Finite Group Databasebeta

← C2×#11736362 #14221304

Subgroups — C2×#11736362 #14221304

Order 1792.

2530 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic7C211yesmaximal, characteristic
2C2×#90208C211yesmaximal, characteristic, complemented
3C2×C7×#418C211yesmaximal, characteristic
4C2×#89987C211yesmaximal, characteristic
5C2×#89893C211yesmaximal, characteristic, complemented
6C2×#89890C211yesmaximal, characteristic, complemented
7C2×#89767C211yesmaximal, characteristic, complemented
8#11736362C2116yesmaximal, direct product factor
9C2×#89535C212yesmaximal, complemented
10C2×#89538C212yesmaximal, complemented
11C2×#89533C212yesmaximal, complemented
12C2×#89531C212yesmaximal, complemented
13C23×C7×D4C2211yescharacteristic
14C24×Dic7—21no—
15C22×C7×M16C2211yescharacteristic
16C2×C7×#117C2211yescharacteristic
17C22×C22⋊Dic7—21no—
18C2×D4×Dic7—21no—
19C2×D4×Dic7—21no—
20C2×D4×Dic7—24no—
21C2×D4×Dic7C2218yes—
22C2×D4×Dic7C2212yes—
23C2×D4×Dic7C2212yes—
24C2×#6489C2211yescharacteristic
25C22×C4⋊Dic7C2211yescharacteristic
26C22×C4×Dic7C2211yescharacteristic
27C22×#1136C2211yescharacteristic
28#90208—24no—
29#90208C2218yescomplemented
30C2×#6453C2211yescharacteristic, complemented
31C2×#6453C2211yescharacteristic, complemented
32C2×#6451C2211yescharacteristic, complemented
33C2×#6451C2211yescharacteristic, complemented
34C2×#6441—21no—
35C2×#6441—21no—
36C2×C4×Dic14—21no—
37C2×C4×Dic14—21no—
38C7×#418C2218yes—
39C2×C7×#102C2212yes—
40C2×C7×#102C2212yes—
41#89987C2218yes—
42C2×#6406C2212yes—
43C2×#6406C2212yes—
44#89893C2218yescomplemented
45#89890C2218yescomplemented
46C2×#6392C2212yescomplemented
47C2×#6391C2212yescomplemented
48C2×Dic7⋊C8C2212yescomplemented
49C2×Dic7⋊C8C2212yescomplemented
50#89767C2218yescomplemented
51C2×#6382C2212yescomplemented
52C2×#6381C2212yescomplemented
53#89538C22116yescomplemented
54#89535C22116yescomplemented
55#89533C22116yescomplemented
56#89531C22116yescomplemented
57C2×#3077—71nomaximal, Sylow
58C25×C7—21no—
59C22×C7×D4—21no—
60C22×C7×D4—22no—
61C22×C7×D4—24no—
62C22×C7×D4Q818yescomplemented
63C22×C7×D4C2312yes—
64C22×C7×D4C2314yes—
65C23×C4×C7C2311yescharacteristic
66C23×Dic7—21no—
67C23×Dic7—28no—
68C23×Dic7—22no—
69C23×Dic7—22no—
70C23×Dic7—48no—
71C23×Dic7D411yescharacteristic
72C23×Dic7D411yescharacteristic
73C22×Dic14—21no—
74C22×Dic14—21no—
75C2×C7×M16—24no—
76C2×C7×M16C2314yes—
77C22×C7×C8C2312yes—
78C7×#117—22no—
79C7×#117C2314yes—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—21no—
83C2×C42×C7—21no—
84C2×C22⋊Dic7—21no—
85C2×C22⋊Dic7—22no—
86C2×C22⋊Dic7—22no—
87C2×C22⋊Dic7—28no—
88D4×Dic7—28no—
89D4×Dic7—28no—
90D4×Dic7—24no—
91D4×Dic7—24no—
92D4×Dic7C2318yes—
93D4×Dic7C2318yes—
94#6489—22no—
95#6489C2314yes—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—22no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4⋊Dic7C2312yes—
102C2×C4⋊Dic7C2311yescharacteristic
103C2×C4⋊Dic7C2314yes—
104C2×Dic7⋊C4—21no—
105C2×Dic7⋊C4—21no—
106C2×Dic7⋊C4—21no—
107C2×Dic7⋊C4—21no—
108C2×C4×Dic7—21no—
109C2×C4×Dic7D411yescharacteristic
110C2×C4×Dic7D411yescharacteristic
111C2×C4×Dic7D414yes—
112C2×C4×Dic7C2311yescharacteristic
113C2×C4×Dic7C2311yescharacteristic
114C2×C4×Dic7C2314yes—
115C2×#1136—24no—
116C2×#1136C2314yes—
117C22×C7⋊C8C2312yes—
118#6453—22no—
119#6453—22no—
120#6453C2314yescomplemented
121#6453C2314yescomplemented
122#6451—22no—
123#6451—22no—
124#6451C2314yescomplemented
125#6451C2314yescomplemented
126#6441—28no—
127#6441—28no—
128C4×Dic14—28no—
129C4×Dic14—28no—
130C7×#102C2318yes—
131C7×#102C2318yes—
132#6406C2318yes—
133#6406C2318yes—
134#6392C2318yescomplemented
135#6391C2318yescomplemented
136Dic7⋊C8C2318yescomplemented
137Dic7⋊C8C2318yescomplemented
138#6382C2318yescomplemented
139#6381C2318yescomplemented
140C2×#533—71no—
141C22×C4×D4—71no—
142#3077—716no—
143C2×#478—72no—
144C2×#478—72no—
145C2×#476—72no—
146C2×#474—72no—
147C2×#428—71no—
148C2×#428—71no—
149C2×#423—71no—
150C2×#418—71no—
151C2×#418D711yescharacteristic
152C24×C7—28no—
153C24×C7—22no—
154C24×C7—24no—
155C24×C7—44no—
156C24×C7—44no—
157C24×C7C2×D411yescharacteristic
158C2×C7×D4—216no—
159C2×C7×D4—28no—
160C2×C7×D4—24no—
161C2×C7×D4—22no—
162C2×C7×D4—24no—
163C2×C7×D4—28no—
164C2×C7×D4C2×Q818yes—
165C2×C7×D4C2×Q8116yescomplemented
166C2×C7×D4C2414yes—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×Q814yes—
172C22×C4×C7#4114yes—
173C22×C4×C7C2411yescharacteristic
174C22×C4×C7C2412yes—
175C22×Dic7—24no—
176C22×Dic7—21no—
177C22×Dic7—21no—
178C22×Dic7—24no—
179C22×Dic7—22no—
180C22×Dic7—28no—
181C22×Dic7—28no—
182C22×Dic7—42no—
183C22×Dic7—48no—
184C22×Dic7—432no—
185C22×Dic7—44no—
186C22×Dic7—48no—
187C22×Dic7C2×D414yes—
188C22×Dic7C2×D411yescharacteristic
189C22×Dic7C2×D411yescharacteristic
190C22×Dic7C2×D411yescharacteristic
191C22×Dic7C2×D414yes—
192C22×Dic7C2×D411yescharacteristic
193C2×Dic14—24no—
194C2×Dic14—24no—
195C2×Dic14—44no—
196C2×Dic14—44no—
197C7×M16—28no—
198C2×C7×C8—24no—
199C2×C7×C8C2414yes—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4C2412yes—
203C7×C4⋊C4C2412yes—
204C7×C22⋊C4—24no—
205C42×C7—24no—
206C22⋊Dic7—24no—
207C22⋊Dic7—28no—
208C22⋊Dic7—28no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—24no—
212C4⋊Dic7—22no—
213C4⋊Dic7—24no—
214C4⋊Dic7—24no—
215C4⋊Dic7C2412yes—
216C4⋊Dic7C2412yes—
217C4⋊Dic7C2414yes—
218Dic7⋊C4—24no—
219Dic7⋊C4—24no—
220Dic7⋊C4—24no—
221Dic7⋊C4—24no—
222C4×Dic7—24no—
223C4×Dic7C2×D414yes—
224C4×Dic7C2×D414yes—
225C4×Dic7C2×D412yes—
226C4×Dic7C2×D412yes—
227C4×Dic7C2412yes—
228C4×Dic7C2412yes—
229#1136—28no—
230C2×C7⋊C8—24no—
231C2×C7⋊C8C2414yes—
232C23×D4C2×D711yescharacteristic
233C24×C4—141no—
234C22×M16—71no—
235C22×M16C2×D711yescharacteristic
236#533—78no—
237#533—144no—
238C2×C4⋊Q8—71no—
239C2×C4⋊Q8—71no—
240C2×#125—71no—
241C2×#125—71no—
242C2×C22⋊Q8—141no—
243C2×C22⋊Q8—141no—
244C2×C4×Q8—141no—
245C2×C4×Q8—141no—
246C2×C4×D4—72no—
247C2×C4×D4—78no—
248C2×C4×D4—72no—
249C2×C4×D4—141no—
250C2×C4×D4—141no—
251C2×C4×D4—144no—
252C2×#117—71no—
253C2×#117C2×D711yescharacteristic
254C22×C4⋊C4—71no—
255C22×C22⋊C4—141no—
256C22×C42—71no—
257#478—716no—
258#478—716no—
259#476—716no—
260#474—716no—
261#428—78no—
262#428—78no—
263C2×#107—72no—
264C2×#107—72no—
265C2×#106—72no—
266C2×#106—72no—
267#423—78no—
268C2×C4⋊C8—72no—
269C2×C4⋊C8—72no—
270#418—78no—
271#418C2×D718yes—
272C2×#102—72no—
273C2×#102—72no—
274C2×#102C2×D712yes—
275C2×#102C2×D712yes—
276C23×C7—22no—
277C23×C7—22no—
278C23×C7—28no—
279C23×C7—216no—
280C23×C7—24no—
281C23×C7—48no—
282C23×C7—432no—
283C23×C7—48no—
284C23×C7—44no—
285C23×C7—44no—
286C23×C7C22⋊Q814yes—
287C23×C7C22×D412yes—
288C23×C7C22×D411yescharacteristic
289C7×D4—28no—
290C7×D4—216no—
291C7×D4—24no—
292C7×D4C22×Q818yes—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7C22⋊Q818yes—
298C2×C4×C7C22⋊Q814yes—
299C2×C4×C7C22×D412yes—
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×Q812yes—
302C2×C4×C7C22×Q814yes—
303C2×C4×C7C2×#4112yes—
304C2×C4×C7C2×#4114yes—
305C2×C4×C7C2511yescharacteristic
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—28no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—24no—
313C2×Dic7—44no—
314C2×Dic7—416no—
315C2×Dic7—48no—
316C2×Dic7—42no—
317C2×Dic7—44no—
318C2×Dic7—42no—
319C2×Dic7—416no—
320C2×Dic7AGL(1,ℤ/8)14yes—
321C2×Dic7C22×D412yes—
322C2×Dic7C22×D412yes—
323C2×Dic7C22×D412yes—
324C2×Dic7C22×D412yes—
325Dic14—48no—
326Dic14—48no—
327C7×C8—24no—
328C7⋊C8—24no—
329C25—21no—
330C22×Q8—141no—
331C22×Q8—141no—
332C22×D4—22no—
333C22×D4—21no—
334C22×D4—24no—
335C22×D4Dic1418yescomplemented
336C22×D4C22×D712yes—
337C22×D4C22×D714yes—
338C23×C4—71no—
339C23×C4—71no—
340C23×C4—142no—
341C23×C4—148no—
342C23×C4—142no—
343C23×C4—141no—
344C23×C4—288no—
345C23×C4C22×D711yescharacteristic
346C2×M16—24no—
347C2×M16—74no—
348C2×M16—144no—
349C2×M16C22×D714yes—
350C22×C8—72no—
351C22×C8C22×D712yes—
352C4⋊Q8—74no—
353C4⋊Q8—74no—
354C4⋊Q8—142no—
355C4⋊Q8—142no—
356#125—74no—
357#125—74no—
358#125—142no—
359#125—142no—
360C22⋊Q8—148no—
361C22⋊Q8—148no—
362C4×Q8—148no—
363C4×Q8—148no—
364C4×D4—78no—
365C4×D4—78no—
366C4×D4—148no—
367C4×D4—148no—
368C4×D4—144no—
369C4×D4—144no—
370#117—22no—
371#117—74no—
372#117—142no—
373#117C22×D714yes—
374C2×C4⋊C4—72no—
375C2×C4⋊C4—74no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—142no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4C22×D711yescharacteristic
387C2×C4⋊C4C22×D711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—142no—
390C2×C22⋊C4—142no—
391C2×C22⋊C4—148no—
392C2×C22⋊C4—141no—
393C2×C42—21no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—74no—
397C2×C42—71no—
398C2×C42—71no—
399C2×C42—74no—
400C2×C42—141no—
401#107—78no—
402#107—78no—
403#106—78no—
404#106—78no—
405C4⋊C8—78no—
406C4⋊C8—78no—
407#102—78no—
408#102—78no—
409#102C22×D718yes—
410#102C22×D718yes—
411C22×C7—28no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—28no—
415C22×C7—432no—
416C22×C7—416no—
417C22×C7—44no—
418C22×C7—44no—
419C22×C7C2×C22⋊Q814yes—
420C22×C7C2×C22⋊Q812yes—
421C22×C7C2×AGL(1,ℤ/8)12yes—
422C22×C7#57512yes—
423C22×C7C23×D411yescharacteristic
424C4×C7—42no—
425C4×C7—42no—
426C4×C7C2×C22⋊Q814yes—
427C4×C7C2×C22⋊Q812yes—
428C4×C7C23×Q811yescharacteristic
429C4×C7C22×#4111yescharacteristic
430Dic7—22no—
431Dic7—22no—
432Dic7—22no—
433Dic7—42no—
434Dic7—44no—
435Dic7—42no—
436Dic7C2×AGL(1,ℤ/8)14yes—
437C24—28no—
438C24—24no—
439C24—22no—
440C24—44no—
441C24—44no—
442C24D4×D711yescharacteristic
443C2×Q8—144no—
444C2×Q8—144no—
445C2×Q8—284no—
446C2×Q8—284no—
447C2×D4—216no—
448C2×D4—28no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—28no—
453C2×D4C2×Dic1418yes—
454C2×D4C2×Dic14116yescomplemented
455C2×D4C23×D714yes—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—71no—
459C22×C4—74no—
460C22×C4—71no—
461C22×C4—71no—
462C22×C4—71no—
463C22×C4—74no—
464C22×C4—144no—
465C22×C4—148no—
466C22×C4—148no—
467C22×C4—144no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—142no—
471C22×C4—288no—
472C22×C4—282no—
473C22×C4—2832no—
474C22×C4—288no—
475C22×C4—284no—
476C22×C4C2×Dic1414yes—
477C22×C4D4×D712yes—
478C22×C4D4×D711yescharacteristic
479C22×C4#115914yes—
480C22×C4C23×D712yes—
481C22×C4C23×D711yescharacteristic
482M16—28no—
483M16—148no—
484C2×C8—24no—
485C2×C8—74no—
486C2×C8—144no—
487C2×C8C23×D714yes—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—72no—
491C4⋊C4—72no—
492C4⋊C4—74no—
493C4⋊C4—144no—
494C4⋊C4—144no—
495C4⋊C4—144no—
496C4⋊C4—144no—
497C4⋊C4—144no—
498C4⋊C4—144no—
499C4⋊C4—144no—
500C4⋊C4—142no—
501C4⋊C4—141no—
502C4⋊C4—141no—
503C4⋊C4C23×D712yes—
504C4⋊C4C23×D712yes—
505C22⋊C4—24no—
506C22⋊C4—148no—
507C22⋊C4—148no—
508C22⋊C4—144no—
509C42—24no—
510C42—74no—
511C42—74no—
512C42—72no—
513C42—72no—
514C42—72no—
515C42—72no—
516C42—144no—
517C2×C7—24no—
518C2×C7—48no—
519C2×C7#307714yescomplemented
520C2×C7C22×C22⋊Q811yescharacteristic
521C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
522C2×C7C2×#57511yescharacteristic
523C23—22no—
524C23—22no—
525C23—28no—
526C23—216no—
527C23—24no—
528C23—48no—
529C23—48no—
530C23—432no—
531C23—44no—
532C23—44no—
533C23#644114yes—
534C23C2×D4×D712yes—
535C23C2×D4×D711yescharacteristic
536Q8—288no—
537Q8—288no—
538D4—28no—
539D4—216no—
540D4—24no—
541D4C22×Dic1418yes—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—74no—
547C2×C4—72no—
548C2×C4—72no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—148no—
552C2×C4—142no—
553C2×C4—142no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—144no—
558C2×C4—2816no—
559C2×C4—284no—
560C2×C4—2816no—
561C2×C4—288no—
562C2×C4—284no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4#644118yes—
566C2×C4#644114yes—
567C2×C4C22×Dic1412yes—
568C2×C4C22×Dic1414yes—
569C2×C4C2×D4×D712yes—
570C2×C4C2×D4×D711yescharacteristic
571C2×C4C2×#115912yes—
572C2×C4C2×#115914yes—
573C2×C4C24×D711yescharacteristic
574C8—24no—
575C8—144no—
576C7C2×#307711yescharacteristic, complemented, Sylow
577C22—22no—
578C22—22no—
579C22—28no—
580C22—28no—
581C22—432no—
582C22—44no—
583C22—44no—
584C22—416no—
585C22C2×#644114yes—
586C22C2×#644112yes—
587C22D7×AGL(1,ℤ/8)12yes—
588C22#9046412yes—
589C22C22×D4×D711yescharacteristic
590C4—42no—
591C4—42no—
592C4—74no—
593C4—142no—
594C4—142no—
595C4—142no—
596C4—284no—
597C4—282no—
598C4—282no—
599C4C2×#644112yes—
600C4C2×#644114yes—
601C4C23×Dic1411yescharacteristic
602C4C22×#115911yescharacteristic
603C2—24no—
604C2—48no—
605C2#1173636214yesdirect product factor
606C2C22×#644111yescharacteristic
607C2C2×D7×AGL(1,ℤ/8)11yescharacteristic
608C2C2×#9046411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#418C2
socleC23×C7C22×D4
prime radicalC23×C7×D4C22
hypercenterC2×#418D7
hypocenterC7C2×#3077
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×#90208C2semidirect
5. C2×#89893C2semidirect
6. C2×#89890C2semidirect
7. C2×#89767C2semidirect
8. #11736362C2direct (N × H)
9. C2×#89535C2semidirect
10. C2×#89538C2semidirect
11. C2×#89533C2semidirect
12. C2×#89531C2semidirect
29. #90208C22semidirect
30. C2×#6453C22semidirect
31. C2×#6453C22semidirect
32. C2×#6451C22semidirect
33. C2×#6451C22semidirect
44. #89893C22semidirect
45. #89890C22semidirect
46. C2×#6392C22semidirect
47. C2×#6391C22semidirect
48. C2×Dic7⋊C8C22semidirect
49. C2×Dic7⋊C8C22semidirect
50. #89767C22semidirect
51. C2×#6382C22semidirect
52. C2×#6381C22semidirect
53. #89538C22semidirect
54. #89535C22semidirect
55. #89533C22semidirect
56. #89531C22semidirect
62. C22×C7×D4Q8semidirect
120. #6453C23semidirect
121. #6453C23semidirect
124. #6451C23semidirect
125. #6451C23semidirect
134. #6392C23semidirect
135. #6391C23semidirect
136. Dic7⋊C8C23semidirect
137. Dic7⋊C8C23semidirect
138. #6382C23semidirect
139. #6381C23semidirect
165. C2×C7×D4C2×Q8semidirect
335. C22×D4Dic14semidirect
454. C2×D4C2×Dic14semidirect
519. C2×C7#3077semidirect
576. C7C2×#3077semidirect
605. C2#11736362direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Dic72
C2×#902082
C2×C7×#4182
C2×#899872
C2×#898932
C2×#898902
C2×#897672
#117363622
C2×#895352
C2×#895382
C2×#895332
C2×#895312
C2×#30777