Finite Group Databasebeta

← #13276346

Subgroups — #13276346

Order 1792.

2374 conjugacy classes of subgroups.

The subgroup lattice has 616 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×#89749C211yesmaximal, characteristic, complemented
5C2×C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
6#11729317C211yesmaximal, characteristic, complemented
7#11729301C214yesmaximal, complemented
8#11729298C214yesmaximal, 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×C4×Dic7C2211yescharacteristic
24C22×C4×Dic7C2211yescharacteristic
25C7×#390—24no—
26#89993—24no—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2214yes—
31C2×C7⋊#95C2214yes—
32#89749—24no—
33C4×Dic7⋊C4—24no—
34#13941—71nomaximal, Sylow
35C25×C7C2312yes—
36C22×C7×D4—22no—
37C22×C7×D4—22no—
38C22×C7×D4—22no—
39C22×C7×D4—22no—
40C22×C7×D4C2×C414yes—
41C22×C7×D4D414yes—
42C22×C7×D4Q814yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×Dic7—24no—
47C23×Dic7—24no—
48C23×Dic7—44no—
49C23×Dic7—44no—
50C23×Dic7C2×C418yes—
51C23×Dic7C2312yes—
52C23×Dic7C2312yes—
53C23×Dic7C2312yes—
54C23×Dic7C2311yescharacteristic
55C23×Dic7C2311yescharacteristic
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C22⋊C4—28no—
60C2×C7×C22⋊C4—28no—
61C2×C7×C22⋊C4—44no—
62C2×C42×C7—22no—
63C2×C42×C7—22no—
64C2×C42×C7—22no—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—24no—
67C2×C22⋊Dic7—28no—
68C2×C22⋊Dic7—28no—
69C2×C22⋊Dic7—44no—
70C2×C22⋊Dic7C2×C418yes—
71D4×Dic7—232no—
72C2×C4⋊Dic7—22no—
73C2×C4⋊Dic7—22no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7—22no—
76C2×C4⋊Dic7—22no—
77C2×C4⋊Dic7C2×C414yes—
78C2×Dic7⋊C4—24no—
79C2×Dic7⋊C4—24no—
80C2×Dic7⋊C4—24no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7—22no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7C2×C414yes—
85C2×C4×Dic7D414yes—
86C2×C4×Dic7D414yes—
87C7⋊#95—24no—
88C7⋊#95—24no—
89C7⋊#95—24no—
90C7⋊#95—44no—
91C7⋊#95—44no—
92C22×C4×D4—71no—
93C2×#390—71no—
94C2×#390D711yescharacteristic
95C2×#382—71no—
96C2×C4×C4⋊C4—71no—
97#1439—74no—
98C4⋊#95—71no—
99C4⋊#95—71no—
100C22⋊#95—74no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—24no—
104C24×C7—24no—
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—24no—
108C24×C7—24no—
109C24×C7C22×C414yes—
110C24×C7C2×D414yes—
111C24×C7C2×Q814yes—
112C24×C7C2411yescharacteristic
113C2×C7×D4—28no—
114C2×C7×D4—28no—
115C2×C7×D4—28no—
116C2×C7×D4—28no—
117C2×C7×D4—28no—
118C2×C7×D4—28no—
119C2×C7×D4C4⋊C4116yescomplemented
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—24no—
129C22×C4×C7—22no—
130C22×C4×C7—24no—
131C22×C4×C7C22×C412yes—
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2×D412yes—
134C22×C4×C7C2×D412yes—
135C22×C4×C7C2×Q812yes—
136C22×C4×C7#4114yes—
137C22×C4×C7#4112yes—
138C22×C4×C7#4112yes—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—24no—
151C22×Dic7—24no—
152C22×Dic7—22no—
153C22×Dic7—216no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—24no—
158C22×Dic7—24no—
159C22×Dic7—416no—
160C22×Dic7—48no—
161C22×Dic7—416no—
162C22×Dic7—48no—
163C22×Dic7C22×C414yes—
164C22×Dic7C22×C412yes—
165C22×Dic7C22×C412yes—
166C22×Dic7C2×D412yes—
167C22×Dic7C2×D412yes—
168C22×Dic7C2×D412yes—
169C22×Dic7C2×D412yes—
170C22×Dic7#4114yes—
171C7×C4⋊C4—44no—
172C7×C22⋊C4—416no—
173C42×C7—44no—
174C22⋊Dic7—216no—
175C22⋊Dic7—416no—
176C4⋊Dic7—28no—
177C4⋊Dic7—44no—
178Dic7⋊C4—216no—
179C4×Dic7—44no—
180C4×Dic7C22⋊C4116yescomplemented
181C23×D4C2×D711yescharacteristic
182C24×C4—72no—
183C2×C4×D4—78no—
184C2×C4×D4—144no—
185C2×C4×D4—144no—
186C22×C4⋊C4—71no—
187C22×C4⋊C4—72no—
188C22×C4⋊C4—71no—
189C22×C4⋊C4C2×D711yescharacteristic
190C22×C22⋊C4—72no—
191C22×C22⋊C4—74no—
192C22×C22⋊C4C2×D714yes—
193C22×C42—71no—
194C22×C42—71no—
195C22×C42C2×D711yescharacteristic
196#390—24no—
197#390—144no—
198#382—144no—
199C4×C4⋊C4—144no—
200C2×#95—72no—
201C2×#95—72no—
202C2×#95—72no—
203C2×#95—74no—
204C2×#95—74no—
205C23×C7—24no—
206C23×C7—24no—
207C23×C7—28no—
208C23×C7—28no—
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—24no—
213C23×C7—28no—
214C23×C7—28no—
215C23×C7—28no—
216C23×C7—28no—
217C23×C7—44no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—44no—
222C23×C7—42no—
223C23×C7—44no—
224C23×C7—44no—
225C23×C7—44no—
226C23×C7—44no—
227C23×C7C2×C4⋊C418yes—
228C23×C7C23×C411yescharacteristic
229C23×C7C22×D411yescharacteristic
230C23×C7C22×D411yescharacteristic
231C23×C7C22×D411yescharacteristic
232C23×C7C22×Q811yescharacteristic
233C23×C7C2×#4111yescharacteristic
234C23×C7C2×#4111yescharacteristic
235C23×C7C2×#4112yes—
236C23×C7C2×#4112yes—
237C23×C7H(5,𝔽2)12yes—
238C23×C7H(5,𝔽2)12yes—
239C7×D4—232no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—24no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—44no—
251C2×C4×C7—44no—
252C2×C4×C7C2×C22⋊C414yes—
253C2×C4×C7C2×C4⋊C414yes—
254C2×C4×C7#11714yes—
255C2×C4×C7#12114yes—
256C2×C4×C7#12114yes—
257C2×C4×C7C22⋊Q814yes—
258C2×C4×C7C22⋊Q814yes—
259C2×Dic7—24no—
260C2×Dic7—24no—
261C2×Dic7—24no—
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—24no—
266C2×Dic7—22no—
267C2×Dic7—24no—
268C2×Dic7—28no—
269C2×Dic7—24no—
270C2×Dic7—24no—
271C2×Dic7—24no—
272C2×Dic7—432no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—44no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7—44no—
284C2×Dic7—44no—
285C2×Dic7C2×C22⋊C414yes—
286C2×Dic7C2×C22⋊C414yes—
287C25C22×D712yes—
288C22×D4—22no—
289C22×D4—22no—
290C22×D4—22no—
291C22×D4—22no—
292C22×D4Dic1414yes—
293C22×D4C4×D714yes—
294C22×D4C22⋊D714yes—
295C23×C4—71no—
296C23×C4—71no—
297C23×C4—72no—
298C23×C4—72no—
299C23×C4—78no—
300C23×C4—72no—
301C23×C4—144no—
302C23×C4—144no—
303C23×C4—284no—
304C23×C4—284no—
305C23×C4C22×D712yes—
306C23×C4C22×D712yes—
307C23×C4C22×D711yescharacteristic
308C4×D4—1432no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—74no—
313C2×C4⋊C4—144no—
314C2×C4⋊C4—144no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—144no—
318C2×C4⋊C4—142no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—78no—
325C2×C22⋊C4—144no—
326C2×C22⋊C4—144no—
327C2×C22⋊C4—148no—
328C2×C22⋊C4—148no—
329C2×C22⋊C4—284no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—74no—
334C2×C42—74no—
335C2×C42—74no—
336C2×C42—142no—
337C2×C42—142no—
338C2×C42—142no—
339#95—144no—
340#95—144no—
341#95—144no—
342#95—284no—
343#95—284no—
344C22×C7—216no—
345C22×C7—28no—
346C22×C7—44no—
347C22×C7—44no—
348C22×C7—44no—
349C22×C7—48no—
350C22×C7—48no—
351C22×C7—48no—
352C22×C7—44no—
353C22×C7—44no—
354C22×C7—44no—
355C22×C7C22×C22⋊C411yescharacteristic
356C22×C7C22×C4⋊C411yescharacteristic
357C22×C7C2×#11711yescharacteristic
358C22×C7C4×#4112yes—
359C22×C7#51812yes—
360C22×C7C2×#12111yescharacteristic
361C22×C7C2×#12111yescharacteristic
362C22×C7C2×C22⋊Q811yescharacteristic
363C22×C7C2×C22⋊Q811yescharacteristic
364C22×C7#52512yes—
365C22×C7#53211yescharacteristic
366C22×C7#53211yescharacteristic
367C22×C7#53511yescharacteristic
368C22×C7#53512yes—
369C22×C7#53511yescharacteristic
370C22×C7#54612yes—
371C22×C7#54612yes—
372C22×C7#55312yes—
373C22×C7#55312yes—
374C22×C7#55412yes—
375C22×C7#55412yes—
376C22×C7#55912yes—
377C22×C7#55912yes—
378C4×C7—44no—
379C4×C7—82no—
380C4×C7#38018yes—
381Dic7—24no—
382Dic7—24no—
383Dic7—44no—
384Dic7—44no—
385Dic7—82no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—22no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24C2×Dic1414yes—
395C24C2×C4×D714yes—
396C24C2×C22⋊D714yes—
397C24C23×D711yescharacteristic
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4Dic7⋊C4116yescomplemented
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—72no—
417C22×C4—72no—
418C22×C4—72no—
419C22×C4—72no—
420C22×C4—72no—
421C22×C4—72no—
422C22×C4—74no—
423C22×C4—74no—
424C22×C4—1416no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—144no—
436C22×C4—144no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—144no—
442C22×C4—144no—
443C22×C4—142no—
444C22×C4—2816no—
445C22×C4—288no—
446C22×C4—2816no—
447C22×C4—288no—
448C22×C4C2×Dic1412yes—
449C22×C4C2×C4×D712yes—
450C22×C4D4×D712yes—
451C22×C4D4×D712yes—
452C22×C4#115914yes—
453C22×C4#115912yes—
454C22×C4#115912yes—
455C22×C4C2×C22⋊D712yes—
456C4⋊C4—44no—
457C4⋊C4—1416no—
458C4⋊C4—148no—
459C4⋊C4—284no—
460C22⋊C4—416no—
461C22⋊C4—1416no—
462C22⋊C4—2816no—
463C42—44no—
464C42—716no—
465C42—284no—
466C2×C7—48no—
467C2×C7C2×#38011yescharacteristic
468C2×C7#229711yescharacteristic
469C2×C7#230911yescharacteristic
470C2×C7#234912yes—
471C2×C7#235712yes—
472C2×C7#253512yes—
473C2×C7#254411yescharacteristic
474C2×C7#254411yescharacteristic
475C2×C7#256611yescharacteristic
476C2×C7#256611yescharacteristic
477C2×C7#258012yes—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23C2×Dic7⋊C418yes—
501C23C22×Dic1411yescharacteristic
502C23C22×C4×D711yescharacteristic
503C23C2×D4×D711yescharacteristic
504C23C2×D4×D711yescharacteristic
505C23C2×#115911yescharacteristic
506C23C2×#115912yes—
507C23C2×#115911yescharacteristic
508C23#654812yes—
509C23D7×#4112yes—
510C23#655312yes—
511C23C22×C22⋊D711yescharacteristic
512D4—232no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—74no—
526C2×C4—74no—
527C2×C4—142no—
528C2×C4—142no—
529C2×C4—144no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—144no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—144no—
536C2×C4—148no—
537C2×C4—144no—
538C2×C4—144no—
539C2×C4—144no—
540C2×C4—2832no—
541C2×C4—282no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—282no—
545C2×C4—284no—
546C2×C4—282no—
547C2×C4—282no—
548C2×C4—284no—
549C2×C4—282no—
550C2×C4—284no—
551C2×C4—282no—
552C2×C4—282no—
553C2×C4#644014yes—
554C2×C4#644114yes—
555C2×C4#644114yes—
556C2×C4D7×C22⋊C414yes—
557C2×C4C2×Dic7⋊C414yes—
558C2×C4#650214yes—
559C2×C4#650214yes—
560C7#1394111yescharacteristic, complemented, Sylow
561C22—216no—
562C22—28no—
563C22—44no—
564C22—44no—
565C22—48no—
566C22—44no—
567C22—48no—
568C22—44no—
569C22—44no—
570C22—48no—
571C22—44no—
572C22C2×#644011yescharacteristic
573C22C2×#644111yescharacteristic
574C22C2×#644111yescharacteristic
575C22C2×D7×C22⋊C411yescharacteristic
576C22C4×#115912yes—
577C22#9023012yes—
578C22#9023412yes—
579C22#9023612yes—
580C22#9029712yes—
581C22#9029812yes—
582C22#9030512yes—
583C22#9030912yes—
584C22#9035911yescharacteristic
585C22#9036511yescharacteristic
586C22#9037212yes—
587C22#9040412yes—
588C22#9040611yescharacteristic
589C22#9040711yescharacteristic
590C22C22×Dic7⋊C411yescharacteristic
591C22C2×#650211yescharacteristic
592C22C2×#650211yescharacteristic
593C22#9049712yes—
594C22#9052012yes—
595C4—44no—
596C4—82no—
597C4—144no—
598C4—144no—
599C4—284no—
600C4—284no—
601C4—562no—
602C4#8971418yes—
603C2—48no—
604C2#1173373511yescharacteristic
605C2#1173374812yes—
606C2#1173380711yescharacteristic
607C2#1173380811yescharacteristic
608C2#1173380912yes—
609C2#1173387311yescharacteristic
610C2C2×#8971411yescharacteristic
611C2#1173977912yes—
612C2#1174011112yes—
613C2#1174041511yescharacteristic
614C2#1174062311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#390C2
socle, cosocleC24×C7C24
prime radicalC23×C7×D4C22
hypercenterC2×#390D7
hypocenterC7#13941
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
4. C2×#89749C2semidirect
5. C2×C4×Dic7⋊C4C2semidirect
6. #11729317C2semidirect
7. #11729301C2semidirect
8. #11729298C2semidirect
9. #11729289C2semidirect
19. C2×D4×Dic7C4semidirect
119. C2×C7×D4C4⋊C4semidirect
180. C4×Dic7C22⋊C4semidirect
404. C2×D4Dic7⋊C4semidirect
560. C7#13941semidirect

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×#897492
C2×C4×Dic7⋊C42
#117293172
#117293012
#117292982
#117292892
#139417