Finite Group Databasebeta

← #13276039

Subgroups — #13276039

Order 1792.

2574 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic7C211yesmaximal, characteristic
2C2×C7×#390C211yesmaximal, characteristic
3C2×#89993C211yesmaximal, characteristic
4C2×#89748C211yesmaximal, characteristic, complemented
5C2×#89714C214yesmaximal, complemented
6C2×C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
7#11729317C211yesmaximal, characteristic, complemented
8#11729301C214yesmaximal, complemented
9#11729289C211yesmaximal, characteristic, complemented
10C23×C7×D4C2211yescharacteristic
11C24×Dic7C2212yes—
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C22⋊C4C2214yes—
14C22×C42×C7C2211yescharacteristic
15C22×C22⋊Dic7C2214yes—
16C22×C22⋊Dic7C2212yes—
17C2×D4×Dic7—24no—
18C2×D4×Dic7—24no—
19C2×D4×Dic7C418yescomplemented
20C22×C4⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7C2211yescharacteristic
22C22×Dic7⋊C4C2212yes—
23C22×Dic7⋊C4C2212yes—
24C22×Dic7⋊C4C2212yes—
25C22×C4×Dic7C2211yescharacteristic
26C22×C4×Dic7C2211yescharacteristic
27C7×#390—24no—
28#89993—24no—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2214yes—
31C2×C7⋊#95C2214yes—
32#89748—24no—
33#89714—216no—
34C4×Dic7⋊C4—24no—
35#13906—71nomaximal, Sylow
36C25×C7C2312yes—
37C22×C7×D4—22no—
38C22×C7×D4—22no—
39C22×C7×D4—22no—
40C22×C7×D4—22no—
41C22×C7×D4C2×C414yes—
42C22×C7×D4D414yes—
43C22×C7×D4Q814yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—28no—
48C23×Dic7—24no—
49C23×Dic7—24no—
50C23×Dic7—28no—
51C23×Dic7C2×C418yes—
52C23×Dic7C2312yes—
53C23×Dic7C2312yes—
54C23×Dic7C2312yes—
55C23×Dic7C2311yescharacteristic
56C23×Dic7C2311yescharacteristic
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4—22no—
60C2×C7×C22⋊C4—28no—
61C2×C7×C22⋊C4—28no—
62C2×C7×C22⋊C4—44no—
63C2×C42×C7—22no—
64C2×C42×C7—22no—
65C2×C42×C7—22no—
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7—24no—
68C2×C22⋊Dic7—28no—
69C2×C22⋊Dic7—28no—
70C2×C22⋊Dic7—44no—
71C2×C22⋊Dic7C2×C418yes—
72D4×Dic7—232no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7—22no—
78C2×C4⋊Dic7C2×C414yes—
79C2×Dic7⋊C4—24no—
80C2×Dic7⋊C4—24no—
81C2×Dic7⋊C4—28no—
82C2×Dic7⋊C4—24no—
83C2×Dic7⋊C4—24no—
84C2×Dic7⋊C4—44no—
85C2×Dic7⋊C4D418yescomplemented
86C2×C4×Dic7—22no—
87C2×C4×Dic7—22no—
88C2×C4×Dic7—22no—
89C2×C4×Dic7C2×C414yes—
90C2×C4×Dic7D414yes—
91C2×C4×Dic7D414yes—
92C7⋊#95—28no—
93C7⋊#95—24no—
94C7⋊#95—28no—
95C22×C4×D4—71no—
96C2×#390—71no—
97C2×#390D711yescharacteristic
98C2×#384—71no—
99C2×#380—74no—
100C2×C4×C4⋊C4—71no—
101#1439—74no—
102C4⋊#95—71no—
103C4⋊#95—71no—
104C24×C7—24no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—24no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—24no—
111C24×C7—24no—
112C24×C7C22×C414yes—
113C24×C7C2×D414yes—
114C24×C7C2×Q814yes—
115C24×C7C2411yescharacteristic
116C2×C7×D4—28no—
117C2×C7×D4—28no—
118C2×C7×D4—28no—
119C2×C7×D4—28no—
120C2×C7×D4—28no—
121C2×C7×D4—28no—
122C2×C7×D4C4⋊C4116yescomplemented
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—24no—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C2×D414yes—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×Q812yes—
140C22×C4×C7#4112yes—
141C22×C4×C7#4112yes—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—24no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—24no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—216no—
155C22×Dic7—232no—
156C22×Dic7—24no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—24no—
161C22×Dic7—416no—
162C22×Dic7—48no—
163C22×Dic7—48no—
164C22×Dic7C22×C414yes—
165C22×Dic7C22×C412yes—
166C22×Dic7C22×C412yes—
167C22×Dic7C2×D414yes—
168C22×Dic7C2×D412yes—
169C22×Dic7C2×D412yes—
170C22×Dic7C2×D412yes—
171C22×Dic7C2×D412yes—
172C22×Dic7C2×D414yes—
173C22×Dic7#4114yes—
174C7×C4⋊C4—44no—
175C7×C22⋊C4—416no—
176C42×C7—44no—
177C22⋊Dic7—216no—
178C22⋊Dic7—416no—
179C4⋊Dic7—28no—
180C4⋊Dic7—44no—
181Dic7⋊C4—216no—
182Dic7⋊C4—48no—
183Dic7⋊C4—48no—
184C4×Dic7—44no—
185C4×Dic7C22⋊C4116yescomplemented
186C23×D4C2×D711yescharacteristic
187C24×C4—72no—
188C2×C4×D4—78no—
189C2×C4×D4—144no—
190C2×C4×D4—144no—
191C22×C4⋊C4—71no—
192C22×C4⋊C4—72no—
193C22×C4⋊C4—72no—
194C22×C4⋊C4—71no—
195C22×C4⋊C4—72no—
196C22×C4⋊C4C2×D711yescharacteristic
197C22×C22⋊C4—72no—
198C22×C22⋊C4—74no—
199C22×C22⋊C4C2×D714yes—
200C22×C42—71no—
201C22×C42—71no—
202C22×C42C2×D711yescharacteristic
203#390—24no—
204#390—144no—
205#384—144no—
206#380—1416no—
207C4×C4⋊C4—144no—
208C2×#95—74no—
209C2×#95—74no—
210C2×#95—72no—
211C23×C7—24no—
212C23×C7—24no—
213C23×C7—28no—
214C23×C7—28no—
215C23×C7—24no—
216C23×C7—24no—
217C23×C7—24no—
218C23×C7—24no—
219C23×C7—28no—
220C23×C7—28no—
221C23×C7—28no—
222C23×C7—28no—
223C23×C7—44no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—44no—
229C23×C7—44no—
230C23×C7—44no—
231C23×C7—44no—
232C23×C7—44no—
233C23×C7C2×C4⋊C418yes—
234C23×C7C23×C411yescharacteristic
235C23×C7C22×D411yescharacteristic
236C23×C7C22×D411yescharacteristic
237C23×C7C22×D412yes—
238C23×C7C22×D411yescharacteristic
239C23×C7C22×Q811yescharacteristic
240C23×C7C2×#4111yescharacteristic
241C23×C7C2×#4111yescharacteristic
242C23×C7C2×#4112yes—
243C23×C7H(5,𝔽2)12yes—
244C23×C72−1+412yes—
245C7×D4—232no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—44no—
257C2×C4×C7—44no—
258C2×C4×C7C2×C22⋊C414yes—
259C2×C4×C7C2×C4⋊C414yes—
260C2×C4×C7#11714yes—
261C2×C4×C7#12114yes—
262C2×C4×C7#12114yes—
263C2×C4×C7C22⋊Q814yes—
264C2×C4×C7C22⋊Q814yes—
265C2×Dic7—24no—
266C2×Dic7—22no—
267C2×Dic7—22no—
268C2×Dic7—24no—
269C2×Dic7—24no—
270C2×Dic7—28no—
271C2×Dic7—24no—
272C2×Dic7—24no—
273C2×Dic7—24no—
274C2×Dic7—42no—
275C2×Dic7—44no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—432no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—44no—
286C2×Dic7—44no—
287C2×Dic7C2×C22⋊C414yes—
288C2×Dic7C2×C22⋊C414yes—
289C2×Dic7C4×D418yes—
290C2×Dic7#12118yes—
291C2×Dic7#12414yes—
292C2×Dic7#12714yes—
293C25C22×D712yes—
294C22×D4—22no—
295C22×D4—22no—
296C22×D4—22no—
297C22×D4—22no—
298C22×D4Dic1414yes—
299C22×D4C4×D714yes—
300C22×D4C22⋊D714yes—
301C23×C4—71no—
302C23×C4—71no—
303C23×C4—78no—
304C23×C4—72no—
305C23×C4—72no—
306C23×C4—72no—
307C23×C4—144no—
308C23×C4—144no—
309C23×C4—148no—
310C23×C4—148no—
311C23×C4C22×D712yes—
312C23×C4C22×D712yes—
313C23×C4C22×D711yescharacteristic
314C4×D4—1432no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—74no—
319C2×C4⋊C4—78no—
320C2×C4⋊C4—144no—
321C2×C4⋊C4—144no—
322C2×C4⋊C4—142no—
323C2×C4⋊C4—142no—
324C2×C4⋊C4—148no—
325C2×C4⋊C4—144no—
326C2×C4⋊C4—144no—
327C2×C4⋊C4—142no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—142no—
330C2×C4⋊C4—284no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—78no—
335C2×C22⋊C4—144no—
336C2×C22⋊C4—144no—
337C2×C22⋊C4—148no—
338C2×C22⋊C4—148no—
339C2×C22⋊C4—284no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—74no—
344C2×C42—74no—
345C2×C42—74no—
346C2×C42—142no—
347C2×C42—142no—
348C2×C42—142no—
349#95—144no—
350#95—148no—
351#95—148no—
352C22×C7—216no—
353C22×C7—28no—
354C22×C7—44no—
355C22×C7—44no—
356C22×C7—48no—
357C22×C7—48no—
358C22×C7—48no—
359C22×C7—44no—
360C22×C7—44no—
361C22×C7—44no—
362C22×C7—44no—
363C22×C7C22×C22⋊C411yescharacteristic
364C22×C7C22×C4⋊C411yescharacteristic
365C22×C7C2×#11711yescharacteristic
366C22×C7C2×C4×D412yes—
367C22×C7#52012yes—
368C22×C7C2×#12111yescharacteristic
369C22×C7C2×#12111yescharacteristic
370C22×C7C2×#12112yes—
371C22×C7C2×C22⋊Q811yescharacteristic
372C22×C7C2×C22⋊Q811yescharacteristic
373C22×C7C2×#12411yescharacteristic
374C22×C7C2×#12711yescharacteristic
375C22×C7#53511yescharacteristic
376C22×C7#53611yescharacteristic
377C22×C7#53712yes—
378C22×C7D4212yes—
379C22×C7#54712yes—
380C22×C7#54712yes—
381C22×C7D4×Q812yes—
382C22×C7#55012yes—
383C22×C7#55412yes—
384C22×C7#55512yes—
385C22×C7#55612yes—
386C4×C7—44no—
387C4×C7—82no—
388C4×C7#38018yescomplemented
389Dic7—24no—
390Dic7—44no—
391Dic7—44no—
392Dic7—82no—
393Dic7#39018yescomplemented
394C24—24no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—24no—
400C24—24no—
401C24—24no—
402C24C2×Dic1414yes—
403C24C2×C4×D714yes—
404C24C2×C22⋊D714yes—
405C24C23×D711yescharacteristic
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4Dic7⋊C4116yescomplemented
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—72no—
425C22×C4—72no—
426C22×C4—72no—
427C22×C4—74no—
428C22×C4—74no—
429C22×C4—72no—
430C22×C4—72no—
431C22×C4—72no—
432C22×C4—74no—
433C22×C4—74no—
434C22×C4—1416no—
435C22×C4—1432no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—144no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—142no—
445C22×C4—144no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—144no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—144no—
452C22×C4—142no—
453C22×C4—288no—
454C22×C4—2816no—
455C22×C4—288no—
456C22×C4C2×Dic1412yes—
457C22×C4C2×C4×D712yes—
458C22×C4D4×D714yes—
459C22×C4D4×D712yes—
460C22×C4D4×D712yes—
461C22×C4#115912yes—
462C22×C4#115912yes—
463C22×C4C2×C22⋊D712yes—
464C4⋊C4—44no—
465C4⋊C4—1416no—
466C4⋊C4—148no—
467C4⋊C4—288no—
468C4⋊C4—288no—
469C4⋊C4—284no—
470C22⋊C4—416no—
471C22⋊C4—1416no—
472C22⋊C4—2816no—
473C42—44no—
474C42—716no—
475C42—284no—
476C2×C7—48no—
477C2×C7C2×#38011yescharacteristic
478C2×C7C2×#39011yescharacteristic
479C2×C7#231111yescharacteristic
480C2×C7D4×C4⋊C412yes—
481C2×C7#235412yes—
482C2×C7#253712yes—
483C2×C7#253812yes—
484C2×C7#254111yescharacteristic
485C2×C7#254211yescharacteristic
486C2×C7#254711yescharacteristic
487C2×C7#256311yescharacteristic
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23C2×Dic7⋊C418yes—
511C23C22×Dic1411yescharacteristic
512C23C22×C4×D711yescharacteristic
513C23C2×D4×D711yescharacteristic
514C23C2×D4×D712yes—
515C23C2×D4×D711yescharacteristic
516C23C2×#115911yescharacteristic
517C23C2×#115911yescharacteristic
518C23#654812yes—
519C23D7×#4112yes—
520C23#655412yes—
521C23C22×C22⋊D711yescharacteristic
522D4—232no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—74no—
536C2×C4—74no—
537C2×C4—78no—
538C2×C4—78no—
539C2×C4—74no—
540C2×C4—74no—
541C2×C4—142no—
542C2×C4—144no—
543C2×C4—144no—
544C2×C4—142no—
545C2×C4—144no—
546C2×C4—148no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—144no—
550C2×C4—2832no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—284no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—282no—
558C2×C4—282no—
559C2×C4—282no—
560C2×C4—284no—
561C2×C4—284no—
562C2×C4—282no—
563C2×C4#644014yes—
564C2×C4#644114yes—
565C2×C4#644114yes—
566C2×C4D7×C22⋊C414yes—
567C2×C4C2×Dic7⋊C414yes—
568C2×C4#650214yes—
569C2×C4#650214yes—
570C7#1390611yescharacteristic, complemented, Sylow
571C22—216no—
572C22—28no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—48no—
580C22—44no—
581C22—48no—
582C22C2×#644011yescharacteristic
583C22C2×#644111yescharacteristic
584C22C2×#644111yescharacteristic
585C22C2×D7×C22⋊C411yescharacteristic
586C22D4×Dic1412yes—
587C22#9023012yes—
588C22C4×D4×D712yes—
589C22#9023712yes—
590C22D7×#12112yes—
591C22#9029912yes—
592C22#9030212yes—
593C22#9030712yes—
594C22#9036212yes—
595C22D7×#12411yescharacteristic
596C22#9036611yescharacteristic
597C22D7×#12711yescharacteristic
598C22#9040611yescharacteristic
599C22#9040912yes—
600C22C22×Dic7⋊C411yescharacteristic
601C22D4×C22⋊D712yes—
602C22C2×#650211yescharacteristic
603C22C2×#650211yescharacteristic
604C22#9051512yes—
605C4—44no—
606C4—78no—
607C4—82no—
608C4—144no—
609C4—284no—
610C4—284no—
611C4—562no—
612C4#8971418yescomplemented
613C2—48no—
614C2#1173373911yescharacteristic
615C2#1173374712yes—
616C2#1173379112yes—
617C2#1173379411yescharacteristic
618C2#1173380611yescharacteristic
619C2D7×#39011yescharacteristic
620C2C2×#8971411yescharacteristic
621C2D4×Dic7⋊C412yes—
622C2#1174011212yes—
623C2#1174041811yescharacteristic
624C2#1174062211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#390C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#390D7
hypocenterC7#13906
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. C2×#89748C2semidirect
5. C2×#89714C2semidirect
6. C2×C4×Dic7⋊C4C2semidirect
7. #11729317C2semidirect
8. #11729301C2semidirect
9. #11729289C2semidirect
19. C2×D4×Dic7C4semidirect
85. C2×Dic7⋊C4D4semidirect
122. C2×C7×D4C4⋊C4semidirect
185. C4×Dic7C22⋊C4semidirect
388. C4×C7#380semidirect
393. Dic7#390semidirect
412. C2×D4Dic7⋊C4semidirect
570. C7#13906semidirect
612. C4#89714semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Dic72
C2×C7×#3902
C2×#899932
C2×#897482
C2×#897142
C2×C4×Dic7⋊C42
#117293172
#117293012
#117292892
#139067