Finite Group Databasebeta

← #13285715

Subgroups — #13285715

Order 1792.

2242 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6460C211yesmaximal, characteristic
2C2×#89759C212yesmaximal
3C2×#89440C211yesmaximal, characteristic
4C2×D7×#95C211yesmaximal, characteristic
5C2×#89428C211yesmaximal, characteristic, complemented
6C7×#1444C211yesmaximal, characteristic, complemented
7#11729320C211yesmaximal, characteristic, complemented
8#11729319C212yesmaximal, complemented
9(C2×D7)⋊#95C212yesmaximal
10#11727178C211yesmaximal, characteristic, complemented
11#11727173C212yesmaximal, complemented
12C23×C4×D7C2211yescharacteristic
13C23×C4×D7C2212yes—
14C23×Dic14C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2212yes—
17C22×(C2×D7)⋊C4C2211yescharacteristic
18C22×(C2×D7)⋊C4C2212yes—
19C22×(C2×D7)⋊C4C2211yescharacteristic
20C22×C4⋊Dic7C2212yes—
21C22×C4⋊Dic7C2212yes—
22C22×C4⋊Dic7C2211yescharacteristic
23C22×Dic7⋊C4C2211yescharacteristic
24C2×#6460—24no—
25C2×#6460—28no—
26C2×C7×#95C2212yes—
27C2×C7×#95C2211yescharacteristic
28C2×C7×#95C2211yescharacteristic
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2211yescharacteristic
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37#89759—28no—
38#89440—24no—
39D7×#95—24no—
40#89428—24no—
41#14758—71nomaximal, Sylow
42C24×D7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2312yes—
53C22×C4×D7—24no—
54C22×C4×D7—24no—
55C22×C4×D7—24no—
56C22×C4×D7—22no—
57C22×C4×D7—24no—
58C22×C4×D7—22no—
59C22×C4×D7—24no—
60C22×C4×D7—42no—
61C22×C4×D7—42no—
62C22×C4×D7—42no—
63C22×C4×D7D414yes—
64C22×C4×D7Q818yescomplemented
65C22×Dic14—22no—
66C22×Dic14—24no—
67C22×Dic14—22no—
68C22×Dic14—22no—
69C22×Dic14—42no—
70C2×C7×C4⋊C4—24no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—24no—
73C2×C7×C4⋊C4—24no—
74C2×C7×C4⋊C4—42no—
75C2×(C2×D7)⋊C4—24no—
76C2×(C2×D7)⋊C4—24no—
77C2×(C2×D7)⋊C4—22no—
78C2×(C2×D7)⋊C4—22no—
79C2×(C2×D7)⋊C4—22no—
80C2×(C2×D7)⋊C4—24no—
81C2×(C2×D7)⋊C4D418yes—
82C2×(C2×D7)⋊C4Q814yescomplemented
83C2×C4⋊Dic7—24no—
84C2×C4⋊Dic7—24no—
85C2×C4⋊Dic7—24no—
86C2×C4⋊Dic7—24no—
87C2×C4⋊Dic7—22no—
88C2×C4⋊Dic7—24no—
89C2×C4⋊Dic7—42no—
90C2×C4⋊Dic7—42no—
91C2×Dic7⋊C4—22no—
92C2×Dic7⋊C4—22no—
93C2×Dic7⋊C4—22no—
94#6460—232no—
95C7×#95—22no—
96C7×#95—24no—
97C7×#95D414yes—
98C7⋊#95—24no—
99C7⋊#95—22no—
100C7⋊#95—24no—
101C7⋊#95—24no—
102C7⋊#95—22no—
103C7⋊#95—24no—
104C7⋊#95—42no—
105C7⋊#95—42no—
106C22×C22⋊Q8—71no—
107C2×#395—71no—
108C2×#385—71no—
109C2×#380—72no—
110C22×#95—71no—
111#1446—71no—
112#1444—72no—
113#1444D711yescharacteristic, complemented
114#1441—72no—
115#1441—71no—
116C22⋊#95—72no—
117C24×C7C2411yescharacteristic
118C23×D7—21no—
119C23×D7—21no—
120C23×D7—22no—
121C23×D7—22no—
122C23×D7—21no—
123C23×D7—21no—
124C23×D7C2×D412yes—
125C23×D7C2×D414yes—
126C23×D7C2×Q812yes—
127C23×D7C2×Q814yes—
128C23×D7#4112yes—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D414yes—
154C22×C4×C7C2×Q812yes—
155C22×C4×C7C2×Q814yes—
156C22×C4×C7#4112yes—
157C22×C4×C7#4114yes—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—22no—
161C22×Dic7—22no—
162C22×Dic7—21no—
163C22×Dic7—22no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—22no—
170C22×Dic7—21no—
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—21no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—22no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—21no—
180C22×Dic7—22no—
181C22×Dic7—21no—
182C22×Dic7—21no—
183C22×Dic7—21no—
184C22×Dic7—22no—
185C22×Dic7C2×D412yes—
186C22×Dic7C2×D414yes—
187C22×Dic7C2×Q812yes—
188C22×Dic7C2×Q814yes—
189C2×C4×D7—28no—
190C2×C4×D7—216no—
191C2×C4×D7—28no—
192C2×C4×D7—216no—
193C2×C4×D7—44no—
194C2×C4×D7—48no—
195C2×C4×D7—48no—
196C2×C4×D7—48no—
197C2×C4×D7—48no—
198C2×C4×D7—48no—
199C2×C4×D7—48no—
200C2×C4×D7—48no—
201C2×Dic14—28no—
202C2×Dic14—48no—
203C2×Dic14—48no—
204C2×Dic14—44no—
205C2×Dic14—44no—
206C7×C4⋊C4—28no—
207C7×C4⋊C4—48no—
208(C2×D7)⋊C4—216no—
209(C2×D7)⋊C4—28no—
210(C2×D7)⋊C4—44no—
211C4⋊Dic7—216no—
212C4⋊Dic7—48no—
213C4⋊Dic7—44no—
214Dic7⋊C4—44no—
215C23×Q8—71no—
216C24×C4—71no—
217C24×C4—72no—
218C2×C22⋊Q8—148no—
219C2×C22⋊Q8—144no—
220C22×C4⋊C4—71no—
221C22×C4⋊C4—72no—
222C22×C4⋊C4—72no—
223C22×C4⋊C4—71no—
224C22×C4⋊C4C2×D711yescharacteristic
225C22×C4⋊C4C2×D712yes—
226C22×C22⋊C4—72no—
227C22×C22⋊C4—71no—
228C22×C22⋊C4—71no—
229#395—144no—
230#385—144no—
231#380—148no—
232C2×#95—72no—
233C2×#95—71no—
234C2×#95—72no—
235C2×#95—72no—
236C2×#95—71no—
237C2×#95—72no—
238C2×#95—72no—
239C2×#95—72no—
240C2×#95—144no—
241C2×#95C2×D712yes—
242C2×#95C2×D711yescharacteristic
243C2×#95C2×D711yescharacteristic
244C23×C7C22×D411yescharacteristic
245C23×C7C22×D411yescharacteristic
246C23×C7C22×D412yes—
247C23×C7C22×Q812yes—
248C23×C7C22×Q811yescharacteristic
249C23×C7C2×#4112yes—
250C23×C7C2×#4111yescharacteristic
251C23×C7C2×#4111yescharacteristic
252C23×C7H(5,𝔽2)11yescharacteristic
253C23×C7H(5,𝔽2)12yes—
254C23×C72−1+411yescharacteristic
255C22×D7—22no—
256C22×D7—24no—
257C22×D7—22no—
258C22×D7—24no—
259C22×D7—22no—
260C22×D7—22no—
261C22×D7—24no—
262C22×D7—24no—
263C22×D7—24no—
264C22×D7—22no—
265C22×D7—24no—
266C22×D7—24no—
267C22×D7—24no—
268C22×D7—22no—
269C22×D7—24no—
270C22×D7—22no—
271C22×D7—22no—
272C22×D7—24no—
273C22×D7#12014yes—
274C22×D7C22⋊Q814yes—
275C22×D7C22⋊Q818yes—
276C22×D7C4⋊Q814yes—
277C22×D7C4⋊Q818yes—
278C2×C4×C7—22no—
279C2×C4×C7—24no—
280C2×C4×C7—22no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—24no—
285C2×C4×C7—24no—
286C2×C4×C7—24no—
287C2×C4×C7—24no—
288C2×C4×C7—24no—
289C2×C4×C7—22no—
290C2×C4×C7—24no—
291C2×C4×C7—24no—
292C2×C4×C7—24no—
293C2×C4×C7—41no—
294C2×C4×C7—42no—
295C2×C4×C7—41no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7#12114yes—
306C2×C4×C7C22⋊Q818yescomplemented
307C2×Dic7—22no—
308C2×Dic7—24no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—24no—
312C2×Dic7—24no—
313C2×Dic7—22no—
314C2×Dic7—24no—
315C2×Dic7—24no—
316C2×Dic7—24no—
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7—41no—
320C2×Dic7—42no—
321C2×Dic7—42no—
322C2×Dic7—41no—
323C2×Dic7—42no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—42no—
327C2×Dic7—42no—
328C2×Dic7—42no—
329C2×Dic7—42no—
330C2×Dic7—41no—
331C2×Dic7—42no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—42no—
337C2×Dic7—42no—
338C4×D7—416no—
339C4×D7—432no—
340Dic14—416no—
341C25—71no—
342C22×Q8—144no—
343C22×Q8—142no—
344C22×Q8—142no—
345C22×Q8—142no—
346C22×Q8—282no—
347C23×C4—72no—
348C23×C4—71no—
349C23×C4—71no—
350C23×C4—72no—
351C23×C4—74no—
352C23×C4—78no—
353C23×C4—71no—
354C23×C4—144no—
355C23×C4—144no—
356C23×C4—144no—
357C23×C4—142no—
358C23×C4—144no—
359C23×C4—144no—
360C23×C4—142no—
361C23×C4—282no—
362C23×C4—282no—
363C23×C4—282no—
364C23×C4C22×D711yescharacteristic
365C23×C4C22×D712yes—
366C23×C4C22×D711yescharacteristic
367C23×C4C22×D711yescharacteristic
368C23×C4C22×D712yes—
369C22⋊Q8—1432no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—24no—
374C2×C4⋊C4—42no—
375C2×C4⋊C4—144no—
376C2×C4⋊C4—144no—
377C2×C4⋊C4—144no—
378C2×C4⋊C4—144no—
379C2×C4⋊C4—142no—
380C2×C4⋊C4—142no—
381C2×C4⋊C4—144no—
382C2×C4⋊C4—142no—
383C2×C4⋊C4—142no—
384C2×C4⋊C4—282no—
385C2×C4⋊C4—282no—
386C2×C22⋊C4—78no—
387C2×C22⋊C4—74no—
388C2×C22⋊C4—144no—
389C2×C22⋊C4—144no—
390C2×C22⋊C4—142no—
391C2×C22⋊C4—142no—
392C2×C22⋊C4—144no—
393C2×C22⋊C4—142no—
394#95—22no—
395#95—24no—
396#95—144no—
397#95—142no—
398#95—142no—
399#95—144no—
400#95—144no—
401#95—144no—
402#95—282no—
403#95—282no—
404#95D2814yes—
405C22×C7C2×#12011yescharacteristic
406C22×C7C2×#12111yescharacteristic
407C22×C7C2×C22⋊Q812yes—
408C22×C7C2×C22⋊Q811yescharacteristic
409C22×C7C2×C22⋊Q812yes—
410C22×C7C2×C4⋊Q811yescharacteristic
411C22×C7C2×C4⋊Q812yes—
412C22×C7#53211yescharacteristic
413C22×C7#53312yes—
414C22×C7#53411yescharacteristic
415C22×C7#53711yescharacteristic
416C22×C7#54312yes—
417C22×C7#54412yes—
418C22×C7D4212yes—
419C22×C7#54611yescharacteristic
420C22×C7#54612yes—
421C22×C7#54612yes—
422C22×C7#54712yes—
423C22×C7D4×Q811yescharacteristic
424C22×C7#55112yes—
425C22×C7#55211yescharacteristic
426C22×C7#55411yescharacteristic
427C22×C7#55412yes—
428C2×D7—24no—
429C2×D7—24no—
430C2×D7—24no—
431C2×D7—24no—
432C2×D7—24no—
433C2×D7—24no—
434C2×D7—28no—
435C2×D7—28no—
436C2×D7—28no—
437C2×D7—28no—
438C2×D7#39518yescomplemented
439C4×C7—24no—
440C4×C7—28no—
441C4×C7—42no—
442C4×C7—44no—
443C4×C7—81no—
444Dic7—44no—
445Dic7—42no—
446Dic7—42no—
447Dic7—44no—
448Dic7—81no—
449C24—74no—
450C24—72no—
451C24—72no—
452C24—74no—
453C24—72no—
454C24—141no—
455C24—142no—
456C24—141no—
457C24—141no—
458C24—142no—
459C24—141no—
460C24C23×D711yescharacteristic
461C2×Q8—148no—
462C2×Q8—288no—
463C2×Q8—288no—
464C2×Q8—284no—
465C2×Q8—284no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—74no—
488C22×C4—72no—
489C22×C4—72no—
490C22×C4—74no—
491C22×C4—148no—
492C22×C4—1416no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—142no—
496C22×C4—142no—
497C22×C4—142no—
498C22×C4—141no—
499C22×C4—142no—
500C22×C4—142no—
501C22×C4—142no—
502C22×C4—142no—
503C22×C4—141no—
504C22×C4—142no—
505C22×C4—141no—
506C22×C4—141no—
507C22×C4—142no—
508C22×C4—141no—
509C22×C4—142no—
510C22×C4—142no—
511C22×C4—142no—
512C22×C4—142no—
513C22×C4—142no—
514C22×C4—141no—
515C22×C4—148no—
516C22×C4—1416no—
517C22×C4—141no—
518C22×C4—142no—
519C22×C4—141no—
520C22×C4—141no—
521C22×C4—142no—
522C22×C4—288no—
523C22×C4—284no—
524C22×C4—288no—
525C22×C4—288no—
526C22×C4—288no—
527C22×C4—288no—
528C22×C4—288no—
529C22×C4—288no—
530C22×C4C2×D2812yes—
531C22×C4C2×D2814yes—
532C22×C4D4×D714yes—
533C22×C4D4×D712yes—
534C22×C4#115914yes—
535C22×C4Q8×D712yes—
536C22×C4Q8×D714yes—
537C22×C4#116112yes—
538C4⋊C4—28no—
539C4⋊C4—48no—
540C4⋊C4—1416no—
541C4⋊C4—288no—
542C4⋊C4—284no—
543C4⋊C4—284no—
544C22⋊C4—148no—
545C22⋊C4—1416no—
546C22⋊C4—284no—
547C2×C7C2×#39511yescharacteristic
548C2×C7#241011yescharacteristic
549C2×C7#241711yescharacteristic
550C2×C7#243011yescharacteristic
551C2×C7#243712yes—
552C2×C7#243812yes—
553C2×C7#244012yes—
554C2×C7#254912yes—
555C2×C7#256411yescharacteristic
556C2×C7#265911yescharacteristic
557C2×C7#285711yescharacteristic
558D7—28no—
559C23—74no—
560C23—78no—
561C23—74no—
562C23—74no—
563C23—78no—
564C23—144no—
565C23—142no—
566C23—142no—
567C23—144no—
568C23—142no—
569C23—142no—
570C23—144no—
571C23—144no—
572C23—142no—
573C23—142no—
574C23—144no—
575C23—142no—
576C23—144no—
577C23—144no—
578C23—144no—
579C23—144no—
580C23—142no—
581C23—144no—
582C23C22×D2811yescharacteristic
583C23C2×D4×D711yescharacteristic
584C23C2×D4×D712yes—
585C23C2×#115912yes—
586C23#654811yescharacteristic
587C23C2×Q8×D712yes—
588C23C2×Q8×D711yescharacteristic
589C23C2×#116111yescharacteristic
590C23D7×#4111yescharacteristic
591C23#655312yes—
592C23#655411yescharacteristic
593Q8—2816no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—142no—
622C2×C4—142no—
623C2×C4—144no—
624C2×C4—144no—
625C2×C4—142no—
626C2×C4—144no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—144no—
630C2×C4—144no—
631C2×C4—144no—
632C2×C4—142no—
633C2×C4—2816no—
634C2×C4—2832no—
635C2×C4—281no—
636C2×C4—281no—
637C2×C4—282no—
638C2×C4—282no—
639C2×C4—282no—
640C2×C4—282no—
641C2×C4—281no—
642C2×C4—282no—
643C2×C4—282no—
644C2×C4—281no—
645C2×C4—282no—
646C2×C4—282no—
647C2×C4—282no—
648C2×C4—282no—
649C2×C4—281no—
650C2×C4—282no—
651C2×C4—281no—
652C2×C4—282no—
653C2×C4—281no—
654C2×C4#645814yes—
655C2×C4#646018yescomplemented
656C7#1475811yescharacteristic, complemented, Sylow
657C22—78no—
658C22—144no—
659C22—144no—
660C22—148no—
661C22—144no—
662C22—144no—
663C22—148no—
664C22—144no—
665C22—148no—
666C22—144no—
667C22—148no—
668C22C2×#645811yescharacteristic
669C22C2×#646012yes—
670C22D4×D2812yes—
671C22#9024511yescharacteristic
672C22Q8×D2811yescharacteristic
673C22D7×#12011yescharacteristic
674C22#9028212yes—
675C22#9029911yescharacteristic
676C22#9030212yes—
677C22D7×C22⋊Q811yescharacteristic
678C22D7×C22⋊Q812yes—
679C22#9032112yes—
680C22#9032512yes—
681C22#9032612yes—
682C22#9034411yescharacteristic
683C22#9034912yes—
684C22#9038012yes—
685C22#9038511yescharacteristic
686C22#9040711yescharacteristic
687C22D7×C4⋊Q811yescharacteristic
688C22D7×C4⋊Q812yes—
689C22#9041912yes—
690C22#9042111yescharacteristic
691C4—24no—
692C4—28no—
693C4—42no—
694C4—44no—
695C4—81no—
696C4—282no—
697C4—284no—
698C4—284no—
699C4—282no—
700C4—561no—
701C2—148no—
702C2#1173182111yescharacteristic
703C2#1173185812yes—
704C2#1173204611yescharacteristic
705C2#1173344412yes—
706C2#1173365711yescharacteristic
707C2#1173420211yescharacteristic
708C2D7×#39511yescharacteristic
709C2#1173469812yes—
710C2#1173539412yes—
711C2#1173541111yescharacteristic
712C2#1173555311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutator, socle, cosocleC24×C7C24
FittingC7×#1444C2
prime radicalC24×D7C23
hypercenter#1444D7
hypocenterC7#14758
LCSG ▷ C24×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
5. C2×#89428C2semidirect
6. C7×#1444C2semidirect
7. #11729320C2semidirect
8. #11729319C2semidirect
10. #11727178C2semidirect
11. #11727173C2semidirect
64. C22×C4×D7Q8semidirect
82. C2×(C2×D7)⋊C4Q8semidirect
113. #1444D7semidirect
306. C2×C4×C7C22⋊Q8semidirect
438. C2×D7#395semidirect
655. C2×C4#6460semidirect
656. C7#14758semidirect

Maximal subgroups

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

subgroupindex
C22×#64602
C2×#897592
C2×#894402
C2×D7×#952
C2×#894282
C7×#14442
#117293202
#117293192
(C2×D7)⋊#952
#117271782
#117271732
#147587