Finite Group Databasebeta

← #13388097

Subgroups — #13388097

Order 1792.

1478 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#532C211yesmaximal, characteristic, complemented
2C2×#89859C211yesmaximal, characteristic, complemented
3C2×#89844C211yesmaximal, characteristic, complemented
4C4×#6493C211yesmaximal, characteristic, complemented
5#11729534C211yesmaximal, characteristic, complemented
6#11729515C211yesmaximal, characteristic, complemented
7#11729438C212yesmaximal, complemented
8#11729433C212yesmaximal, complemented
9#11729367C212yesmaximal, complemented
10#11729360C212yesmaximal, complemented
11#11729284C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C7×#532—24no—
14C7×#532—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#127C2211yescharacteristic
17C2×C7×#124—21no—
18C2×C7×#121—21no—
19C2×C7×#121C2212yes—
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2212yes—
22C22×C42×C7C2211yescharacteristic
23C22×#1141C2212yes—
24C2×#6493C2211yescharacteristic
25C2×C4×C22⋊D7—21no—
26C2×C4×D28C2211yescharacteristic
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29C2×#6406C2212yes—
30#89985—21no—
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2212yes—
33#89859—24no—
34#89844—24no—
35C2×#6383C2212yes—
36C2×#6381C2212yes—
37#89712—21no—
38#89711C2211yescharacteristic
39C4×(C2×D7)⋊C4—21no—
40C4×C4⋊Dic7C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×C4⋊(C7⋊C8)C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89319C2212yes—
47#24150—71nomaximal, Sylow
48C2×C7×#41—28no—
49C2×C7×#41—44no—
50C2×C7×#41—44no—
51C22×C7×Q8—21no—
52C22×C7×D4—22no—
53C22×C7×D4—21no—
54C22×C7×D4—21no—
55C22×C7×D4C2312yes—
56C23×C4×C7—22no—
57C23×C4×C7D411yescharacteristic
58C23×C4×C7D411yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C22×C22⋊D7—21no—
61C22×D28C2311yescharacteristic
62C22×C4×D7—21no—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8D414yescomplemented
65C7×#127—22no—
66C7×#127D414yescomplemented
67C7×#124—28no—
68C7×#121—28no—
69C7×#121—24no—
70C7×#121—24no—
71C4×C7×D4—28no—
72C4×C7×D4—24no—
73C4×C7×D4—24no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2312yes—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C42×C7—22no—
79C2×C42×C7—22no—
80C2×C42×C7D411yescharacteristic
81C2×C42×C7D411yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C22⋊Dic7—21no—
85C2×#1141—24no—
86C2×#1141—24no—
87C2×#1141—42no—
88#6493—22no—
89#6493—22no—
90C4×C22⋊D7—44no—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×(C2×D7)⋊C4—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C22×C7⋊C8C2312yes—
101C22×C7⋊C8C2312yes—
102C4×D28—22no—
103C4×D28—22no—
104#6406—24no—
105#6406—24no—
106#6406—42no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C22⋊(C7⋊C8)—42no—
111#6383—24no—
112#6381—24no—
113C4⋊(C7⋊C8)—22no—
114C4⋊(C7⋊C8)—22no—
115C2×#532D711yescharacteristic, complemented
116C2×#474—71no—
117C2×#459—71no—
118C4×#121—71no—
119#2033—72no—
120#1999—72no—
121#1988—71no—
122#1982—71no—
123#1879—72no—
124#1862—72no—
125#1797—71no—
126C24×C7—22no—
127C24×C7C2×D411yescharacteristic
128C23×D7—21no—
129C7×#41—432no—
130C2×C7×Q8—24no—
131C2×C7×Q8—42no—
132C2×C7×Q8—42no—
133C2×C7×D4—28no—
134C2×C7×D4—24no—
135C2×C7×D4—24no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C2×C7×D4—44no—
144C2×C7×D4—44no—
145C2×C7×D4—42no—
146C2×C7×D4—42no—
147C2×C7×D4C2×D414yes—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—28no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—41no—
160C22×C4×C7—41no—
161C22×C4×C7—44no—
162C22×C4×C7—44no—
163C22×C4×C7D814yescomplemented
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2411yescharacteristic
170C2×C22⋊D7—24no—
171C2×C22⋊D7—42no—
172C2×C22⋊D7—42no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C2×D28—21no—
177C2×D28—21no—
178C2×D28—21no—
179C2×D28—21no—
180C2×D28—21no—
181C2×D28#4112yescomplemented
182C2×C4×D7—42no—
183C2×C4×D7—42no—
184C2×C4×D7—42no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4C2×D414yes—
188C7×C22⋊C4—28no—
189C7×C22⋊C4—28no—
190C42×C7—22no—
191C42×C7—22no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7C2×D412yes—
195C42×C7C2×D412yes—
196C22⋊Dic7—42no—
197#1141—48no—
198(C2×D7)⋊C4—24no—
199(C2×D7)⋊C4—42no—
200(C2×D7)⋊C4—42no—
201(C2×D7)⋊C4—42no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7#4112yescomplemented
208Dic7⋊C4—42no—
209C4×Dic7—42no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8—22no—
214C2×C7⋊C8—22no—
215C2×C7⋊C8—22no—
216C22×#41—21no—
217C22×D8—72no—
218#532—24no—
219#532—24no—
220C2×C4⋊Q8C2×D711yescharacteristic
221C2×#127C2×D711yescharacteristic
222C2×#124—21no—
223C2×#121—21no—
224C2×#121—71no—
225C2×#121C2×D712yes—
226C2×C4×D4—21no—
227C2×C4×D4—71no—
228C2×C4×D4—141no—
229C2×C4×D4C2×D712yes—
230C22×C42C2×D711yescharacteristic
231#474—144no—
232#459—144no—
233C2×#107—72no—
234C2×C4⋊C8—71no—
235C2×C4⋊C8—71no—
236C2×#102—72no—
237C2×#102—72no—
238C2×#102—72no—
239C2×#102—72no—
240C2×C22⋊C8—72no—
241#390—71no—
242#389—71no—
243#388—141no—
244#386—141no—
245#380—71no—
246C4×C4⋊C4—71no—
247C4×C22⋊C4—141no—
248#340—72no—
249#340—72no—
250#336—72no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—24no—
258C23×C7—42no—
259C23×C7—42no—
260C23×C7—42no—
261C23×C7—42no—
262C23×C7—42no—
263C23×C7—42no—
264C23×C7C2×D812yes—
265C23×C7C22×D411yescharacteristic
266C22×D7—22no—
267C22×D7—41no—
268C22×D7—41no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C7×Q8—48no—
274C7×D4—24no—
275C7×D4—24no—
276C7×D4—48no—
277C7×D4—416no—
278C7×D4—42no—
279C7×D4—42no—
280C7×D4—48no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—28no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—24no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—44no—
294C2×C4×C7—416no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7#12012yes—
302C2×C4×C7#12012yes—
303C2×C4×C7C2×D812yes—
304C2×C4×C7C2×D812yes—
305C2×C4×C7AGL(1,ℤ/8)12yes—
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C22×D411yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7H(5,𝔽2)11yescharacteristic
311C2×C4×C72−1+411yescharacteristic
312C22⋊D7—48no—
313C2×Dic7—22no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322D28—22no—
323D28—22no—
324D28—41no—
325D28—41no—
326C4×D7—84no—
327C7⋊C8—42no—
328C7⋊C8—42no—
329C2×#41—28no—
330C2×#41—44no—
331C2×#41—44no—
332C22×Q8—21no—
333C22×D4—22no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—71no—
337C22×D4—141no—
338C22×D4C22×D712yes—
339C23×C4—22no—
340C23×C4—141no—
341C23×C4C22⋊D711yescharacteristic
342C23×C4C22⋊D711yescharacteristic
343C23×C4C22×D711yescharacteristic
344C2×D8—144no—
345C2×D8—144no—
346C2×D8—282no—
347C22×C8—72no—
348C22×C8—72no—
349C4⋊Q8—22no—
350C4⋊Q8C22⋊D714yescomplemented
351#127—22no—
352#127C22⋊D714yescomplemented
353#124—28no—
354#121—28no—
355#121—24no—
356#121—24no—
357#121—142no—
358#121—142no—
359C4×D4—28no—
360C4×D4—24no—
361C4×D4—24no—
362C4×D4—142no—
363C4×D4—142no—
364C4×D4—284no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4C22×D712yes—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—141no—
374C2×C22⋊C4—141no—
375C2×C22⋊C4—141no—
376C2×C22⋊C4—141no—
377C2×C22⋊C4—141no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—141no—
381C2×C42C22⋊D711yescharacteristic
382C2×C42C22⋊D711yescharacteristic
383C2×C42C22×D711yescharacteristic
384C2×C42C22×D711yescharacteristic
385#107—144no—
386C4⋊C8—142no—
387C4⋊C8—142no—
388#102—144no—
389#102—144no—
390#102—144no—
391#102—282no—
392C22⋊C8—282no—
393#95—141no—
394#95—141no—
395#95—141no—
396C22×C7—22no—
397C22×C7—22no—
398C22×C7—22no—
399C22×C7—41no—
400C22×C7—41no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—44no—
404C22×C7—44no—
405C22×C7—44no—
406C22×C7—82no—
407C22×C7—82no—
408C22×C7—82no—
409C22×C7—82no—
410C22×C7C2×#12011yescharacteristic
411C22×C7#52511yescharacteristic
412C22×C7#53711yescharacteristic
413C22×C7C22×D811yescharacteristic
414C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
415C22×C7#57511yescharacteristic
416C22×C7#57511yescharacteristic
417C2×D7—42no—
418C2×D7—42no—
419C2×D7—42no—
420C2×D7—81no—
421C2×D7—81no—
422C2×D7—81no—
423C2×D7—81no—
424C4×C7—22no—
425C4×C7—22no—
426C4×C7—41no—
427C4×C7—44no—
428C4×C7—41no—
429C4×C7#44714yes—
430C4×C7#54611yescharacteristic
431C4×C7#54711yescharacteristic
432C4×C7#54811yescharacteristic
433C4×C7#55011yescharacteristic
434Dic7—42no—
435Dic7—81no—
436Dic7—81no—
437C24—22no—
438C24—141no—
439C24C2×C22⋊D711yescharacteristic
440#41—432no—
441C2×Q8—24no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—28no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—72no—
459C2×D4—144no—
460C2×D4—141no—
461C2×D4—141no—
462C2×D4—141no—
463C2×D4—141no—
464C2×D4—141no—
465C2×D4—282no—
466C2×D4—282no—
467C2×D4C2×C22⋊D714yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—141no—
486C22×C4—282no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4#114114yescomplemented
490C22×C4C2×C22⋊D711yescharacteristic
491C22×C4C2×C22⋊D711yescharacteristic
492C22×C4C2×C22⋊D711yescharacteristic
493C22×C4C2×C22⋊D711yescharacteristic
494C22×C4C2×C22⋊D711yescharacteristic
495C22×C4C23×D711yescharacteristic
496D8—288no—
497C2×C8—142no—
498C2×C8—142no—
499C2×C8—142no—
500C2×C8—142no—
501C2×C8—142no—
502C2×C8—142no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—72no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—282no—
512C4⋊C4C2×C22⋊D714yes—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—144no—
516C22⋊C4—282no—
517C22⋊C4—282no—
518C22⋊C4—282no—
519C22⋊C4—282no—
520C42—22no—
521C42—22no—
522C42—21no—
523C42—21no—
524C42—282no—
525C42C2×C22⋊D712yes—
526C42C2×C22⋊D712yes—
527C2×C7—42no—
528C2×C7—84no—
529C2×C7#240711yescharacteristic
530C2×C7C2×#44711yescharacteristic
531C2×C7#300511yescharacteristic
532C2×C7#310011yescharacteristic
533C2×C7#311511yescharacteristic
534C2×C7#324511yescharacteristic
535C2×C7#325111yescharacteristic
536D7—82no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—24no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—142no—
551C23—281no—
552C23—281no—
553C23—281no—
554C23—281no—
555C23—281no—
556C23—281no—
557C23C2×#114112yes—
558C23C22×C22⋊D711yescharacteristic
559Q8—48no—
560D4—24no—
561D4—24no—
562D4—48no—
563D4—416no—
564D4—48no—
565D4—42no—
566D4—42no—
567D4—142no—
568D4—142no—
569D4—288no—
570D4—281no—
571D4—281no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—28no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—416no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—142no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—564no—
602C2×C4C2×#114112yes—
603C2×C4C2×#114112yes—
604C2×C4#649512yes—
605C2×C4#651612yes—
606C2×C4#651612yes—
607C2×C4D7×#4111yescharacteristic
608C2×C4D7×#4111yescharacteristic
609C2×C4#655311yescharacteristic
610C2×C4#655411yescharacteristic
611C2×C4C22×C22⋊D711yescharacteristic
612C2×C4C22×C22⋊D711yescharacteristic
613C8—282no—
614C8—282no—
615C7#2415011yescharacteristic, complemented, Sylow
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—282no—
631C22—282no—
632C22—282no—
633C22—561no—
634C22—561no—
635C22—561no—
636C22—561no—
637C22C22×#114111yescharacteristic
638C22C2×#649511yescharacteristic
639C22#9051311yescharacteristic
640C22#9051311yescharacteristic
641C22#9051911yescharacteristic
642C22#9052411yescharacteristic
643C22C2×#651611yescharacteristic
644C4—22no—
645C4—22no—
646C4—44no—
647C4—41no—
648C4—41no—
649C4—282no—
650C4—561no—
651C4—561no—
652C4#8998914yes—
653C4#9051511yescharacteristic
654C4#9052011yescharacteristic
655C4#9052211yescharacteristic
656C4#9052511yescharacteristic
657C2—42no—
658C2—84no—
659C2—562no—
660C2#1174074311yescharacteristic
661C2#1174074611yescharacteristic
662C2#1174075511yescharacteristic
663C2#1174075711yescharacteristic
664C2C2×#8998911yescharacteristic
665C2#1174152611yescharacteristic
666C2#1174154211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#532C2
socleC23×C7C22×D4
hypercenterC2×#532D7
hypocenterC7#24150
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C7×#532C2semidirect
2. C2×#89859C2semidirect
3. C2×#89844C2semidirect
4. C4×#6493C2semidirect
5. #11729534C2semidirect
6. #11729515C2semidirect
7. #11729438C2semidirect
8. #11729433C2semidirect
9. #11729367C2semidirect
10. #11729360C2semidirect
11. #11729284C2semidirect
64. C7×C4⋊Q8D4semidirect
66. C7×#127D4semidirect
115. C2×#532D7semidirect
163. C22×C4×C7D8semidirect
181. C2×D28#41semidirect
207. C4⋊Dic7#41semidirect
350. C4⋊Q8C22⋊D7semidirect
352. #127C22⋊D7semidirect
489. C22×C4#1141semidirect
615. C7#24150semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5322
C2×#898592
C2×#898442
C4×#64932
#117295342
#117295152
#117294382
#117294332
#117293672
#117293602
#117292842
#241507