Finite Group Databasebeta

← C2×#11740222 #13525062

Subgroups — C2×#11740222 #13525062

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×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×#6509C211yesmaximal, characteristic, complemented
3C22×Q8×Dic7C211yesmaximal, characteristic, complemented
4C22×C4×C22⋊D7C211yesmaximal, characteristic
5C2×#89984C211yesmaximal, characteristic, complemented
6#11740222C2116yesmaximal, direct product factor
7C2×#89759C211yesmaximal, characteristic, complemented
8C2×#89756C212yesmaximal, complemented
9C2×#89762C212yesmaximal, complemented
10C2×#89749C211yesmaximal, characteristic, complemented
11C2×#89733C212yesmaximal
12C2×#89717C212yesmaximal, complemented
13C23×C7×Q8C2211yescharacteristic
14C24×C4×C7C2211yescharacteristic
15C23×C22⋊D7C2211yescharacteristic
16C23×C4×D7C2211yescharacteristic
17C2×C7×C22⋊Q8—28no—
18C2×C7×C22⋊Q8C2218yescomplemented
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C7×C4⋊C4C2212yes—
21C22×C7×C22⋊C4C2212yes—
22C22×C22⋊Dic7C2212yes—
23C22×C22⋊Dic7C2211yescharacteristic
24C2×#6509—28no—
25C2×#6509C2218yescomplemented
26C2×Q8×Dic7—28no—
27C2×Q8×Dic7C2218yescomplemented
28C2×C4×C22⋊D7—28no—
29C2×C4×C22⋊D7C2218yes—
30C22×(C2×D7)⋊C4C2211yescharacteristic
31C22×(C2×D7)⋊C4C2212yes—
32C22×(C2×D7)⋊C4C2212yes—
33C22×C4⋊Dic7C2211yescharacteristic
34C22×C4⋊Dic7C2211yescharacteristic
35C22×C4⋊Dic7C2212yes—
36C22×Dic7⋊C4C2211yescharacteristic
37C22×Dic7⋊C4C2212yes—
38C22×C4×Dic7C2212yes—
39C22×C4×Dic7C2211yescharacteristic
40#89984C2218yescomplemented
41C2×C7⋊#95C2212yes—
42C2×C7⋊#95C2212yes—
43C2×C7⋊#95C2212yes—
44C2×C7⋊#95C2212yes—
45#89762C22116yescomplemented
46#89759C2218yescomplemented
47#89756C22116yescomplemented
48#89749C2218yescomplemented
49#89733C22116yes—
50#89717C22116yescomplemented
51C2×#2536—71nomaximal, Sylow
52C25×C7C2311yescharacteristic
53C24×D7C2311yescharacteristic
54C22×C7×Q8—24no—
55C22×C7×Q8—24no—
56C22×C7×Q8D418yescomplemented
57C22×C7×Q8C2314yes—
58C23×C4×C7—24no—
59C23×C4×C7—24no—
60C23×C4×C7—24no—
61C23×C4×C7C2311yescharacteristic
62C23×C4×C7C2312yes—
63C23×C4×C7C2314yes—
64C23×C4×C7C2312yes—
65C23×C4×C7C2311yescharacteristic
66C22×C22⋊D7—24no—
67C22×C22⋊D7—24no—
68C22×C22⋊D7—24no—
69C22×C22⋊D7C2314yes—
70C23×Dic7C2312yes—
71C23×Dic7C2311yescharacteristic
72C23×Dic7C2312yes—
73C23×Dic7C2311yescharacteristic
74C23×Dic7C2311yescharacteristic
75C22×C4×D7—24no—
76C22×C4×D7—24no—
77C22×C4×D7—24no—
78C22×C4×D7C2314yes—
79C7×C22⋊Q8—232no—
80C2×C7×C4⋊C4—24no—
81C2×C7×C4⋊C4—28no—
82C2×C7×C4⋊C4C2314yes—
83C2×C7×C4⋊C4C2318yes—
84C2×C7×C22⋊C4—28no—
85C2×C7×C22⋊C4C2318yes—
86C2×C22⋊Dic7—28no—
87C2×C22⋊Dic7—24no—
88C2×C22⋊Dic7C2314yes—
89C2×C22⋊Dic7C2318yes—
90#6509—232no—
91Q8×Dic7—232no—
92C4×C22⋊D7—232no—
93C2×(C2×D7)⋊C4—28no—
94C2×(C2×D7)⋊C4—28no—
95C2×(C2×D7)⋊C4—24no—
96C2×(C2×D7)⋊C4C2314yes—
97C2×(C2×D7)⋊C4C2318yes—
98C2×(C2×D7)⋊C4C2318yes—
99C2×C4⋊Dic7—24no—
100C2×C4⋊Dic7—24no—
101C2×C4⋊Dic7—28no—
102C2×C4⋊Dic7C2314yes—
103C2×C4⋊Dic7C2314yes—
104C2×C4⋊Dic7C2318yes—
105C2×Dic7⋊C4—28no—
106C2×Dic7⋊C4—24no—
107C2×Dic7⋊C4C2314yes—
108C2×Dic7⋊C4C2318yes—
109C2×C4×Dic7—28no—
110C2×C4×Dic7D418yes—
111C2×C4×Dic7C2318yes—
112C2×C4×Dic7C2314yes—
113C7⋊#95C2318yes—
114C7⋊#95C2318yes—
115C7⋊#95C2318yes—
116C7⋊#95C2318yes—
117C22×C22⋊Q8—71no—
118C22×C22⋊Q8D711yescharacteristic, complemented
119C22×C4×Q8—71no—
120C22×C4×D4—71no—
121#2536—716no—
122C2×#398—72no—
123C2×#394—72no—
124C2×#388—72no—
125C2×#388—72no—
126C2×#382—71no—
127C2×#380—71no—
128C2×#380—71no—
129C24×C7—22no—
130C24×C7—22no—
131C24×C7—22no—
132C24×C7—22no—
133C24×C7—22no—
134C24×C7—22no—
135C24×C7#4114yes—
136C24×C7C2412yes—
137C24×C7C2411yescharacteristic
138C23×D7—22no—
139C23×D7—22no—
140C23×D7—22no—
141C23×D7—22no—
142C23×D7—22no—
143C23×D7—22no—
144C23×D7#4114yes—
145C23×D7C2412yes—
146C2×C7×Q8—232no—
147C2×C7×Q8—28no—
148C2×C7×Q8—28no—
149C2×C7×Q8C2×D4116yescomplemented
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7—22no—
153C22×C4×C7—24no—
154C22×C4×C7—22no—
155C22×C4×C7—24no—
156C22×C4×C7—22no—
157C22×C4×C7—24no—
158C22×C4×C7—28no—
159C22×C4×C7—28no—
160C22×C4×C7—28no—
161C22×C4×C7—416no—
162C22×C4×C7C2×D418yes—
163C22×C4×C7C2×D414yes—
164C22×C4×C7C2×D414yes—
165C22×C4×C7#4114yes—
166C22×C4×C7C2414yes—
167C22×C4×C7C2412yes—
168C22×C4×C7C2414yes—
169C22×C4×C7C2412yes—
170C2×C22⋊D7—232no—
171C2×C22⋊D7—28no—
172C2×C22⋊D7—28no—
173C2×C22⋊D7—28no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—24no—
179C22×Dic7—22no—
180C22×Dic7—24no—
181C22×Dic7—22no—
182C22×Dic7—24no—
183C22×Dic7—24no—
184C22×Dic7—24no—
185C22×Dic7—22no—
186C22×Dic7—24no—
187C22×Dic7C2×D414yes—
188C22×Dic7C2×D414yes—
189C22×Dic7C2414yes—
190C22×Dic7C2414yes—
191C22×Dic7C2412yes—
192C22×Dic7C2412yes—
193C22×Dic7C2412yes—
194C2×C4×D7—28no—
195C2×C4×D7—28no—
196C2×C4×D7—28no—
197C2×C4×D7—416no—
198C7×C4⋊C4—28no—
199C7×C4⋊C4—216no—
200C7×C22⋊C4—216no—
201C22⋊Dic7—216no—
202C22⋊Dic7—28no—
203(C2×D7)⋊C4—216no—
204(C2×D7)⋊C4—216no—
205(C2×D7)⋊C4—28no—
206C4⋊Dic7—28no—
207C4⋊Dic7—28no—
208C4⋊Dic7—216no—
209Dic7⋊C4—216no—
210Dic7⋊C4—28no—
211C4×Dic7—216no—
212C4×Dic7C2×D4116yes—
213C23×Q8C2×D711yescharacteristic
214C23×D4—71no—
215C24×C4—71no—
216C24×C4C2×D711yescharacteristic
217C2×C22⋊Q8—28no—
218C2×C22⋊Q8—78no—
219C2×C22⋊Q8—148no—
220C2×C22⋊Q8C2×D718yescomplemented
221C2×C4×Q8—78no—
222C2×C4×Q8—148no—
223C2×C4×D4—78no—
224C2×C4×D4—148no—
225C22×C4⋊C4—72no—
226C22×C4⋊C4—71no—
227C22×C4⋊C4—72no—
228C22×C4⋊C4—71no—
229C22×C4⋊C4—71no—
230C22×C4⋊C4C2×D712yes—
231C22×C4⋊C4C2×D711yescharacteristic
232C22×C22⋊C4—72no—
233C22×C22⋊C4—71no—
234C22×C22⋊C4—72no—
235C22×C22⋊C4—72no—
236C22×C22⋊C4—71no—
237C22×C22⋊C4C2×D712yes—
238C22×C42—71no—
239C22×C42—72no—
240#398—716no—
241#394—716no—
242#388—716no—
243#388—716no—
244#382—78no—
245#380—78no—
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—22no—
255C23×C7—22no—
256C23×C7—22no—
257C23×C7—22no—
258C23×C7—22no—
259C23×C7—22no—
260C23×C7—44no—
261C23×C7—44no—
262C23×C7—44no—
263C23×C7—44no—
264C23×C7C22×D412yes—
265C23×C7C22×D412yes—
266C23×C7C2×#4112yes—
267C23×C7C2×#4114yes—
268C23×C7C2×#4112yes—
269C23×C7C2×#4112yes—
270C23×C72−1+412yes—
271C23×C72−1+412yes—
272C23×C7C2511yescharacteristic
273C22×D7—28no—
274C22×D7—28no—
275C22×D7—28no—
276C22×D7—22no—
277C22×D7—22no—
278C22×D7—22no—
279C22×D7—22no—
280C22×D7—22no—
281C22×D7—22no—
282C22×D7—44no—
283C22×D7—44no—
284C22×D7—44no—
285C22×D7—44no—
286C22×D7C2×#4114yes—
287C7×Q8—232no—
288C2×C4×C7—216no—
289C2×C4×C7—24no—
290C2×C4×C7—24no—
291C2×C4×C7—22no—
292C2×C4×C7—24no—
293C2×C4×C7—22no—
294C2×C4×C7—24no—
295C2×C4×C7—22no—
296C2×C4×C7—24no—
297C2×C4×C7—44no—
298C2×C4×C7—48no—
299C2×C4×C7—416no—
300C2×C4×C7#121116yes—
301C2×C4×C7C22×D414yes—
302C2×C4×C7C22×D418yes—
303C2×C4×C7C22×D414yes—
304C2×C4×C7C2×#4114yes—
305C22⋊D7—232no—
306C2×Dic7—28no—
307C2×Dic7—28no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—22no—
312C2×Dic7—24no—
313C2×Dic7—22no—
314C2×Dic7—24no—
315C2×Dic7—22no—
316C2×Dic7—24no—
317C2×Dic7—24no—
318C2×Dic7—24no—
319C2×Dic7—22no—
320C2×Dic7—24no—
321C2×Dic7—44no—
322C2×Dic7—48no—
323C2×Dic7—48no—
324C2×Dic7C22×D414yes—
325C2×Dic7C22×D414yes—
326C4×D7—416no—
327C25—71no—
328C25C22×D711yescharacteristic
329C22×Q8—24no—
330C22×Q8—24no—
331C22×Q8C22⋊D718yescomplemented
332C22×Q8C22×D714yes—
333C22×D4—74no—
334C22×D4—144no—
335C22×D4—144no—
336C22×D4—144no—
337C23×C4—24no—
338C23×C4—24no—
339C23×C4—24no—
340C23×C4—71no—
341C23×C4—71no—
342C23×C4—72no—
343C23×C4—72no—
344C23×C4—71no—
345C23×C4—74no—
346C23×C4—144no—
347C23×C4—144no—
348C23×C4—144no—
349C23×C4C22×D714yes—
350C23×C4C22×D712yes—
351C23×C4C22×D711yescharacteristic
352C23×C4C22×D712yes—
353C23×C4C22×D711yescharacteristic
354C22⋊Q8—232no—
355C22⋊Q8—1432no—
356C4×Q8—1432no—
357C4×D4—1432no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—28no—
360C2×C4⋊C4—74no—
361C2×C4⋊C4—74no—
362C2×C4⋊C4—78no—
363C2×C4⋊C4—78no—
364C2×C4⋊C4—74no—
365C2×C4⋊C4—144no—
366C2×C4⋊C4—148no—
367C2×C4⋊C4—144no—
368C2×C4⋊C4—148no—
369C2×C4⋊C4—144no—
370C2×C4⋊C4C22×D714yes—
371C2×C4⋊C4C22×D718yes—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—74no—
374C2×C22⋊C4—78no—
375C2×C22⋊C4—74no—
376C2×C22⋊C4—78no—
377C2×C22⋊C4—78no—
378C2×C22⋊C4—144no—
379C2×C22⋊C4—148no—
380C2×C22⋊C4—144no—
381C2×C22⋊C4—148no—
382C2×C22⋊C4—148no—
383C2×C22⋊C4C22×D718yes—
384C2×C42—78no—
385C2×C42—74no—
386C2×C42—78no—
387C2×C42—148no—
388#95—78no—
389#95—78no—
390#95—78no—
391#95—78no—
392C22×C7—24no—
393C22×C7—24no—
394C22×C7—24no—
395C22×C7—42no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—416no—
400C22×C7C2×#12114yes—
401C22×C7#52514yes—
402C22×C7#53614yes—
403C22×C7#54814yes—
404C22×C7#54814yes—
405C22×C7#55214yes—
406C22×C7#55214yes—
407C22×C7C23×D411yescharacteristic
408C22×C7C23×D411yescharacteristic
409C22×C7C22×#4111yescharacteristic
410C22×C7C22×#4111yescharacteristic
411C22×C7C22×#4111yescharacteristic
412C22×C7C2×2−1+411yescharacteristic
413C22×C7C2×2−1+411yescharacteristic
414C2×D7—24no—
415C2×D7—24no—
416C2×D7—24no—
417C2×D7—42no—
418C2×D7—42no—
419C2×D7—42no—
420C2×D7—42no—
421C2×D7—416no—
422C4×C7—28no—
423C4×C7—42no—
424C4×C7—44no—
425C4×C7C2×#12118yes—
426Dic7—24no—
427Dic7—24no—
428Dic7—42no—
429Dic7—44no—
430Dic7—44no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—74no—
438C24—72no—
439C24—142no—
440C24—142no—
441C24—142no—
442C24—142no—
443C24—142no—
444C24—142no—
445C24#116114yes—
446C24C23×D712yes—
447C24C23×D711yescharacteristic
448C2×Q8—232no—
449C2×Q8—28no—
450C2×Q8—28no—
451C2×Q8C2×C22⋊D7116yescomplemented
452C2×D4—148no—
453C2×D4—148no—
454C2×D4—1432no—
455C2×D4—148no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—416no—
468C22×C4—74no—
469C22×C4—74no—
470C22×C4—72no—
471C22×C4—72no—
472C22×C4—74no—
473C22×C4—74no—
474C22×C4—72no—
475C22×C4—142no—
476C22×C4—142no—
477C22×C4—142no—
478C22×C4—142no—
479C22×C4—144no—
480C22×C4—142no—
481C22×C4—142no—
482C22×C4—144no—
483C22×C4—144no—
484C22×C4—144no—
485C22×C4—144no—
486C22×C4—144no—
487C22×C4—142no—
488C22×C4—148no—
489C22×C4—148no—
490C22×C4—148no—
491C22×C4—2816no—
492C22×C4D4×D714yes—
493C22×C4#115914yes—
494C22×C4C2×C22⋊D718yes—
495C22×C4C2×C22⋊D714yes—
496C22×C4C23×D714yes—
497C22×C4C23×D712yes—
498C22×C4C23×D714yes—
499C22×C4C23×D712yes—
500C4⋊C4—28no—
501C4⋊C4—216no—
502C4⋊C4—148no—
503C4⋊C4—1416no—
504C4⋊C4—148no—
505C4⋊C4—1416no—
506C4⋊C4—148no—
507C22⋊C4—216no—
508C22⋊C4—148no—
509C22⋊C4—1416no—
510C22⋊C4—148no—
511C22⋊C4—1416no—
512C22⋊C4—1416no—
513C42—716no—
514C42—1416no—
515C2×C7—44no—
516C2×C7#253618yescomplemented
517C2×C7C22×#12111yescharacteristic
518C2×C7C2×#52511yescharacteristic
519C2×C7C2×#53611yescharacteristic
520C2×C7C2×#54811yescharacteristic
521C2×C7C2×#54811yescharacteristic
522C2×C7C2×#55211yescharacteristic
523C2×C7C2×#55211yescharacteristic
524D7—44no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—74no—
539C23—148no—
540C23—148no—
541C23—148no—
542C23—142no—
543C23—142no—
544C23—142no—
545C23—142no—
546C23—142no—
547C23—142no—
548C23—284no—
549C23—284no—
550C23—284no—
551C23—284no—
552C23C2×D4×D712yes—
553C23C2×#115912yes—
554C23C2×#116114yes—
555C23C2×#116112yes—
556C23#655112yes—
557C23D7×#4112yes—
558C23#655412yes—
559C23C22×C22⋊D712yes—
560C23C24×D711yescharacteristic
561Q8—232no—
562D4—1432no—
563C2×C4—216no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—416no—
575C2×C4—74no—
576C2×C4—74no—
577C2×C4—148no—
578C2×C4—148no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—142no—
583C2×C4—144no—
584C2×C4—142no—
585C2×C4—142no—
586C2×C4—144no—
587C2×C4—144no—
588C2×C4—144no—
589C2×C4—144no—
590C2×C4—144no—
591C2×C4—142no—
592C2×C4—284no—
593C2×C4—288no—
594C2×C4—288no—
595C2×C4—2816no—
596C2×C4#6502116yes—
597C2×C4C2×D4×D714yes—
598C2×C4C2×#115914yes—
599C2×C4C22×C22⋊D718yes—
600C2×C4C22×C22⋊D714yes—
601C7C2×#253611yescharacteristic, complemented, Sylow
602C22—24no—
603C22—24no—
604C22—24no—
605C22—42no—
606C22—416no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—144no—
611C22—144no—
612C22—144no—
613C22—282no—
614C22—282no—
615C22—2816no—
616C22—282no—
617C22—282no—
618C22#9031814yes—
619C22#9031914yes—
620C22#9033014yes—
621C22#9033314yes—
622C22C2×#650214yes—
623C22#9050814yes—
624C22#9052214yes—
625C22C22×D4×D711yescharacteristic
626C22C22×#115911yescharacteristic
627C22C22×#116111yescharacteristic
628C22C2×#655111yescharacteristic
629C22C2×D7×#4111yescharacteristic
630C22C2×#655411yescharacteristic
631C22C23×C22⋊D711yescharacteristic
632C4—28no—
633C4—42no—
634C4—44no—
635C4—144no—
636C4—144no—
637C4—282no—
638C4—284no—
639C4—284no—
640C4C2×#650218yes—
641C2—44no—
642C2—284no—
643C2#1174022218yesdirect product factor
644C2C2×#9031811yescharacteristic
645C2C2×#9033311yescharacteristic
646C2C2×#9031911yescharacteristic
647C2C2×#9033011yescharacteristic
648C2C22×#650211yescharacteristic
649C2C2×#9050811yescharacteristic
650C2C2×#9052211yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC22×C7×C22⋊Q8C2
socleC24×C7C24
prime radicalC23×C22⋊D7C22
hypercenterC22×C22⋊Q8D7
hypocenterC7C2×#2536
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C7×C22⋊Q8C2semidirect
2. C22×#6509C2semidirect
3. C22×Q8×Dic7C2semidirect
5. C2×#89984C2semidirect
6. #11740222C2direct (N × H)
7. C2×#89759C2semidirect
8. C2×#89756C2semidirect
9. C2×#89762C2semidirect
10. C2×#89749C2semidirect
12. C2×#89717C2semidirect
18. C2×C7×C22⋊Q8C22semidirect
25. C2×#6509C22semidirect
27. C2×Q8×Dic7C22semidirect
40. #89984C22semidirect
45. #89762C22semidirect
46. #89759C22semidirect
47. #89756C22semidirect
48. #89749C22semidirect
50. #89717C22semidirect
56. C22×C7×Q8D4semidirect
118. C22×C22⋊Q8D7semidirect
149. C2×C7×Q8C2×D4semidirect
220. C2×C22⋊Q8C2×D7semidirect
331. C22×Q8C22⋊D7semidirect
451. C2×Q8C2×C22⋊D7semidirect
516. C2×C7#2536semidirect
601. C7C2×#2536semidirect
643. C2#11740222direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C22×#65092
C22×Q8×Dic72
C22×C4×C22⋊D72
C2×#899842
#117402222
C2×#897592
C2×#897562
C2×#897622
C2×#897492
C2×#897332
C2×#897172
C2×#25367