Finite Group Databasebeta

← C2×#11727886 #13382717

Subgroups — C2×#11727886 #13382717

Order 1792.

906 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#413C211yesmaximal, characteristic
2C2×#90008C211yesmaximal, characteristic
3C2×#90005C211yesmaximal, characteristic
4C2×#89891C211yesmaximal, characteristic, complemented
5C2×#89354C211yesmaximal, characteristic, complemented
6C2×#89353C211yesmaximal, characteristic, complemented
7C2×#89352C211yesmaximal, characteristic, complemented
8#11727886C218yesmaximal, direct product factor
9C22×C7×#41C2211yescharacteristic
10C22×C7×M16C2211yescharacteristic
11C22×C7×M16C2211yescharacteristic
12C2×#6489C2211yescharacteristic
13C2×#6489C2211yescharacteristic
14C22×#1136C2211yescharacteristic
15C23×C7⋊C8C2211yescharacteristic
16C7×#413—22no—
17C7×#413C2214yes—
18C2×C7×#101—21no—
19C2×C7×#100—21no—
20#90008—22no—
21#90008C2214yes—
22C2×#6411—22no—
23#90005—22no—
24#90005C2214yes—
25C2×#6409—21no—
26C2×#6406—21no—
27#89891—22no—
28#89891C2214yescomplemented
29C2×Dic7⋊C8—22no—
30#89354C2214yescomplemented
31#89353C2214yescomplemented
32#89352C2214yescomplemented
33C2×#2080—71nomaximal, Sylow
34C2×C7×#41—21no—
35C2×C7×#41—21no—
36C2×C7×#41—21no—
37C2×C7×#41—21no—
38C2×C7×#41—21no—
39C2×C7×#41—21no—
40C2×C7×#41—21no—
41C2×C7×#41—21no—
42C2×C7×#41—21no—
43C2×C7×#41—21no—
44C2×C7×#41D412yes—
45C2×C7×#41C2312yes—
46C22×C7×Q8D411yescharacteristic
47C22×C7×D4—21no—
48C22×C7×D4D411yescharacteristic
49C23×C4×C7—21no—
50C23×C4×C7C2311yescharacteristic
51C2×C7×M16—21no—
52C2×C7×M16—21no—
53C2×C7×M16—21no—
54C2×C7×M16—21no—
55C2×C7×M16—21no—
56C2×C7×M16—21no—
57C2×C7×M16—21no—
58C2×C7×M16—21no—
59C2×C7×M16—21no—
60C2×C7×M16—21no—
61C2×C7×M16C2312yes—
62C2×C7×M16C2312yes—
63C22×C7×C8—21no—
64C22×C7×C8—21no—
65C2×C22⋊Dic7—21no—
66C2×C22⋊Dic7—21no—
67#6489—21no—
68#6489—21no—
69#6489—21no—
70#6489—21no—
71#6489—21no—
72#6489D412yes—
73#6489C2312yes—
74#6489C2312yes—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4×Dic7—21no—
78C2×C4×Dic7D412yes—
79C2×#1136—21no—
80C2×#1136—21no—
81C2×#1136—21no—
82C2×#1136—21no—
83C2×#1136—21no—
84C2×#1136C2312yes—
85C22×C7⋊C8—21no—
86C22×C7⋊C8—21no—
87C22×C7⋊C8—21no—
88C22×C7⋊C8—21no—
89C22×C7⋊C8—21no—
90C22×C7⋊C8—21no—
91C22×C7⋊C8—21no—
92C22×C7⋊C8C2312yes—
93C7×#101—24no—
94C7×#100—24no—
95#6411—28no—
96#6409—24no—
97#6406—24no—
98Dic7⋊C8—28no—
99C2×#424—71no—
100C2×#421—71no—
101C2×#416—71no—
102C2×#413D711yescharacteristic
103#2080—78no—
104C2×#344—71no—
105C2×#341—71no—
106C2×H(3,ℤ/4)—71no—
107C24×C7—21no—
108C24×C7#4111yescharacteristic
109C7×#41—22no—
110C7×#41—22no—
111C7×#41—22no—
112C7×#41—22no—
113C7×#41—22no—
114C7×#41—22no—
115C7×#41—22no—
116C7×#41—22no—
117C7×#41—41no—
118C7×#41—41no—
119C7×#41—41no—
120C7×#41—41no—
121C7×#41—41no—
122C7×#41—41no—
123C7×#41—41no—
124C7×#41—41no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8C2×D412yes—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—41no—
139C2×C7×D4—41no—
140C2×C7×D4—41no—
141C2×C7×D4—41no—
142C2×C7×D4C2×D412yes—
143C22×C4×C7—21no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—41no—
147C22×C4×C7—41no—
148C22×C4×C7—41no—
149C22×C4×C7—41no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7#4111yescharacteristic
155C22×C4×C7#4111yescharacteristic
156C22×C4×C7#4111yescharacteristic
157C22×C4×C7#4111yescharacteristic
158C22×C4×C7#4111yescharacteristic
159C22×C4×C7#4111yescharacteristic
160C22×C4×C7#4111yescharacteristic
161C22×C4×C7#4111yescharacteristic
162C22×C4×C7#4111yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—22no—
165C22×Dic7—41no—
166C7×M16—22no—
167C7×M16—22no—
168C7×M16—22no—
169C7×M16—22no—
170C7×M16—41no—
171C7×M16—41no—
172C7×M16—41no—
173C7×M16—41no—
174C2×C7×C8—22no—
175C2×C7×C8—22no—
176C2×C7×C8—41no—
177C2×C7×C8—41no—
178C2×C7×C8—41no—
179C2×C7×C8—41no—
180C22⋊Dic7—22no—
181C22⋊Dic7—22no—
182C22⋊Dic7—22no—
183C22⋊Dic7—41no—
184C4⋊Dic7—22no—
185C4⋊Dic7—22no—
186C4⋊Dic7—22no—
187C4⋊Dic7—41no—
188C4×Dic7—22no—
189C4×Dic7—22no—
190C4×Dic7—41no—
191C4×Dic7C2×D414yes—
192#1136—22no—
193#1136—22no—
194#1136—41no—
195#1136—41no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—41no—
201C2×C7⋊C8—41no—
202C2×C7⋊C8—41no—
203C2×C7⋊C8—41no—
204C2×C7⋊C8—41no—
205C2×C7⋊C8—41no—
206C22×#41C2×D711yescharacteristic
207C22×M16—71no—
208C22×M16C2×D711yescharacteristic
209C22×M16C2×D711yescharacteristic
210C23×C8—71no—
211C2×#117—71no—
212C2×#117—71no—
213#424—74no—
214#424—142no—
215C2×C4⋊C8—142no—
216#421—74no—
217#421—142no—
218C2×#104—142no—
219#416—74no—
220#416—142no—
221C2×#103—141no—
222C2×#102—141no—
223#413—22no—
224#413C2×D714yes—
225C2×#101—21no—
226C2×#100—21no—
227#344—74no—
228#341—74no—
229H(3,ℤ/4)—74no—
230C23×C7—22no—
231C23×C7—42no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—82no—
235C23×C7#12311yescharacteristic
236C23×C7#12311yescharacteristic
237C23×C7#12311yescharacteristic
238C23×C7#12311yescharacteristic
239C23×C7#12611yescharacteristic
240C23×C7#12611yescharacteristic
241C23×C7C2×#4111yescharacteristic
242C7×Q8—22no—
243C7×Q8—22no—
244C7×Q8—41no—
245C7×Q8—41no—
246C7×D4—22no—
247C7×D4—22no—
248C7×D4—42no—
249C7×D4—42no—
250C7×D4—42no—
251C7×D4—42no—
252C7×D4—41no—
253C7×D4—41no—
254C2×C4×C7—22no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7#12111yescharacteristic
262C2×C4×C7#12111yescharacteristic
263C2×C4×C7#12111yescharacteristic
264C2×C4×C7#12111yescharacteristic
265C2×C4×C7#12311yescharacteristic
266C2×C4×C7#12311yescharacteristic
267C2×C4×C7#12311yescharacteristic
268C2×C4×C7#12311yescharacteristic
269C2×C4×C7#12311yescharacteristic
270C2×C4×C7#12311yescharacteristic
271C2×C4×C7#12311yescharacteristic
272C2×C4×C7#12311yescharacteristic
273C2×C4×C7#12411yescharacteristic
274C2×C4×C7#12411yescharacteristic
275C2×C4×C7#12411yescharacteristic
276C2×C4×C7#12411yescharacteristic
277C2×C4×C7#12611yescharacteristic
278C2×C4×C7#12611yescharacteristic
279C2×C4×C7#12611yescharacteristic
280C2×C4×C7#12611yescharacteristic
281C2×C4×C7#12611yescharacteristic
282C2×C4×C7#12611yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C2×C4×C7C22×D411yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×Dic7—24no—
290C2×Dic7—44no—
291C2×Dic7—42no—
292C2×Dic7—82no—
293C7×C8—42no—
294C7×C8—42no—
295C7⋊C8—42no—
296C7⋊C8—42no—
297C2×#41—21no—
298C2×#41—21no—
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—21no—
302C2×#41—21no—
303C2×#41—21no—
304C2×#41—21no—
305C2×#41—21no—
306C2×#41—21no—
307C2×#41C22⋊D712yes—
308C2×#41C22×D712yes—
309C22×Q8C22⋊D711yescharacteristic
310C22×D4—21no—
311C22×D4C22⋊D711yescharacteristic
312C23×C4—21no—
313C23×C4C22×D711yescharacteristic
314C2×M16—21no—
315C2×M16—21no—
316C2×M16—21no—
317C2×M16—21no—
318C2×M16—21no—
319C2×M16—21no—
320C2×M16—21no—
321C2×M16—21no—
322C2×M16—21no—
323C2×M16—21no—
324C2×M16—72no—
325C2×M16—141no—
326C2×M16—141no—
327C2×M16—141no—
328C2×M16—141no—
329C2×M16—141no—
330C2×M16C22×D712yes—
331C2×M16C22×D712yes—
332C22×C8—21no—
333C22×C8—21no—
334C22×C8—72no—
335C22×C8—141no—
336C22×C8—141no—
337C22×C8—141no—
338C22×C8—141no—
339C22×C8—141no—
340C22×C8—141no—
341C22×C8—141no—
342#117—72no—
343#117—72no—
344#117—72no—
345#117—141no—
346#117—141no—
347#117—141no—
348#117—141no—
349#117—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C42—72no—
355C2×C42—141no—
356C4⋊C8—148no—
357#104—148no—
358#103—144no—
359#102—144no—
360#101—24no—
361#100—24no—
362C22×C7—22no—
363C22×C7—22no—
364C22×C7—22no—
365C22×C7—41no—
366C22×C7—41no—
367C22×C7—41no—
368C22×C7—41no—
369C22×C7—42no—
370C22×C7—82no—
371C22×C7—82no—
372C22×C7—82no—
373C22×C7#39711yescharacteristic
374C22×C7#39711yescharacteristic
375C22×C7#39711yescharacteristic
376C22×C7#39711yescharacteristic
377C22×C7C2×#12311yescharacteristic
378C22×C7C2×#12311yescharacteristic
379C22×C7C2×#12611yescharacteristic
380C4×C7—42no—
381C4×C7#39711yescharacteristic
382C4×C7#39711yescharacteristic
383C4×C7#39711yescharacteristic
384C4×C7#39711yescharacteristic
385C4×C7C2×#12111yescharacteristic
386C4×C7C2×#12311yescharacteristic
387C4×C7C2×#12411yescharacteristic
388C4×C7C2×#12611yescharacteristic
389Dic7—44no—
390Dic7—82no—
391C24—21no—
392C24#115711yescharacteristic
393#41—22no—
394#41—22no—
395#41—22no—
396#41—22no—
397#41—22no—
398#41—22no—
399#41—22no—
400#41—22no—
401#41—41no—
402#41—41no—
403#41—41no—
404#41—41no—
405#41—41no—
406#41—41no—
407#41—41no—
408#41—41no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8C2×C22⋊D712yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4C2×C22⋊D712yes—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—142no—
435C22×C4—281no—
436C22×C4#115711yescharacteristic
437C22×C4#115711yescharacteristic
438C22×C4#115711yescharacteristic
439C22×C4D4×D711yescharacteristic
440C22×C4D4×D711yescharacteristic
441C22×C4#115911yescharacteristic
442C22×C4#115911yescharacteristic
443C22×C4#115911yescharacteristic
444C22×C4#115911yescharacteristic
445C22×C4#115911yescharacteristic
446C22×C4#115911yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D711yescharacteristic
449C22×C4C23×D711yescharacteristic
450M16—22no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—142no—
459M16—142no—
460M16—281no—
461M16—281no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—142no—
469C2×C8—142no—
470C2×C8—142no—
471C2×C8—142no—
472C2×C8—281no—
473C2×C8—281no—
474C2×C8—281no—
475C2×C8—281no—
476C2×C8—281no—
477C2×C8—281no—
478C4⋊C4—142no—
479C4⋊C4—142no—
480C4⋊C4—142no—
481C4⋊C4—281no—
482C22⋊C4—142no—
483C22⋊C4—142no—
484C22⋊C4—142no—
485C22⋊C4—281no—
486C42—74no—
487C42—142no—
488C42—142no—
489C42—281no—
490C2×C7—21no—
491C2×C7—21no—
492C2×C7—21no—
493C2×C7—21no—
494C2×C7—21no—
495C2×C7—21no—
496C2×C7—82no—
497C2×C7#208012yescomplemented
498C2×C7C2×#39711yescharacteristic
499C23—22no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—82no—
504C23#644211yescharacteristic
505C23#644211yescharacteristic
506C23#644611yescharacteristic
507C23#644611yescharacteristic
508C23#649211yescharacteristic
509C23#649211yescharacteristic
510C23C2×#115711yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—41no—
514Q8—41no—
515D4—22no—
516D4—22no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—41no—
522D4—41no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—144no—
531C2×C4—284no—
532C2×C4—282no—
533C2×C4—562no—
534C2×C4#644211yescharacteristic
535C2×C4#644211yescharacteristic
536C2×C4#644211yescharacteristic
537C2×C4#644211yescharacteristic
538C2×C4#644211yescharacteristic
539C2×C4#644211yescharacteristic
540C2×C4#644611yescharacteristic
541C2×C4#644611yescharacteristic
542C2×C4#644811yescharacteristic
543C2×C4#644811yescharacteristic
544C2×C4#644811yescharacteristic
545C2×C4#644811yescharacteristic
546C2×C4#649211yescharacteristic
547C2×C4#649211yescharacteristic
548C2×C4#649811yescharacteristic
549C2×C4#649811yescharacteristic
550C2×C4#649811yescharacteristic
551C2×C4#649811yescharacteristic
552C2×C4#650211yescharacteristic
553C2×C4#650211yescharacteristic
554C2×C4#650211yescharacteristic
555C2×C4#650211yescharacteristic
556C2×C4C2×#115711yescharacteristic
557C2×C4C2×D4×D711yescharacteristic
558C2×C4C2×#115911yescharacteristic
559C2×C4C2×#115911yescharacteristic
560C2×C4C2×#115911yescharacteristic
561C2×C4C22×C22⋊D711yescharacteristic
562C8—42no—
563C8—42no—
564C8—282no—
565C8—282no—
566C7C2×#208011yescharacteristic, complemented, Sylow
567C22—22no—
568C22—22no—
569C22—22no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22#8972411yescharacteristic
579C22#8972411yescharacteristic
580C22#8972411yescharacteristic
581C22#8972411yescharacteristic
582C22C2×#644211yescharacteristic
583C22C2×#644611yescharacteristic
584C22C2×#649211yescharacteristic
585C4—42no—
586C4—284no—
587C4—562no—
588C4#8972411yescharacteristic
589C4#8972411yescharacteristic
590C4#8972411yescharacteristic
591C4#8972411yescharacteristic
592C4C2×#644211yescharacteristic
593C4C2×#644811yescharacteristic
594C4C2×#649811yescharacteristic
595C4C2×#650211yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—82no—
603C2#1172788612yesdirect product factor
604C2C2×#8972411yescharacteristic
subgroupG/H
centerC2×C4#6448
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#413C2
socleC22×C7#397
prime radicalC22×C7×#41C22
hypercenterC2×#413D7
hypocenterC7C2×#2080
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
4. C2×#89891C2semidirect
5. C2×#89354C2semidirect
6. C2×#89353C2semidirect
7. C2×#89352C2semidirect
8. #11727886C2direct (N × H)
28. #89891C22semidirect
30. #89354C22semidirect
31. #89353C22semidirect
32. #89352C22semidirect
497. C2×C7#2080semidirect
566. C7C2×#2080semidirect
603. C2#11727886direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4132
C2×#900082
C2×#900052
C2×#898912
C2×#893542
C2×#893532
C2×#893522
#117278862
C2×#20807