Finite Group Databasebeta

← C2×#11740304 #13522853

Subgroups — C2×#11740304 #13522853

Order 1792.

4222 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#123C211yesmaximal, characteristic, complemented
2C22×#6498C211yesmaximal, characteristic, complemented
3C22×#6502C211yesmaximal, characteristic
4C22×#6492C211yesmaximal, characteristic, complemented
5C23×Dic7⋊C4C211yesmaximal, characteristic, complemented
6C22×#6449C211yesmaximal, characteristic, complemented
7#11740304C2116yesmaximal, direct product factor
8C2×#89764C212yesmaximal, complemented
9C2×#89748C212yesmaximal, complemented
10C2×#89721C212yesmaximal, complemented
11C2×#89729C212yesmaximal, complemented
12C2×C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
13C23×C7×D4C2211yescharacteristic
14C24×C4×C7C2211yescharacteristic
15C23×C22⋊D7C2211yescharacteristic
16C23×C22⋊D7C2211yescharacteristic
17C24×Dic7C2211yescharacteristic
18C24×Dic7C2211yescharacteristic
19C2×C7×#123—28no—
20C2×C7×#123C2218yescomplemented
21C22×C7×C4⋊C4C2212yes—
22C22×C7×C22⋊C4C2212yes—
23C22×C7×C22⋊C4C2211yescharacteristic
24C22×C22⋊Dic7C2211yescharacteristic
25C22×C22⋊Dic7C2212yes—
26C22×C22⋊Dic7C2211yescharacteristic
27C2×#6502—28no—
28C2×#6502C2218yes—
29C2×#6498—28no—
30C2×#6498C2218yescomplemented
31C2×#6492—28no—
32C2×#6492C2218yescomplemented
33C22×(C2×D7)⋊C4C2212yes—
34C22×(C2×D7)⋊C4C2212yes—
35C22×(C2×D7)⋊C4C2211yescharacteristic
36C22×C4⋊Dic7C2212yes—
37C22×Dic7⋊C4—28no—
38C22×Dic7⋊C4C2212yes—
39C22×Dic7⋊C4C2212yes—
40C22×Dic7⋊C4C2212yes—
41C22×Dic7⋊C4C2211yescharacteristic
42C22×Dic7⋊C4C2212yes—
43C22×Dic7⋊C4C2218yescomplemented
44C22×C4×Dic7C2212yes—
45C2×#6449—28no—
46C2×#6449C2218yescomplemented
47C2×C7⋊#95C2212yes—
48#89764C22116yescomplemented
49#89748C22116yescomplemented
50#89729C22116yescomplemented
51#89721C22116yescomplemented
52C22⋊C4×Dic7C2218yescomplemented
53C2×#2501—71nomaximal, Sylow
54C25×C7C2311yescharacteristic
55C25×C7C2311yescharacteristic
56C24×D7C2311yescharacteristic
57C22×C7×D4—24no—
58C22×C7×D4—24no—
59C22×C7×D4—24no—
60C22×C7×D4C2314yes—
61C23×C4×C7—24no—
62C23×C4×C7—24no—
63C23×C4×C7—24no—
64C23×C4×C7C2311yescharacteristic
65C23×C4×C7C2312yes—
66C23×C4×C7C2314yes—
67C23×C4×C7C2312yes—
68C22×C22⋊D7—24no—
69C22×C22⋊D7—24no—
70C22×C22⋊D7—24no—
71C22×C22⋊D7—24no—
72C22×C22⋊D7—28no—
73C22×C22⋊D7C2314yes—
74C22×C22⋊D7C2314yes—
75C23×Dic7—28no—
76C23×Dic7—24no—
77C23×Dic7—24no—
78C23×Dic7—24no—
79C23×Dic7D418yes—
80C23×Dic7C2314yes—
81C23×Dic7C2311yescharacteristic
82C23×Dic7C2312yes—
83C23×Dic7C2312yes—
84C23×Dic7C2311yescharacteristic
85C23×Dic7C2314yes—
86C23×Dic7C2311yescharacteristic
87C7×#123—232no—
88C2×C7×C4⋊C4—28no—
89C2×C7×C4⋊C4C2318yes—
90C2×C7×C22⋊C4—28no—
91C2×C7×C22⋊C4D418yescomplemented
92C2×C7×C22⋊C4C2318yes—
93C2×C7×C22⋊C4C2314yes—
94C2×C22⋊Dic7—28no—
95C2×C22⋊Dic7—24no—
96C2×C22⋊Dic7D418yescomplemented
97C2×C22⋊Dic7C2318yes—
98C2×C22⋊Dic7C2314yes—
99C2×C22⋊Dic7C2314yes—
100#6502—232no—
101#6498—232no—
102#6492—232no—
103C2×(C2×D7)⋊C4—28no—
104C2×(C2×D7)⋊C4—28no—
105C2×(C2×D7)⋊C4—24no—
106C2×(C2×D7)⋊C4C2318yes—
107C2×(C2×D7)⋊C4C2318yes—
108C2×(C2×D7)⋊C4C2314yes—
109C2×C4⋊Dic7—28no—
110C2×C4⋊Dic7C2318yes—
111C2×Dic7⋊C4—28no—
112C2×Dic7⋊C4—28no—
113C2×Dic7⋊C4—232no—
114C2×Dic7⋊C4—28no—
115C2×Dic7⋊C4—28no—
116C2×Dic7⋊C4—24no—
117C2×Dic7⋊C4C2318yes—
118C2×Dic7⋊C4C2318yes—
119C2×Dic7⋊C4C2318yes—
120C2×Dic7⋊C4C2318yes—
121C2×Dic7⋊C4C2314yes—
122C2×C4×Dic7—28no—
123C2×C4×Dic7C2318yes—
124#6449—232no—
125C7⋊#95C2318yes—
126C22×#123—71no—
127C22×#123—71no—
128C22×#123—71no—
129C22×#123D711yescharacteristic, complemented
130C22×#121—71no—
131C23×C4⋊C4—71no—
132#2501—716no—
133C2×#399—72no—
134C2×#399—72no—
135C2×#390—72no—
136C2×#384—72no—
137C2×C4×C22⋊C4—71no—
138C24×C7—24no—
139C24×C7—24no—
140C24×C7—22no—
141C24×C7—22no—
142C24×C7—22no—
143C24×C7—24no—
144C24×C7—24no—
145C24×C7C2×D414yes—
146C24×C7C2×D414yes—
147C24×C7C2×D414yes—
148C24×C7C2412yes—
149C24×C7C2412yes—
150C24×C7C2411yescharacteristic
151C23×D7—22no—
152C23×D7—22no—
153C23×D7—22no—
154C23×D7—24no—
155C23×D7—24no—
156C23×D7C2412yes—
157C2×C7×D4—28no—
158C2×C7×D4—28no—
159C2×C7×D4—28no—
160C2×C7×D4—416no—
161C22×C4×C7—24no—
162C22×C4×C7—24no—
163C22×C4×C7—28no—
164C22×C4×C7—24no—
165C22×C4×C7—22no—
166C22×C4×C7—24no—
167C22×C4×C7—22no—
168C22×C4×C7—28no—
169C22×C4×C7—24no—
170C22×C4×C7—22no—
171C22×C4×C7—28no—
172C22×C4×C7—416no—
173C22×C4×C7C2×D418yes—
174C22×C4×C7C2414yes—
175C22×C4×C7C2412yes—
176C22×C4×C7C2414yes—
177C2×C22⋊D7—28no—
178C2×C22⋊D7—28no—
179C2×C22⋊D7—28no—
180C2×C22⋊D7—28no—
181C2×C22⋊D7—216no—
182C2×C22⋊D7—416no—
183C2×C22⋊D7—416no—
184C22×Dic7—24no—
185C22×Dic7—24no—
186C22×Dic7—24no—
187C22×Dic7—232no—
188C22×Dic7—216no—
189C22×Dic7—28no—
190C22×Dic7—28no—
191C22×Dic7—24no—
192C22×Dic7—22no—
193C22×Dic7—24no—
194C22×Dic7—22no—
195C22×Dic7—22no—
196C22×Dic7—24no—
197C22×Dic7—28no—
198C22×Dic7—416no—
199C22×Dic7C2×D418yes—
200C22×Dic7C2×D414yes—
201C22×Dic7C2×D4116yes—
202C22×Dic7C2×D414yes—
203C22×Dic7#4118yes—
204C22×Dic7C2412yes—
205C22×Dic7C2414yes—
206C22×Dic7C2412yes—
207C22×Dic7C2414yes—
208C22×Dic7C2412yes—
209C7×C4⋊C4—216no—
210C7×C22⋊C4—216no—
211C7×C22⋊C4C2×D4116yescomplemented
212C22⋊Dic7—216no—
213C22⋊Dic7—28no—
214C22⋊Dic7C2×D4116yescomplemented
215(C2×D7)⋊C4—216no—
216(C2×D7)⋊C4—216no—
217(C2×D7)⋊C4—28no—
218C4⋊Dic7—216no—
219Dic7⋊C4—216no—
220Dic7⋊C4—216no—
221Dic7⋊C4—216no—
222Dic7⋊C4—216no—
223Dic7⋊C4—28no—
224C4×Dic7—216no—
225C23×D4—71no—
226C23×D4—71no—
227C23×D4C2×D711yescharacteristic
228C24×C4—71no—
229C24×C4—71no—
230C24×C4C2×D711yescharacteristic
231C2×#123—28no—
232C2×#123—78no—
233C2×#123—78no—
234C2×#123—78no—
235C2×#123—148no—
236C2×#123—148no—
237C2×#123—148no—
238C2×#123C2×D718yescomplemented
239C2×#121—78no—
240C2×#121—148no—
241C22×C4⋊C4—72no—
242C22×C4⋊C4—72no—
243C22×C4⋊C4—72no—
244C22×C4⋊C4—72no—
245C22×C4⋊C4—72no—
246C22×C4⋊C4—78no—
247C22×C4⋊C4—71no—
248C22×C4⋊C4—148no—
249C22×C4⋊C4C2×D712yes—
250C22×C22⋊C4—71no—
251C22×C22⋊C4—71no—
252C22×C22⋊C4—72no—
253C22×C22⋊C4—71no—
254C22×C22⋊C4—72no—
255C22×C22⋊C4—72no—
256C22×C22⋊C4C2×D712yes—
257C22×C22⋊C4C2×D711yescharacteristic
258C22×C42—72no—
259#399—716no—
260#399—716no—
261#390—716no—
262#384—716no—
263C4×C22⋊C4—78no—
264C2×#95—72no—
265C23×C7—28no—
266C23×C7—28no—
267C23×C7—216no—
268C23×C7—216no—
269C23×C7—24no—
270C23×C7—24no—
271C23×C7—22no—
272C23×C7—22no—
273C23×C7—22no—
274C23×C7—24no—
275C23×C7—24no—
276C23×C7—44no—
277C23×C7—44no—
278C23×C7—44no—
279C23×C7—48no—
280C23×C7—48no—
281C23×C7#127116yes—
282C23×C7C22×D414yes—
283C23×C7C22×D414yes—
284C23×C7C22×D412yes—
285C23×C7C22×D412yes—
286C23×C7C22×D414yes—
287C23×C7C22×D412yes—
288C23×C7C2×#4114yes—
289C23×C72−1+414yes—
290C23×C7C2511yescharacteristic
291C22×D7—22no—
292C22×D7—22no—
293C22×D7—22no—
294C22×D7—24no—
295C22×D7—24no—
296C22×D7—44no—
297C22×D7—44no—
298C22×D7—44no—
299C22×D7—48no—
300C22×D7—48no—
301C7×D4—416no—
302C2×C4×C7—216no—
303C2×C4×C7—24no—
304C2×C4×C7—24no—
305C2×C4×C7—24no—
306C2×C4×C7—22no—
307C2×C4×C7—24no—
308C2×C4×C7—22no—
309C2×C4×C7—24no—
310C2×C4×C7—22no—
311C2×C4×C7—416no—
312C2×C4×C7—48no—
313C2×C4×C7—44no—
314C2×C4×C7C22×D418yes—
315C22⋊D7—416no—
316C22⋊D7—416no—
317C2×Dic7—28no—
318C2×Dic7—216no—
319C2×Dic7—24no—
320C2×Dic7—24no—
321C2×Dic7—24no—
322C2×Dic7—232no—
323C2×Dic7—24no—
324C2×Dic7—22no—
325C2×Dic7—24no—
326C2×Dic7—22no—
327C2×Dic7—22no—
328C2×Dic7—24no—
329C2×Dic7—416no—
330C2×Dic7—48no—
331C2×Dic7—44no—
332C2×Dic7#123116yescomplemented
333C2×Dic7C22×D418yes—
334C2×Dic7C22×D414yes—
335C2×Dic7C22×D414yes—
336C2×Dic7C2×#4118yes—
337C25—71no—
338C25C22×D711yescharacteristic
339C25C22×D711yescharacteristic
340C22×D4—24no—
341C22×D4—24no—
342C22×D4—24no—
343C22×D4—74no—
344C22×D4—74no—
345C22×D4—144no—
346C22×D4—148no—
347C22×D4—144no—
348C22×D4—144no—
349C22×D4—144no—
350C22×D4C22×D714yes—
351C23×C4—24no—
352C23×C4—24no—
353C23×C4—24no—
354C23×C4—71no—
355C23×C4—78no—
356C23×C4—74no—
357C23×C4—72no—
358C23×C4—71no—
359C23×C4—74no—
360C23×C4—72no—
361C23×C4—71no—
362C23×C4—148no—
363C23×C4—144no—
364C23×C4—144no—
365C23×C4—144no—
366C23×C4C22×D714yes—
367C23×C4C22×D712yes—
368C23×C4C22×D711yescharacteristic
369C23×C4C22×D712yes—
370#123—232no—
371#123—1432no—
372#123—1432no—
373#123—1432no—
374#121—1432no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—78no—
377C2×C4⋊C4—78no—
378C2×C4⋊C4—74no—
379C2×C4⋊C4—78no—
380C2×C4⋊C4—78no—
381C2×C4⋊C4—78no—
382C2×C4⋊C4—148no—
383C2×C4⋊C4—148no—
384C2×C4⋊C4—1432no—
385C2×C4⋊C4—144no—
386C2×C4⋊C4—148no—
387C2×C4⋊C4—148no—
388C2×C4⋊C4—148no—
389C2×C4⋊C4C22×D718yes—
390C2×C22⋊C4—28no—
391C2×C22⋊C4—78no—
392C2×C22⋊C4—74no—
393C2×C22⋊C4—74no—
394C2×C22⋊C4—78no—
395C2×C22⋊C4—74no—
396C2×C22⋊C4—78no—
397C2×C22⋊C4—78no—
398C2×C22⋊C4—144no—
399C2×C22⋊C4—148no—
400C2×C22⋊C4—144no—
401C2×C22⋊C4—148no—
402C2×C22⋊C4—148no—
403C2×C22⋊C4C22⋊D718yescomplemented
404C2×C22⋊C4C22×D718yes—
405C2×C22⋊C4C22×D714yes—
406C2×C42—78no—
407C2×C42—148no—
408#95—78no—
409C22×C7—24no—
410C22×C7—24no—
411C22×C7—28no—
412C22×C7—28no—
413C22×C7—232no—
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—44no—
418C22×C7—44no—
419C22×C7—88no—
420C22×C7C2×#12314yes—
421C22×C7C2×#12718yes—
422C22×C7C2×#12714yes—
423C22×C7#53614yes—
424C22×C7#54718yes—
425C22×C7#54718yes—
426C22×C7C23×D411yescharacteristic
427C22×C7C23×D411yescharacteristic
428C22×C7C23×D411yescharacteristic
429C22×C7C22×#4112yes—
430C22×C7C2×2−1+412yes—
431C2×D7—42no—
432C2×D7—42no—
433C2×D7—42no—
434C2×D7—44no—
435C2×D7—44no—
436C2×D7—88no—
437C4×C7—28no—
438C4×C7—44no—
439C4×C7—42no—
440Dic7—24no—
441Dic7—28no—
442Dic7—44no—
443Dic7—42no—
444Dic7C2×#12318yescomplemented
445C24—24no—
446C24—24no—
447C24—22no—
448C24—22no—
449C24—24no—
450C24—24no—
451C24—22no—
452C24—72no—
453C24—142no—
454C24—142no—
455C24—144no—
456C24—144no—
457C24—142no—
458C24D4×D714yes—
459C24D4×D714yes—
460C24C2×C22⋊D714yes—
461C24C23×D712yes—
462C24C23×D712yes—
463C24C23×D711yescharacteristic
464C2×D4—28no—
465C2×D4—28no—
466C2×D4—28no—
467C2×D4—416no—
468C2×D4—148no—
469C2×D4—1416no—
470C2×D4—148no—
471C2×D4—148no—
472C2×D4—148no—
473C2×D4—2816no—
474C2×D4—2816no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—28no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—28no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—28no—
486C22×C4—416no—
487C22×C4—74no—
488C22×C4—78no—
489C22×C4—72no—
490C22×C4—78no—
491C22×C4—74no—
492C22×C4—716no—
493C22×C4—72no—
494C22×C4—74no—
495C22×C4—74no—
496C22×C4—72no—
497C22×C4—1432no—
498C22×C4—1416no—
499C22×C4—144no—
500C22×C4—144no—
501C22×C4—144no—
502C22×C4—144no—
503C22×C4—142no—
504C22×C4—148no—
505C22×C4—144no—
506C22×C4—142no—
507C22×C4—148no—
508C22×C4—144no—
509C22×C4—142no—
510C22×C4—148no—
511C22×C4—2816no—
512C22×C4C2×C22⋊D718yes—
513C22×C4C23×D714yes—
514C22×C4C23×D712yes—
515C22×C4C23×D714yes—
516C4⋊C4—216no—
517C4⋊C4—1416no—
518C4⋊C4—1416no—
519C4⋊C4—148no—
520C4⋊C4—1416no—
521C4⋊C4—1416no—
522C4⋊C4—1416no—
523C22⋊C4—216no—
524C22⋊C4—716no—
525C22⋊C4—148no—
526C22⋊C4—1416no—
527C22⋊C4—148no—
528C22⋊C4—1416no—
529C22⋊C4—1416no—
530C22⋊C4C2×C22⋊D7116yescomplemented
531C42—1416no—
532C2×C7—28no—
533C2×C7—82no—
534C2×C7#250118yescomplemented
535C2×C7C22×#12311yescharacteristic
536C2×C7C22×#12711yescharacteristic
537C2×C7C2×#53611yescharacteristic
538C2×C7C2×#54712yes—
539C2×C7C2×#54712yes—
540D7—82no—
541C23—216no—
542C23—28no—
543C23—28no—
544C23—216no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—22no—
552C23—44no—
553C23—48no—
554C23—44no—
555C23—44no—
556C23—48no—
557C23—142no—
558C23—142no—
559C23—144no—
560C23—144no—
561C23—142no—
562C23—284no—
563C23—288no—
564C23—284no—
565C23—284no—
566C23—288no—
567C23#6503116yes—
568C23C2×D4×D714yes—
569C23C2×D4×D714yes—
570C23C2×D4×D712yes—
571C23C2×D4×D712yes—
572C23D7×#4114yes—
573C23#655414yes—
574C23C22×C22⋊D714yes—
575C23C22×C22⋊D712yes—
576C23C24×D711yescharacteristic
577D4—416no—
578D4—2816no—
579D4—2816no—
580C2×C4—216no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—416no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—716no—
593C2×C4—74no—
594C2×C4—78no—
595C2×C4—78no—
596C2×C4—74no—
597C2×C4—1432no—
598C2×C4—148no—
599C2×C4—1416no—
600C2×C4—144no—
601C2×C4—144no—
602C2×C4—144no—
603C2×C4—144no—
604C2×C4—142no—
605C2×C4—144no—
606C2×C4—142no—
607C2×C4—144no—
608C2×C4—142no—
609C2×C4—2816no—
610C2×C4—288no—
611C2×C4—284no—
612C2×C4C22×C22⋊D718yes—
613C7C2×#250111yescharacteristic, complemented, Sylow
614C22—28no—
615C22—24no—
616C22—24no—
617C22—28no—
618C22—232no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—88no—
625C22—282no—
626C22—284no—
627C22—282no—
628C22—282no—
629C22—284no—
630C22—568no—
631C22#9033914yes—
632C22D7×#12314yes—
633C22#9034618yes—
634C22C2×#650318yes—
635C22C2×#650314yes—
636C22#9051518yes—
637C22C22×D4×D711yescharacteristic
638C22C22×D4×D711yescharacteristic
639C22C2×D7×#4112yes—
640C22C2×#655412yes—
641C22C23×C22⋊D711yescharacteristic
642C4—28no—
643C4—44no—
644C4—42no—
645C4—78no—
646C4—144no—
647C4—148no—
648C4—284no—
649C4—282no—
650C2—28no—
651C2—82no—
652C2—562no—
653C2#1174030418yesdirect product factor
654C2C2×#9033911yescharacteristic
655C2C2×D7×#12311yescharacteristic
656C2C2×#9034612yes—
657C2C22×#650311yescharacteristic
658C2C2×#9051512yes—
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC22×C7×#123C2
socleC24×C7C24
prime radicalC22×#6502C2
hypercenterC22×#123D7
hypocenterC7C2×#2501
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C22×C7×#123C2semidirect
2. C22×#6498C2semidirect
4. C22×#6492C2semidirect
5. C23×Dic7⋊C4C2semidirect
6. C22×#6449C2semidirect
7. #11740304C2direct (N × H)
8. C2×#89764C2semidirect
9. C2×#89748C2semidirect
10. C2×#89721C2semidirect
11. C2×#89729C2semidirect
12. C2×C22⋊C4×Dic7C2semidirect
20. C2×C7×#123C22semidirect
30. C2×#6498C22semidirect
32. C2×#6492C22semidirect
43. C22×Dic7⋊C4C22semidirect
46. C2×#6449C22semidirect
48. #89764C22semidirect
49. #89748C22semidirect
50. #89729C22semidirect
51. #89721C22semidirect
52. C22⋊C4×Dic7C22semidirect
91. C2×C7×C22⋊C4D4semidirect
96. C2×C22⋊Dic7D4semidirect
129. C22×#123D7semidirect
211. C7×C22⋊C4C2×D4semidirect
214. C22⋊Dic7C2×D4semidirect
238. C2×#123C2×D7semidirect
332. C2×Dic7#123semidirect
403. C2×C22⋊C4C22⋊D7semidirect
444. Dic7C2×#123semidirect
530. C22⋊C4C2×C22⋊D7semidirect
534. C2×C7#2501semidirect
613. C7C2×#2501semidirect
653. C2#11740304direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1232
C22×#64982
C22×#65022
C22×#64922
C23×Dic7⋊C42
C22×#64492
#117403042
C2×#897642
C2×#897482
C2×#897212
C2×#897292
C2×C22⋊C4×Dic72
C2×#25017