Finite Group Databasebeta

← C2×#11727722 #13319651

Subgroups — C2×#11727722 #13319651

Order 1792.

1738 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D7C211yesmaximal, characteristic
2C2×C7×#407C211yesmaximal, characteristic, complemented
3C2×#89979C211yesmaximal, characteristic, complemented
4#11727722C218yesmaximal, direct product factor
5C2×#89264C212yesmaximal, complemented
6C2×#89263C212yesmaximal, complemented
7C24×C4×C7C2211yescharacteristic
8C23×C22⋊D7C411yescharacteristic
9C23×C4×D7C411yescharacteristic
10C22×C7×M16—21no—
11C22×C22⋊Dic7C411yescharacteristic
12C2×C4×C22⋊D7—21no—
13C2×C4×C22⋊D7—21no—
14C2×C4×C22⋊D7—22no—
15C2×C4×C22⋊D7—21no—
16C2×C4×C22⋊D7—21no—
17C2×C4×C22⋊D7—22no—
18C2×C4×C22⋊D7C414yes—
19C2×C4×C22⋊D7C2214yes—
20C22×(C2×D7)⋊C4C2211yescharacteristic
21C22×Dic7⋊C4C2211yescharacteristic
22C22×C4×Dic7C411yescharacteristic
23C22×#1136—21no—
24C7×#407—22no—
25C7×#407C2214yescomplemented
26C2×C7×C22⋊C8—21no—
27C2×C7×C22⋊C8C2212yes—
28#89979—22no—
29#89979C2214yescomplemented
30C2×C22⋊(C7⋊C8)—21no—
31C2×C22⋊(C7⋊C8)C2212yes—
32#89264C2218yescomplemented
33#89263C2218yescomplemented
34C2×#1465—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—21no—
50C22×C22⋊D7—22no—
51C22×C22⋊D7—24no—
52C22×C22⋊D7—24no—
53C22×C22⋊D7C2×C414yes—
54C23×Dic7C2×C411yescharacteristic
55C23×Dic7C2×C411yescharacteristic
56C23×Dic7C2×C411yescharacteristic
57C22×C4×D7—21no—
58C22×C4×D7—22no—
59C22×C4×D7—21no—
60C22×C4×D7—44no—
61C22×C4×D7C2×C414yes—
62C2×C7×M16—24no—
63C2×C7×M16—44no—
64C22×C7×C8—22no—
65C2×C22⋊Dic7—21no—
66C2×C22⋊Dic7—21no—
67C2×C22⋊Dic7—22no—
68C2×C22⋊Dic7C2×C414yes—
69C4×C22⋊D7—24no—
70C4×C22⋊D7—24no—
71C4×C22⋊D7—24no—
72C4×C22⋊D7—24no—
73C4×C22⋊D7—24no—
74C4×C22⋊D7—24no—
75C4×C22⋊D7—24no—
76C4×C22⋊D7—24no—
77C2×(C2×D7)⋊C4—21no—
78C2×(C2×D7)⋊C4—21no—
79C2×(C2×D7)⋊C4—21no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4C2×C412yes—
82C2×(C2×D7)⋊C4C2312yes—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4C2×C412yes—
88C2×Dic7⋊C4C2312yes—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—22no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2×C414yes—
93C2×#1136—24no—
94C2×#1136—44no—
95C22×C7⋊C8—22no—
96C7×C22⋊C8—24no—
97C7×C22⋊C8—22no—
98C7×C22⋊C8C2314yes—
99C22⋊(C7⋊C8)—24no—
100C22⋊(C7⋊C8)—22no—
101C22⋊(C7⋊C8)C2314yes—
102C22×C4×D4—71no—
103C2×#407—71no—
104C2×#407D711yescharacteristic, complemented
105#1465—78no—
106C2×#324—72no—
107C2×#323—72no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—42no—
111C24×C7—42no—
112C24×C7C22⋊C411yescharacteristic
113C24×C7C22⋊C411yescharacteristic
114C24×C7C22⋊C412yes—
115C24×C7C22×C411yescharacteristic
116C24×C7C22×C412yes—
117C23×D7—21no—
118C23×D7—21no—
119C23×D7—22no—
120C23×D7—42no—
121C23×D7—42no—
122C23×D7M1614yescomplemented
123C23×D7C22×C412yes—
124C22×C4×C7—24no—
125C22×C4×C7—24no—
126C22×C4×C7—24no—
127C22×C4×C7—44no—
128C22×C4×C7—44no—
129C22×C4×C7—44no—
130C22×C4×C7—44no—
131C22×C4×C7—42no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7C22⋊C411yescharacteristic
135C22×C4×C7C22⋊C412yes—
136C22×C4×C7C22⋊C412yes—
137C22×C4×C7C22⋊C412yes—
138C22×C4×C7C22⋊C412yes—
139C22×C4×C7C22⋊C412yes—
140C22×C4×C7C22⋊C411yescharacteristic
141C22×C4×C7C22×C412yes—
142C22×C4×C7C22×C411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2411yescharacteristic
152C2×C22⋊D7—216no—
153C2×C22⋊D7—28no—
154C2×C22⋊D7—28no—
155C2×C22⋊D7—24no—
156C2×C22⋊D7—24no—
157C2×C22⋊D7—28no—
158C2×C22⋊D7—28no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—22no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—42no—
167C22×Dic7—42no—
168C22×Dic7M1614yescomplemented
169C22×Dic7C22×C412yes—
170C22×Dic7C22×C412yes—
171C22×Dic7C22×C412yes—
172C2×C4×D7—24no—
173C2×C4×D7—24no—
174C2×C4×D7—44no—
175C2×C4×D7—48no—
176C2×C4×D7—44no—
177C2×C4×D7—48no—
178C7×M16—48no—
179C2×C7×C8—24no—
180C2×C7×C8—44no—
181C22⋊Dic7—24no—
182C22⋊Dic7—24no—
183(C2×D7)⋊C4—22no—
184(C2×D7)⋊C4—22no—
185(C2×D7)⋊C4—22no—
186(C2×D7)⋊C4—22no—
187Dic7⋊C4—22no—
188Dic7⋊C4—22no—
189Dic7⋊C4—22no—
190Dic7⋊C4—22no—
191C4×Dic7—24no—
192C4×Dic7—24no—
193#1136—48no—
194C2×C7⋊C8—24no—
195C2×C7⋊C8—44no—
196C23×D4—71no—
197C24×C4—71no—
198C24×C4C2×D711yescharacteristic
199C22×M16—21no—
200C22×M16—141no—
201C2×C4×D4—74no—
202C2×C4×D4—74no—
203C2×C4×D4—141no—
204C2×C4×D4—141no—
205C2×C4×D4—142no—
206C2×C4×D4—141no—
207C2×C4×D4—141no—
208C2×C4×D4—142no—
209C22×C4⋊C4—71no—
210C22×C22⋊C4—71no—
211C22×C22⋊C4—71no—
212C22×C42—71no—
213#407—22no—
214#407—74no—
215#407—142no—
216#407C2×D714yescomplemented
217C2×C22⋊C8—21no—
218C2×C22⋊C8—72no—
219C2×C22⋊C8—141no—
220C2×C22⋊C8C2×D712yes—
221#324—78no—
222#323—78no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—24no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—48no—
239C23×C7—48no—
240C23×C7C2×C22⋊C412yes—
241C23×C7C2×C22⋊C412yes—
242C23×C7C2×C22⋊C411yescharacteristic
243C23×C7C2×C22⋊C411yescharacteristic
244C23×C7C2×M1612yes—
245C23×C7C2×M1612yes—
246C23×C7C23×C411yescharacteristic
247C22×D7—22no—
248C22×D7—22no—
249C22×D7—24no—
250C22×D7—24no—
251C22×D7—42no—
252C22×D7—42no—
253C22×D7—48no—
254C22×D7—42no—
255C22×D7—42no—
256C22×D7—42no—
257C22×D7—48no—
258C22×D7—42no—
259C22×D7C2×M1614yescomplemented
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7#9914yes—
278C2×C4×C7C2×C22⋊C412yes—
279C2×C4×C7C2×C22⋊C411yescharacteristic
280C2×C4×C7C2×C22⋊C411yescharacteristic
281C2×C4×C7C2×C22⋊C412yes—
282C2×C4×C7C22×D411yescharacteristic
283C2×C4×C7C22×D411yescharacteristic
284C22⋊D7—216no—
285C22⋊D7—216no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—24no—
289C2×Dic7—24no—
290C2×Dic7—22no—
291C2×Dic7—22no—
292C2×Dic7—42no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—42no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7C2×M1614yescomplemented
299C4×D7—48no—
300C4×D7—48no—
301C7×C8—44no—
302C7⋊C8—44no—
303C25—71no—
304C25C4×D711yescharacteristic
305C22×D4—74no—
306C22×D4—144no—
307C22×D4—144no—
308C22×D4—142no—
309C22×D4—141no—
310C22×D4—141no—
311C23×C4—44no—
312C23×C4—71no—
313C23×C4—71no—
314C23×C4—71no—
315C23×C4—74no—
316C23×C4—142no—
317C23×C4—141no—
318C23×C4—141no—
319C23×C4—284no—
320C23×C4C4×D711yescharacteristic
321C23×C4C4×D712yes—
322C23×C4D2811yescharacteristic
323C23×C4D2811yescharacteristic
324C23×C4D2812yes—
325C23×C4C22⋊D711yescharacteristic
326C23×C4C22⋊D711yescharacteristic
327C23×C4C22⋊D712yes—
328C23×C4C22×D711yescharacteristic
329C23×C4C22×D712yes—
330C2×M16—24no—
331C2×M16—44no—
332C2×M16—144no—
333C2×M16—284no—
334C22×C8—22no—
335C22×C8—142no—
336C4×D4—144no—
337C4×D4—144no—
338C4×D4—144no—
339C4×D4—144no—
340C4×D4—144no—
341C4×D4—144no—
342C4×D4—144no—
343C4×D4—144no—
344C2×C4⋊C4—72no—
345C2×C4⋊C4—72no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C22⋊C4—74no—
351C2×C22⋊C4—72no—
352C2×C22⋊C4—72no—
353C2×C22⋊C4—142no—
354C2×C22⋊C4—141no—
355C2×C22⋊C4—141no—
356C2×C22⋊C4—141no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C42—74no—
361C2×C42—142no—
362C2×C42—141no—
363C2×C42—141no—
364C22⋊C8—24no—
365C22⋊C8—22no—
366C22⋊C8—74no—
367C22⋊C8—142no—
368C22⋊C8—144no—
369C22⋊C8C22×D714yes—
370C22×C7—21no—
371C22×C7—21no—
372C22×C7—21no—
373C22×C7—21no—
374C22×C7—24no—
375C22×C7—24no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—44no—
383C22×C7—44no—
384C22×C7—48no—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7#40712yes—
388C22×C7#40712yes—
389C22×C7C2×#9912yes—
390C22×C7#41312yes—
391C22×C7C22×C22⋊C411yescharacteristic
392C22×C7C22×M1611yescharacteristic
393C22×C7C22×M1611yescharacteristic
394C2×D7—24no—
395C2×D7—44no—
396C2×D7—48no—
397C2×D7—42no—
398C2×D7—42no—
399C2×D7—48no—
400C2×D7—44no—
401C4×C7—21no—
402C4×C7—21no—
403C4×C7—22no—
404C4×C7—21no—
405C4×C7—21no—
406C4×C7C2×#9914yes—
407Dic7—24no—
408Dic7—42no—
409Dic7—42no—
410C24—22no—
411C24—22no—
412C24—42no—
413C24—42no—
414C24—74no—
415C24—72no—
416C24—142no—
417C24—141no—
418C24—141no—
419C24—282no—
420C24—282no—
421C24(C2×D7)⋊C411yescharacteristic
422C24(C2×D7)⋊C411yescharacteristic
423C24(C2×D7)⋊C412yes—
424C24C2×C4×D711yescharacteristic
425C24C2×C4×D712yes—
426C2×D4—148no—
427C2×D4—148no—
428C2×D4—1416no—
429C2×D4—148no—
430C2×D4—148no—
431C2×D4—144no—
432C2×D4—144no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—74no—
444C22×C4—72no—
445C22×C4—72no—
446C22×C4—72no—
447C22×C4—142no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—144no—
455C22×C4—144no—
456C22×C4—284no—
457C22×C4—288no—
458C22×C4—288no—
459C22×C4—284no—
460C22×C4—282no—
461C22×C4—282no—
462C22×C4(C2×D7)⋊C412yes—
463C22×C4(C2×D7)⋊C412yes—
464C22×C4(C2×D7)⋊C412yes—
465C22×C4(C2×D7)⋊C411yescharacteristic
466C22×C4(C2×D7)⋊C412yes—
467C22×C4(C2×D7)⋊C412yes—
468C22×C4(C2×D7)⋊C411yescharacteristic
469C22×C4C2×C4×D712yes—
470C22×C4C2×C4×D711yescharacteristic
471C22×C4C2×D2812yes—
472C22×C4C2×D2811yescharacteristic
473C22×C4C2×D2811yescharacteristic
474C22×C4C2×D2812yes—
475C22×C4C2×C22⋊D712yes—
476C22×C4C2×C22⋊D711yescharacteristic
477C22×C4C2×C22⋊D711yescharacteristic
478C22×C4C2×C22⋊D712yes—
479C22×C4C23×D711yescharacteristic
480M16—48no—
481M16—288no—
482C2×C8—24no—
483C2×C8—44no—
484C2×C8—144no—
485C2×C8—284no—
486C4⋊C4—142no—
487C4⋊C4—142no—
488C4⋊C4—142no—
489C4⋊C4—142no—
490C22⋊C4—144no—
491C22⋊C4—144no—
492C22⋊C4—142no—
493C22⋊C4—142no—
494C22⋊C4—142no—
495C22⋊C4—142no—
496C42—144no—
497C42—144no—
498C2×C7—21no—
499C2×C7—21no—
500C2×C7—22no—
501C2×C7—42no—
502C2×C7—42no—
503C2×C7#146514yescomplemented
504C2×C7C2×#40711yescharacteristic
505C2×C7C22×#9911yescharacteristic
506C2×C7C2×#41311yescharacteristic
507D7—44no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—48no—
524C23—48no—
525C23—74no—
526C23—144no—
527C23—144no—
528C23—142no—
529C23—142no—
530C23—282no—
531C23—282no—
532C23—288no—
533C23—282no—
534C23—282no—
535C23—288no—
536C23—282no—
537C23—282no—
538C23D7×M1612yes—
539C23D7×M1612yes—
540C23C2×(C2×D7)⋊C412yes—
541C23C2×(C2×D7)⋊C412yes—
542C23C2×(C2×D7)⋊C411yescharacteristic
543C23C2×(C2×D7)⋊C411yescharacteristic
544C23C22×C4×D711yescharacteristic
545D4—1416no—
546D4—1416no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—74no—
565C2×C4—144no—
566C2×C4—144no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—288no—
572C2×C4—282no—
573C2×C4—282no—
574C2×C4—282no—
575C2×C4—282no—
576C2×C4—288no—
577C2×C4—282no—
578C2×C4—282no—
579C2×C4#638014yes—
580C2×C4C2×(C2×D7)⋊C412yes—
581C2×C4C2×(C2×D7)⋊C411yescharacteristic
582C2×C4C2×(C2×D7)⋊C411yescharacteristic
583C2×C4C2×(C2×D7)⋊C412yes—
584C2×C4C22×D2811yescharacteristic
585C2×C4C22×C22⋊D711yescharacteristic
586C8—44no—
587C8—284no—
588C7C2×#146511yescharacteristic, complemented, Sylow
589C22—21no—
590C22—21no—
591C22—24no—
592C22—24no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—48no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—144no—
607C22—284no—
608C22—288no—
609C22—284no—
610C22—282no—
611C22—288no—
612C22—282no—
613C22C2×#638012yes—
614C22#8989812yes—
615C22#8989812yes—
616C22#8990412yes—
617C22C2×D7×M1611yescharacteristic
618C22C2×D7×M1611yescharacteristic
619C22C22×(C2×D7)⋊C411yescharacteristic
620C4—21no—
621C4—21no—
622C4—22no—
623C4—21no—
624C4—21no—
625C4—144no—
626C4—282no—
627C4—282no—
628C4C2×#638014yes—
629C2—22no—
630C2—21no—
631C2—21no—
632C2—42no—
633C2—42no—
634C2—284no—
635C2#1172772214yesdirect product factor
636C2C22×#638011yescharacteristic
637C2C2×#8989811yescharacteristic
638C2C2×#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×#1465
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. C2×#89979C2semidirect
4. #11727722C2direct (N × H)
5. C2×#89264C2semidirect
6. C2×#89263C2semidirect
25. C7×#407C22semidirect
29. #89979C22semidirect
32. #89264C22semidirect
33. #89263C22semidirect
104. C2×#407D7semidirect
122. C23×D7M16semidirect
168. C22×Dic7M16semidirect
216. #407C2×D7semidirect
259. C22×D7C2×M16semidirect
298. C2×Dic7C2×M16semidirect
503. C2×C7#1465semidirect
588. C7C2×#1465semidirect
635. C2#11727722direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D72
C2×C7×#4072
C2×#899792
#117277222
C2×#892642
C2×#892632
C2×#14657