Finite Group Databasebeta

← #13276025

Subgroups — #13276025

Order 1792.

2278 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic7⋊C4C211yesmaximal, characteristic
2C2×C7×#381C211yesmaximal, characteristic
3C2×#89983C211yesmaximal, characteristic
4C2×#89715C211yesmaximal, characteristic
5C2×#89715C211yesmaximal, characteristic
6C2×#89714C211yesmaximal, characteristic
7C2×C22⋊C4×Dic7C211yesmaximal, characteristic
8#11727175C214yesmaximal, complemented
9Dic7⋊#95C214yesmaximal, complemented
10C24×C4×C7C2211yescharacteristic
11C24×Dic7C2211yescharacteristic
12C24×Dic7C2211yescharacteristic
13C22×C7×C22⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C22⋊Dic7C2211yescharacteristic
16C22×C22⋊Dic7C2211yescharacteristic
17C22×Dic7⋊C4—24no—
18C22×Dic7⋊C4—24no—
19C22×Dic7⋊C4C418yescomplemented
20C22×Dic7⋊C4C2212yes—
21C22×Dic7⋊C4C2212yes—
22C22×Dic7⋊C4C2212yes—
23C22×C4×Dic7C2212yes—
24C7×#381—24no—
25C2×C7×#95C2214yes—
26#89983—24no—
27C2×C7⋊#95C2214yes—
28C2×C7⋊#95C2214yes—
29C2×C7⋊#95C2214yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32#89715—24no—
33#89715—24no—
34#89714—24no—
35C22⋊C4×Dic7—24no—
36#13904—71nomaximal, Sylow
37C25×C7C2311yescharacteristic
38C23×C4×C7—22no—
39C23×C4×C7—22no—
40C23×C4×C7—22no—
41C23×C4×C7—22no—
42C23×C4×C7—22no—
43C23×C4×C7—22no—
44C23×C4×C7C2×C414yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2312yes—
47C23×C4×C7C2312yes—
48C23×Dic7—28no—
49C23×Dic7—22no—
50C23×Dic7—22no—
51C23×Dic7—22no—
52C23×Dic7—22no—
53C23×Dic7—22no—
54C23×Dic7—22no—
55C23×Dic7C2×C414yes—
56C23×Dic7C2×C414yes—
57C23×Dic7D414yes—
58C23×Dic7D414yes—
59C23×Dic7C2312yes—
60C23×Dic7C2312yes—
61C23×Dic7C2312yes—
62C23×Dic7C2311yescharacteristic
63C23×Dic7C2311yescharacteristic
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—22no—
67C2×C7×C22⋊C4—22no—
68C2×C7×C22⋊C4—22no—
69C2×C7×C22⋊C4D414yes—
70C2×C22⋊Dic7—22no—
71C2×C22⋊Dic7—22no—
72C2×C22⋊Dic7—22no—
73C2×C22⋊Dic7—22no—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7D414yes—
76C2×Dic7⋊C4—232no—
77C2×Dic7⋊C4—24no—
78C2×Dic7⋊C4—24no—
79C2×Dic7⋊C4—24no—
80C2×Dic7⋊C4—24no—
81C2×Dic7⋊C4—24no—
82C2×Dic7⋊C4—28no—
83C2×Dic7⋊C4C2×C418yes—
84C2×Dic7⋊C4C2×C418yes—
85C2×C4×Dic7—24no—
86C2×C4×Dic7—28no—
87C7×#95—28no—
88C7⋊#95—28no—
89C7⋊#95—24no—
90C7⋊#95—24no—
91C7⋊#95—28no—
92C7⋊#95—28no—
93C23×C4⋊C4—71no—
94C2×#381—71no—
95C2×#381—71no—
96C2×#381—71no—
97C2×#381D711yescharacteristic
98C2×#380—71no—
99C2×C4×C22⋊C4—71no—
100#1445—74no—
101C4⋊#95—74no—
102C24×C7—24no—
103C24×C7—24no—
104C24×C7C22×C412yes—
105C24×C7C2×D412yes—
106C24×C7C2×D412yes—
107C24×C7C2×D412yes—
108C24×C7C2×D412yes—
109C24×C7#4112yes—
110C24×C7#4112yes—
111C24×C7C2411yescharacteristic
112C22×C4×C7—28no—
113C22×C4×C7—28no—
114C22×C4×C7—28no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—24no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—24no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—24no—
129C22×C4×C7—24no—
130C22×C4×C7—44no—
131C22×C4×C7—44no—
132C22×C4×C7—44no—
133C22×C4×C7—44no—
134C22×C4×C7C22×C414yes—
135C22×C4×C7C2×D414yes—
136C22×Dic7—24no—
137C22×Dic7—216no—
138C22×Dic7—216no—
139C22×Dic7—216no—
140C22×Dic7—28no—
141C22×Dic7—28no—
142C22×Dic7—28no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—24no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—24no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—24no—
157C22×Dic7—24no—
158C22×Dic7—44no—
159C22×Dic7—44no—
160C22×Dic7—44no—
161C22×Dic7—44no—
162C22×Dic7C22⋊C4116yes—
163C22×Dic7C22×C414yes—
164C22×Dic7C22×C412yes—
165C22×Dic7C22×C412yes—
166C22×Dic7C2×D414yes—
167C22×Dic7C2×D412yes—
168C22×Dic7C2×D412yes—
169C22×Dic7C2×D412yes—
170C22×Dic7C2×D412yes—
171C22×Dic7#4118yes—
172C7×C22⋊C4—28no—
173C7×C22⋊C4—44no—
174C22⋊Dic7—28no—
175C22⋊Dic7—44no—
176Dic7⋊C4—216no—
177Dic7⋊C4—216no—
178Dic7⋊C4—48no—
179C4×Dic7—48no—
180C24×C4—71no—
181C24×C4—71no—
182C24×C4C2×D711yescharacteristic
183C22×C4⋊C4—78no—
184C22×C4⋊C4—72no—
185C22×C4⋊C4—72no—
186C22×C4⋊C4—72no—
187C22×C4⋊C4—144no—
188C22×C4⋊C4—144no—
189C22×C22⋊C4—71no—
190C22×C22⋊C4—71no—
191C22×C22⋊C4C2×D711yescharacteristic
192C22×C22⋊C4C2×D711yescharacteristic
193C22×C42—72no—
194#381—24no—
195#381—144no—
196#381—144no—
197#381—144no—
198#380—144no—
199C4×C22⋊C4—144no—
200C2×#95—74no—
201C2×#95—74no—
202C2×#95—74no—
203C2×#95—72no—
204C2×#95—72no—
205C2×#95C2×D714yes—
206C23×C7—24no—
207C23×C7—24no—
208C23×C7—24no—
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—28no—
213C23×C7—28no—
214C23×C7—28no—
215C23×C7—28no—
216C23×C7C2×C22⋊C414yes—
217C23×C7C4×D414yes—
218C23×C7C4×D414yes—
219C23×C7#12114yes—
220C23×C7#12114yes—
221C23×C7#12414yes—
222C23×C7#12714yes—
223C23×C7C23×C411yescharacteristic
224C23×C7C22×D411yescharacteristic
225C23×C7C22×D411yescharacteristic
226C23×C7C22×D411yescharacteristic
227C23×C7C22×D411yescharacteristic
228C23×C7C2×#4111yescharacteristic
229C23×C7C2×#4111yescharacteristic
230C23×C7C2×#4114yes—
231C23×C72−1+414yes—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—28no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—44no—
244C2×C4×C7—416no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—44no—
249C2×C4×C7—44no—
250C2×Dic7—24no—
251C2×Dic7—24no—
252C2×Dic7—24no—
253C2×Dic7—232no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—24no—
257C2×Dic7—24no—
258C2×Dic7—28no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—44no—
266C2×Dic7—416no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—44no—
271C2×Dic7—44no—
272C2×Dic7C2×C22⋊C414yes—
273C2×Dic7C2×C22⋊C414yes—
274C2×Dic7#11718yes—
275C2×Dic7#12318yes—
276C2×Dic7#12318yes—
277C25C22×D711yescharacteristic
278C23×C4—22no—
279C23×C4—22no—
280C23×C4—22no—
281C23×C4—22no—
282C23×C4—22no—
283C23×C4—22no—
284C23×C4—71no—
285C23×C4—74no—
286C23×C4—74no—
287C23×C4—74no—
288C23×C4—71no—
289C23×C4—74no—
290C23×C4—72no—
291C23×C4—72no—
292C23×C4—72no—
293C23×C4—148no—
294C23×C4—142no—
295C23×C4—142no—
296C23×C4—142no—
297C23×C4—142no—
298C23×C4—142no—
299C23×C4—142no—
300C23×C4C4×D714yes—
301C23×C4C22×D712yes—
302C23×C4C22×D712yes—
303C23×C4C22×D712yes—
304C2×C4⋊C4—78no—
305C2×C4⋊C4—78no—
306C2×C4⋊C4—1432no—
307C2×C4⋊C4—144no—
308C2×C4⋊C4—144no—
309C2×C4⋊C4—144no—
310C2×C4⋊C4—144no—
311C2×C4⋊C4—148no—
312C2×C4⋊C4—144no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—74no—
319C2×C22⋊C4—142no—
320C2×C22⋊C4—142no—
321C2×C22⋊C4—142no—
322C2×C22⋊C4—142no—
323C2×C22⋊C4—142no—
324C2×C22⋊C4C22⋊D714yes—
325C2×C42—144no—
326C2×C42—148no—
327#95—28no—
328#95—144no—
329#95—144no—
330#95—148no—
331#95—148no—
332#95—148no—
333C22×C7—24no—
334C22×C7—28no—
335C22×C7—28no—
336C22×C7—28no—
337C22×C7—28no—
338C22×C7—28no—
339C22×C7—28no—
340C22×C7—24no—
341C22×C7#39018yes—
342C22×C7C22×C22⋊C411yescharacteristic
343C22×C7C2×#11712yes—
344C22×C7C2×C4×D411yescharacteristic
345C22×C7C2×C4×D411yescharacteristic
346C22×C7#51912yes—
347C22×C7C2×#12111yescharacteristic
348C22×C7C2×#12111yescharacteristic
349C22×C7C2×#12312yes—
350C22×C7C2×#12312yes—
351C22×C7C2×#12411yescharacteristic
352C22×C7C2×#12711yescharacteristic
353C22×C7#53612yes—
354C22×C7#53612yes—
355C22×C7#54714yes—
356C22×C7#54714yes—
357C22×C7#55214yes—
358C22×C7#55214yes—
359C4×C7—44no—
360C4×C7—44no—
361C4×C7—82no—
362Dic7—24no—
363Dic7—44no—
364Dic7—44no—
365Dic7—82no—
366Dic7#38118yescomplemented
367C24—24no—
368C24—24no—
369C24C2×C4×D712yes—
370C24#115712yes—
371C24D4×D712yes—
372C24D4×D712yes—
373C24D4×D712yes—
374C24#115912yes—
375C24C2×C22⋊D712yes—
376C24C23×D711yescharacteristic
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—72no—
400C22×C4—72no—
401C22×C4—72no—
402C22×C4—78no—
403C22×C4—716no—
404C22×C4—72no—
405C22×C4—72no—
406C22×C4—72no—
407C22×C4—74no—
408C22×C4—74no—
409C22×C4—1416no—
410C22×C4—1416no—
411C22×C4—1416no—
412C22×C4—144no—
413C22×C4—148no—
414C22×C4—148no—
415C22×C4—148no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4—144no—
424C22×C4—144no—
425C22×C4—142no—
426C22×C4—142no—
427C22×C4—142no—
428C22×C4—144no—
429C22×C4—144no—
430C22×C4—142no—
431C22×C4—284no—
432C22×C4—284no—
433C22×C4—284no—
434C22×C4—284no—
435C22×C4C2×C4×D714yes—
436C22×C4C2×C22⋊D714yes—
437C4⋊C4—1416no—
438C4⋊C4—1416no—
439C4⋊C4—288no—
440C22⋊C4—28no—
441C22⋊C4—44no—
442C22⋊C4—148no—
443C22⋊C4—284no—
444C42—288no—
445C2×C7—28no—
446C2×C7C2×#38111yescharacteristic
447C2×C7C2×#39011yescharacteristic
448C2×C7#231011yescharacteristic
449C2×C7#234412yes—
450C2×C7#234412yes—
451C2×C7#248612yes—
452C2×C7#248612yes—
453C2×C7#250112yes—
454C2×C7#250312yes—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—28no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23D7×C22⋊C414yes—
466C23(C22⋊D7)⋊C414yes—
467C23#644714yes—
468C23#644814yes—
469C23C4×C22⋊D714yes—
470C23#650214yes—
471C23#650314yes—
472C23C22×C4×D711yescharacteristic
473C23C2×#115711yescharacteristic
474C23C2×D4×D711yescharacteristic
475C23C2×D4×D711yescharacteristic
476C23C2×D4×D711yescharacteristic
477C23C2×#115911yescharacteristic
478C23D7×#4114yes—
479C23#655414yes—
480C23C22×C22⋊D711yescharacteristic
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—416no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—74no—
500C2×C4—78no—
501C2×C4—78no—
502C2×C4—78no—
503C2×C4—74no—
504C2×C4—1432no—
505C2×C4—144no—
506C2×C4—144no—
507C2×C4—144no—
508C2×C4—144no—
509C2×C4—144no—
510C2×C4—144no—
511C2×C4—144no—
512C2×C4—148no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—284no—
516C2×C4—282no—
517C2×C4—282no—
518C2×C4—2816no—
519C2×C4—282no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—284no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—284no—
526C7#1390411yescharacteristic, complemented, Sylow
527C22—28no—
528C22—24no—
529C22—28no—
530C22—28no—
531C22—28no—
532C22—24no—
533C22—28no—
534C22—28no—
535C22#8972918yes—
536C22C2×D7×C22⋊C411yescharacteristic
537C22C2×(C22⋊D7)⋊C411yescharacteristic
538C22C2×#644711yescharacteristic
539C22C2×#644811yescharacteristic
540C22#9020712yes—
541C22D7×#11712yes—
542C22#9022214yes—
543C22#9033912yes—
544C22#9033912yes—
545C22#9034114yes—
546C22D7×#12312yes—
547C22D7×#12312yes—
548C22#9034614yes—
549C22C2×C4×C22⋊D711yescharacteristic
550C22C2×#650211yescharacteristic
551C22C2×#650311yescharacteristic
552C22#9051514yes—
553C4—44no—
554C4—44no—
555C4—78no—
556C4—82no—
557C4—144no—
558C4—284no—
559C4—284no—
560C4—562no—
561C2—28no—
562C2#1173149811yescharacteristic
563C2#1173150512yes—
564C2#1173152012yes—
565C2#1173152412yes—
566C2D7×#38111yescharacteristic
567C2C2×#8972911yescharacteristic
568C2#1173968312yes—
569C2#1174030112yes—
570C2#1174030412yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#381C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#381D7
hypocenterC7#13904
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
8. #11727175C2semidirect
9. Dic7⋊#95C2semidirect
19. C22×Dic7⋊C4C4semidirect
366. Dic7#381semidirect
526. C7#13904semidirect

Maximal subgroups

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

subgroupindex
C23×Dic7⋊C42
C2×C7×#3812
C2×#899832
C2×#897152
C2×#897142
C2×C22⋊C4×Dic72
#117271752
Dic7⋊#952
#139047