Finite Group Databasebeta

← C2×#11737475 #14221985

Subgroups — C2×#11737475 #14221985

Order 1792.

2398 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic7C211yesmaximal, characteristic, complemented
2C2×#90287C211yesmaximal, characteristic
3C2×C7×#448C211yesmaximal, characteristic, complemented
4C2×#89999C211yesmaximal, characteristic, complemented
5C2×#89935C212yesmaximal, complemented
6C2×#89936C212yesmaximal, complemented
7C2×#89805C211yesmaximal, characteristic, complemented
8#11737475C2116yesmaximal, direct product factor
9C2×#89566C212yesmaximal, complemented
10C2×#89589C212yesmaximal, complemented
11C2×#89508C211yesmaximal, characteristic, complemented
12C2×#89491C211yesmaximal, characteristic, complemented
13C23×C7×Q8C2211yescharacteristic
14C22×#1159—21no—
15C22×C7×SD16C2212yes—
16C2×C7×#121C2211yescharacteristic
17C2×Q8×Dic7—21no—
18C2×Q8×Dic7—21no—
19C2×Q8×Dic7—24no—
20C2×Q8×Dic7C2212yes—
21C2×Q8×Dic7C2212yes—
22C2×Q8×Dic7C2218yescomplemented
23C22×#1143C2212yes—
24C2×#6503C2211yescharacteristic
25C2×#6502—21no—
26C2×D4×Dic7C2211yescharacteristic
27C2×#6493C2211yescharacteristic
28C22×C4⋊Dic7—21no—
29C22×C4⋊Dic7C2211yescharacteristic
30C22×C4×Dic7—21no—
31C22×C4×Dic7C2211yescharacteristic
32#90287—24no—
33#90287C2218yes—
34C2×#6456C2211yescharacteristic
35C2×#6451C2211yescharacteristic
36C2×#6448—21no—
37C2×(C22⋊D7)⋊C4—21no—
38C7×#448C2218yescomplemented
39C2×C7×#103C2212yes—
40C2×C7×C22⋊C8C2211yescharacteristic
41#89999C2218yescomplemented
42C2×#6409C2212yes—
43C2×#6406C2212yes—
44C2×C22⋊(C7⋊C8)C2211yescharacteristic
45#89936C22116yescomplemented
46#89935C22116yescomplemented
47C2×#6395C2212yes—
48C2×#6392C2212yes—
49C2×#6389C2212yes—
50C2×Dic7⋊C8C2212yes—
51#89805C2218yescomplemented
52C2×#6381C2212yes—
53#89589C22116yescomplemented
54#89566C22116yescomplemented
55#89508C2218yescomplemented
56#89491C2218yescomplemented
57C2×#3112—71nomaximal, Sylow
58C22×C7×Q8—28no—
59C22×C7×Q8—21no—
60C22×C7×Q8—22no—
61C22×C7×Q8C2314yes—
62C22×C7×Q8C2312yes—
63C22×C7×D4—21no—
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—21no—
66C23×C4×C7C2311yescharacteristic
67C22×C22⋊D7—21no—
68C22×C22⋊D7—21no—
69C22×C22⋊D7—21no—
70C23×Dic7—21no—
71C23×Dic7—21no—
72C23×Dic7D411yescharacteristic
73C23×Dic7D411yescharacteristic
74C2×#1159—28no—
75C2×#1159—48no—
76C22×D28C2311yescharacteristic
77C22×C4×D7—21no—
78C22×Dic14—21no—
79C2×C7×SD16—28no—
80C2×C7×SD16C2318yes—
81C22×C7×C8C2312yes—
82C7×#121—22no—
83C7×#121C2314yes—
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C22⋊C4—21no—
86C2×C22⋊Dic7—21no—
87Q8×Dic7—24no—
88Q8×Dic7—24no—
89Q8×Dic7—28no—
90Q8×Dic7—28no—
91Q8×Dic7C2318yes—
92Q8×Dic7C2318yes—
93C2×#1143—28no—
94C2×#1143C2318yes—
95#6503—22no—
96#6503C2314yes—
97#6502—28no—
98D4×Dic7—22no—
99D4×Dic7C2314yes—
100#6493—22no—
101#6493C2314yes—
102C2×(C2×D7)⋊C4—21no—
103C2×(C2×D7)⋊C4—21no—
104C2×C4⋊Dic7—21no—
105C2×C4⋊Dic7—22no—
106C2×C4⋊Dic7—22no—
107C2×C4⋊Dic7—22no—
108C2×C4⋊Dic7—28no—
109C2×C4⋊Dic7C2311yescharacteristic
110C2×C4⋊Dic7C2312yes—
111C2×C4⋊Dic7C2314yes—
112C2×Dic7⋊C4—21no—
113C2×C4×Dic7—22no—
114C2×C4×Dic7—22no—
115C2×C4×Dic7—28no—
116C2×C4×Dic7D411yescharacteristic
117C2×C4×Dic7D411yescharacteristic
118C2×C4×Dic7D414yes—
119C2×C4×Dic7C2311yescharacteristic
120C2×C4×Dic7C2311yescharacteristic
121C2×C4×Dic7C2314yes—
122C22×C7⋊C8C2312yes—
123#6456—22no—
124#6456C2314yes—
125#6451—22no—
126#6451C2314yes—
127#6448—28no—
128(C22⋊D7)⋊C4—28no—
129C7×#103C2318yes—
130C7×C22⋊C8C2314yes—
131#6409C2318yes—
132#6406C2318yes—
133C22⋊(C7⋊C8)C2314yes—
134#6395C2318yes—
135#6392C2318yes—
136#6389C2318yes—
137Dic7⋊C8C2318yes—
138#6381C2318yes—
139C2×#532—71no—
140C22×C4×Q8—71no—
141#3112—716no—
142C2×#480—71no—
143C2×#480—71no—
144C2×#477—72no—
145C2×#460—72no—
146C2×#448—71no—
147C2×#448D711yescharacteristic, complemented
148C2×#440—72no—
149C2×#440—72no—
150C2×#432—71no—
151C24×C7—21no—
152C24×C7C2×D411yescharacteristic
153C23×D7—21no—
154C2×C7×Q8—216no—
155C2×C7×Q8—28no—
156C2×C7×Q8—216no—
157C2×C7×Q8—24no—
158C2×C7×Q8—22no—
159C2×C7×Q8—24no—
160C2×C7×Q8#4118yescomplemented
161C2×C7×Q8C2414yes—
162C2×C7×D4—24no—
163C2×C7×D4—21no—
164C2×C7×D4—24no—
165C2×C7×D4—44no—
166C2×C7×D4C2412yes—
167C22×C4×C7—28no—
168C22×C4×C7—24no—
169C22×C4×C7—22no—
170C22×C4×C7—24no—
171C22×C4×C7—21no—
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D412yes—
174C22×C4×C7C2412yes—
175C22×C4×C7C2411yescharacteristic
176C2×C22⋊D7—24no—
177C2×C22⋊D7—24no—
178C2×C22⋊D7—24no—
179C2×C22⋊D7—44no—
180C2×C22⋊D7—44no—
181C2×C22⋊D7—44no—
182C22×Dic7—24no—
183C22×Dic7—24no—
184C22×Dic7—24no—
185C22×Dic7—22no—
186C22×Dic7—21no—
187C22×Dic7—24no—
188C22×Dic7—44no—
189C22×Dic7—44no—
190C22×Dic7C2×D411yescharacteristic
191C22×Dic7C2×D414yes—
192C22×Dic7C2×D411yescharacteristic
193C22×Dic7C2×D411yescharacteristic
194C22×Dic7C2×D414yes—
195C22×Dic7C2×D411yescharacteristic
196#1159—432no—
197C2×D28—24no—
198C2×D28—21no—
199C2×D28C2412yes—
200C2×C4×D7—24no—
201C2×C4×D7—44no—
202C2×Dic14—24no—
203C2×Dic14—44no—
204C7×SD16—216no—
205C2×C7×C8—24no—
206C2×C7×C8C2414yes—
207C7×C4⋊C4—21no—
208C7×C4⋊C4C2412yes—
209C7×C22⋊C4—24no—
210C22⋊Dic7—24no—
211#1143—216no—
212(C2×D7)⋊C4—24no—
213(C2×D7)⋊C4—24no—
214C4⋊Dic7—21no—
215C4⋊Dic7—24no—
216C4⋊Dic7—22no—
217C4⋊Dic7—28no—
218C4⋊Dic7—28no—
219C4⋊Dic7C2414yes—
220C4⋊Dic7C2412yes—
221Dic7⋊C4—24no—
222C4×Dic7—28no—
223C4×Dic7—28no—
224C4×Dic7C2×D414yes—
225C4×Dic7C2×D414yes—
226C4×Dic7C2×D412yes—
227C4×Dic7C2×D412yes—
228C4×Dic7C2412yes—
229C4×Dic7C2412yes—
230C2×C7⋊C8—24no—
231C2×C7⋊C8C2414yes—
232C22×#41—141no—
233C23×Q8C2×D711yescharacteristic
234C22×SD16—72no—
235C22×SD16C2×D712yes—
236#532—78no—
237#532—144no—
238C2×C4⋊Q8—71no—
239C2×#127—71no—
240C2×#124—141no—
241C2×#121—71no—
242C2×#121—141no—
243C2×#121C2×D711yescharacteristic
244C2×C4×Q8—72no—
245C2×C4×Q8—78no—
246C2×C4×Q8—72no—
247C2×C4×Q8—141no—
248C2×C4×Q8—141no—
249C2×C4×Q8—144no—
250C2×C4×D4—71no—
251C2×C4×D4—71no—
252C2×C4×D4—141no—
253C22×C4⋊C4—71no—
254C22×C4⋊C4—141no—
255C22×C42—71no—
256C22×C42—141no—
257#480—78no—
258#480—78no—
259#477—716no—
260#460—716no—
261#448—78no—
262#448C2×D718yescomplemented
263#440—716no—
264#440—716no—
265#432—78no—
266C2×#107—72no—
267C2×#107—72no—
268C2×C4⋊C8—72no—
269C2×#103—72no—
270C2×#103C2×D712yes—
271C2×#102—72no—
272C2×#102—72no—
273C2×C22⋊C8—71no—
274C2×C22⋊C8C2×D711yescharacteristic
275C2×#97—72no—
276C23×C7—22no—
277C23×C7—22no—
278C23×C7—22no—
279C23×C7—42no—
280C23×C7—42no—
281C23×C7—42no—
282C23×C7AGL(1,ℤ/8)14yes—
283C23×C7C22×D412yes—
284C23×C7C22×D411yescharacteristic
285C22×D7—22no—
286C22×D7—42no—
287C22×D7—42no—
288C22×D7—42no—
289C7×Q8—216no—
290C7×Q8—28no—
291C7×Q8—24no—
292C7×Q8C2×#4118yescomplemented
293C7×D4—24no—
294C7×D4—48no—
295C7×D4—42no—
296C2×C4×C7—216no—
297C2×C4×C7—28no—
298C2×C4×C7—22no—
299C2×C4×C7—24no—
300C2×C4×C7—24no—
301C2×C4×C7—24no—
302C2×C4×C7—22no—
303C2×C4×C7—42no—
304C2×C4×C7C22×D412yes—
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C2×#4114yes—
307C2×C4×C7C2511yescharacteristic
308C22⋊D7—48no—
309C22⋊D7—48no—
310C22⋊D7—48no—
311C2×Dic7—22no—
312C2×Dic7—22no—
313C2×Dic7—28no—
314C2×Dic7—22no—
315C2×Dic7—28no—
316C2×Dic7—24no—
317C2×Dic7—22no—
318C2×Dic7—48no—
319C2×Dic7—42no—
320C2×Dic7—48no—
321C2×Dic7—44no—
322C2×Dic7#13714yes—
323C2×Dic7C22×D412yes—
324C2×Dic7C22×D412yes—
325C2×Dic7C22×D412yes—
326C2×Dic7C22×D412yes—
327D28—24no—
328D28—42no—
329C4×D7—48no—
330Dic14—48no—
331C7×C8—24no—
332C7⋊C8—24no—
333C2×#41—148no—
334C2×#41—288no—
335C22×Q8—28no—
336C22×Q8—22no—
337C22×Q8—21no—
338C22×Q8—141no—
339C22×Q8C22×D714yes—
340C22×Q8C22×D712yes—
341C22×D4—21no—
342C22×D4—71no—
343C22×D4—141no—
344C22×D4—141no—
345C22×D4—141no—
346C22×D4C22×D711yescharacteristic
347C23×C4—21no—
348C23×C4—71no—
349C23×C4—71no—
350C23×C4—141no—
351C23×C4—141no—
352C23×C4—141no—
353C23×C4C22×D711yescharacteristic
354C2×SD16—28no—
355C2×SD16—78no—
356C2×SD16—148no—
357C2×SD16C22×D718yes—
358C22×C8—72no—
359C22×C8C22×D712yes—
360C4⋊Q8—74no—
361C4⋊Q8—142no—
362#127—74no—
363#127—142no—
364#124—148no—
365#121—22no—
366#121—74no—
367#121—148no—
368#121—142no—
369#121C22×D714yes—
370C4×Q8—78no—
371C4×Q8—78no—
372C4×Q8—148no—
373C4×Q8—144no—
374C4×Q8—148no—
375C4×Q8—144no—
376C4×D4—74no—
377C4×D4—74no—
378C4×D4—148no—
379C4×D4—142no—
380C4×D4—142no—
381C2×C4⋊C4—72no—
382C2×C4⋊C4—74no—
383C2×C4⋊C4—71no—
384C2×C4⋊C4—148no—
385C2×C4⋊C4—142no—
386C2×C4⋊C4—142no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—142no—
389C2×C4⋊C4—141no—
390C2×C4⋊C4C22×D711yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—141no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4—141no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—74no—
398C2×C42—71no—
399C2×C42—71no—
400C2×C42—74no—
401C2×C42—148no—
402C2×C42—142no—
403C2×C42—142no—
404#107—78no—
405#107—78no—
406C4⋊C8—78no—
407#103—78no—
408#103C22×D718yes—
409#102—78no—
410#102—78no—
411C22⋊C8—74no—
412C22⋊C8C22×D714yes—
413#97—78no—
414C22×C7—28no—
415C22×C7—22no—
416C22×C7—22no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—84no—
421C22×C7#52512yes—
422C22×C7C2×AGL(1,ℤ/8)14yes—
423C22×C7C2×AGL(1,ℤ/8)12yes—
424C22×C7C2×#13712yes—
425C22×C7C23×D411yescharacteristic
426C2×D7—42no—
427C2×D7—42no—
428C2×D7—42no—
429C2×D7—84no—
430C4×C7—28no—
431C4×C7—22no—
432C4×C7—42no—
433C4×C7#52512yes—
434C4×C7C22×#4112yes—
435Dic7—22no—
436Dic7—22no—
437Dic7—22no—
438Dic7—42no—
439Dic7—44no—
440Dic7C2×#13714yes—
441C24—21no—
442C24—141no—
443C24D4×D711yescharacteristic
444#41—2832no—
445C2×Q8—216no—
446C2×Q8—28no—
447C2×Q8—216no—
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—144no—
452C2×Q8—284no—
453C2×Q8#115918yescomplemented
454C2×Q8C23×D714yes—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—24no—
458C2×D4—44no—
459C2×D4—72no—
460C2×D4—144no—
461C2×D4—144no—
462C2×D4—144no—
463C2×D4—141no—
464C2×D4—144no—
465C2×D4—284no—
466C2×D4—284no—
467C2×D4—284no—
468C2×D4C23×D712yes—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—71no—
475C22×C4—74no—
476C22×C4—74no—
477C22×C4—71no—
478C22×C4—71no—
479C22×C4—71no—
480C22×C4—144no—
481C22×C4—144no—
482C22×C4—144no—
483C22×C4—144no—
484C22×C4—144no—
485C22×C4—142no—
486C22×C4—141no—
487C22×C4—284no—
488C22×C4—284no—
489C22×C4—284no—
490C22×C4D4×D712yes—
491C22×C4D4×D711yescharacteristic
492C22×C4C23×D712yes—
493C22×C4C23×D711yescharacteristic
494SD16—216no—
495SD16—1416no—
496C2×C8—24no—
497C2×C8—74no—
498C2×C8—144no—
499C2×C8C23×D714yes—
500C4⋊C4—21no—
501C4⋊C4—74no—
502C4⋊C4—72no—
503C4⋊C4—144no—
504C4⋊C4—148no—
505C4⋊C4—148no—
506C4⋊C4—144no—
507C4⋊C4—142no—
508C4⋊C4—141no—
509C4⋊C4C23×D712yes—
510C22⋊C4—24no—
511C22⋊C4—144no—
512C22⋊C4—144no—
513C22⋊C4—144no—
514C42—74no—
515C42—72no—
516C42—74no—
517C42—72no—
518C42—72no—
519C42—72no—
520C42—148no—
521C42—148no—
522C2×C7—24no—
523C2×C7—82no—
524C2×C7#311214yescomplemented
525C2×C7C2×#52511yescharacteristic
526C2×C7C22×AGL(1,ℤ/8)11yescharacteristic
527C2×C7C22×#13711yescharacteristic
528D7—82no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—142no—
536C23—282no—
537C23—282no—
538C23—282no—
539C23#647714yes—
540C23C2×D4×D712yes—
541C23C2×D4×D711yescharacteristic
542Q8—216no—
543Q8—28no—
544Q8—24no—
545Q8—288no—
546Q8C2×#115918yescomplemented
547D4—24no—
548D4—48no—
549D4—42no—
550D4—144no—
551D4—288no—
552D4—288no—
553D4—288no—
554D4—282no—
555C2×C4—216no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—74no—
564C2×C4—72no—
565C2×C4—72no—
566C2×C4—72no—
567C2×C4—72no—
568C2×C4—148no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—148no—
572C2×C4—142no—
573C2×C4—144no—
574C2×C4—142no—
575C2×C4—288no—
576C2×C4—288no—
577C2×C4—288no—
578C2×C4—284no—
579C2×C4—282no—
580C2×C4C2×D4×D712yes—
581C2×C4C2×D4×D711yescharacteristic
582C2×C4C2×#115914yes—
583C2×C4C24×D711yescharacteristic
584C8—24no—
585C8—144no—
586C7C2×#311211yescharacteristic, complemented, Sylow
587C22—22no—
588C22—22no—
589C22—28no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—84no—
594C22—282no—
595C22—282no—
596C22—282no—
597C22—564no—
598C22#9027712yes—
599C22C2×#647714yes—
600C22C2×#647712yes—
601C22D7×#13712yes—
602C22C22×D4×D711yescharacteristic
603C4—28no—
604C4—22no—
605C4—42no—
606C4—74no—
607C4—142no—
608C4—142no—
609C4—142no—
610C4—282no—
611C4—284no—
612C4#9027712yes—
613C4C22×#115912yes—
614C2—24no—
615C2—82no—
616C2—562no—
617C2#1173747514yesdirect product factor
618C2C2×#9027711yescharacteristic
619C2C22×#647711yescharacteristic
620C2C2×D7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#448C2
socleC23×C7C22×D4
prime radicalC2×#90287C2
hypercenterC2×#448D7
hypocenterC7C2×#3112
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic7C2semidirect
3. C2×C7×#448C2semidirect
4. C2×#89999C2semidirect
5. C2×#89935C2semidirect
6. C2×#89936C2semidirect
7. C2×#89805C2semidirect
8. #11737475C2direct (N × H)
9. C2×#89566C2semidirect
10. C2×#89589C2semidirect
11. C2×#89508C2semidirect
12. C2×#89491C2semidirect
22. C2×Q8×Dic7C22semidirect
38. C7×#448C22semidirect
41. #89999C22semidirect
45. #89936C22semidirect
46. #89935C22semidirect
51. #89805C22semidirect
53. #89589C22semidirect
54. #89566C22semidirect
55. #89508C22semidirect
56. #89491C22semidirect
147. C2×#448D7semidirect
160. C2×C7×Q8#41semidirect
262. #448C2×D7semidirect
292. C7×Q8C2×#41semidirect
453. C2×Q8#1159semidirect
524. C2×C7#3112semidirect
546. Q8C2×#1159semidirect
586. C7C2×#3112semidirect
617. C2#11737475direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×Dic72
C2×#902872
C2×C7×#4482
C2×#899992
C2×#899352
C2×#899362
C2×#898052
#117374752
C2×#895662
C2×#895892
C2×#895082
C2×#894912
C2×#31127