Finite Group Databasebeta

← C2×#11740863 #14221641

Subgroups — C2×#11740863 #14221641

Order 1792.

2234 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#537C211yesmaximal, characteristic
2C2×#90518C211yesmaximal, characteristic, complemented
3C2×#90006C211yesmaximal, characteristic, complemented
4C2×#90004C211yesmaximal, characteristic, complemented
5C2×#90005C211yesmaximal, characteristic, complemented
6#11740863C2116yesmaximal, direct product factor
7C2×#89816C212yesmaximal, complemented
8C2×#89817C212yesmaximal, complemented
9C2×#89807C212yesmaximal, complemented
10C2×#89805C212yesmaximal, complemented
11C2×#89736C211yesmaximal, characteristic, complemented
12C22×#6381C211yesmaximal, characteristic, complemented
13C22×C7×#41C2211yescharacteristic
14C22×C7×#41C2211yescharacteristic
15C7×#537—24no—
16C7×#537C2218yes—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8C2212yes—
19C2×C7×#121—21no—
20C2×C7×#121—22no—
21C2×C7×#121C2212yes—
22C22×C7×C4⋊C4C2211yescharacteristic
23#90518—24no—
24#90518C2218yescomplemented
25C2×Q8×Dic7—21no—
26C2×D4×Dic7—22no—
27C2×D4×Dic7—21no—
28C2×#6489C2212yes—
29C2×#6489C2211yescharacteristic
30C22×C4⋊Dic7C2212yes—
31C22×C4⋊Dic7C2211yescharacteristic
32C22×#1136C2211yescharacteristic
33C23×C7⋊C8C2211yescharacteristic
34#90006C2218yescomplemented
35#90005C2218yescomplemented
36#90004C2218yescomplemented
37C2×#6409C2211yescharacteristic
38C2×#6409C2212yes—
39C2×#6409C2211yescharacteristic
40C2×#6406C2211yescharacteristic
41C2×#6406C2211yescharacteristic
42C2×#6406C2212yes—
43C2×C22⋊(C7⋊C8)C2212yes—
44C2×C22⋊(C7⋊C8)C2212yes—
45#89817C22116yescomplemented
46#89816C22116yescomplemented
47#89807C22116yescomplemented
48#89805C22116yescomplemented
49#89736C2218yescomplemented
50C2×#6382C2212yes—
51C2×#6381C2212yes—
52C2×#6381C2212yes—
53C2×#6381C2212yes—
54C2×#6381C2218yescomplemented
55C2×#3098—71nomaximal, Sylow
56C2×C7×#41—22no—
57C2×C7×#41—24no—
58C2×C7×#41—24no—
59C2×C7×#41—24no—
60C2×C7×#41—24no—
61C2×C7×#41D414yes—
62C2×C7×#41C2314yes—
63C2×C7×#41C2314yes—
64C22×C7×Q8D411yescharacteristic
65C22×C7×Q8C2311yescharacteristic
66C22×C7×D4—21no—
67C22×C7×D4—22no—
68C22×C7×D4D412yes—
69C22×C7×D4D411yescharacteristic
70C22×C7×D4C2311yescharacteristic
71C23×C4×C7—21no—
72C23×C4×C7—21no—
73C23×C4×C7C2312yes—
74C23×C4×C7C2311yescharacteristic
75C23×Dic7—22no—
76C23×Dic7—21no—
77C7×C22⋊Q8—28no—
78C7×C22⋊Q8—24no—
79C7×C22⋊Q8C2318yes—
80C7×#121—28no—
81C7×#121—216no—
82C7×#121—24no—
83C7×#121C2318yes—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4—22no—
87C2×C7×C4⋊C4C2312yes—
88C2×C7×C4⋊C4C2314yes—
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—22no—
91C2×C22⋊Dic7—22no—
92C2×C22⋊Dic7—21no—
93Q8×Dic7—28no—
94D4×Dic7—216no—
95D4×Dic7—28no—
96#6489—24no—
97#6489—22no—
98#6489C2314yes—
99#6489C2318yes—
100C2×C4⋊Dic7—22no—
101C2×C4⋊Dic7—21no—
102C2×C4⋊Dic7—24no—
103C2×C4⋊Dic7—22no—
104C2×C4⋊Dic7C2311yescharacteristic
105C2×C4⋊Dic7C2312yes—
106C2×C4⋊Dic7C2314yes—
107C2×C4⋊Dic7C2318yes—
108C2×C4⋊Dic7C2311yescharacteristic
109C2×C4×Dic7—22no—
110C2×C4×Dic7—21no—
111C2×#1136—24no—
112C2×#1136C2314yes—
113C22×C7⋊C8—24no—
114C22×C7⋊C8C2311yescharacteristic
115C22×C7⋊C8C2311yescharacteristic
116C22×C7⋊C8C2314yes—
117C22×C7⋊C8C2312yes—
118#6409C2318yes—
119#6409C2314yes—
120#6409C2314yes—
121#6406C2318yes—
122#6406C2314yes—
123#6406C2314yes—
124C22⋊(C7⋊C8)C2318yes—
125C22⋊(C7⋊C8)C2318yes—
126#6382C2318yes—
127#6381C2318yes—
128#6381C2318yes—
129#6381C2318yes—
130C2×#537D711yescharacteristic
131C2×#520—71no—
132#3098—716no—
133C2×#484—72no—
134C2×#485—72no—
135C2×#482—72no—
136C2×#480—72no—
137C2×#428—71no—
138C22×#107—71no—
139C2×#417—71no—
140C2×#416—71no—
141C2×#408—71no—
142C24×C7—21no—
143C24×C7C2×D412yes—
144C24×C7C2×D411yescharacteristic
145C7×#41—216no—
146C7×#41—28no—
147C7×#41—28no—
148C7×#41—28no—
149C7×#41—28no—
150C7×#41—48no—
151C2×C7×Q8—24no—
152C2×C7×Q8—21no—
153C2×C7×Q8—22no—
154C2×C7×Q8—22no—
155C2×C7×Q8C2×D414yes—
156C2×C7×Q8C2412yes—
157C2×C7×D4—28no—
158C2×C7×D4—24no—
159C2×C7×D4—28no—
160C2×C7×D4—24no—
161C2×C7×D4—21no—
162C2×C7×D4—22no—
163C2×C7×D4—22no—
164C2×C7×D4—44no—
165C2×C7×D4—84no—
166C2×C7×D4C2×D418yes—
167C2×C7×D4C2×D414yes—
168C2×C7×D4C2412yes—
169C22×C4×C7—24no—
170C22×C4×C7—24no—
171C22×C4×C7—22no—
172C22×C4×C7—21no—
173C22×C4×C7—24no—
174C22×C4×C7—24no—
175C22×C4×C7—44no—
176C22×C4×C7—82no—
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D414yes—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D412yes—
181C22×C4×C7#4114yes—
182C22×C4×C7#4114yes—
183C22×C4×C7C2414yes—
184C22×C4×C7C2411yescharacteristic
185C22×C4×C7C2412yes—
186C22×Dic7—21no—
187C22×Dic7—22no—
188C22×Dic7—21no—
189C22×Dic7—28no—
190C22×Dic7—24no—
191C22×Dic7—82no—
192C22×Dic7—84no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—28no—
196C7×C4⋊C4C2414yes—
197C7×C22⋊C4—28no—
198C7×C22⋊C4—28no—
199C22⋊Dic7—28no—
200C22⋊Dic7—24no—
201C4⋊Dic7—21no—
202C4⋊Dic7—22no—
203C4⋊Dic7—28no—
204C4⋊Dic7—24no—
205C4⋊Dic7—21no—
206C4⋊Dic7C2414yes—
207C4⋊Dic7C2412yes—
208C4⋊Dic7C2412yes—
209C4×Dic7—28no—
210C4×Dic7—24no—
211#1136—28no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8—22no—
214C2×C7⋊C8—28no—
215C2×C7⋊C8—24no—
216C2×C7⋊C8C2412yes—
217C2×C7⋊C8C2412yes—
218C2×C7⋊C8C2414yes—
219C22×#41C2×D711yescharacteristic
220C22×#41C2×D711yescharacteristic
221C22×M16—71no—
222C23×C8—71no—
223#537—24no—
224#537C2×D718yes—
225C2×C22⋊Q8—21no—
226C2×C22⋊Q8C2×D712yes—
227C2×#121—21no—
228C2×#121—22no—
229C2×#121C2×D712yes—
230#520—78no—
231#520—144no—
232C2×C4×Q8—141no—
233C2×C4×D4—142no—
234C2×C4×D4—141no—
235C2×#117—72no—
236C2×#117—71no—
237C22×C4⋊C4—72no—
238C22×C4⋊C4—71no—
239C22×C4⋊C4C2×D711yescharacteristic
240#485—716no—
241#484—716no—
242#482—716no—
243#480—716no—
244#428—78no—
245C2×#107—72no—
246C2×#107—72no—
247C2×#107—72no—
248C2×#107—78no—
249C2×#106—72no—
250#417—78no—
251#416—78no—
252C2×#103—71no—
253C2×#103—72no—
254C2×#103—71no—
255C2×#102—71no—
256C2×#102—72no—
257C2×#102—71no—
258#408—78no—
259C2×C22⋊C8—72no—
260C2×C22⋊C8—72no—
261C23×C7—22no—
262C23×C7—22no—
263C23×C7—24no—
264C23×C7—24no—
265C23×C7—24no—
266C23×C7—22no—
267C23×C7—42no—
268C23×C7—42no—
269C23×C7—42no—
270C23×C7#12314yes—
271C23×C7C22×D414yes—
272C23×C7C22×D412yes—
273C23×C7C22×D411yescharacteristic
274C7×Q8—28no—
275C7×Q8—22no—
276C7×Q8—22no—
277C7×Q8—42no—
278C7×D4—216no—
279C7×D4—28no—
280C7×D4—22no—
281C7×D4—22no—
282C7×D4—48no—
283C7×D4—42no—
284C7×D4—88no—
285C2×C4×C7—24no—
286C2×C4×C7—28no—
287C2×C4×C7—24no—
288C2×C4×C7—24no—
289C2×C4×C7—24no—
290C2×C4×C7—22no—
291C2×C4×C7—42no—
292C2×C4×C7—44no—
293C2×C4×C7—48no—
294C2×C4×C7—84no—
295C2×C4×C7#12314yes—
296C2×C4×C7#12318yes—
297C2×C4×C7C22×D414yes—
298C2×C4×C7C22×D412yes—
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C2×#4112yes—
301C2×C4×C7C2×#4112yes—
302C2×C4×C7C2×#4114yes—
303C2×C4×C7C2×#4114yes—
304C2×C4×C7C2511yescharacteristic
305C2×Dic7—22no—
306C2×Dic7—24no—
307C2×Dic7—22no—
308C2×Dic7—44no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—84no—
312C2×Dic7—88no—
313C7⋊C8—22no—
314C7⋊C8—22no—
315C7⋊C8—24no—
316C2×#41—24no—
317C2×#41—24no—
318C2×#41—22no—
319C2×#41—24no—
320C2×#41—24no—
321C2×#41C22⋊D714yes—
322C2×#41C22×D714yes—
323C2×#41C22×D714yes—
324C22×Q8C22⋊D711yescharacteristic
325C22×Q8C22×D711yescharacteristic
326C22×D4—21no—
327C22×D4—22no—
328C22×D4C22⋊D712yes—
329C22×D4C22⋊D711yescharacteristic
330C22×D4C22×D711yescharacteristic
331C23×C4—21no—
332C23×C4—21no—
333C23×C4—142no—
334C23×C4—141no—
335C23×C4C22×D712yes—
336C23×C4C22×D711yescharacteristic
337C2×M16—74no—
338C2×M16—144no—
339C22×C8—71no—
340C22×C8—71no—
341C22×C8—74no—
342C22×C8—72no—
343C22×C8—144no—
344C22⋊Q8—28no—
345C22⋊Q8—24no—
346C22⋊Q8C22×D718yes—
347#121—28no—
348#121—216no—
349#121—24no—
350#121C22×D718yes—
351C4×Q8—148no—
352C4×D4—1416no—
353C4×D4—148no—
354#117—74no—
355#117—78no—
356#117—144no—
357#117—142no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—72no—
363C2×C4⋊C4—74no—
364C2×C4⋊C4—78no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—142no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—144no—
369C2×C4⋊C4—142no—
370C2×C4⋊C4C22×D712yes—
371C2×C4⋊C4C22×D714yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—142no—
375C2×C22⋊C4—141no—
376C2×C42—142no—
377C2×C42—141no—
378#107—78no—
379#107—78no—
380#107—78no—
381#106—78no—
382#103—74no—
383#103—74no—
384#103—78no—
385#102—74no—
386#102—74no—
387#102—78no—
388C22⋊C8—78no—
389C22⋊C8—78no—
390C22×C7—28no—
391C22×C7—22no—
392C22×C7—22no—
393C22×C7—24no—
394C22×C7—24no—
395C22×C7—24no—
396C22×C7—48no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—84no—
401C22×C7C2×#12314yes—
402C22×C7C2×#12312yes—
403C22×C7#57612yes—
404C22×C7#57812yes—
405C22×C7C23×D411yescharacteristic
406C4×C7—24no—
407C4×C7—42no—
408C4×C7—44no—
409C4×C7C2×#12312yes—
410C4×C7C2×#12314yes—
411C4×C7C22×#4111yescharacteristic
412C4×C7C22×#4111yescharacteristic
413Dic7—44no—
414Dic7—42no—
415Dic7—42no—
416C24—21no—
417C24C2×C22⋊D712yes—
418C24C2×C22⋊D711yescharacteristic
419#41—216no—
420#41—28no—
421#41—28no—
422#41—28no—
423#41—28no—
424#41—48no—
425C2×Q8—24no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8C2×C22⋊D714yes—
430C2×Q8C23×D712yes—
431C2×D4—28no—
432C2×D4—24no—
433C2×D4—28no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—44no—
439C2×D4—84no—
440C2×D4C2×C22⋊D718yes—
441C2×D4C2×C22⋊D714yes—
442C2×D4C23×D712yes—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—44no—
450C22×C4—82no—
451C22×C4—141no—
452C22×C4—142no—
453C22×C4—141no—
454C22×C4—148no—
455C22×C4—144no—
456C22×C4—562no—
457C22×C4—564no—
458C22×C4#115914yes—
459C22×C4#115914yes—
460C22×C4C2×C22⋊D714yes—
461C22×C4C2×C22⋊D712yes—
462C22×C4C2×C22⋊D712yes—
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C23×D714yes—
465C22×C4C23×D712yes—
466C22×C4C23×D711yescharacteristic
467M16—148no—
468C2×C8—74no—
469C2×C8—72no—
470C2×C8—72no—
471C2×C8—142no—
472C2×C8—142no—
473C2×C8—148no—
474C2×C8—144no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—28no—
478C4⋊C4—72no—
479C4⋊C4—72no—
480C4⋊C4—74no—
481C4⋊C4—141no—
482C4⋊C4—142no—
483C4⋊C4—148no—
484C4⋊C4—144no—
485C4⋊C4—141no—
486C4⋊C4C23×D714yes—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—148no—
490C22⋊C4—144no—
491C42—148no—
492C42—144no—
493C2×C7—24no—
494C2×C7—44no—
495C2×C7—82no—
496C2×C7#309814yescomplemented
497C2×C7C22×#12311yescharacteristic
498C2×C7C2×#57611yescharacteristic
499C2×C7C2×#57811yescharacteristic
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23#649814yes—
510C23C22×C22⋊D714yes—
511C23C22×C22⋊D712yes—
512C23C22×C22⋊D711yescharacteristic
513Q8—28no—
514Q8—22no—
515Q8—22no—
516Q8—42no—
517D4—216no—
518D4—28no—
519D4—22no—
520D4—22no—
521D4—48no—
522D4—42no—
523D4—88no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—84no—
534C2×C4—142no—
535C2×C4—144no—
536C2×C4—142no—
537C2×C4—284no—
538C2×C4—282no—
539C2×C4—282no—
540C2×C4—564no—
541C2×C4—568no—
542C2×C4#649814yes—
543C2×C4#649818yes—
544C2×C4C2×#115912yes—
545C2×C4C2×#115912yes—
546C2×C4C2×#115914yes—
547C2×C4C2×#115914yes—
548C2×C4C22×C22⋊D714yes—
549C2×C4C22×C22⋊D712yes—
550C2×C4C22×C22⋊D711yescharacteristic
551C2×C4C24×D711yescharacteristic
552C8—142no—
553C8—142no—
554C8—144no—
555C7C2×#309811yescharacteristic, complemented, Sylow
556C22—22no—
557C22—22no—
558C22—28no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—48no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—84no—
567C22C2×#649814yes—
568C22C2×#649812yes—
569C22#9052612yes—
570C22#9052912yes—
571C22C23×C22⋊D711yescharacteristic
572C4—24no—
573C4—42no—
574C4—44no—
575C4—284no—
576C4—282no—
577C4—282no—
578C4C2×#649812yes—
579C4C2×#649814yes—
580C4C22×#115911yescharacteristic
581C4C22×#115911yescharacteristic
582C2—24no—
583C2—44no—
584C2—82no—
585C2#1174086314yesdirect product factor
586C2C22×#649811yescharacteristic
587C2C2×#9052611yescharacteristic
588C2C2×#9052911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
Fitting, prime radicalC2×C7×#537C2
socleC23×C7C22×D4
hypercenterC2×#537D7
hypocenterC7C2×#3098
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
2. C2×#90518C2semidirect
3. C2×#90006C2semidirect
4. C2×#90004C2semidirect
5. C2×#90005C2semidirect
6. #11740863C2direct (N × H)
7. C2×#89816C2semidirect
8. C2×#89817C2semidirect
9. C2×#89807C2semidirect
10. C2×#89805C2semidirect
11. C2×#89736C2semidirect
12. C22×#6381C2semidirect
24. #90518C22semidirect
34. #90006C22semidirect
35. #90005C22semidirect
36. #90004C22semidirect
45. #89817C22semidirect
46. #89816C22semidirect
47. #89807C22semidirect
48. #89805C22semidirect
49. #89736C22semidirect
54. C2×#6381C22semidirect
496. C2×C7#3098semidirect
555. C7C2×#3098semidirect
585. C2#11740863direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5372
C2×#905182
C2×#900062
C2×#900042
C2×#900052
#117408632
C2×#898162
C2×#898172
C2×#898072
C2×#898052
C2×#897362
C22×#63812
C2×#30987