Finite Group Databasebeta

← C2×#11729622 #13380694

Subgroups — C2×#11729622 #13380694

Order 1792.

1998 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C22×#6514C211yesmaximal, characteristic
3C22×#6507C211yesmaximal, characteristic, complemented
4C22×#6411C211yesmaximal, characteristic
5C22×#6410C211yesmaximal, characteristic, complemented
6C2×#89776C211yesmaximal, characteristic, complemented
7#11729622C218yesmaximal, direct product factor
8C2×#89327C211yesmaximal, characteristic, complemented
9C22×C7×#41C2211yescharacteristic
10C23×C7×Q8C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C7×#536—22no—
13C7×#536—22no—
14C7×#536—22no—
15C7×#536C2214yes—
16C2×C7×C4⋊Q8—21no—
17C2×C7×#124—21no—
18C2×C7×#123—22no—
19C2×C7×C22⋊Q8—22no—
20C2×C7×#117C2211yescharacteristic
21C2×#6514—22no—
22C2×#6514—24no—
23C2×#6514—24no—
24C2×#6514C2214yes—
25C2×#6507—22no—
26C2×#6507—24no—
27C2×#6507—24no—
28C2×#6507C2214yescomplemented
29C22×#1144—21no—
30C22×#1142—21no—
31C22×Dic7⋊C4—22no—
32C22×C4×Dic7C2211yescharacteristic
33C22×#1136C2211yescharacteristic
34C22×#1136C2211yescharacteristic
35C2×#6411—24no—
36C2×#6411C2214yes—
37C2×#6410—24no—
38C2×#6410C2214yescomplemented
39#89776—22no—
40#89776C2214yescomplemented
41C2×#6384—22no—
42#89327C2214yescomplemented
43C2×#2005—71nomaximal, Sylow
44C2×C7×#41—21no—
45C2×C7×#41—21no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41D412yes—
51C2×C7×#41C2312yes—
52C22×C7×Q8—22no—
53C22×C7×Q8—22no—
54C22×C7×Q8—22no—
55C22×C7×Q8—21no—
56C22×C7×Q8—21no—
57C22×C7×Q8—21no—
58C22×C7×Q8—21no—
59C22×C7×Q8—22no—
60C22×C7×Q8D411yescharacteristic
61C22×C7×Q8D412yes—
62C22×C7×Q8C2312yes—
63C22×C7×D4—21no—
64C22×C7×D4D411yescharacteristic
65C23×C4×C7—21no—
66C23×C4×C7D412yes—
67C23×C4×C7C2311yescharacteristic
68C23×Dic7—21no—
69C23×Dic7—21no—
70C22×Dic14—21no—
71C22×Dic14—21no—
72C22×Dic14—22no—
73C22×Dic14—21no—
74C22×Dic14—21no—
75C22×Dic14—22no—
76C22×Dic14—21no—
77C22×Dic14—22no—
78C22×Dic14D414yes—
79C22×Dic14C2312yes—
80C7×C4⋊Q8—24no—
81C7×C4⋊Q8—24no—
82C7×#124—24no—
83C7×#124—24no—
84C7×#123—28no—
85C7×#123—28no—
86C7×C22⋊Q8—28no—
87C7×C22⋊Q8—28no—
88C7×#117—21no—
89C7×#117—21no—
90C7×#117D412yes—
91C7×#117C2312yes—
92C2×C7×C4⋊C4—24no—
93C2×C7×C4⋊C4D412yes—
94C2×C7×C22⋊C4—24no—
95C2×C7×C22⋊C4—21no—
96C2×C42×C7—21no—
97#6514—216no—
98#6514—28no—
99#6514—28no—
100#6507—216no—
101#6507—28no—
102#6507—28no—
103C2×#1144—28no—
104C2×#1144—24no—
105C2×#1142—28no—
106C2×#1142—24no—
107C2×Dic7⋊C4—216no—
108C2×Dic7⋊C4—28no—
109C2×C4×Dic7—21no—
110C2×C4×Dic7—22no—
111C2×C4×Dic7D414yes—
112C2×C4×Dic7C2312yes—
113C2×#1136—22no—
114C2×#1136—21no—
115C2×#1136—21no—
116C2×#1136—22no—
117C2×#1136—22no—
118C2×#1136D414yes—
119C2×#1136C2312yes—
120C2×#1136C2312yes—
121C22×C7⋊C8—21no—
122C22×C7⋊C8—21no—
123#6411—28no—
124#6410—28no—
125#6384—28no—
126C22×#137—71no—
127C2×#536D711yescharacteristic
128C22×C4⋊Q8—71no—
129C22×#104—71no—
130C2×#419—71no—
131C22×#101—71no—
132#2005—78no—
133C2×#339—71no—
134C24×C7—21no—
135C24×C7C2×D411yescharacteristic
136C7×#41—24no—
137C7×#41—24no—
138C7×#41—22no—
139C7×#41—22no—
140C7×#41—22no—
141C7×#41—22no—
142C7×#41—42no—
143C7×#41—42no—
144C7×#41—42no—
145C7×#41—42no—
146C2×C7×Q8—24no—
147C2×C7×Q8—22no—
148C2×C7×Q8—22no—
149C2×C7×Q8—21no—
150C2×C7×Q8—24no—
151C2×C7×Q8—22no—
152C2×C7×Q8—22no—
153C2×C7×Q8—22no—
154C2×C7×Q8—22no—
155C2×C7×Q8—22no—
156C2×C7×Q8—22no—
157C2×C7×Q8—22no—
158C2×C7×Q8—22no—
159C2×C7×Q8—42no—
160C2×C7×Q8—42no—
161C2×C7×Q8—42no—
162C2×C7×Q8—42no—
163C2×C7×Q8—42no—
164C2×C7×Q8—42no—
165C2×C7×Q8—42no—
166C2×C7×Q8—42no—
167C2×C7×Q8C2×D412yes—
168C2×C7×D4—22no—
169C2×C7×D4—22no—
170C2×C7×D4—22no—
171C2×C7×D4—21no—
172C2×C7×D4—22no—
173C2×C7×D4—42no—
174C2×C7×D4—42no—
175C2×C7×D4C2×D412yes—
176C22×C4×C7—24no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—21no—
180C22×C4×C7—22no—
181C22×C4×C7—24no—
182C22×C4×C7—22no—
183C22×C4×C7—22no—
184C22×C4×C7—42no—
185C22×C4×C7—42no—
186C22×C4×C7C2×D414yes—
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D412yes—
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D412yes—
192C22×C4×C7C2×D412yes—
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D411yescharacteristic
195C22×C4×C7#4112yes—
196C22×C4×C7C2411yescharacteristic
197C22×Dic7—28no—
198C22×Dic7—24no—
199C22×Dic7—22no—
200C22×Dic7—24no—
201C22×Dic7—22no—
202C22×Dic7—41no—
203C22×Dic7—41no—
204C22×Dic7—42no—
205C2×Dic14—24no—
206C2×Dic14—24no—
207C2×Dic14—24no—
208C2×Dic14—24no—
209C2×Dic14—22no—
210C2×Dic14—22no—
211C2×Dic14—22no—
212C2×Dic14—22no—
213C2×Dic14—24no—
214C2×Dic14—22no—
215C2×Dic14—41no—
216C2×Dic14—41no—
217C2×Dic14—42no—
218C2×Dic14—42no—
219C2×Dic14—42no—
220C2×Dic14—42no—
221C2×Dic14—42no—
222C2×Dic14C2×D414yes—
223C7×C4⋊C4—216no—
224C7×C4⋊C4—22no—
225C7×C4⋊C4C2×D414yes—
226C7×C22⋊C4—216no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C42×C7—22no—
230C42×C7—22no—
231#1144—216no—
232#1142—216no—
233Dic7⋊C4—232no—
234C4×Dic7—24no—
235C4×Dic7—22no—
236C4×Dic7C2×D414yes—
237#1136—24no—
238#1136—22no—
239#1136—22no—
240#1136—22no—
241#1136—42no—
242#1136C2×D414yes—
243C2×C7⋊C8—24no—
244C2×C7⋊C8—22no—
245C2×C7⋊C8—22no—
246C2×C7⋊C8—42no—
247C22×#41C2×D711yescharacteristic
248C23×Q8—71no—
249C23×Q8C2×D711yescharacteristic
250C2×#137—74no—
251C2×#137—144no—
252C2×#137—144no—
253C2×#137—142no—
254C22×Q16—141no—
255C22×SD16—141no—
256C22×M16—71no—
257C22×M16—71no—
258#536—22no—
259#536—22no—
260#536—22no—
261#536C2×D714yes—
262C2×C4⋊Q8—21no—
263C2×C4⋊Q8—74no—
264C2×C4⋊Q8—144no—
265C2×C4⋊Q8—144no—
266C2×C4⋊Q8—142no—
267C2×#124—21no—
268C2×#123—22no—
269C2×C22⋊Q8—22no—
270C2×#117C2×D711yescharacteristic
271C22×C4⋊C4—142no—
272C22×C42—71no—
273C2×#104—74no—
274C2×#104—144no—
275#419—74no—
276#419—142no—
277C2×#103—142no—
278C2×#101—74no—
279C2×#101—144no—
280#339—74no—
281C23×C7—24no—
282C23×C7—22no—
283C23×C7—42no—
284C23×C7—44no—
285C23×C7#12012yes—
286C23×C7#12011yescharacteristic
287C23×C7#12011yescharacteristic
288C23×C7#12112yes—
289C23×C7C22×D411yescharacteristic
290C7×Q8—22no—
291C7×Q8—22no—
292C7×Q8—42no—
293C7×Q8—42no—
294C7×Q8—42no—
295C7×Q8—42no—
296C7×Q8—44no—
297C7×Q8—44no—
298C7×Q8—42no—
299C7×D4—22no—
300C7×D4—22no—
301C7×D4—42no—
302C7×D4—42no—
303C7×D4—44no—
304C7×D4—42no—
305C2×C4×C7—24no—
306C2×C4×C7—24no—
307C2×C4×C7—24no—
308C2×C4×C7—22no—
309C2×C4×C7—24no—
310C2×C4×C7—44no—
311C2×C4×C7—44no—
312C2×C4×C7—42no—
313C2×C4×C7—44no—
314C2×C4×C7—42no—
315C2×C4×C7—42no—
316C2×C4×C7—44no—
317C2×C4×C7#12012yes—
318C2×C4×C7#12012yes—
319C2×C4×C7#12012yes—
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12112yes—
323C2×C4×C7#12112yes—
324C2×C4×C7#12112yes—
325C2×C4×C7#12112yes—
326C2×C4×C7#12112yes—
327C2×C4×C7#12712yes—
328C2×C4×C7#12712yes—
329C2×C4×C7C22×D411yescharacteristic
330C2×C4×C7C22×D411yescharacteristic
331C2×C4×C7C22×D411yescharacteristic
332C2×C4×C7C22×D411yescharacteristic
333C2×C4×C7C22×D411yescharacteristic
334C2×C4×C7C2×#4111yescharacteristic
335C2×Dic7—28no—
336C2×Dic7—216no—
337C2×Dic7—24no—
338C2×Dic7—24no—
339C2×Dic7—42no—
340C2×Dic7—42no—
341C2×Dic7—44no—
342C2×Dic7—44no—
343Dic14—24no—
344Dic14—24no—
345Dic14—42no—
346Dic14—42no—
347Dic14—44no—
348Dic14—44no—
349C7⋊C8—24no—
350C7⋊C8—42no—
351C2×#41—21no—
352C2×#41—21no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41C22⋊D712yes—
358C2×#41C22×D712yes—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—22no—
363C22×Q8—21no—
364C22×Q8—21no—
365C22×Q8—21no—
366C22×Q8—21no—
367C22×Q8—74no—
368C22×Q8—72no—
369C22×Q8—142no—
370C22×Q8—141no—
371C22×Q8—142no—
372C22×Q8—142no—
373C22×Q8—141no—
374C22×Q8—141no—
375C22×Q8—141no—
376C22×Q8—141no—
377C22×Q8C22⋊D711yescharacteristic
378C22×Q8C22⋊D712yes—
379C22×Q8C22×D712yes—
380C22×D4—21no—
381C22×D4C22⋊D711yescharacteristic
382C23×C4—21no—
383C23×C4—141no—
384C23×C4—141no—
385C23×C4C22⋊D712yes—
386C23×C4C22×D711yescharacteristic
387#137—148no—
388#137—148no—
389#137—1416no—
390C2×Q16—148no—
391C2×Q16—144no—
392C2×SD16—148no—
393C2×SD16—144no—
394C2×M16—74no—
395C2×M16—72no—
396C2×M16—72no—
397C2×M16—142no—
398C2×M16—141no—
399C2×M16—141no—
400C2×M16—142no—
401C2×M16—142no—
402C22×C8—141no—
403C22×C8—141no—
404C4⋊Q8—24no—
405C4⋊Q8—24no—
406C4⋊Q8—148no—
407C4⋊Q8—148no—
408C4⋊Q8—1416no—
409#124—24no—
410#124—24no—
411#123—28no—
412#123—28no—
413C22⋊Q8—28no—
414C22⋊Q8—28no—
415#117—21no—
416#117—21no—
417#117C22⋊D712yes—
418#117C22×D712yes—
419C2×C4⋊C4—24no—
420C2×C4⋊C4—1416no—
421C2×C4⋊C4—148no—
422C2×C4⋊C4C22⋊D712yes—
423C2×C22⋊C4—24no—
424C2×C22⋊C4—21no—
425C2×C42—21no—
426C2×C42—74no—
427C2×C42—72no—
428C2×C42—142no—
429C2×C42—141no—
430#104—148no—
431#103—148no—
432#101—148no—
433C22×C7—22no—
434C22×C7—22no—
435C22×C7—24no—
436C22×C7—24no—
437C22×C7—42no—
438C22×C7—44no—
439C22×C7—84no—
440C22×C7#39212yes—
441C22×C7#39212yes—
442C22×C7#45614yes—
443C22×C7C2×#12011yescharacteristic
444C22×C7C2×#12011yescharacteristic
445C22×C7C2×#12111yescharacteristic
446C4×C7—44no—
447C4×C7—42no—
448C4×C7—44no—
449C4×C7#39212yes—
450C4×C7#39212yes—
451C4×C7C2×#12011yescharacteristic
452C4×C7C2×#12111yescharacteristic
453C4×C7C2×#12111yescharacteristic
454C4×C7C2×#12711yescharacteristic
455Dic7—28no—
456Dic7—44no—
457C24—21no—
458C24C2×C22⋊D711yescharacteristic
459#41—24no—
460#41—24no—
461#41—22no—
462#41—22no—
463#41—22no—
464#41—22no—
465#41—42no—
466#41—42no—
467#41—42no—
468#41—42no—
469C2×Q8—24no—
470C2×Q8—24no—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—21no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—42no—
486C2×Q8—42no—
487C2×Q8—42no—
488C2×Q8—42no—
489C2×Q8—42no—
490C2×Q8—74no—
491C2×Q8—144no—
492C2×Q8—142no—
493C2×Q8—144no—
494C2×Q8—144no—
495C2×Q8—144no—
496C2×Q8—144no—
497C2×Q8—142no—
498C2×Q8—142no—
499C2×Q8—142no—
500C2×Q8—142no—
501C2×Q8—282no—
502C2×Q8—282no—
503C2×Q8—282no—
504C2×Q8—282no—
505C2×Q8—282no—
506C2×Q8—281no—
507C2×Q8—281no—
508C2×Q8C2×C22⋊D712yes—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—22no—
512C2×D4—22no—
513C2×D4—21no—
514C2×D4—42no—
515C2×D4—42no—
516C2×D4C2×C22⋊D712yes—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—21no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—42no—
526C22×C4—42no—
527C22×C4—148no—
528C22×C4—144no—
529C22×C4—142no—
530C22×C4—144no—
531C22×C4—142no—
532C22×C4—282no—
533C22×C4—281no—
534C22×C4—281no—
535C22×C4D4×D712yes—
536C22×C4D4×D712yes—
537C22×C4D4×D712yes—
538C22×C4#115912yes—
539C22×C4C2×C22⋊D714yes—
540C22×C4C2×C22⋊D711yescharacteristic
541C22×C4C2×C22⋊D711yescharacteristic
542C22×C4C2×C22⋊D711yescharacteristic
543C22×C4C2×C22⋊D711yescharacteristic
544C22×C4C2×C22⋊D711yescharacteristic
545C22×C4C23×D711yescharacteristic
546Q16—1416no—
547SD16—1416no—
548M16—74no—
549M16—144no—
550M16—142no—
551M16—142no—
552M16—142no—
553M16—282no—
554C2×C8—144no—
555C2×C8—142no—
556C2×C8—142no—
557C2×C8—282no—
558C4⋊C4—216no—
559C4⋊C4—22no—
560C4⋊C4—1432no—
561C4⋊C4C2×C22⋊D714yes—
562C22⋊C4—216no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C42—22no—
566C42—22no—
567C42—74no—
568C42—144no—
569C42—142no—
570C2×C7—21no—
571C2×C7—21no—
572C2×C7—22no—
573C2×C7—82no—
574C2×C7#200514yescomplemented
575C2×C7C2×#39211yescharacteristic
576C2×C7C2×#45612yes—
577C23—24no—
578C23—22no—
579C23—42no—
580C23—44no—
581C23#650012yes—
582C23#650212yes—
583C23#651611yescharacteristic
584C23#651611yescharacteristic
585C23C22×C22⋊D711yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—44no—
593Q8—44no—
594Q8—42no—
595Q8—144no—
596Q8—144no—
597Q8—282no—
598Q8—282no—
599Q8—284no—
600Q8—284no—
601D4—22no—
602D4—22no—
603D4—42no—
604D4—42no—
605D4—44no—
606D4—42no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—148no—
620C2×C4—1416no—
621C2×C4—144no—
622C2×C4—144no—
623C2×C4—282no—
624C2×C4—284no—
625C2×C4—284no—
626C2×C4—282no—
627C2×C4#650012yes—
628C2×C4#650012yes—
629C2×C4#650012yes—
630C2×C4#650112yes—
631C2×C4#650112yes—
632C2×C4#650212yes—
633C2×C4#650212yes—
634C2×C4#650212yes—
635C2×C4#650312yes—
636C2×C4#650312yes—
637C2×C4#651611yescharacteristic
638C2×C4#651611yescharacteristic
639C2×C4C2×D4×D711yescharacteristic
640C2×C4C2×D4×D711yescharacteristic
641C2×C4C2×D4×D711yescharacteristic
642C2×C4C2×#115911yescharacteristic
643C2×C4C22×C22⋊D711yescharacteristic
644C2×C4C22×C22⋊D711yescharacteristic
645C8—144no—
646C8—282no—
647C7C2×#200511yescharacteristic, complemented, Sylow
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—44no—
654C22—84no—
655C22#8999512yes—
656C22#8999512yes—
657C22#9001814yes—
658C22C2×#650011yescharacteristic
659C22C2×#650211yescharacteristic
660C22C2×#651611yescharacteristic
661C4—44no—
662C4—42no—
663C4—44no—
664C4—148no—
665C4—284no—
666C4#8999512yes—
667C4#8999512yes—
668C4C2×#650011yescharacteristic
669C4C2×#650111yescharacteristic
670C4C2×#650211yescharacteristic
671C4C2×#650311yescharacteristic
672C2—21no—
673C2—21no—
674C2—22no—
675C2—82no—
676C2#1172962214yesdirect product factor
677C2C2×#8999511yescharacteristic
678C2C2×#9001812yes—
subgroupG/H
centerC23#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#536C2
socleC23×C7#120
prime radicalC22×C7×#41C22
hypercenterC2×#536D7
hypocenterC7C2×#2005
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. C22×#6507C2semidirect
5. C22×#6410C2semidirect
6. C2×#89776C2semidirect
7. #11729622C2direct (N × H)
8. C2×#89327C2semidirect
28. C2×#6507C22semidirect
38. C2×#6410C22semidirect
40. #89776C22semidirect
42. #89327C22semidirect
574. C2×C7#2005semidirect
647. C7C2×#2005semidirect
676. C2#11729622direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C22×#65142
C22×#65072
C22×#64112
C22×#64102
C2×#897762
#117296222
C2×#893272
C2×#20057