Finite Group Databasebeta

← C2×#11736684 #14221306

Subgroups — C2×#11736684 #14221306

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
1C2×#90208C211yesmaximal, characteristic, complemented
2C22×#6456C211yesmaximal, characteristic
3C2×C7×#428C211yesmaximal, characteristic, complemented
4C2×#89903C211yesmaximal, characteristic
5C2×#89890C211yesmaximal, characteristic, complemented
6C2×#89773C211yesmaximal, characteristic
7C2×#89736C211yesmaximal, characteristic, complemented
8#11736684C2116yesmaximal, direct product factor
9C2×#89663C212yesmaximal, complemented
10C2×#89662C212yesmaximal, complemented
11C2×#89643C212yesmaximal, complemented
12C2×#89642C212yesmaximal, complemented
13C23×D28C2211yescharacteristic
14C23×C4×D7—21no—
15C22×C7×M16C2211yescharacteristic
16C2×C7×#117C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×(C2×D7)⋊C4—21no—
19C2×#6489C2211yescharacteristic
20C22×C4×Dic7C2211yescharacteristic
21C22×#1136C2211yescharacteristic
22#90208—24no—
23#90208C2218yescomplemented
24C2×#6456—21no—
25C2×#6456—21no—
26C2×#6456—24no—
27C2×#6456C2212yes—
28C2×#6456C2212yes—
29C2×#6456C2218yes—
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×#428C2218yescomplemented
39C2×C7×#107C2212yescomplemented
40C2×C7×#106C2212yescomplemented
41#89903C2218yes—
42#89890C2218yescomplemented
43C2×#6395C2212yes—
44C2×#6395C2212yes—
45C2×Dic7⋊C8C2212yescomplemented
46C2×Dic7⋊C8C2212yescomplemented
47#89773C2218yes—
48C2×#6383C2212yes—
49C2×#6383C2212yes—
50#89736C2218yescomplemented
51C2×#6382C2212yescomplemented
52C2×#6381C2212yescomplemented
53#89663C22116yescomplemented
54#89662C22116yescomplemented
55#89643C22116yescomplemented
56#89642C22116yescomplemented
57C2×#3077—71nomaximal, Sylow
58C24×D7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C23×Dic7D411yescharacteristic
62C23×Dic7D411yescharacteristic
63C22×D28—21no—
64C22×D28—22no—
65C22×D28—24no—
66C22×D28Q818yescomplemented
67C22×D28C2312yes—
68C22×D28C2314yes—
69C22×C4×D7—28no—
70C22×C4×D7—22no—
71C22×C4×D7—22no—
72C22×C4×D7—48no—
73C22×Dic14—21no—
74C22×Dic14—21no—
75C2×C7×M16—24no—
76C2×C7×M16C2314yes—
77C22×C7×C8C2312yes—
78C7×#117—22no—
79C7×#117C2314yes—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2312yes—
85C2×C7×C4⋊C4C2314yes—
86C2×C7×C22⋊C4—21no—
87C2×C42×C7—21no—
88C2×C22⋊Dic7—21no—
89C2×(C2×D7)⋊C4—22no—
90C2×(C2×D7)⋊C4—22no—
91C2×(C2×D7)⋊C4—28no—
92#6489—22no—
93#6489C2314yes—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×Dic7⋊C4—21no—
101C2×Dic7⋊C4—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7D411yescharacteristic
104C2×C4×Dic7D411yescharacteristic
105C2×C4×Dic7D414yes—
106C2×C4×Dic7C2311yescharacteristic
107C2×C4×Dic7C2311yescharacteristic
108C2×C4×Dic7C2314yes—
109C2×#1136—24no—
110C2×#1136C2314yes—
111C22×C7⋊C8C2312yes—
112#6456—28no—
113#6456—28no—
114#6456—24no—
115#6456—24no—
116#6456C2318yes—
117#6456C2318yes—
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×#107C2318yescomplemented
131C7×#106C2318yescomplemented
132#6395C2318yes—
133#6395C2318yes—
134Dic7⋊C8C2318yescomplemented
135Dic7⋊C8C2318yescomplemented
136#6383C2318yes—
137#6383C2318yes—
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×#428D711yescharacteristic, complemented
149C2×#423—71no—
150C2×#418—71no—
151C2×#418—71no—
152C24×C7C2×D411yescharacteristic
153C23×D7—28no—
154C23×D7—22no—
155C23×D7—24no—
156C23×D7—44no—
157C23×D7—44no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—24no—
161C22×C4×C7—22no—
162C22×C4×C7—44no—
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×Q814yes—
166C22×C4×C7#4114yes—
167C22×C4×C7C2411yescharacteristic
168C22×C4×C7C2412yes—
169C22×Dic7—24no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—42no—
173C22×Dic7C2×D414yes—
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C22×Dic7C2×D414yes—
178C22×Dic7C2×D411yescharacteristic
179C2×D28—216no—
180C2×D28—28no—
181C2×D28—22no—
182C2×D28—24no—
183C2×D28—24no—
184C2×D28—28no—
185C2×D28C2×Q818yes—
186C2×D28C2×Q8116yescomplemented
187C2×D28C2414yes—
188C2×C4×D7—28no—
189C2×C4×D7—28no—
190C2×C4×D7—48no—
191C2×C4×D7—432no—
192C2×C4×D7—48no—
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⋊C4—24no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4C2412yes—
205C7×C4⋊C4C2414yes—
206C7×C4⋊C4C2412yes—
207C7×C22⋊C4—24no—
208C42×C7—24no—
209C22⋊Dic7—24no—
210(C2×D7)⋊C4—28no—
211(C2×D7)⋊C4—28no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—24no—
215C4⋊Dic7—24no—
216C4⋊Dic7C2412yes—
217C4⋊Dic7C2412yes—
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×D4—71no—
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⋊C4C2×D711yescharacteristic
255C22×C22⋊C4—141no—
256C22×C42—71no—
257#478—716no—
258#478—716no—
259#476—716no—
260#474—716no—
261#428—78no—
262#428C2×D718yescomplemented
263C2×#107—72no—
264C2×#107C2×D712yescomplemented
265C2×#106—72no—
266C2×#106C2×D712yescomplemented
267#423—78no—
268C2×C4⋊C8—72no—
269C2×C4⋊C8—72no—
270#418—78no—
271#418—78no—
272C2×#102—72no—
273C2×#102—72no—
274C2×#102—72no—
275C2×#102—72no—
276C23×C7—22no—
277C23×C7—22no—
278C23×C7C22⋊Q814yes—
279C23×C7C22×D412yes—
280C23×C7C22×D411yescharacteristic
281C22×D7—28no—
282C22×D7—216no—
283C22×D7—24no—
284C22×D7—48no—
285C22×D7—432no—
286C22×D7—48no—
287C22×D7—44no—
288C22×D7—44no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—24no—
292C2×C4×C7—48no—
293C2×C4×C7—42no—
294C2×C4×C7—44no—
295C2×C4×C7—42no—
296C2×C4×C7C22⋊Q818yes—
297C2×C4×C7C22⋊Q814yes—
298C2×C4×C7C22×D412yes—
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C22×Q812yes—
301C2×C4×C7C22×Q814yes—
302C2×C4×C7C2×#4112yes—
303C2×C4×C7C2×#4114yes—
304C2×C4×C7C2511yescharacteristic
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—28no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—44no—
312C2×Dic7—42no—
313C2×Dic7—42no—
314C2×Dic7AGL(1,ℤ/8)14yes—
315C2×Dic7C22×D412yes—
316C2×Dic7C22×D412yes—
317C2×Dic7C22×D412yes—
318C2×Dic7C22×D412yes—
319D28—28no—
320D28—216no—
321D28—24no—
322D28C22×Q818yes—
323C4×D7—416no—
324C4×D7—416no—
325Dic14—48no—
326Dic14—48no—
327C7×C8—24no—
328C7⋊C8—24no—
329C25—141no—
330C22×Q8—141no—
331C22×Q8—141no—
332C22×D4—78no—
333C22×D4—72no—
334C22×D4—74no—
335C22×D4—144no—
336C22×D4—142no—
337C22×D4—141no—
338C23×C4—21no—
339C23×C4—71no—
340C23×C4—71no—
341C23×C4—148no—
342C23×C4—142no—
343C23×C4—142no—
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—22no—
375C2×C4⋊C4—21no—
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⋊C4C22×D712yes—
385C2×C4⋊C4C22×D714yes—
386C2×C4⋊C4C22×D711yescharacteristic
387C2×C4⋊C4C22×D711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—148no—
390C2×C22⋊C4—142no—
391C2×C22⋊C4—142no—
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#107C22×D718yescomplemented
403#106—78no—
404#106C22×D718yescomplemented
405C4⋊C8—78no—
406C4⋊C8—78no—
407#102—78no—
408#102—78no—
409#102—78no—
410#102—78no—
411C22×C7—28no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7C2×C22⋊Q814yes—
415C22×C7C2×C22⋊Q812yes—
416C22×C7C2×AGL(1,ℤ/8)12yes—
417C22×C7#57512yes—
418C22×C7C23×D411yescharacteristic
419C2×D7—28no—
420C2×D7—432no—
421C2×D7—416no—
422C2×D7—44no—
423C2×D7—44no—
424C4×C7—42no—
425C4×C7—44no—
426C4×C7—42no—
427C4×C7C2×C22⋊Q812yes—
428C4×C7C2×C22⋊Q814yes—
429C4×C7C23×Q811yescharacteristic
430C4×C7C22×#4111yescharacteristic
431Dic7—22no—
432Dic7—22no—
433Dic7—22no—
434Dic7—42no—
435Dic7—42no—
436Dic7C2×AGL(1,ℤ/8)14yes—
437C24—148no—
438C24—144no—
439C24—142no—
440C24—284no—
441C24—284no—
442C24D4×D711yescharacteristic
443C2×Q8—144no—
444C2×Q8—144no—
445C2×Q8—284no—
446C2×Q8—284no—
447C2×D4—78no—
448C2×D4—716no—
449C2×D4—74no—
450C2×D4—1416no—
451C2×D4—148no—
452C2×D4—144no—
453C2×D4—148no—
454C2×D4—144no—
455C2×D4—142no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—44no—
461C22×C4—71no—
462C22×C4—74no—
463C22×C4—71no—
464C22×C4—71no—
465C22×C4—71no—
466C22×C4—74no—
467C22×C4—148no—
468C22×C4—148no—
469C22×C4—144no—
470C22×C4—141no—
471C22×C4—141no—
472C22×C4—288no—
473C22×C4—2832no—
474C22×C4—288no—
475C22×C4—282no—
476C22×C4#115714yes—
477C22×C4D4×D712yes—
478C22×C4D4×D711yescharacteristic
479C22×C4Q8×D714yes—
480C22×C4C23×D712yes—
481C22×C4C23×D711yescharacteristic
482M16—28no—
483M16—148no—
484C2×C8—24no—
485C2×C8—74no—
486C2×C8—144no—
487C2×C8C23×D714yes—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—72no—
493C4⋊C4—72no—
494C4⋊C4—144no—
495C4⋊C4—144no—
496C4⋊C4—144no—
497C4⋊C4—144no—
498C4⋊C4—144no—
499C4⋊C4—144no—
500C4⋊C4—141no—
501C4⋊C4—141no—
502C4⋊C4C23×D712yes—
503C4⋊C4C23×D712yes—
504C4⋊C4C23×D714yes—
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#307714yescomplemented
519C2×C7C22×C22⋊Q811yescharacteristic
520C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
521C2×C7C2×#57511yescharacteristic
522D7—48no—
523C23—22no—
524C23—22no—
525C23—1416no—
526C23—148no—
527C23—144no—
528C23—288no—
529C23—288no—
530C23—2832no—
531C23—284no—
532C23—284no—
533C23#645914yes—
534C23C2×D4×D712yes—
535C23C2×D4×D711yescharacteristic
536Q8—288no—
537Q8—288no—
538D4—78no—
539D4—148no—
540D4—1416no—
541D4—144no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—74no—
550C2×C4—72no—
551C2×C4—72no—
552C2×C4—72no—
553C2×C4—72no—
554C2×C4—148no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—2816no—
561C2×C4—2816no—
562C2×C4—284no—
563C2×C4—282no—
564C2×C4—282no—
565C2×C4#645918yes—
566C2×C4#645914yes—
567C2×C4C2×#115712yes—
568C2×C4C2×#115714yes—
569C2×C4C2×D4×D712yes—
570C2×C4C2×D4×D711yescharacteristic
571C2×C4C2×Q8×D714yes—
572C2×C4C2×Q8×D712yes—
573C2×C4C24×D711yescharacteristic
574C8—24no—
575C8—144no—
576C7C2×#307711yescharacteristic, complemented, Sylow
577C22—22no—
578C22—22no—
579C22—28no—
580C22—148no—
581C22—2832no—
582C22—284no—
583C22—284no—
584C22—2816no—
585C22C2×#645914yes—
586C22C2×#645912yes—
587C22D7×AGL(1,ℤ/8)12yes—
588C22#9046812yes—
589C22C22×D4×D711yescharacteristic
590C4—44no—
591C4—42no—
592C4—42no—
593C4—74no—
594C4—142no—
595C4—142no—
596C4—142no—
597C4—282no—
598C4—282no—
599C4C2×#645912yes—
600C4C2×#645914yes—
601C4C22×#115711yescharacteristic
602C4C22×Q8×D711yescharacteristic
603C2—24no—
604C2—288no—
605C2#1173668414yesdirect product factor
606C2C22×#645911yescharacteristic
607C2C2×D7×AGL(1,ℤ/8)11yescharacteristic
608C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#428C2
socleC23×C7C22×D4
prime radicalC23×D28C22
hypercenterC2×#428D7
hypocenterC7C2×#3077
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#90208C2semidirect
3. C2×C7×#428C2semidirect
5. C2×#89890C2semidirect
7. C2×#89736C2semidirect
8. #11736684C2direct (N × H)
9. C2×#89663C2semidirect
10. C2×#89662C2semidirect
11. C2×#89643C2semidirect
12. C2×#89642C2semidirect
23. #90208C22semidirect
30. C2×#6453C22semidirect
31. C2×#6453C22semidirect
32. C2×#6451C22semidirect
33. C2×#6451C22semidirect
38. C7×#428C22semidirect
39. C2×C7×#107C22semidirect
40. C2×C7×#106C22semidirect
42. #89890C22semidirect
45. C2×Dic7⋊C8C22semidirect
46. C2×Dic7⋊C8C22semidirect
50. #89736C22semidirect
51. C2×#6382C22semidirect
52. C2×#6381C22semidirect
53. #89663C22semidirect
54. #89662C22semidirect
55. #89643C22semidirect
56. #89642C22semidirect
66. C22×D28Q8semidirect
120. #6453C23semidirect
121. #6453C23semidirect
124. #6451C23semidirect
125. #6451C23semidirect
130. C7×#107C23semidirect
131. C7×#106C23semidirect
134. Dic7⋊C8C23semidirect
135. Dic7⋊C8C23semidirect
138. #6382C23semidirect
139. #6381C23semidirect
148. C2×#428D7semidirect
186. C2×D28C2×Q8semidirect
262. #428C2×D7semidirect
264. C2×#107C2×D7semidirect
266. C2×#106C2×D7semidirect
402. #107C22×D7semidirect
404. #106C22×D7semidirect
518. C2×C7#3077semidirect
576. C7C2×#3077semidirect
605. C2#11736684direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902082
C22×#64562
C2×C7×#4282
C2×#899032
C2×#898902
C2×#897732
C2×#897362
#117366842
C2×#896632
C2×#896622
C2×#896432
C2×#896422
C2×#30777