Finite Group Databasebeta

← C2×#11727723 #13319741

Subgroups — C2×#11727723 #13319741

Order 1792.

1490 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6492C211yesmaximal, characteristic
2C2×C7×#407C211yesmaximal, characteristic, complemented
3C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4#11727723C218yesmaximal, direct product factor
5C2×#89264C212yesmaximal, complemented
6C2×#89263C212yesmaximal, complemented
7C24×C4×C7C2211yescharacteristic
8C23×C22⋊D7C411yescharacteristic
9C22×C7×M16—21no—
10C22×C22⋊Dic7C411yescharacteristic
11C2×#6492—21no—
12C2×#6492—21no—
13C2×#6492—22no—
14C2×#6492—21no—
15C2×#6492—21no—
16C2×#6492—22no—
17C2×#6492C414yes—
18C2×#6492C2214yes—
19C22×(C2×D7)⋊C4C411yescharacteristic
20C22×(C2×D7)⋊C4C2211yescharacteristic
21C22×Dic7⋊C4C411yescharacteristic
22C22×Dic7⋊C4C2211yescharacteristic
23C23×C7⋊C8—21no—
24C7×#407—22no—
25C7×#407C2214yescomplemented
26C2×C7×C22⋊C8—21no—
27C2×C7×C22⋊C8C2212yes—
28C2×C22⋊(C7⋊C8)—22no—
29C2×C22⋊(C7⋊C8)—21no—
30C2×C22⋊(C7⋊C8)C2212yes—
31C2×C22⋊(C7⋊C8)C2214yescomplemented
32#89264C2218yescomplemented
33#89263C2218yescomplemented
34C2×#1468—71nomaximal, Sylow
35C25×C7C2×C411yescharacteristic
36C24×D7C2×C411yescharacteristic
37C23×C4×C7—44no—
38C23×C4×C7C2×C411yescharacteristic
39C23×C4×C7C2×C412yes—
40C23×C4×C7D411yescharacteristic
41C23×C4×C7D411yescharacteristic
42C23×C4×C7D412yes—
43C23×C4×C7D411yescharacteristic
44C23×C4×C7D411yescharacteristic
45C23×C4×C7D412yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C22×C22⋊D7—21no—
49C22×C22⋊D7—22no—
50C22×C22⋊D7—21no—
51C22×C22⋊D7—44no—
52C22×C22⋊D7C2×C414yes—
53C23×Dic7C2×C411yescharacteristic
54C23×Dic7C2×C411yescharacteristic
55C23×Dic7C2×C411yescharacteristic
56C2×C7×M16—24no—
57C2×C7×M16—44no—
58C22×C7×C8—22no—
59C2×C22⋊Dic7—21no—
60C2×C22⋊Dic7—22no—
61C2×C22⋊Dic7—21no—
62C2×C22⋊Dic7C2×C414yes—
63#6492—24no—
64#6492—24no—
65#6492—24no—
66#6492—24no—
67#6492—24no—
68#6492—24no—
69#6492—24no—
70#6492—24no—
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—22no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4C2×C414yes—
79C2×(C2×D7)⋊C4C2×C412yes—
80C2×(C2×D7)⋊C4C2312yes—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—22no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4C2×C414yes—
89C2×Dic7⋊C4C2×C412yes—
90C2×Dic7⋊C4C2312yes—
91C22×C7⋊C8—22no—
92C22×C7⋊C8—24no—
93C22×C7⋊C8—44no—
94C7×C22⋊C8—24no—
95C7×C22⋊C8—22no—
96C7×C22⋊C8C2314yes—
97C22⋊(C7⋊C8)—22no—
98C22⋊(C7⋊C8)—24no—
99C22⋊(C7⋊C8)C2314yes—
100C22×#123—71no—
101C2×#407D711yescharacteristic, complemented
102C22×C22⋊C8—71no—
103#1468—78no—
104C2×#324—72no—
105C2×#323—72no—
106C24×C7—24no—
107C24×C7—24no—
108C24×C7—42no—
109C24×C7C22⋊C411yescharacteristic
110C24×C7C22⋊C412yes—
111C24×C7C22⋊C411yescharacteristic
112C24×C7C22×C412yes—
113C24×C7C22×C411yescharacteristic
114C23×D7—21no—
115C23×D7—21no—
116C23×D7—24no—
117C23×D7—42no—
118C23×D7—42no—
119C23×D7C22×C412yes—
120C22×C4×C7—24no—
121C22×C4×C7—24no—
122C22×C4×C7—42no—
123C22×C4×C7—44no—
124C22×C4×C7—44no—
125C22×C4×C7—42no—
126C22×C4×C7—42no—
127C22×C4×C7—44no—
128C22×C4×C7—42no—
129C22×C4×C7—44no—
130C22×C4×C7C22⋊C411yescharacteristic
131C22×C4×C7C22⋊C411yescharacteristic
132C22×C4×C7C22⋊C412yes—
133C22×C4×C7C22⋊C412yes—
134C22×C4×C7C22⋊C412yes—
135C22×C4×C7C22⋊C412yes—
136C22×C4×C7C22⋊C412yes—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C22×C411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D412yes—
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2411yescharacteristic
148C2×C22⋊D7—24no—
149C2×C22⋊D7—24no—
150C2×C22⋊D7—48no—
151C2×C22⋊D7—44no—
152C2×C22⋊D7—44no—
153C2×C22⋊D7—48no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—24no—
161C22×Dic7—42no—
162C22×Dic7—42no—
163C22×Dic7C22×C412yes—
164C22×Dic7C22×C412yes—
165C22×Dic7C22×C412yes—
166C7×M16—48no—
167C2×C7×C8—24no—
168C2×C7×C8—44no—
169C22⋊Dic7—24no—
170C22⋊Dic7—24no—
171(C2×D7)⋊C4—24no—
172(C2×D7)⋊C4—24no—
173(C2×D7)⋊C4—22no—
174(C2×D7)⋊C4—22no—
175(C2×D7)⋊C4—22no—
176(C2×D7)⋊C4—22no—
177Dic7⋊C4—24no—
178Dic7⋊C4—24no—
179Dic7⋊C4—22no—
180Dic7⋊C4—22no—
181Dic7⋊C4—22no—
182Dic7⋊C4—22no—
183C2×C7⋊C8—24no—
184C2×C7⋊C8—44no—
185C2×C7⋊C8—48no—
186C23×D4—71no—
187C24×C4C2×D711yescharacteristic
188C22×M16—21no—
189C23×C8—141no—
190C2×#123—74no—
191C2×#123—74no—
192C2×#123—141no—
193C2×#123—142no—
194C2×#123—141no—
195C2×#123—141no—
196C2×#123—141no—
197C2×#123—142no—
198C22×C4⋊C4—71no—
199C22×C4⋊C4—71no—
200C22×C22⋊C4—71no—
201C22×C22⋊C4—71no—
202C22×C22⋊C4—71no—
203#407—22no—
204#407C2×D714yescomplemented
205C2×C22⋊C8—21no—
206C2×C22⋊C8—72no—
207C2×C22⋊C8—74no—
208C2×C22⋊C8—142no—
209C2×C22⋊C8—141no—
210C2×C22⋊C8C2×D712yes—
211#324—78no—
212#323—78no—
213C23×C7—22no—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7—22no—
217C23×C7—24no—
218C23×C7—24no—
219C23×C7—22no—
220C23×C7—24no—
221C23×C7—22no—
222C23×C7—24no—
223C23×C7—22no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—416no—
227C23×C7#9914yes—
228C23×C7C2×C22⋊C412yes—
229C23×C7C2×C22⋊C411yescharacteristic
230C23×C7C2×C22⋊C412yes—
231C23×C7C2×C22⋊C411yescharacteristic
232C23×C7#13114yes—
233C23×C7C23×C411yescharacteristic
234C22×D7—22no—
235C22×D7—24no—
236C22×D7—42no—
237C22×D7—42no—
238C22×D7—42no—
239C22×D7—42no—
240C22×D7—42no—
241C22×D7—42no—
242C22×D7—42no—
243C22×D7—42no—
244C22×D7—88no—
245C2×C4×C7—24no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7C2×C22⋊C412yes—
264C2×C4×C7C2×C22⋊C412yes—
265C2×C4×C7C2×C22⋊C411yescharacteristic
266C2×C4×C7C2×C22⋊C411yescharacteristic
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C22×D411yescharacteristic
269C22⋊D7—48no—
270C22⋊D7—48no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—24no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C7×C8—44no—
284C7⋊C8—44no—
285C25—71no—
286C25C4×D711yescharacteristic
287C22×D4—74no—
288C22×D4—142no—
289C22×D4—141no—
290C22×D4—141no—
291C22×D4—284no—
292C23×C4—44no—
293C23×C4—71no—
294C23×C4—71no—
295C23×C4—71no—
296C23×C4C4×D711yescharacteristic
297C23×C4C4×D712yes—
298C23×C4D2811yescharacteristic
299C23×C4D2812yes—
300C23×C4D2811yescharacteristic
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22⋊D711yescharacteristic
303C23×C4C22⋊D712yes—
304C23×C4C22×D711yescharacteristic
305C23×C4C22×D712yes—
306C2×M16—24no—
307C2×M16—44no—
308C22×C8—22no—
309C22×C8—144no—
310C22×C8—142no—
311C22×C8—284no—
312#123—144no—
313#123—144no—
314#123—144no—
315#123—144no—
316#123—144no—
317#123—144no—
318#123—144no—
319#123—144no—
320C2×C4⋊C4—72no—
321C2×C4⋊C4—74no—
322C2×C4⋊C4—72no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4—141no—
329C2×C4⋊C4—141no—
330C2×C22⋊C4—74no—
331C2×C22⋊C4—74no—
332C2×C22⋊C4—72no—
333C2×C22⋊C4—72no—
334C2×C22⋊C4—142no—
335C2×C22⋊C4—141no—
336C2×C22⋊C4—141no—
337C2×C22⋊C4—142no—
338C2×C22⋊C4—141no—
339C2×C22⋊C4—141no—
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C22⋊C8—24no—
345C22⋊C8—22no—
346C22⋊C8—74no—
347C22⋊C8—142no—
348C22⋊C8—144no—
349C22⋊C8C22×D714yes—
350C22×C7—22no—
351C22×C7—21no—
352C22×C7—21no—
353C22×C7—21no—
354C22×C7—21no—
355C22×C7—22no—
356C22×C7—28no—
357C22×C7—28no—
358C22×C7—22no—
359C22×C7—22no—
360C22×C7—22no—
361C22×C7—48no—
362C22×C7—48no—
363C22×C7—44no—
364C22×C7#40812yes—
365C22×C7#40812yes—
366C22×C7C2×#9914yes—
367C22×C7C2×#9912yes—
368C22×C7#41312yes—
369C22×C7C22×C22⋊C411yescharacteristic
370C22×C7C2×#13112yes—
371C2×D7—42no—
372C2×D7—42no—
373C2×D7—42no—
374C2×D7—84no—
375C2×D7—84no—
376C2×D7—84no—
377C4×C7—21no—
378C4×C7—21no—
379C4×C7—22no—
380C4×C7—21no—
381C4×C7—21no—
382C4×C7—22no—
383Dic7—42no—
384Dic7—42no—
385Dic7—42no—
386C24—24no—
387C24—24no—
388C24—42no—
389C24—72no—
390C24—141no—
391C24—141no—
392C24—144no—
393C24—282no—
394C24—282no—
395C24(C2×D7)⋊C411yescharacteristic
396C24(C2×D7)⋊C412yes—
397C24(C2×D7)⋊C411yescharacteristic
398C24C2×C4×D712yes—
399C24C2×C4×D711yescharacteristic
400C2×D4—144no—
401C2×D4—144no—
402C2×D4—288no—
403C2×D4—284no—
404C2×D4—284no—
405C2×D4—288no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—72no—
417C22×C4—72no—
418C22×C4—72no—
419C22×C4—144no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—282no—
427C22×C4—282no—
428C22×C4(C2×D7)⋊C412yes—
429C22×C4(C2×D7)⋊C412yes—
430C22×C4(C2×D7)⋊C411yescharacteristic
431C22×C4(C2×D7)⋊C411yescharacteristic
432C22×C4(C2×D7)⋊C412yes—
433C22×C4(C2×D7)⋊C412yes—
434C22×C4(C2×D7)⋊C412yes—
435C22×C4C2×C4×D712yes—
436C22×C4C2×C4×D711yescharacteristic
437C22×C4C2×D2812yes—
438C22×C4C2×D2811yescharacteristic
439C22×C4C2×D2811yescharacteristic
440C22×C4C2×D2812yes—
441C22×C4C2×C22⋊D712yes—
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D712yes—
445C22×C4C23×D711yescharacteristic
446M16—48no—
447C2×C8—24no—
448C2×C8—44no—
449C2×C8—144no—
450C2×C8—288no—
451C2×C8—284no—
452C4⋊C4—144no—
453C4⋊C4—144no—
454C4⋊C4—142no—
455C4⋊C4—142no—
456C4⋊C4—142no—
457C4⋊C4—142no—
458C22⋊C4—144no—
459C22⋊C4—144no—
460C22⋊C4—144no—
461C22⋊C4—144no—
462C22⋊C4—142no—
463C22⋊C4—142no—
464C22⋊C4—142no—
465C22⋊C4—142no—
466C2×C7—22no—
467C2×C7—22no—
468C2×C7—24no—
469C2×C7—42no—
470C2×C7#146814yescomplemented
471C2×C7C2×#40811yescharacteristic
472C2×C7C22×#9911yescharacteristic
473C2×C7C2×#41311yescharacteristic
474D7—82no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—42no—
487C23—42no—
488C23—416no—
489C23—142no—
490C23—144no—
491C23—282no—
492C23—282no—
493C23—282no—
494C23—282no—
495C23—282no—
496C23—282no—
497C23—282no—
498C23—282no—
499C23—568no—
500C23#638014yes—
501C23#647014yes—
502C23C2×(C2×D7)⋊C412yes—
503C23C2×(C2×D7)⋊C411yescharacteristic
504C23C2×(C2×D7)⋊C412yes—
505C23C2×(C2×D7)⋊C411yescharacteristic
506C23C22×C4×D711yescharacteristic
507D4—288no—
508D4—288no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—144no—
528C2×C4—142no—
529C2×C4—142no—
530C2×C4—142no—
531C2×C4—282no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4—282no—
539C2×C4C2×(C2×D7)⋊C411yescharacteristic
540C2×C4C2×(C2×D7)⋊C411yescharacteristic
541C2×C4C2×(C2×D7)⋊C412yes—
542C2×C4C2×(C2×D7)⋊C412yes—
543C2×C4C22×D2811yescharacteristic
544C2×C4C22×C22⋊D711yescharacteristic
545C8—44no—
546C8—284no—
547C7C2×#146811yescharacteristic, complemented, Sylow
548C22—22no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—28no—
553C22—28no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—282no—
563C22—282no—
564C22—282no—
565C22—564no—
566C22—564no—
567C22—564no—
568C22C2×#638014yes—
569C22C2×#638012yes—
570C22#8990112yes—
571C22#8990112yes—
572C22#8990412yes—
573C22C2×#647012yes—
574C22C22×(C2×D7)⋊C411yescharacteristic
575C4—21no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—21no—
580C4—22no—
581C4—282no—
582C4—282no—
583C4—282no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—562no—
589C2#1172772314yesdirect product factor
590C2C22×#638011yescharacteristic
591C2C2×#8990111yescharacteristic
592C2C2×#8990411yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#407C2
socleC23×C7C2×C22⋊C4
prime radicalC23×C22⋊D7C4
cosocleC22×C4×C7C24
hypercenterC2×#407D7
hypocenterC7C2×#1468
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×C7×#407C2semidirect
3. C22×C22⋊(C7⋊C8)C2semidirect
4. #11727723C2direct (N × H)
5. C2×#89264C2semidirect
6. C2×#89263C2semidirect
25. C7×#407C22semidirect
31. C2×C22⋊(C7⋊C8)C22semidirect
32. #89264C22semidirect
33. #89263C22semidirect
101. C2×#407D7semidirect
204. #407C2×D7semidirect
470. C2×C7#1468semidirect
547. C7C2×#1468semidirect
589. C2#11727723direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64922
C2×C7×#4072
C22×C22⋊(C7⋊C8)2
#117277232
C2×#892642
C2×#892632
C2×#14687