Finite Group Databasebeta

← C2×#11729456 #13319740

Subgroups — C2×#11729456 #13319740

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×C7×#123C211yesmaximal, characteristic
2C2×#89979C211yesmaximal, characteristic, complemented
3C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4#11729456C218yesmaximal, direct product factor
5C2×#89323C212yesmaximal, complemented
6C2×#89320C212yesmaximal, complemented
7C23×C7×D4C411yescharacteristic
8C24×C4×C7C2211yescharacteristic
9C2×C7×#123—21no—
10C2×C7×#123—21no—
11C2×C7×#123—21no—
12C2×C7×#123—21no—
13C2×C7×#123—22no—
14C2×C7×#123—22no—
15C2×C7×#123C414yes—
16C2×C7×#123C2214yes—
17C22×C7×C4⋊C4C411yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C411yescharacteristic
20C22×C7×C22⋊C4C411yescharacteristic
21C22×C7×C22⋊C4C2211yescharacteristic
22C22×#1136—21no—
23C23×C7⋊C8—21no—
24#89979—22no—
25#89979C2214yescomplemented
26C2×C22⋊(C7⋊C8)—22no—
27C2×C22⋊(C7⋊C8)—21no—
28C2×C22⋊(C7⋊C8)—21no—
29C2×C22⋊(C7⋊C8)C2212yes—
30C2×C22⋊(C7⋊C8)C2212yes—
31C2×C22⋊(C7⋊C8)C2214yescomplemented
32#89323C2218yescomplemented
33#89320C2218yescomplemented
34C2×#1468—71nomaximal, Sylow
35C25×C7C2×C411yescharacteristic
36C25×C7C2×C411yescharacteristic
37C22×C7×D4—21no—
38C22×C7×D4—22no—
39C22×C7×D4—21no—
40C22×C7×D4—44no—
41C22×C7×D4C2×C414yes—
42C23×C4×C7—44no—
43C23×C4×C7C2×C411yescharacteristic
44C23×C4×C7C2×C411yescharacteristic
45C23×C4×C7C2×C411yescharacteristic
46C23×C4×C7C2×C411yescharacteristic
47C23×C4×C7C2×C412yes—
48C23×C4×C7D411yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D412yes—
53C23×C4×C7D412yes—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2312yes—
56C7×#123—24no—
57C7×#123—24no—
58C7×#123—24no—
59C7×#123—24no—
60C7×#123—24no—
61C7×#123—24no—
62C7×#123—24no—
63C7×#123—24no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2×C414yes—
72C2×C7×C4⋊C4C2×C412yes—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4C2×C414yes—
85C2×C7×C22⋊C4C2×C414yes—
86C2×C7×C22⋊C4C2×C412yes—
87C2×C7×C22⋊C4C2312yes—
88C2×#1136—24no—
89C2×#1136—44no—
90C22×C7⋊C8—22no—
91C22×C7⋊C8—24no—
92C22×C7⋊C8—22no—
93C22×C7⋊C8—44no—
94C22⋊(C7⋊C8)—22no—
95C22⋊(C7⋊C8)—24no—
96C22⋊(C7⋊C8)—24no—
97C22⋊(C7⋊C8)—22no—
98C22⋊(C7⋊C8)C2314yes—
99C22⋊(C7⋊C8)C2314yes—
100C22×#123D711yescharacteristic
101C2×#407—71no—
102C22×C22⋊C8—71no—
103#1468—78no—
104C2×#324—72no—
105C2×#323—72no—
106C24×C7—24no—
107C24×C7—24no—
108C24×C7—21no—
109C24×C7—21no—
110C24×C7—24no—
111C24×C7—42no—
112C24×C7—42no—
113C24×C7—42no—
114C24×C7C22⋊C411yescharacteristic
115C24×C7C22⋊C412yes—
116C24×C7C22⋊C411yescharacteristic
117C24×C7C22×C412yes—
118C24×C7C22×C411yescharacteristic
119C24×C7C22×C412yes—
120C2×C7×D4—24no—
121C2×C7×D4—24no—
122C2×C7×D4—48no—
123C2×C7×D4—44no—
124C2×C7×D4—44no—
125C2×C7×D4—48no—
126C22×C4×C7—24no—
127C22×C4×C7—24no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—24no—
135C22×C4×C7—42no—
136C22×C4×C7—44no—
137C22×C4×C7—44no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7—44no—
141C22×C4×C7—42no—
142C22×C4×C7—44no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7C22⋊C411yescharacteristic
146C22×C4×C7C22⋊C411yescharacteristic
147C22×C4×C7C22⋊C412yes—
148C22×C4×C7C22⋊C412yes—
149C22×C4×C7C22⋊C412yes—
150C22×C4×C7C22⋊C412yes—
151C22×C4×C7C22⋊C412yes—
152C22×C4×C7C22×C412yes—
153C22×C4×C7C22×C412yes—
154C22×C4×C7C22×C412yes—
155C22×C4×C7C22×C412yes—
156C22×C4×C7C22×C411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2411yescharacteristic
166C7×C4⋊C4—24no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—24no—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180#1136—48no—
181C2×C7⋊C8—24no—
182C2×C7⋊C8—24no—
183C2×C7⋊C8—44no—
184C2×C7⋊C8—48no—
185C2×C7⋊C8—44no—
186C23×D4Dic711yescharacteristic
187C24×C4C2×D711yescharacteristic
188C22×M16—141no—
189C23×C8—141no—
190C2×#123—21no—
191C2×#123—21no—
192C2×#123—22no—
193C2×#123—21no—
194C2×#123—21no—
195C2×#123—22no—
196C2×#123Dic714yes—
197C2×#123C2×D714yes—
198C22×C4⋊C4Dic711yescharacteristic
199C22×C4⋊C4C2×D711yescharacteristic
200C22×C22⋊C4Dic711yescharacteristic
201C22×C22⋊C4Dic711yescharacteristic
202C22×C22⋊C4C2×D711yescharacteristic
203#407—74no—
204#407—142no—
205C2×C22⋊C8—72no—
206C2×C22⋊C8—74no—
207C2×C22⋊C8—72no—
208C2×C22⋊C8—142no—
209C2×C22⋊C8—141no—
210C2×C22⋊C8—141no—
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—22no—
225C23×C7—24no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—416no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—88no—
238C23×C7#9914yes—
239C23×C7C2×C22⋊C412yes—
240C23×C7C2×C22⋊C411yescharacteristic
241C23×C7C2×C22⋊C412yes—
242C23×C7C2×C22⋊C411yescharacteristic
243C23×C7#13114yes—
244C23×C7C23×C411yescharacteristic
245C7×D4—48no—
246C7×D4—48no—
247C2×C4×C7—24no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7C2×C22⋊C412yes—
278C2×C4×C7C2×C22⋊C412yes—
279C2×C4×C7C2×C22⋊C411yescharacteristic
280C2×C4×C7C2×C22⋊C411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C22×D411yescharacteristic
283C7⋊C8—44no—
284C7⋊C8—44no—
285C25C2×Dic711yescharacteristic
286C25C2×Dic711yescharacteristic
287C22×D4—22no—
288C22×D4—21no—
289C22×D4—21no—
290C22×D4—44no—
291C22×D4C2×Dic714yes—
292C23×C4—44no—
293C23×C4C2×Dic711yescharacteristic
294C23×C4C2×Dic711yescharacteristic
295C23×C4C2×Dic711yescharacteristic
296C23×C4C2×Dic711yescharacteristic
297C23×C4C2×Dic712yes—
298C23×C4C22⋊D711yescharacteristic
299C23×C4C22⋊D711yescharacteristic
300C23×C4C22⋊D712yes—
301C23×C4C22⋊D711yescharacteristic
302C23×C4C22⋊D711yescharacteristic
303C23×C4C22⋊D712yes—
304C23×C4C22×D711yescharacteristic
305C23×C4C22×D712yes—
306C2×M16—144no—
307C2×M16—284no—
308C22×C8—144no—
309C22×C8—142no—
310C22×C8—142no—
311C22×C8—284no—
312#123—24no—
313#123—24no—
314#123—24no—
315#123—24no—
316#123—24no—
317#123—24no—
318#123—24no—
319#123—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C2×Dic714yes—
328C2×C4⋊C4C2×Dic712yes—
329C2×C4⋊C4C22×D712yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4C2×Dic714yes—
341C2×C22⋊C4C2×Dic714yes—
342C2×C22⋊C4C2×Dic712yes—
343C2×C22⋊C4C22×D712yes—
344C22⋊C8—74no—
345C22⋊C8—74no—
346C22⋊C8—142no—
347C22⋊C8—144no—
348C22⋊C8—142no—
349C22⋊C8—144no—
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—42no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—84no—
368C22×C7—84no—
369C22×C7—84no—
370C22×C7#40812yes—
371C22×C7#40812yes—
372C22×C7C2×#9914yes—
373C22×C7C2×#9912yes—
374C22×C7#41312yes—
375C22×C7C22×C22⋊C411yescharacteristic
376C22×C7C2×#13112yes—
377C4×C7—21no—
378C4×C7—21no—
379C4×C7—22no—
380C4×C7—21no—
381C4×C7—21no—
382C4×C7—22no—
383C4×C7—42no—
384C4×C7—42no—
385C4×C7—42no—
386C24—24no—
387C24—24no—
388C24—21no—
389C24—21no—
390C24—24no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22⋊Dic711yescharacteristic
395C24C22⋊Dic712yes—
396C24C22⋊Dic711yescharacteristic
397C24C22×Dic712yes—
398C24C22×Dic711yescharacteristic
399C24C22×Dic712yes—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—48no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C22⋊Dic712yes—
426C22×C4C22⋊Dic712yes—
427C22×C4C22⋊Dic711yescharacteristic
428C22×C4C22⋊Dic711yescharacteristic
429C22×C4C22⋊Dic712yes—
430C22×C4C22⋊Dic712yes—
431C22×C4C22⋊Dic712yes—
432C22×C4C22×Dic712yes—
433C22×C4C22×Dic712yes—
434C22×C4C22×Dic712yes—
435C22×C4C22×Dic712yes—
436C22×C4C22×Dic711yescharacteristic
437C22×C4C2×C22⋊D712yes—
438C22×C4C2×C22⋊D712yes—
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D712yes—
444C22×C4C2×C22⋊D712yes—
445C22×C4C23×D711yescharacteristic
446M16—288no—
447C2×C8—144no—
448C2×C8—144no—
449C2×C8—288no—
450C2×C8—284no—
451C2×C8—284no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C2×C7—22no—
467C2×C7—22no—
468C2×C7—24no—
469C2×C7—42no—
470C2×C7—82no—
471C2×C7#146814yescomplemented
472C2×C7C2×#40811yescharacteristic
473C2×C7C22×#9911yescharacteristic
474C2×C7C2×#41311yescharacteristic
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—416no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—88no—
500C23#640814yes—
501C23#651114yes—
502C23C2×C22⋊Dic712yes—
503C23C2×C22⋊Dic712yes—
504C23C2×C22⋊Dic711yescharacteristic
505C23C2×C22⋊Dic711yescharacteristic
506C23C23×Dic711yescharacteristic
507D4—48no—
508D4—48no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4C2×C22⋊Dic711yescharacteristic
540C2×C4C2×C22⋊Dic711yescharacteristic
541C2×C4C2×C22⋊Dic712yes—
542C2×C4C2×C22⋊Dic712yes—
543C2×C4C22×C22⋊D711yescharacteristic
544C2×C4C22×C22⋊D711yescharacteristic
545C8—284no—
546C8—284no—
547C7C2×#146811yescharacteristic, complemented, Sylow
548C22—21no—
549C22—21no—
550C22—28no—
551C22—28no—
552C22—22no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—48no—
560C22—48no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22C2×#640814yes—
569C22C2×#640812yes—
570C22#9000612yes—
571C22#9000612yes—
572C22#9000912yes—
573C22C2×#651112yes—
574C22C22×C22⋊Dic711yescharacteristic
575C4—21no—
576C4—21no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—42no—
584C2—22no—
585C2—22no—
586C2—24no—
587C2—42no—
588C2—82no—
589C2#1172945614yesdirect product factor
590C2C22×#640811yescharacteristic
591C2C2×#9000611yescharacteristic
592C2C2×#9000911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C7×#123C2
socleC23×C7C2×C22⋊C4
prime radicalC23×C7×D4C4
cosocleC22×C4×C7C24
hypercenterC22×#123D7
hypocenterC7C2×#1468
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. C2×#89979C2semidirect
3. C22×C22⋊(C7⋊C8)C2semidirect
4. #11729456C2direct (N × H)
5. C2×#89323C2semidirect
6. C2×#89320C2semidirect
25. #89979C22semidirect
31. C2×C22⋊(C7⋊C8)C22semidirect
32. #89323C22semidirect
33. #89320C22semidirect
471. C2×C7#1468semidirect
547. C7C2×#1468semidirect
589. C2#11729456direct (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
C2×#899792
C22×C22⋊(C7⋊C8)2
#117294562
C2×#893232
C2×#893202
C2×#14687