7C2×C22⋊Q811yescharacteristic
385C2×D7—24no—
386C2×D7—48no—
387C2×D7—42no—
388C2×D7—42no—
389C4×C7—44no—
390C4×C7#38512yes—
391C4×C7#38512yes—
392C4×C7C22×C4⋊C411yescharacteristic
393C4×C7C2×C4×D411yescharacteristic
394C4×C7C2×C22⋊Q811yescharacteristic
395C4×C7C2×#12311yescharacteristic
396Dic7—28no—
397Dic7—24no—
398Dic7—42no—
399Dic7—42no—
400Dic7—82no—
401C24—141no—
402C24C2×D2811yescharacteristic
403#41—78no—
404#41—74no—
405#41—74no—
406#41—144no—
407#41—144no—
408#41—144no—
409#41—144no—
410#41—144no—
411#41—142no—
412#41—142no—
413C2×Q8—74no—
414C2×Q8—72no—
415C2×Q8—141no—
416C2×Q8—142no—
417C2×D4—74no—
418C2×D4—72no—
419C2×D4—144no—
420C2×D4—142no—
421C2×D4—144no—
422C2×D4—142no—
423C2×D4—141no—
424C2×D4—142no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—148no—
429C22×C4—144no—
430C22×C4—144no—
431C22×C4—144no—
432C22×C4—141no—
433C22×C4—142no—
434C22×C4—144no—
435C22×C4—142no—
436C22×C4—144no—
437C22×C4—142no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—142no—
441C22×C4—284no—
442C22×C4—281no—
443C22×C4—281no—
444C22×C4—564no—
445C22×C4C2×C4×D712yes—
446C22×C4C2×D2814yes—
447C22×C4C2×D2811yescharacteristic
448C22×C4#115712yes—
449C22×C4D4×D711yescharacteristic
450C22×C4D4×D711yescharacteristic
451C22×C4D4×D711yescharacteristic
452C22×C4D4×D711yescharacteristic
453C22×C4#115912yes—
454C22×C4Q8×D712yes—
455C22×C4C23×D711yescharacteristic
456M16—24no—
457M16—22no—
458M16—22no—
459M16—22no—
460M16—42no—
461M16C2×C4×D714yescomplemented
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—42no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—72no—
471C4⋊C4—72no—
472C4⋊C4—148no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—142no—
476C4⋊C4—142no—
477C4⋊C4—2816no—
478C4⋊C4—281no—
479C4⋊C4—281no—
480C22⋊C4—148no—
481C22⋊C4—142no—
482C22⋊C4—142no—
483C42—74no—
484C42—148no—
485C42—144no—
486C42—142no—
487C42—142no—
488C42—142no—
489C2×C7—21no—
490C2×C7—21no—
491C2×C7—22no—
492C2×C7#185914yescomplemented
493C2×C7C2×#38511yescharacteristic
494C2×C7C2×#45311yescharacteristic
495C2×C7C2×#45611yescharacteristic
496D7—44no—
497C23—22no—
498C23—22no—
499C23—142no—
500C23—144no—
501C23—282no—
502C23—282no—
503C23C4×D2812yes—
504C23#644511yescharacteristic
505C23#644511yescharacteristic
506C23#646012yes—
507C23C22×D2811yescharacteristic
508Q8—74no—
509Q8—142no—
510Q8—144no—
511D4—74no—
512D4—144no—
513D4—144no—
514D4—148no—
515D4—142no—
516D4—144no—
517C2×C4—24no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—82no—
521C2×C4—144no—
522C2×C4—144no—
523C2×C4—1416no—
524C2×C4—144no—
525C2×C4—142no—
526C2×C4—144no—
527C2×C4—144no—
528C2×C4—144no—
529C2×C4—148no—
530C2×C4—142no—
531C2×C4—142no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4—284no—
535C2×C4—282no—
536C2×C4—568no—
537C2×C4—562no—
538C2×C4—562no—
539C2×C4C4×D2812yes—
540C2×C4(C22⋊D7)⋊C412yes—
541C2×C4(C22⋊D7)⋊C412yes—
542C2×C4#644511yescharacteristic
543C2×C4#644511yescharacteristic
544C2×C4#644612yes—
545C2×C4#644612yes—
546C2×C4D7×C4⋊C412yes—
547C2×C4D7×C4⋊C412yes—
548C2×C4#645912yes—
549C2×C4#645912yes—
550C2×C4#646012yes—
551C2×C4C22×C4×D711yescharacteristic
552C2×C4C2×#115711yescharacteristic
553C2×C4C2×D4×D711yescharacteristic
554C2×C4C2×D4×D711yescharacteristic
555C2×C4C2×#115911yescharacteristic
556C2×C4C2×Q8×D711yescharacteristic
557C8—24no—
558C8—42no—
559C7C2×#185911yescharacteristic, complemented, Sylow
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—144no—
567C22—282no—
568C22—282no—
569C22—288no—
570C22#8944012yes—
571C22#8944012yes—
572C22#8951912yes—
573C22#8952412yes—
574C22C2×C4×D2811yescharacteristic
575C22C2×#644511yescharacteristic
576C22C2×#646011yescharacteristic
577C4—44no—
578C4—148no—
579C4—144no—
580C4—282no—
581C4—282no—
582C4—562no—
583C4#8944012yes—
584C4#8944012yes—
585C4C2×(C22⋊D7)⋊C411yescharacteristic
586C4C2×#644611yescharacteristic
587C4C2×D7×C4⋊C411yescharacteristic
588C4C2×#645911yescharacteristic
589C2—21no—
590C2—21no—
591C2—22no—
592C2—284no—
593C2#1172675614yesdirect product factor
594C2C2×#8944011yescharacteristic
595C2C2×#8951911yescharacteristic
596C2C2×#8952411yescharacteristic
subgroupG/H
centerC23#6445
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#343C2
socleC23×C7#120
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenterC2×#343D7
hypocenterC7C2×#1859
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. C2×#89893C2semidirect
5. C2×#89748C2semidirect
6. C2×C7×#343C2semidirect
7. C2×#89350C2semidirect
8. #11726756C2direct (N × H)
26. C2×#6399C4semidirect
31. #89893C22semidirect
37. #89748C22semidirect
38. C7×#343C22semidirect
39. #89350C22semidirect
63. C2×C7×M16C2×C4semidirect
97. C2×C4×Dic7C2×C4semidirect
107. #6399C2×C4semidirect
118. C2×#343D7semidirect
158. #1157C4⋊C4semidirect
163. C2×D28C4⋊C4semidirect
173. C2×Dic14C4⋊C4semidirect
180. C7×M16C22×C4semidirect
207. C4×Dic7C22×C4semidirect
237. #343C2×D7semidirect
291. D28C2×C4⋊C4semidirect
298. Dic14C2×C4⋊C4semidirect
323. C2×M16C4×D7semidirect
461. M16C2×C4×D7semidirect
492. C2×C7#1859semidirect
559. C7C2×#1859semidirect
593. C2#11726756direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#901952
C22×#63992
C2×#899052
C2×#898932
C2×#897482
C2×C7×#3432
C2×#893502
#117267562
C2×#18597