Finite Group Databasebeta

← #13502635

Subgroups — #13502635

Order 1792.

1468 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×#41C211yesmaximal, characteristic
2C2×#90011C212yesmaximal
3C2×#89803C211yesmaximal, characteristic, complemented
4C2×#89791C211yesmaximal, characteristic, complemented
5#11739521C211yesmaximal, characteristic, complemented
6#11729386C212yesmaximal, complemented
7#11729380C211yesmaximal, characteristic, complemented
8#11729366C212yesmaximal, complemented
9#11729358C211yesmaximal, characteristic, complemented
10#11729339C212yesmaximal, complemented
11#11729284C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C4×C7×#41—24no—
14C4×C7×#41—24no—
15C2×C4×C7×Q8C2211yescharacteristic
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×D4—21no—
18C2×C4×C7×D4—21no—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C2×C7×#117—21no—
22C2×C7×#117C2211yescharacteristic
23C22×C42×C7—21no—
24C22×C42×C7C2211yescharacteristic
25C22×#1144C2212yes—
26C22×#1142C2212yes—
27C2×#6487C2212yes—
28C2×#6432C2211yescharacteristic
29#90011—28no—
30C2×#6409C2212yes—
31C2×#6406C2212yes—
32#89984—21no—
33#89984—21no—
34C2×C22⋊(C7⋊C8)C2212yes—
35#89803—24no—
36#89791—24no—
37C2×#6384C2212yes—
38C2×#6384C2212yes—
39#89707C2211yescharacteristic
40#89701C2212yes—
41#89700—21no—
42#89700—21no—
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45#89324C2212yes—
46#89324C2212yes—
47#89319C2212yes—
48#32907—71nomaximal, Sylow
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—22no—
53C2×C7×#41—22no—
54C2×C7×#41D414yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7—21no—
61C23×C4×C7D411yescharacteristic
62C23×C4×C7D411yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C22×Dic14C2312yes—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8D414yes—
69C4×C7×D4—28no—
70C4×C7×D4—28no—
71C4×C7×D4—22no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4D414yes—
75C7×#117—28no—
76C7×#117—22no—
77C7×#117—22no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C42×C7—28no—
86C2×C42×C7—21no—
87C2×C42×C7—21no—
88C2×C42×C7—21no—
89C2×C42×C7—22no—
90C2×C42×C7—22no—
91C2×C42×C7D411yescharacteristic
92C2×C42×C7D411yescharacteristic
93C2×C42×C7C2311yescharacteristic
94C2×C42×C7C2311yescharacteristic
95C2×C22⋊Dic7—22no—
96C2×#1144—24no—
97C2×#1144—24no—
98C2×#1144—24no—
99C2×#1142—24no—
100C2×#1142—24no—
101C2×#1142—24no—
102C2×C4⋊Dic7—21no—
103C2×C4⋊Dic7—21no—
104C2×C4⋊Dic7—21no—
105C2×C4⋊Dic7—21no—
106C2×C4⋊Dic7C2312yes—
107#6487—24no—
108#6487—24no—
109C2×Dic7⋊C4—22no—
110C22×C7⋊C8C2312yes—
111C22×C7⋊C8C2312yes—
112#6432—22no—
113#6432—22no—
114#6409—24no—
115#6406—24no—
116C7⋊#95—22no—
117C7⋊#95—22no—
118C22⋊(C7⋊C8)—24no—
119#6384—24no—
120#6384—24no—
121C4⋊(C7⋊C8)—22no—
122C4⋊(C7⋊C8)—22no—
123C2×C4×#41D711yescharacteristic
124C2×#462—71no—
125C2×#461—71no—
126C2×#452—72no—
127#2857—71no—
128#1878—72no—
129#1877—72no—
130#1873—72no—
131#1803—71no—
132#1802—71no—
133#1797—71no—
134C24×C7—21no—
135C24×C7C2×D411yescharacteristic
136C7×#41—28no—
137C7×#41—28no—
138C7×#41—28no—
139C7×#41—28no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8C2×D412yes—
144C2×C7×Q8C2×D412yes—
145C2×C7×Q8#4112yes—
146C2×C7×D4—24no—
147C2×C7×D4—24no—
148C2×C7×D4—24no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4C2×D412yes—
153C2×C7×D4C2×D412yes—
154C2×C7×D4#4112yes—
155C22×C4×C7—24no—
156C22×C4×C7—24no—
157C22×C4×C7—21no—
158C22×C4×C7—24no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—24no—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—24no—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—22no—
172C22×C4×C7—41no—
173C22×C4×C7—42no—
174C22×C4×C7—42no—
175C22×C4×C7—42no—
176C22×C4×C7—42no—
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D412yes—
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7#4112yes—
184C22×C4×C7C2411yescharacteristic
185C22×Dic7—22no—
186C22×Dic7—22no—
187C2×Dic14—22no—
188C2×Dic14—22no—
189C2×Dic14—22no—
190C2×Dic14—22no—
191C2×Dic14—22no—
192C2×Dic14—22no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4C2×D412yes—
198C7×C4⋊C4C2×D412yes—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—24no—
202C42×C7—24no—
203C42×C7—24no—
204C42×C7—22no—
205C42×C7—24no—
206C42×C7—22no—
207C42×C7—21no—
208C42×C7—21no—
209C42×C7C2×D412yes—
210C42×C7C2×D412yes—
211C22⋊Dic7—44no—
212#1144—48no—
213#1142—48no—
214C4⋊Dic7—22no—
215C4⋊Dic7—22no—
216C4⋊Dic7—42no—
217C4⋊Dic7—42no—
218C4⋊Dic7—42no—
219C4⋊Dic7—42no—
220Dic7⋊C4—44no—
221C2×C7⋊C8—22no—
222C2×C7⋊C8—22no—
223C2×C7⋊C8—22no—
224C2×C7⋊C8—22no—
225C2×C7⋊C8—22no—
226C2×C7⋊C8—22no—
227C22×#41C2×D711yescharacteristic
228C22×Q16—72no—
229C22×SD16—72no—
230C2×C4⋊Q8—71no—
231C2×C22⋊Q8—72no—
232C4×#41—24no—
233C4×#41—24no—
234C2×C4×Q8C2×D711yescharacteristic
235C2×C4×Q8C2×D711yescharacteristic
236C2×C4×D4—21no—
237C2×C4×D4—21no—
238C2×C4×D4C2×D711yescharacteristic
239C2×C4×D4C2×D711yescharacteristic
240C2×#117—21no—
241C2×#117C2×D711yescharacteristic
242C22×C42—21no—
243C22×C42C2×D711yescharacteristic
244#462—144no—
245#461—144no—
246#452—148no—
247C2×C4⋊C8—71no—
248C2×C4⋊C8—71no—
249C2×#103—72no—
250C2×#103—72no—
251C2×#103—72no—
252C2×#102—72no—
253C2×C22⋊C8—72no—
254#391—72no—
255#389—141no—
256#389—141no—
257#384—71no—
258#380—141no—
259#380—141no—
260#340—72no—
261#340—72no—
262#336—72no—
263C23×C7—21no—
264C23×C7—22no—
265C23×C7—22no—
266C23×C7—21no—
267C23×C7—22no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7#12112yes—
277C23×C7C22×D411yescharacteristic
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—22no—
281C7×Q8#12114yes—
282C7×D4—216no—
283C7×D4—22no—
284C7×D4—22no—
285C7×D4—22no—
286C7×D4#12114yes—
287C2×C4×C7—216no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—22no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—44no—
307C2×C4×C7—48no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—44no—
313C2×C4×C7—48no—
314C2×C4×C7#12012yes—
315C2×C4×C7#12012yes—
316C2×C4×C7#12112yes—
317C2×C4×C7#13512yes—
318C2×C4×C7#13712yes—
319C2×C4×C7#13712yes—
320C2×C4×C7C22×D411yescharacteristic
321C2×C4×C7C22×D411yescharacteristic
322C2×C4×C7C22×D411yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×Dic7—42no—
327C2×Dic7—42no—
328C2×Dic7—42no—
329C2×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7—42no—
332Dic14—42no—
333Dic14—42no—
334Dic14—42no—
335Dic14—42no—
336C7⋊C8—42no—
337C7⋊C8—42no—
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41D2814yes—
344C22×Q8—72no—
345C22×Q8C22×D711yescharacteristic
346C22×D4—21no—
347C22×D4C22×D711yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—21no—
351C23×C4C22⋊D711yescharacteristic
352C23×C4C22⋊D711yescharacteristic
353C23×C4C22×D711yescharacteristic
354C2×Q16—144no—
355C2×Q16—144no—
356C2×Q16—144no—
357C2×SD16—144no—
358C2×SD16—144no—
359C2×SD16—144no—
360C22×C8—72no—
361C22×C8—72no—
362C4⋊Q8—142no—
363C4⋊Q8—142no—
364C22⋊Q8—144no—
365C22⋊Q8—144no—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×Q8—22no—
369C4×Q8C22⋊D714yes—
370C4×D4—28no—
371C4×D4—28no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—22no—
375C4×D4C22⋊D714yes—
376#117—28no—
377#117—22no—
378#117—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—72no—
382C2×C4⋊C4—142no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4C22×D711yescharacteristic
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—142no—
393C2×C42—28no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42C22⋊D711yescharacteristic
400C2×C42C22⋊D711yescharacteristic
401C2×C42C22×D711yescharacteristic
402C2×C42C22×D711yescharacteristic
403C4⋊C8—142no—
404C4⋊C8—142no—
405#103—144no—
406#103—144no—
407#103—144no—
408#102—144no—
409C22⋊C8—144no—
410#95—142no—
411#95—142no—
412C22×C7—22no—
413C22×C7—24no—
414C22×C7—22no—
415C22×C7—22no—
416C22×C7—41no—
417C22×C7—42no—
418C22×C7—41no—
419C22×C7—42no—
420C22×C7—41no—
421C22×C7—42no—
422C22×C7—41no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7C2×#12011yescharacteristic
427C22×C7C2×#12111yescharacteristic
428C22×C7#52511yescharacteristic
429C22×C7C2×#13511yescharacteristic
430C22×C7C2×#13711yescharacteristic
431C22×C7C2×#13711yescharacteristic
432C22×C7#57511yescharacteristic
433C4×C7—22no—
434C4×C7—24no—
435C4×C7—22no—
436C4×C7—22no—
437C4×C7—41no—
438C4×C7—42no—
439C4×C7—42no—
440C4×C7#45214yescomplemented
441C4×C7C2×#12111yescharacteristic
442C4×C7C2×#12111yescharacteristic
443C4×C7#52511yescharacteristic
444C4×C7#52511yescharacteristic
445Dic7—82no—
446Dic7—82no—
447C24—21no—
448C24C2×C22⋊D711yescharacteristic
449#41—28no—
450#41—28no—
451#41—28no—
452#41—28no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—142no—
457C2×Q8—142no—
458C2×Q8—142no—
459C2×Q8—142no—
460C2×Q8—142no—
461C2×Q8—142no—
462C2×Q8C2×D2812yes—
463C2×Q8#115712yes—
464C2×Q8C2×C22⋊D712yes—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4C2×D2812yes—
472C2×D4#115712yes—
473C2×D4C2×C22⋊D712yes—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—41no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4C2×D2812yes—
499C22×C4#115712yes—
500C22×C4C2×C22⋊D711yescharacteristic
501C22×C4C2×C22⋊D711yescharacteristic
502C22×C4C2×C22⋊D711yescharacteristic
503C22×C4C2×C22⋊D711yescharacteristic
504C22×C4C2×C22⋊D711yescharacteristic
505C22×C4C23×D711yescharacteristic
506Q16—288no—
507SD16—288no—
508C2×C8—142no—
509C2×C8—142no—
510C2×C8—142no—
511C2×C8—142no—
512C2×C8—142no—
513C2×C8—142no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—142no—
519C4⋊C4—142no—
520C4⋊C4—282no—
521C4⋊C4—284no—
522C4⋊C4—282no—
523C4⋊C4—282no—
524C4⋊C4—282no—
525C4⋊C4C2×C22⋊D712yes—
526C4⋊C4C2×C22⋊D712yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—284no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—24no—
535C42—22no—
536C42—21no—
537C42—21no—
538C42C2×C22⋊D712yes—
539C42C2×C22⋊D712yes—
540C2×C7—42no—
541C2×C7—44no—
542C2×C7#284711yescharacteristic
543C2×C7C2×#45211yescharacteristic
544C2×C7#300711yescharacteristic
545C2×C7#303511yescharacteristic
546C2×C7#303911yescharacteristic
547C2×C7#310711yescharacteristic
548C2×C7#311611yescharacteristic
549C23—22no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#649312yes—
563C23C22×C22⋊D711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—282no—
568Q8—282no—
569Q8—282no—
570Q8—282no—
571Q8#649314yes—
572D4—216no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4#649314yes—
577C2×C4—216no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—282no—
605C2×C4—282no—
606C2×C4—282no—
607C2×C4—282no—
608C2×C4—282no—
609C2×C4—282no—
610C2×C4#649312yes—
611C2×C4#651312yes—
612C2×C4#651412yes—
613C2×C4#651412yes—
614C2×C4#651612yes—
615C2×C4#651612yes—
616C2×C4C22×D2811yescharacteristic
617C2×C4C2×#115711yescharacteristic
618C2×C4C2×#115711yescharacteristic
619C2×C4C2×#115711yescharacteristic
620C2×C4C22×C22⋊D711yescharacteristic
621C2×C4C22×C22⋊D711yescharacteristic
622C8—282no—
623C8—282no—
624C7#3290711yescharacteristic, complemented, Sylow
625C22—22no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22C2×#649311yescharacteristic
640C22#9048211yescharacteristic
641C22C2×#651311yescharacteristic
642C22#9051311yescharacteristic
643C22C2×#651411yescharacteristic
644C22C2×#651411yescharacteristic
645C22C2×#651611yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—22no—
650C4—41no—
651C4—42no—
652C4—42no—
653C4—562no—
654C4—562no—
655C4#9001114yescomplemented
656C4C2×#649311yescharacteristic
657C4C2×#649311yescharacteristic
658C4#9048211yescharacteristic
659C4#9048211yescharacteristic
660C2—42no—
661C2—44no—
662C2#1173997011yescharacteristic
663C2#1173997811yescharacteristic
664C2#1173997911yescharacteristic
665C2#1173998011yescharacteristic
666C2#1174147711yescharacteristic
667C2C2×#9001111yescharacteristic
668C2#1174152911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×#41C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
hypercenterC2×C4×#41D7
hypocenterC7#32907
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
3. C2×#89803C2semidirect
4. C2×#89791C2semidirect
5. #11739521C2semidirect
6. #11729386C2semidirect
7. #11729380C2semidirect
8. #11729366C2semidirect
9. #11729358C2semidirect
10. #11729339C2semidirect
11. #11729284C2semidirect
440. C4×C7#452semidirect
624. C7#32907semidirect
655. C4#90011semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×#412
C2×#900112
C2×#898032
C2×#897912
#117395212
#117293862
#117293802
#117293662
#117293582
#117293392
#117292842
#329077