Finite Group Databasebeta

← C2×#11739796 #13522210

Subgroups — C2×#11739796 #13522210

Order 1792.

3526 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×D4C211yesmaximal, characteristic
2C22×#6487C211yesmaximal, characteristic, complemented
3C22×#6487C211yesmaximal, characteristic, complemented
4C22×#6489C211yesmaximal, characteristic, complemented
5C2×#89994C212yesmaximal
6C2×#89984C211yesmaximal, characteristic, complemented
7#11739796C2116yesmaximal, direct product factor
8C2×#89744C212yesmaximal, complemented
9C2×#89724C212yesmaximal, complemented
10C2×#89717C212yesmaximal, complemented
11C2×#89701C211yesmaximal, characteristic, complemented
12C2×#89700C211yesmaximal, characteristic, complemented
13C23×C7×D4C2211yescharacteristic
14C24×C4×C7C2211yescharacteristic
15C24×C4×C7C2211yescharacteristic
16C23×Dic14C2211yescharacteristic
17C2×C4×C7×D4—28no—
18C2×C4×C7×D4C2218yes—
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4C2211yescharacteristic
21C22×C7×C22⋊C4C2211yescharacteristic
22C22×C42×C7C2211yescharacteristic
23C22×C22⋊Dic7C2212yes—
24C22×C22⋊Dic7C2212yes—
25C22×C22⋊Dic7C2212yes—
26C22×C22⋊Dic7C2212yes—
27C2×#6489—28no—
28C2×#6489C2218yescomplemented
29C22×C4⋊Dic7C2211yescharacteristic
30C22×C4⋊Dic7C2211yescharacteristic
31C22×C4⋊Dic7C2211yescharacteristic
32C2×#6487—28no—
33C2×#6487—28no—
34C2×#6487C2218yescomplemented
35C2×#6487C2218yescomplemented
36C22×Dic7⋊C4C2212yes—
37C22×Dic7⋊C4C2212yes—
38C22×Dic7⋊C4C2212yes—
39C22×C4×Dic7C2212yes—
40#89994C22116yes—
41#89984C2218yescomplemented
42C2×C7⋊#95C2212yes—
43C2×C7⋊#95C2212yes—
44C2×C7⋊#95C2212yes—
45C2×C7⋊#95C2212yes—
46#89744C22116yescomplemented
47#89724C22116yescomplemented
48#89717C22116yescomplemented
49#89701C2218yescomplemented
50#89700C2218yescomplemented
51C2×#2415—71nomaximal, Sylow
52C25×C7C2311yescharacteristic
53C25×C7C2311yescharacteristic
54C22×C7×D4—24no—
55C22×C7×D4—24no—
56C22×C7×D4—24no—
57C22×C7×D4C2314yes—
58C23×C4×C7—24no—
59C23×C4×C7—24no—
60C23×C4×C7—24no—
61C23×C4×C7—24no—
62C23×C4×C7—24no—
63C23×C4×C7D418yes—
64C23×C4×C7C2311yescharacteristic
65C23×C4×C7C2314yes—
66C23×C4×C7C2311yescharacteristic
67C23×C4×C7C2311yescharacteristic
68C23×C4×C7C2314yes—
69C23×C4×C7C2311yescharacteristic
70C23×C4×C7C2311yescharacteristic
71C23×Dic7C2312yes—
72C23×Dic7C2312yes—
73C23×Dic7C2312yes—
74C23×Dic7C2312yes—
75C22×Dic14—24no—
76C22×Dic14—24no—
77C22×Dic14—24no—
78C22×Dic14C2314yes—
79C4×C7×D4—232no—
80C2×C7×C4⋊C4—24no—
81C2×C7×C4⋊C4C2314yes—
82C2×C7×C22⋊C4—24no—
83C2×C7×C22⋊C4—24no—
84C2×C7×C22⋊C4C2314yes—
85C2×C7×C22⋊C4C2314yes—
86C2×C42×C7—24no—
87C2×C42×C7C2314yes—
88C2×C22⋊Dic7—28no—
89C2×C22⋊Dic7—28no—
90C2×C22⋊Dic7—28no—
91C2×C22⋊Dic7—28no—
92C2×C22⋊Dic7C2318yes—
93C2×C22⋊Dic7C2318yes—
94C2×C22⋊Dic7C2318yes—
95C2×C22⋊Dic7C2318yes—
96#6489—232no—
97C2×C4⋊Dic7—24no—
98C2×C4⋊Dic7—24no—
99C2×C4⋊Dic7D418yescomplemented
100C2×C4⋊Dic7C2314yes—
101C2×C4⋊Dic7C2314yes—
102C2×C4⋊Dic7C2314yes—
103#6487—232no—
104#6487—232no—
105C2×Dic7⋊C4—28no—
106C2×Dic7⋊C4—28no—
107C2×Dic7⋊C4—28no—
108C2×Dic7⋊C4C2318yes—
109C2×Dic7⋊C4C2318yes—
110C2×Dic7⋊C4C2318yes—
111C2×C4×Dic7—28no—
112C2×C4×Dic7C2318yes—
113C7⋊#95C2318yes—
114C7⋊#95C2318yes—
115C7⋊#95C2318yes—
116C7⋊#95C2318yes—
117C22×C22⋊Q8—71no—
118C22×C22⋊Q8—71no—
119C22×C4×D4D711yescharacteristic
120C22×#117—71no—
121#2415—716no—
122C2×#397—72no—
123C2×#394—72no—
124C2×#391—71no—
125C2×#389—71no—
126C2×#389—72no—
127C2×#388—72no—
128C2×#380—71no—
129C24×C7—22no—
130C24×C7—22no—
131C24×C7—22no—
132C24×C7—22no—
133C24×C7—22no—
134C24×C7—22no—
135C24×C7—22no—
136C24×C7—22no—
137C24×C7—22no—
138C24×C7—22no—
139C24×C7—22no—
140C24×C7—22no—
141C24×C7C2×D414yes—
142C24×C7#4114yes—
143C24×C7C2412yes—
144C24×C7C2412yes—
145C24×C7C2411yescharacteristic
146C2×C7×D4—232no—
147C2×C7×D4—28no—
148C2×C7×D4—28no—
149C2×C7×D4—28no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—232no—
155C22×C4×C7—28no—
156C22×C4×C7—28no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—28no—
164C22×C4×C7—28no—
165C22×C4×C7—28no—
166C22×C4×C7—416no—
167C22×C4×C7C2×D414yes—
168C22×C4×C7C2×D4116yes—
169C22×C4×C7C2×D414yes—
170C22×C4×C7C2×D414yes—
171C22×C4×C7#4114yes—
172C22×C4×C7#4114yes—
173C22×C4×C7C2412yes—
174C22×C4×C7C2412yes—
175C22×C4×C7C2412yes—
176C22×C4×C7C2412yes—
177C22×C4×C7C2412yes—
178C22×Dic7—24no—
179C22×Dic7—24no—
180C22×Dic7—24no—
181C22×Dic7—24no—
182C22×Dic7—24no—
183C22×Dic7—24no—
184C22×Dic7—24no—
185C22×Dic7—24no—
186C22×Dic7—24no—
187C22×Dic7—24no—
188C22×Dic7—24no—
189C22×Dic7C2×D418yes—
190C22×Dic7C2414yes—
191C22×Dic7C2414yes—
192C22×Dic7C2414yes—
193C22×Dic7C2414yes—
194C2×Dic14—28no—
195C2×Dic14—28no—
196C2×Dic14—28no—
197C2×Dic14—416no—
198C7×C4⋊C4—28no—
199C7×C22⋊C4—28no—
200C7×C22⋊C4—28no—
201C42×C7—28no—
202C22⋊Dic7—216no—
203C22⋊Dic7—216no—
204C22⋊Dic7—216no—
205C22⋊Dic7—216no—
206C4⋊Dic7—28no—
207C4⋊Dic7—28no—
208C4⋊Dic7C2×D4116yescomplemented
209Dic7⋊C4—216no—
210Dic7⋊C4—216no—
211Dic7⋊C4—216no—
212C4×Dic7—216no—
213C23×Q8—71no—
214C23×D4C2×D711yescharacteristic
215C24×C4C2×D711yescharacteristic
216C24×C4C2×D711yescharacteristic
217C2×C22⋊Q8—78no—
218C2×C22⋊Q8—78no—
219C2×C22⋊Q8—148no—
220C2×C22⋊Q8—148no—
221C2×C4×D4—28no—
222C2×C4×D4C2×D718yes—
223C2×#117—78no—
224C2×#117—148no—
225C22×C4⋊C4—71no—
226C22×C4⋊C4—72no—
227C22×C4⋊C4—72no—
228C22×C4⋊C4—71no—
229C22×C4⋊C4—72no—
230C22×C4⋊C4—71no—
231C22×C4⋊C4C2×D711yescharacteristic
232C22×C22⋊C4—72no—
233C22×C22⋊C4—72no—
234C22×C22⋊C4—72no—
235C22×C22⋊C4—72no—
236C22×C22⋊C4C2×D711yescharacteristic
237C22×C22⋊C4C2×D711yescharacteristic
238C22×C42—72no—
239C22×C42C2×D711yescharacteristic
240#397—716no—
241#394—716no—
242#391—78no—
243#389—716no—
244#389—78no—
245#388—716no—
246#380—78no—
247C2×#95—72no—
248C2×#95—72no—
249C2×#95—72no—
250C2×#95—72no—
251C23×C7—28no—
252C23×C7—28no—
253C23×C7—28no—
254C23×C7—28no—
255C23×C7—28no—
256C23×C7—28no—
257C23×C7—22no—
258C23×C7—22no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—22no—
262C23×C7—22no—
263C23×C7—22no—
264C23×C7—22no—
265C23×C7—22no—
266C23×C7—22no—
267C23×C7—22no—
268C23×C7—22no—
269C23×C7—44no—
270C23×C7—44no—
271C23×C7—44no—
272C23×C7—44no—
273C23×C7—44no—
274C23×C7—44no—
275C23×C7—44no—
276C23×C7—44no—
277C23×C7C22×D412yes—
278C23×C7C22×D414yes—
279C23×C7C22×D412yes—
280C23×C7C2×#4114yes—
281C23×C7C2×#4112yes—
282C23×C7C2×#4112yes—
283C23×C7C2×#4112yes—
284C23×C72−1+412yes—
285C23×C72−1+412yes—
286C23×C7C2511yescharacteristic
287C7×D4—232no—
288C2×C4×C7—28no—
289C2×C4×C7—28no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—232no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—44no—
302C2×C4×C7—44no—
303C2×C4×C7—416no—
304C2×C4×C7#121116yes—
305C2×C4×C7C22×D414yes—
306C2×C4×C7C22×D414yes—
307C2×C4×C7C22×D414yes—
308C2×C4×C7C2×#4114yes—
309C2×C4×C7C2×#4114yes—
310C2×Dic7—216no—
311C2×Dic7—24no—
312C2×Dic7—24no—
313C2×Dic7—24no—
314C2×Dic7—24no—
315C2×Dic7—24no—
316C2×Dic7—24no—
317C2×Dic7—24no—
318C2×Dic7—24no—
319C2×Dic7—24no—
320C2×Dic7—24no—
321C2×Dic7—24no—
322C2×Dic7—48no—
323C2×Dic7—48no—
324C2×Dic7—48no—
325C2×Dic7C22×D418yes—
326Dic14—416no—
327C25C22×D711yescharacteristic
328C25C22×D711yescharacteristic
329C22×Q8—74no—
330C22×Q8—144no—
331C22×Q8—144no—
332C22×Q8—144no—
333C22×D4—24no—
334C22×D4—24no—
335C22×D4—24no—
336C22×D4C22×D714yes—
337C23×C4—24no—
338C23×C4—24no—
339C23×C4—24no—
340C23×C4—24no—
341C23×C4—24no—
342C23×C4—72no—
343C23×C4—72no—
344C23×C4—72no—
345C23×C4—72no—
346C23×C4C22⋊D718yes—
347C23×C4C22×D714yes—
348C23×C4C22×D711yescharacteristic
349C23×C4C22×D711yescharacteristic
350C23×C4C22×D711yescharacteristic
351C23×C4C22×D714yes—
352C23×C4C22×D711yescharacteristic
353C23×C4C22×D711yescharacteristic
354C22⋊Q8—1432no—
355C22⋊Q8—1432no—
356C4×D4—232no—
357#117—1432no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—78no—
360C2×C4⋊C4—74no—
361C2×C4⋊C4—74no—
362C2×C4⋊C4—78no—
363C2×C4⋊C4—78no—
364C2×C4⋊C4—74no—
365C2×C4⋊C4—78no—
366C2×C4⋊C4—148no—
367C2×C4⋊C4—148no—
368C2×C4⋊C4—144no—
369C2×C4⋊C4—148no—
370C2×C4⋊C4—144no—
371C2×C4⋊C4C22×D714yes—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—78no—
375C2×C22⋊C4—78no—
376C2×C22⋊C4—78no—
377C2×C22⋊C4—78no—
378C2×C22⋊C4—148no—
379C2×C22⋊C4—148no—
380C2×C22⋊C4—148no—
381C2×C22⋊C4—148no—
382C2×C22⋊C4C22×D714yes—
383C2×C22⋊C4C22×D714yes—
384C2×C42—24no—
385C2×C42—78no—
386C2×C42—148no—
387C2×C42C22×D714yes—
388#95—78no—
389#95—78no—
390#95—78no—
391#95—78no—
392C22×C7—24no—
393C22×C7—24no—
394C22×C7—24no—
395C22×C7—24no—
396C22×C7—24no—
397C22×C7—24no—
398C22×C7—42no—
399C22×C7—416no—
400C22×C7—416no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7C2×#12114yes—
409C22×C7#52514yes—
410C22×C7#53614yes—
411C22×C7#54714yes—
412C22×C7#54814yes—
413C22×C7#55214yes—
414C22×C7#55614yes—
415C22×C7C23×D411yescharacteristic
416C22×C7C23×D411yescharacteristic
417C22×C7C22×#4111yescharacteristic
418C22×C7C22×#4111yescharacteristic
419C22×C7C22×#4111yescharacteristic
420C22×C7C2×2−1+411yescharacteristic
421C22×C7C2×2−1+411yescharacteristic
422C4×C7—24no—
423C4×C7—24no—
424C4×C7—42no—
425C4×C7—42no—
426C4×C7C2×#12118yes—
427Dic7—28no—
428Dic7—44no—
429Dic7—44no—
430Dic7—44no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24#115714yes—
444C24C2×C22⋊D714yes—
445C24C23×D712yes—
446C24C23×D712yes—
447C24C23×D711yescharacteristic
448C2×Q8—148no—
449C2×Q8—148no—
450C2×Q8—148no—
451C2×Q8—2816no—
452C2×D4—232no—
453C2×D4—28no—
454C2×D4—28no—
455C2×D4—28no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—232no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—416no—
473C22×C4—78no—
474C22×C4—74no—
475C22×C4—74no—
476C22×C4—74no—
477C22×C4—74no—
478C22×C4—144no—
479C22×C4—144no—
480C22×C4—144no—
481C22×C4—144no—
482C22×C4—144no—
483C22×C4—144no—
484C22×C4—144no—
485C22×C4—144no—
486C22×C4—144no—
487C22×C4—144no—
488C22×C4—144no—
489C22×C4#115714yes—
490C22×C4D4×D714yes—
491C22×C4#115914yes—
492C22×C4C2×C22⋊D714yes—
493C22×C4C2×C22⋊D7116yes—
494C22×C4C2×C22⋊D714yes—
495C22×C4C23×D712yes—
496C22×C4C23×D712yes—
497C22×C4C23×D712yes—
498C22×C4C23×D712yes—
499C22×C4C23×D712yes—
500C4⋊C4—28no—
501C4⋊C4—716no—
502C4⋊C4—1416no—
503C4⋊C4—1416no—
504C4⋊C4—148no—
505C4⋊C4—1416no—
506C4⋊C4—148no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—1416no—
510C22⋊C4—1416no—
511C22⋊C4—1416no—
512C22⋊C4—1416no—
513C42—28no—
514C42—1416no—
515C2×C7—44no—
516C2×C7—44no—
517C2×C7#241518yescomplemented
518C2×C7C22×#12111yescharacteristic
519C2×C7C2×#52511yescharacteristic
520C2×C7C2×#53611yescharacteristic
521C2×C7C2×#54711yescharacteristic
522C2×C7C2×#54811yescharacteristic
523C2×C7C2×#55211yescharacteristic
524C2×C7C2×#55611yescharacteristic
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C2×#115714yes—
552C23C2×#115712yes—
553C23C2×#115712yes—
554C23C2×D4×D712yes—
555C23C2×#115912yes—
556C23#655412yes—
557C23#655412yes—
558C23C22×C22⋊D714yes—
559C23C22×C22⋊D712yes—
560C23C24×D711yescharacteristic
561Q8—2816no—
562D4—232no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—232no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—416no—
579C2×C4—78no—
580C2×C4—1416no—
581C2×C4—144no—
582C2×C4—144no—
583C2×C4—144no—
584C2×C4—144no—
585C2×C4—144no—
586C2×C4—144no—
587C2×C4—144no—
588C2×C4—144no—
589C2×C4—144no—
590C2×C4—144no—
591C2×C4—144no—
592C2×C4—288no—
593C2×C4—288no—
594C2×C4—288no—
595C2×C4#6502116yes—
596C2×C4C2×#115714yes—
597C2×C4C2×D4×D714yes—
598C2×C4C2×#115914yes—
599C2×C4C22×C22⋊D714yes—
600C2×C4C22×C22⋊D714yes—
601C7C2×#241511yescharacteristic, complemented, Sylow
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—416no—
617C22—416no—
618C22#9023214yes—
619C22#9023314yes—
620C22#9024214yes—
621C22#9024314yes—
622C22#9048214yes—
623C22C2×#650214yes—
624C22#9051614yes—
625C22C22×#115711yescharacteristic
626C22C22×#115711yescharacteristic
627C22C22×D4×D711yescharacteristic
628C22C22×#115911yescharacteristic
629C22C2×#655411yescharacteristic
630C22C2×#655411yescharacteristic
631C22C23×C22⋊D711yescharacteristic
632C4—24no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—148no—
637C4—284no—
638C4—284no—
639C4—284no—
640C4C2×#650218yes—
641C2—44no—
642C2—44no—
643C2#1173979618yesdirect product factor
644C2C2×#9024311yescharacteristic
645C2C2×#9023211yescharacteristic
646C2C2×#9023311yescharacteristic
647C2C2×#9024211yescharacteristic
648C2C2×#9048211yescharacteristic
649C2C22×#650211yescharacteristic
650C2C2×#9051611yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC22×C4×C7×D4C2
socleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC22×C4×D4D7
hypocenterC7C2×#2415
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6487C2semidirect
3. C22×#6487C2semidirect
4. C22×#6489C2semidirect
6. C2×#89984C2semidirect
7. #11739796C2direct (N × H)
8. C2×#89744C2semidirect
9. C2×#89724C2semidirect
10. C2×#89717C2semidirect
11. C2×#89701C2semidirect
12. C2×#89700C2semidirect
28. C2×#6489C22semidirect
34. C2×#6487C22semidirect
35. C2×#6487C22semidirect
41. #89984C22semidirect
46. #89744C22semidirect
47. #89724C22semidirect
48. #89717C22semidirect
49. #89701C22semidirect
50. #89700C22semidirect
99. C2×C4⋊Dic7D4semidirect
208. C4⋊Dic7C2×D4semidirect
517. C2×C7#2415semidirect
601. C7C2×#2415semidirect
643. C2#11739796direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C7×D42
C22×#64872
C22×#64892
C2×#899942
C2×#899842
#117397962
C2×#897442
C2×#897242
C2×#897172
C2×#897012
C2×#897002
C2×#24157