Finite Group Databasebeta

← C2×#11739368 #14226004

Subgroups — C2×#11739368 #14226004

Order 1792.

1678 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90380C211yesmaximal, characteristic
2C2×#90388C211yesmaximal, characteristic
3C2×C7×#499C211yesmaximal, characteristic, complemented
4C2×#89837C211yesmaximal, characteristic, complemented
5C2×#89841C211yesmaximal, characteristic, complemented
6#11739368C2116yesmaximal, direct product factor
7C2×#89649C211yesmaximal, characteristic, complemented
8C2×#89647C211yesmaximal, characteristic, complemented
9C2×#89656C211yesmaximal, characteristic
10C2×#89660C211yesmaximal, characteristic
11C2×#89629C211yesmaximal, characteristic, complemented
12C2×#89626C211yesmaximal, characteristic, complemented
13C2×#89634C211yesmaximal, characteristic
14C2×#89639C211yesmaximal, characteristic
15C2×#89453C211yesmaximal, characteristic, complemented
16C2×#89462C211yesmaximal, characteristic
17C2×C7×#125C2211yescharacteristic
18C2×C7×#125C2211yescharacteristic
19#90388—24no—
20#90388C2218yes—
21#90380—24no—
22#90380C2218yes—
23C2×#6461—21no—
24C2×#6461—21no—
25C2×#6460—21no—
26C2×#6460C2211yescharacteristic
27C2×#6460C2211yescharacteristic
28C2×#6459—21no—
29C2×#6455C2211yescharacteristic
30C2×D7×C4⋊C4C2211yescharacteristic
31C2×#6453—21no—
32C2×#6453—21no—
33C2×#6453C2211yescharacteristic
34C2×#6452—21no—
35C2×#6451—21no—
36C2×#6451C2211yescharacteristic
37C2×#6450C2211yescharacteristic
38C2×#6450C2211yescharacteristic
39C2×#6435C2211yescharacteristic
40C2×#6432C2211yescharacteristic
41C2×#6432C2211yescharacteristic
42C7×#499C2218yescomplemented
43C2×C7×#107C2211yescharacteristic
44C2×C7×#107C2211yescharacteristic
45C2×C7×#106C2211yescharacteristic
46C2×C7×#106C2211yescharacteristic
47C2×C4×C7×C8C2211yescharacteristic
48C2×(C2×D7)⋊C8C2211yescharacteristic
49C2×(C2×D7)⋊C8C2211yescharacteristic
50C2×#6390C2211yescharacteristic
51C2×#6390C2211yescharacteristic
52C2×#6390C2211yescharacteristic
53C2×#6390C2211yescharacteristic
54C2×Dic7⋊C8C2211yescharacteristic
55C2×Dic7⋊C8C2211yescharacteristic
56#89841C2218yescomplemented
57#89837C2218yescomplemented
58C2×#6384C2211yescharacteristic
59C2×#6384C2211yescharacteristic
60C2×#6384C2211yescharacteristic
61C2×#6384C2211yescharacteristic
62C2×#6382C2211yescharacteristic
63C2×#6382C2211yescharacteristic
64C2×#6381C2211yescharacteristic
65C2×#6381C2211yescharacteristic
66C2×#6377C2211yescharacteristic
67#89660C2218yes—
68#89656C2218yes—
69#89649C2218yescomplemented
70#89647C2218yescomplemented
71#89639C2218yes—
72#89634C2218yes—
73#89629C2218yescomplemented
74#89626C2218yescomplemented
75#89462C2218yes—
76#89453C2218yescomplemented
77C2×#3242—71nomaximal, Sylow
78C22×C4×D7—21no—
79C22×C4×D7C2311yescharacteristic
80C22×Dic14—21no—
81C22×Dic14—21no—
82C22×Dic14C2311yescharacteristic
83C22×Dic14C2311yescharacteristic
84C22×C7×C8C2311yescharacteristic
85C22×C7×C8C2311yescharacteristic
86C7×#125—22no—
87C7×#125—22no—
88C7×#125C2314yes—
89C7×#125C2314yes—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C7×C4⋊C4C2311yescharacteristic
97C2×C7×C4⋊C4C2311yescharacteristic
98C2×C42×C7C2311yescharacteristic
99C2×(C2×D7)⋊C4—21no—
100C2×(C2×D7)⋊C4—21no—
101C2×(C2×D7)⋊C4—21no—
102C2×(C2×D7)⋊C4D411yescharacteristic
103C2×(C2×D7)⋊C4D411yescharacteristic
104C2×C4⋊Dic7—21no—
105C2×C4⋊Dic7—21no—
106C2×C4⋊Dic7—21no—
107C2×C4⋊Dic7—21no—
108C2×C4⋊Dic7—21no—
109C2×C4⋊Dic7—21no—
110C2×C4⋊Dic7C2311yescharacteristic
111C2×C4⋊Dic7C2311yescharacteristic
112C2×Dic7⋊C4—21no—
113C2×Dic7⋊C4—21no—
114C2×Dic7⋊C4—21no—
115C2×Dic7⋊C4—21no—
116C2×Dic7⋊C4—21no—
117C2×Dic7⋊C4D411yescharacteristic
118C2×Dic7⋊C4D411yescharacteristic
119C2×C4×Dic7—21no—
120C2×C4×Dic7—21no—
121C2×C4×Dic7—21no—
122C2×C4×Dic7C2311yescharacteristic
123C22×C7⋊C8C2311yescharacteristic
124C22×C7⋊C8C2311yescharacteristic
125#6461—28no—
126#6461—28no—
127#6460—22no—
128#6460—22no—
129#6460—28no—
130#6460C2314yes—
131#6460C2314yes—
132#6459—28no—
133#6455—22no—
134#6455C2314yes—
135D7×C4⋊C4—22no—
136D7×C4⋊C4C2314yes—
137#6453—28no—
138#6453—28no—
139#6453—22no—
140#6453C2314yes—
141#6452—28no—
142#6451—28no—
143#6451—22no—
144#6451C2314yes—
145#6450—22no—
146#6450—22no—
147#6450C2314yes—
148#6450C2314yes—
149#6435D414yes—
150#6435C2314yes—
151#6432—22no—
152#6432—22no—
153#6432C2314yes—
154#6432C2314yes—
155C7×#107C2314yes—
156C7×#107C2314yes—
157C7×#106C2314yes—
158C7×#106C2314yes—
159C4×C7×C8C2314yes—
160(C2×D7)⋊C8C2314yes—
161(C2×D7)⋊C8C2314yes—
162#6390C2314yes—
163#6390C2314yes—
164#6390C2314yes—
165#6390C2314yes—
166Dic7⋊C8C2314yes—
167Dic7⋊C8C2314yes—
168#6384C2314yes—
169#6384C2314yes—
170#6384C2314yes—
171#6384C2314yes—
172#6382C2314yes—
173#6382C2314yes—
174#6381C2314yes—
175#6381C2314yes—
176#6377C2314yes—
177C2×#544—71no—
178C2×#541—71no—
179#3242—716no—
180C2×#501—71no—
181C2×#499D711yescharacteristic, complemented
182C2×#491—71no—
183C2×#487—71no—
184C2×#484—71no—
185C2×#485—71no—
186C2×#485—71no—
187C2×#483—71no—
188C2×#479—71no—
189C2×#479—71no—
190C2×#477—71no—
191C2×#475—71no—
192C2×#433—71no—
193C23×D7C2×D411yescharacteristic
194C22×C4×C7—21no—
195C22×C4×C7—21no—
196C22×C4×C7—21no—
197C22×C4×C7—21no—
198C22×C4×C7C2×D411yescharacteristic
199C22×C4×C7C2×D411yescharacteristic
200C22×C4×C7C2411yescharacteristic
201C22×Dic7—21no—
202C22×Dic7—21no—
203C22×Dic7—21no—
204C22×Dic7—21no—
205C22×Dic7C2×D411yescharacteristic
206C22×Dic7C2×D411yescharacteristic
207C22×Dic7C2×D411yescharacteristic
208C2×C4×D7—22no—
209C2×C4×D7—22no—
210C2×C4×D7—24no—
211C2×C4×D7—82no—
212C2×C4×D7C2×D412yes—
213C2×C4×D7C2412yes—
214C2×Dic14—22no—
215C2×Dic14—22no—
216C2×Dic14—22no—
217C2×Dic14—22no—
218C2×Dic14—21no—
219C2×Dic14—21no—
220C2×Dic14—24no—
221C2×Dic14—24no—
222C2×Dic14—82no—
223C2×Dic14—82no—
224C2×Dic14C2412yes—
225C2×Dic14C2412yes—
226C2×C7×C8—22no—
227C2×C7×C8—22no—
228C2×C7×C8C2412yes—
229C2×C7×C8C2412yes—
230C7×C4⋊C4—21no—
231C7×C4⋊C4—21no—
232C7×C4⋊C4—21no—
233C7×C4⋊C4—24no—
234C7×C4⋊C4—24no—
235C7×C4⋊C4—21no—
236C7×C4⋊C4—24no—
237C7×C4⋊C4—24no—
238C7×C4⋊C4C2412yes—
239C7×C4⋊C4C2412yes—
240C7×C4⋊C4C2412yes—
241C7×C4⋊C4C2412yes—
242C42×C7C2×D412yes—
243C42×C7C2412yes—
244(C2×D7)⋊C4—24no—
245(C2×D7)⋊C4—24no—
246(C2×D7)⋊C4—24no—
247(C2×D7)⋊C4C2×D414yes—
248(C2×D7)⋊C4C2×D414yes—
249C4⋊Dic7—21no—
250C4⋊Dic7—21no—
251C4⋊Dic7—24no—
252C4⋊Dic7—24no—
253C4⋊Dic7—24no—
254C4⋊Dic7—24no—
255C4⋊Dic7—24no—
256C4⋊Dic7—24no—
257C4⋊Dic7C2412yes—
258C4⋊Dic7C2412yes—
259Dic7⋊C4—24no—
260Dic7⋊C4—24no—
261Dic7⋊C4—24no—
262Dic7⋊C4—24no—
263Dic7⋊C4—24no—
264Dic7⋊C4C2×D414yes—
265Dic7⋊C4C2×D414yes—
266C4×Dic7—24no—
267C4×Dic7—24no—
268C4×Dic7—24no—
269C4×Dic7C2×D412yes—
270C4×Dic7C2412yes—
271C2×C7⋊C8—22no—
272C2×C7⋊C8—22no—
273C2×C7⋊C8C2412yes—
274C2×C7⋊C8C2412yes—
275#544—78no—
276#544—144no—
277#541—78no—
278#541—144no—
279C2×C4⋊Q8—71no—
280C2×C4⋊Q8—71no—
281C2×C4⋊Q8—71no—
282C2×C4⋊Q8—141no—
283C2×#126—141no—
284C2×#126—141no—
285C2×#125—71no—
286C2×#125—141no—
287C2×#125—141no—
288C2×#125—141no—
289C2×#125C2×D711yescharacteristic
290C2×#125C2×D711yescharacteristic
291C2×C22⋊Q8—71no—
292C2×C22⋊Q8—71no—
293C2×C22⋊Q8—141no—
294C2×C22⋊Q8—141no—
295C2×C4×Q8—71no—
296C2×C4×Q8—71no—
297C2×#117—71no—
298C2×#117—71no—
299C22×C4⋊C4—71no—
300#501—78no—
301#499C2×D718yescomplemented
302#491—78no—
303#487—78no—
304#485—78no—
305#485—78no—
306#484—78no—
307#483—78no—
308#479—78no—
309#479—78no—
310#477—78no—
311#475—78no—
312#433—78no—
313C2×#107—71no—
314C2×#107—71no—
315C2×#107C2×D711yescharacteristic
316C2×#107C2×D711yescharacteristic
317C2×#106—71no—
318C2×#106—71no—
319C2×#106C2×D711yescharacteristic
320C2×#106C2×D711yescharacteristic
321C2×C4⋊C8—71no—
322C2×C4⋊C8—71no—
323C2×#103—71no—
324C2×#103—71no—
325C2×#103—71no—
326C2×#103—71no—
327C2×#103—71no—
328C2×#103—71no—
329C2×#103—71no—
330C2×#103—71no—
331C2×C22⋊C8—71no—
332C2×C22⋊C8—71no—
333C2×#97—71no—
334C2×C4×C8C2×D711yescharacteristic
335C23×C7C22×D411yescharacteristic
336C22×D7—22no—
337C22×D7—22no—
338C22×D7—22no—
339C22×D7C22×D412yes—
340C2×C4×C7—22no—
341C2×C4×C7—22no—
342C2×C4×C7—22no—
343C2×C4×C7—22no—
344C2×C4×C7—22no—
345C2×C4×C7—22no—
346C2×C4×C7—42no—
347C2×C4×C7—42no—
348C2×C4×C7—42no—
349C2×C4×C7—42no—
350C2×C4×C7C22×D412yes—
351C2×C4×C7C22×D412yes—
352C2×C4×C7C22×D411yescharacteristic
353C2×C4×C72−1+412yes—
354C2×C4×C72−1+412yes—
355C2×C4×C7C2511yescharacteristic
356C2×Dic7—22no—
357C2×Dic7—22no—
358C2×Dic7—22no—
359C2×Dic7—22no—
360C2×Dic7—22no—
361C2×Dic7—22no—
362C2×Dic7—22no—
363C2×Dic7—42no—
364C2×Dic7—42no—
365C2×Dic7—42no—
366C2×Dic7—42no—
367C2×Dic7C22×D412yes—
368C2×Dic7C22×D412yes—
369C2×Dic7C22×D412yes—
370C4×D7—24no—
371C4×D7—22no—
372C4×D7—22no—
373C4×D7—84no—
374Dic14—22no—
375Dic14—22no—
376Dic14—22no—
377Dic14—22no—
378Dic14—42no—
379Dic14—42no—
380Dic14—84no—
381Dic14—84no—
382C7×C8—22no—
383C7×C8—22no—
384C7⋊C8—22no—
385C7⋊C8—22no—
386C22×Q8—71no—
387C22×Q8—71no—
388C22×Q8—141no—
389C22×Q8—141no—
390C23×C4—71no—
391C23×C4—141no—
392C22×C8—71no—
393C22×C8—71no—
394C22×C8C22×D711yescharacteristic
395C22×C8C22×D711yescharacteristic
396C4⋊Q8—74no—
397C4⋊Q8—74no—
398C4⋊Q8—74no—
399C4⋊Q8—142no—
400C4⋊Q8—148no—
401C4⋊Q8—142no—
402C4⋊Q8—142no—
403#126—148no—
404#126—148no—
405#125—22no—
406#125—22no—
407#125—74no—
408#125—142no—
409#125—148no—
410#125—148no—
411#125—148no—
412#125C22×D714yes—
413#125C22×D714yes—
414C22⋊Q8—74no—
415C22⋊Q8—74no—
416C22⋊Q8—148no—
417C22⋊Q8—148no—
418C22⋊Q8—142no—
419C22⋊Q8—142no—
420C4×Q8—74no—
421C4×Q8—74no—
422C4×Q8—142no—
423C4×Q8—142no—
424#117—74no—
425#117—74no—
426#117—74no—
427#117—142no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—71no—
433C2×C4⋊C4—71no—
434C2×C4⋊C4—71no—
435C2×C4⋊C4—71no—
436C2×C4⋊C4—74no—
437C2×C4⋊C4—142no—
438C2×C4⋊C4—141no—
439C2×C4⋊C4—141no—
440C2×C4⋊C4—141no—
441C2×C4⋊C4—141no—
442C2×C4⋊C4—141no—
443C2×C4⋊C4—141no—
444C2×C4⋊C4—141no—
445C2×C4⋊C4—141no—
446C2×C4⋊C4—141no—
447C2×C4⋊C4—141no—
448C2×C4⋊C4—141no—
449C2×C4⋊C4C22×D711yescharacteristic
450C2×C4⋊C4C22×D711yescharacteristic
451C2×C4⋊C4C22×D711yescharacteristic
452C2×C4⋊C4C22×D711yescharacteristic
453C2×C22⋊C4—71no—
454C2×C22⋊C4—71no—
455C2×C22⋊C4—141no—
456C2×C22⋊C4—141no—
457C2×C22⋊C4—141no—
458C2×C42—71no—
459C2×C42—141no—
460C2×C42—141no—
461C2×C42—141no—
462C2×C42C22×D711yescharacteristic
463#107—74no—
464#107—74no—
465#107C22×D714yes—
466#107C22×D714yes—
467#106—74no—
468#106—74no—
469#106C22×D714yes—
470#106C22×D714yes—
471C4⋊C8—74no—
472C4⋊C8—74no—
473#103—74no—
474#103—74no—
475#103—74no—
476#103—74no—
477#103—74no—
478#103—74no—
479#103—74no—
480#103—74no—
481C22⋊C8—74no—
482C22⋊C8—74no—
483#97—74no—
484C4×C8C22×D714yes—
485C22×C7#53612yes—
486C22×C7#57712yes—
487C22×C7#57812yes—
488C22×C7C23×D411yescharacteristic
489C2×D7—22no—
490C2×D7—22no—
491C2×D7—22no—
492C2×D7—44no—
493C4×C7—22no—
494C4×C7—22no—
495C4×C7—42no—
496C4×C7—42no—
497C4×C7—42no—
498C4×C7—42no—
499C4×C7#53612yes—
500C4×C7C2×2−1+411yescharacteristic
501C4×C7C2×2−1+411yescharacteristic
502Dic7—22no—
503Dic7—22no—
504Dic7—22no—
505Dic7—42no—
506Dic7—42no—
507Dic7—42no—
508Dic7—42no—
509C24—71no—
510C2×Q8—72no—
511C2×Q8—72no—
512C2×Q8—144no—
513C2×Q8—141no—
514C2×Q8—141no—
515C2×Q8—144no—
516C2×Q8—142no—
517C2×Q8—142no—
518C2×Q8—142no—
519C2×Q8—142no—
520C2×Q8—562no—
521C2×Q8—562no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—71no—
527C22×C4—71no—
528C22×C4—71no—
529C22×C4—72no—
530C22×C4—72no—
531C22×C4—144no—
532C22×C4—142no—
533C22×C4—142no—
534C22×C4—141no—
535C22×C4—141no—
536C22×C4—141no—
537C22×C4—141no—
538C22×C4—562no—
539C22×C4D4×D711yescharacteristic
540C22×C4D4×D711yescharacteristic
541C22×C4C23×D711yescharacteristic
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—72no—
545C2×C8—72no—
546C2×C8—142no—
547C2×C8—142no—
548C2×C8C23×D712yes—
549C2×C8C23×D712yes—
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—24no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—74no—
559C4⋊C4—74no—
560C4⋊C4—72no—
561C4⋊C4—72no—
562C4⋊C4—144no—
563C4⋊C4—144no—
564C4⋊C4—144no—
565C4⋊C4—144no—
566C4⋊C4—144no—
567C4⋊C4—144no—
568C4⋊C4—144no—
569C4⋊C4—144no—
570C4⋊C4—141no—
571C4⋊C4—141no—
572C4⋊C4—144no—
573C4⋊C4—144no—
574C4⋊C4—144no—
575C4⋊C4C23×D712yes—
576C4⋊C4C23×D712yes—
577C4⋊C4C23×D712yes—
578C4⋊C4C23×D712yes—
579C22⋊C4—74no—
580C22⋊C4—74no—
581C22⋊C4—144no—
582C22⋊C4—144no—
583C22⋊C4—144no—
584C42—72no—
585C42—72no—
586C42—144no—
587C42—144no—
588C42—144no—
589C42D4×D712yes—
590C42C23×D712yes—
591C2×C7#324214yescomplemented
592C2×C7C2×#53611yescharacteristic
593C2×C7C2×#57711yescharacteristic
594C2×C7C2×#57811yescharacteristic
595D7—42no—
596C23—72no—
597C23—142no—
598C23—142no—
599C23—142no—
600C23C2×D4×D711yescharacteristic
601Q8—142no—
602Q8—142no—
603Q8—142no—
604Q8—142no—
605Q8—282no—
606Q8—282no—
607Q8—564no—
608Q8—564no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—72no—
620C2×C4—72no—
621C2×C4—72no—
622C2×C4—142no—
623C2×C4—142no—
624C2×C4—142no—
625C2×C4—144no—
626C2×C4—142no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—142no—
630C2×C4—142no—
631C2×C4—142no—
632C2×C4—282no—
633C2×C4—282no—
634C2×C4—282no—
635C2×C4—282no—
636C2×C4—564no—
637C2×C4C2×D4×D712yes—
638C2×C4C2×D4×D712yes—
639C2×C4C2×D4×D711yescharacteristic
640C2×C4#655112yes—
641C2×C4#655112yes—
642C2×C4C24×D711yescharacteristic
643C8—22no—
644C8—22no—
645C8—142no—
646C8—142no—
647C7C2×#324211yescharacteristic, complemented, Sylow
648C22—142no—
649C22—142no—
650C22—142no—
651C22—284no—
652C22#9041512yes—
653C22#9046112yes—
654C22#9046212yes—
655C22C22×D4×D711yescharacteristic
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—142no—
663C4—142no—
664C4—142no—
665C4—282no—
666C4—282no—
667C4—282no—
668C4—282no—
669C4#9041512yes—
670C4C2×#655111yescharacteristic
671C4C2×#655111yescharacteristic
672C2—282no—
673C2#1173936814yesdirect product factor
674C2C2×#9041511yescharacteristic
675C2C2×#9046111yescharacteristic
676C2C2×#9046211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#499C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#499D7
hypocenterC7C2×#3242
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
3. C2×C7×#499C2semidirect
4. C2×#89837C2semidirect
5. C2×#89841C2semidirect
6. #11739368C2direct (N × H)
7. C2×#89649C2semidirect
8. C2×#89647C2semidirect
11. C2×#89629C2semidirect
12. C2×#89626C2semidirect
15. C2×#89453C2semidirect
42. C7×#499C22semidirect
56. #89841C22semidirect
57. #89837C22semidirect
69. #89649C22semidirect
70. #89647C22semidirect
73. #89629C22semidirect
74. #89626C22semidirect
76. #89453C22semidirect
181. C2×#499D7semidirect
301. #499C2×D7semidirect
591. C2×C7#3242semidirect
647. C7C2×#3242semidirect
673. C2#11739368direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#903802
C2×#903882
C2×C7×#4992
C2×#898372
C2×#898412
#117393682
C2×#896492
C2×#896472
C2×#896562
C2×#896602
C2×#896292
C2×#896262
C2×#896342
C2×#896392
C2×#894532
C2×#894622
C2×#32427