Finite Group Databasebeta

← #13319673

Subgroups — #13319673

Order 1792.

1738 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C7×D4C211yesmaximal, characteristic
2C2×#89979C212yesmaximal, complemented
3#11729356C214yesmaximal, complemented
4#11729354C214yesmaximal, complemented
5#11729306C212yesmaximal, complemented
6#11729293C212yesmaximal, complemented
7C23×C7×D4C411yescharacteristic
8C24×C4×C7C411yescharacteristic
9C24×C4×C7C2211yescharacteristic
10C2×C4×C7×D4—22no—
11C2×C4×C7×D4—22no—
12C2×C4×C7×D4—22no—
13C2×C4×C7×D4—22no—
14C2×C4×C7×D4C412yes—
15C2×C4×C7×D4C412yes—
16C2×C4×C7×D4C2212yes—
17C2×C4×C7×D4C2212yes—
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4C411yescharacteristic
20C22×C7×C22⋊C4C2211yescharacteristic
21C22×C42×C7C411yescharacteristic
22C22×#1136—22no—
23#89979—22no—
24#89979—22no—
25#89979C2214yes—
26#89979C2214yescomplemented
27C2×C22⋊(C7⋊C8)—22no—
28C2×C22⋊(C7⋊C8)C2214yescomplemented
29#89323C2218yescomplemented
30#89320C2218yescomplemented
31#17362—71nomaximal, Sylow
32C25×C7C2×C411yescharacteristic
33C25×C7C2×C411yescharacteristic
34C22×C7×D4—22no—
35C22×C7×D4—22no—
36C22×C7×D4—28no—
37C22×C7×D4C2×C412yes—
38C22×C7×D4C2×C412yes—
39C23×C4×C7—21no—
40C23×C4×C7—21no—
41C23×C4×C7—21no—
42C23×C4×C7—21no—
43C23×C4×C7—44no—
44C23×C4×C7—44no—
45C23×C4×C7C2×C411yescharacteristic
46C23×C4×C7C2×C411yescharacteristic
47C23×C4×C7C2×C411yescharacteristic
48C23×C4×C7C2×C411yescharacteristic
49C23×C4×C7C2×C412yes—
50C23×C4×C7C2×C412yes—
51C23×C4×C7C2×C411yescharacteristic
52C23×C4×C7C2×C411yescharacteristic
53C23×C4×C7D412yes—
54C23×C4×C7D412yes—
55C23×C4×C7D412yes—
56C23×C4×C7D412yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C4×C7×D4—24no—
61C4×C7×D4—24no—
62C4×C7×D4—24no—
63C4×C7×D4—24no—
64C4×C7×D4—24no—
65C4×C7×D4—24no—
66C4×C7×D4—24no—
67C4×C7×D4—24no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4C2×C412yes—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—22no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4C2×C412yes—
77C2×C7×C22⋊C4C2×C412yes—
78C2×C7×C22⋊C4C2×C412yes—
79C2×C7×C22⋊C4C2312yes—
80C2×C42×C7—21no—
81C2×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7C2×C412yes—
85C2×C42×C7C2×C412yes—
86C2×#1136—24no—
87C2×#1136—24no—
88C2×#1136—48no—
89C22×C7⋊C8—24no—
90C22⋊(C7⋊C8)—24no—
91C22⋊(C7⋊C8)—24no—
92C22⋊(C7⋊C8)—24no—
93C22⋊(C7⋊C8)C2318yes—
94C22×C4×D4D711yescharacteristic
95C2×#407—72no—
96#1465—74no—
97#1463—72no—
98#1462—72no—
99#1459—74no—
100C24×C7—21no—
101C24×C7—21no—
102C24×C7—24no—
103C24×C7—24no—
104C24×C7—42no—
105C24×C7—42no—
106C24×C7—42no—
107C24×C7—42no—
108C24×C7C22⋊C412yes—
109C24×C7C22⋊C412yes—
110C24×C7C22×C411yescharacteristic
111C24×C7C22×C411yescharacteristic
112C24×C7C22×C412yes—
113C24×C7C22×C411yescharacteristic
114C2×C7×D4—216no—
115C2×C7×D4—216no—
116C2×C7×D4—24no—
117C2×C7×D4—24no—
118C2×C7×D4—24no—
119C2×C7×D4—24no—
120C2×C7×D4—24no—
121C2×C7×D4M1618yescomplemented
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—28no—
125C22×C4×C7—24no—
126C22×C4×C7—24no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—42no—
136C22×C4×C7—42no—
137C22×C4×C7—44no—
138C22×C4×C7—44no—
139C22×C4×C7—42no—
140C22×C4×C7—42no—
141C22×C4×C7—44no—
142C22×C4×C7—42no—
143C22×C4×C7—44no—
144C22×C4×C7—44no—
145C22×C4×C7—44no—
146C22×C4×C7—44no—
147C22×C4×C7—44no—
148C22×C4×C7—44no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7C22⋊C411yescharacteristic
152C22×C4×C7C22⋊C411yescharacteristic
153C22×C4×C7C22⋊C411yescharacteristic
154C22×C4×C7C22⋊C412yes—
155C22×C4×C7C22⋊C412yes—
156C22×C4×C7C22⋊C412yes—
157C22×C4×C7C22⋊C412yes—
158C22×C4×C7C22⋊C411yescharacteristic
159C22×C4×C7C22×C411yescharacteristic
160C22×C4×C7C22×C411yescharacteristic
161C22×C4×C7C22×C412yes—
162C22×C4×C7C22×C412yes—
163C22×C4×C7C22×C412yes—
164C22×C4×C7C22×C411yescharacteristic
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2411yescharacteristic
172C7×C4⋊C4—24no—
173C7×C4⋊C4—24no—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—24no—
177C7×C22⋊C4—24no—
178C42×C7—22no—
179C42×C7—22no—
180C42×C7—22no—
181C42×C7—22no—
182#1136—48no—
183#1136—48no—
184C2×C7⋊C8—28no—
185C2×C7⋊C8—48no—
186C23×D4Dic711yescharacteristic
187C24×C4Dic711yescharacteristic
188C24×C4C2×D711yescharacteristic
189C22×M16—142no—
190C2×C4×D4—22no—
191C2×C4×D4—22no—
192C2×C4×D4—22no—
193C2×C4×D4—22no—
194C2×C4×D4Dic712yes—
195C2×C4×D4Dic712yes—
196C2×C4×D4C2×D712yes—
197C2×C4×D4C2×D712yes—
198C22×C4⋊C4C2×D711yescharacteristic
199C22×C22⋊C4Dic711yescharacteristic
200C22×C22⋊C4C2×D711yescharacteristic
201C22×C42Dic711yescharacteristic
202#407—74no—
203#407—74no—
204#407—142no—
205#407—142no—
206C2×C22⋊C8—74no—
207C2×C22⋊C8—142no—
208#324—78no—
209#323—78no—
210C23×C7—22no—
211C23×C7—22no—
212C23×C7—22no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—22no—
216C23×C7—22no—
217C23×C7—22no—
218C23×C7—24no—
219C23×C7—24no—
220C23×C7—24no—
221C23×C7—24no—
222C23×C7—42no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—42no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—416no—
232C23×C7—416no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7C2×C22⋊C412yes—
237C23×C7C2×C22⋊C412yes—
238C23×C7C2×C22⋊C411yescharacteristic
239C23×C7C2×C22⋊C411yescharacteristic
240C23×C7C2×M1614yes—
241C23×C7C2×M1614yescomplemented
242C23×C7C23×C411yescharacteristic
243C7×D4—216no—
244C7×D4—216no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
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—24no—
258C2×C4×C7—24no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—41no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—44no—
269C2×C4×C7—44no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—44no—
275C2×C4×C7—44no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7#9914yes—
279C2×C4×C7C2×C22⋊C411yescharacteristic
280C2×C4×C7C2×C22⋊C411yescharacteristic
281C2×C4×C7C2×C22⋊C412yes—
282C2×C4×C7C2×C22⋊C411yescharacteristic
283C2×C4×C7C2×C22⋊C411yescharacteristic
284C2×C4×C7C2×M1614yescomplemented
285C2×C4×C7C22×D412yes—
286C7⋊C8—48no—
287C25C2×Dic711yescharacteristic
288C25C2×Dic711yescharacteristic
289C22×D4—22no—
290C22×D4—22no—
291C22×D4—28no—
292C22×D4C2×Dic712yes—
293C22×D4C2×Dic712yes—
294C23×C4—21no—
295C23×C4—21no—
296C23×C4—21no—
297C23×C4—21no—
298C23×C4—44no—
299C23×C4—44no—
300C23×C4C2×Dic711yescharacteristic
301C23×C4C2×Dic711yescharacteristic
302C23×C4C2×Dic711yescharacteristic
303C23×C4C2×Dic711yescharacteristic
304C23×C4C2×Dic712yes—
305C23×C4C2×Dic712yes—
306C23×C4C2×Dic711yescharacteristic
307C23×C4C2×Dic711yescharacteristic
308C23×C4C22⋊D712yes—
309C23×C4C22⋊D712yes—
310C23×C4C22⋊D712yes—
311C23×C4C22⋊D712yes—
312C23×C4C22×D711yescharacteristic
313C23×C4C22×D711yescharacteristic
314C23×C4C22×D711yescharacteristic
315C2×M16—144no—
316C2×M16—144no—
317C2×M16—288no—
318C22×C8—144no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4C2×Dic712yes—
330C2×C4⋊C4C22×D712yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×Dic712yes—
336C2×C22⋊C4C2×Dic712yes—
337C2×C22⋊C4C2×Dic712yes—
338C2×C22⋊C4C22×D712yes—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42C2×Dic712yes—
344C2×C42C2×Dic712yes—
345C22⋊C8—78no—
346C22⋊C8—144no—
347C22⋊C8—144no—
348C22⋊C8—144no—
349C22×C7—28no—
350C22×C7—21no—
351C22×C7—21no—
352C22×C7—21no—
353C22×C7—21no—
354C22×C7—22no—
355C22×C7—21no—
356C22×C7—21no—
357C22×C7—22no—
358C22×C7—22no—
359C22×C7—24no—
360C22×C7—22no—
361C22×C7—22no—
362C22×C7—44no—
363C22×C7—48no—
364C22×C7—48no—
365C22×C7—44no—
366C22×C7—44no—
367C22×C7—44no—
368C22×C7—48no—
369C22×C7—48no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7#40714yes—
373C22×C7C2×#9912yes—
374C22×C7#41312yes—
375C22×C7C22×C22⋊C411yescharacteristic
376C22×C7C22×M1612yes—
377C4×C7—24no—
378C4×C7—22no—
379C4×C7—21no—
380C4×C7—21no—
381C4×C7—22no—
382C4×C7—42no—
383C4×C7—42no—
384C4×C7C2×#9912yes—
385C4×C7C2×#9912yes—
386C24—24no—
387C24—24no—
388C24—21no—
389C24—21no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22⋊Dic712yes—
395C24C22⋊Dic712yes—
396C24C22×Dic711yescharacteristic
397C24C22×Dic712yes—
398C24C22×Dic711yescharacteristic
399C24C22×Dic711yescharacteristic
400C2×D4—216no—
401C2×D4—216no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4#113618yescomplemented
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4C22⋊Dic711yescharacteristic
438C22×C4C22⋊Dic711yescharacteristic
439C22×C4C22⋊Dic712yes—
440C22×C4C22⋊Dic711yescharacteristic
441C22×C4C22⋊Dic712yes—
442C22×C4C22⋊Dic712yes—
443C22×C4C22⋊Dic712yes—
444C22×C4C22⋊Dic711yescharacteristic
445C22×C4C22×Dic712yes—
446C22×C4C22×Dic711yescharacteristic
447C22×C4C22×Dic711yescharacteristic
448C22×C4C22×Dic712yes—
449C22×C4C22×Dic712yes—
450C22×C4C22×Dic711yescharacteristic
451C22×C4C2×C22⋊D712yes—
452C22×C4C2×C22⋊D712yes—
453C22×C4C2×C22⋊D712yes—
454C22×C4C2×C22⋊D712yes—
455C22×C4C2×C22⋊D712yes—
456C22×C4C2×C22⋊D712yes—
457C22×C4C23×D711yescharacteristic
458M16—288no—
459M16—288no—
460C2×C8—148no—
461C2×C8—288no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C2×C7—22no—
473C2×C7—22no—
474C2×C7—22no—
475C2×C7—24no—
476C2×C7—42no—
477C2×C7—44no—
478C2×C7C2×#40711yescharacteristic
479C2×C7C22×#9911yescharacteristic
480C2×C7C2×#41311yescharacteristic
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—416no—
503C23—416no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C2×#113614yes—
508C23C2×#113614yescomplemented
509C23C2×C22⋊Dic712yes—
510C23C2×C22⋊Dic712yes—
511C23C2×C22⋊Dic711yescharacteristic
512C23C2×C22⋊Dic711yescharacteristic
513C23C23×Dic711yescharacteristic
514D4—216no—
515D4—216no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4#640814yes—
550C2×C4C2×#113614yescomplemented
551C2×C4C2×C22⋊Dic711yescharacteristic
552C2×C4C2×C22⋊Dic711yescharacteristic
553C2×C4C2×C22⋊Dic712yes—
554C2×C4C2×C22⋊Dic711yescharacteristic
555C2×C4C2×C22⋊Dic711yescharacteristic
556C2×C4C22×C22⋊D712yes—
557C8—288no—
558C7#1736211yescharacteristic, complemented, Sylow
559C22—28no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—48no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—48no—
579C22—48no—
580C22—42no—
581C22—42no—
582C22#8997914yes—
583C22C2×#640812yes—
584C22#9000912yes—
585C22C22×#113612yes—
586C22C22×C22⋊Dic711yescharacteristic
587C4—24no—
588C4—22no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4C2×#640812yes—
595C4C2×#640812yes—
596C2—22no—
597C2—22no—
598C2—22no—
599C2—24no—
600C2—42no—
601C2—44no—
602C2C2×#8997911yescharacteristic
603C2C22×#640811yescharacteristic
604C2C2×#9000911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊Dic7
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C4×C7×D4C2
socleC22×C7C22×C22⋊C4
prime radicalC23×C7×D4C4
cosocleC22×C4×C7C24
hypercenterC22×C4×D4D7
hypocenterC7#17362
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#89979C2semidirect
3. #11729356C2semidirect
4. #11729354C2semidirect
5. #11729306C2semidirect
6. #11729293C2semidirect
26. #89979C22semidirect
28. C2×C22⋊(C7⋊C8)C22semidirect
29. #89323C22semidirect
30. #89320C22semidirect
121. C2×C7×D4M16semidirect
241. C23×C7C2×M16semidirect
284. C2×C4×C7C2×M16semidirect
407. C2×D4#1136semidirect
508. C23C2×#1136semidirect
550. C2×C4C2×#1136semidirect
558. C7#17362semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C7×D42
C2×#899792
#117293562
#117293542
#117293062
#117292932
#173627