Finite Group Databasebeta

← #13280863

Subgroups — #13280863

Order 1792.

1346 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#395C211yesmaximal, characteristic
2C2×#90002C211yesmaximal, characteristic
3C2×#89753C211yesmaximal, characteristic
4#11729888C211yesmaximal, characteristic
5#11729319C211yesmaximal, characteristic
6#11729319C212yesmaximal
7#11729317C212yesmaximal
8#11727173C211yesmaximal, characteristic
9#11727166C212yesmaximal
10#11727155C212yesmaximal
11Dic7×#95C211yesmaximal, characteristic
12C23×C7×Q8C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C4⋊Dic7C2212yes—
16C22×C4⋊Dic7C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19C22×C4×Dic7C2212yes—
20C7×#395C418yescomplemented
21C2×C7×#95C2211yescharacteristic
22C2×C7×#95C2212yes—
23#90002—24no—
24C2×C7⋊#95C2212yes—
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2211yescharacteristic
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37#89753—24no—
38#14327—71nomaximal, Sylow
39C22×C7×Q8—22no—
40C22×C7×Q8—22no—
41C22×C7×Q8—42no—
42C22×C7×Q8—42no—
43C22×C7×Q8C2×C414yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C23×C4×C7C2312yes—
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2312yes—
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2311yescharacteristic
53C23×Dic7C2312yes—
54C23×Dic7C2311yescharacteristic
55C2×C7×C4⋊C4—22no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—42no—
58C2×C7×C4⋊C4—42no—
59C2×C7×C4⋊C4C2×C414yes—
60C2×C7×C4⋊C4C2×C418yes—
61C2×C4⋊Dic7—22no—
62C2×C4⋊Dic7—24no—
63C2×C4⋊Dic7—22no—
64C2×C4⋊Dic7—42no—
65C2×C4⋊Dic7—42no—
66C2×C4⋊Dic7—42no—
67C2×C4⋊Dic7—42no—
68C2×C4×Dic7—42no—
69C2×C4×Dic7—42no—
70C2×C4×Dic7—42no—
71C2×C4×Dic7D414yes—
72C2×C4×Dic7Q818yescomplemented
73C7×#95C2×C418yes—
74C7×#95C2×C414yes—
75C7⋊#95—22no—
76C7⋊#95—22no—
77C7⋊#95—24no—
78C7⋊#95—24no—
79C7⋊#95—24no—
80C7⋊#95—24no—
81C7⋊#95—22no—
82C7⋊#95—22no—
83C7⋊#95—22no—
84C7⋊#95—42no—
85C7⋊#95—42no—
86C7⋊#95—42no—
87C7⋊#95—42no—
88C2×#395D711yescharacteristic
89C2×#391—71no—
90C2×#387—71no—
91#1446—71no—
92#1444—71no—
93#1442—72no—
94#1441—72no—
95#1441—71no—
96C4⋊#95—72no—
97#1434—72no—
98C4×#95—71no—
99C24×C7C2411yescharacteristic
100C2×C7×Q8—28no—
101C2×C7×Q8—48no—
102C2×C7×Q8—44no—
103C2×C7×Q8—44no—
104C2×C7×Q8—48no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—21no—
114C22×C4×C7—21no—
115C22×C4×C7—21no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
127C22×C4×C7C22×C412yes—
128C22×C4×C7C22×C414yes—
129C22×C4×C7C22×C412yes—
130C22×C4×C7C22×C412yes—
131C22×C4×C7C22×C414yes—
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2×Q814yes—
134C22×C4×C7#4112yes—
135C22×C4×C7#4114yes—
136C22×Dic7—22no—
137C22×Dic7—21no—
138C22×Dic7—21no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—22no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—22no—
166C22×Dic7C2×D412yes—
167C22×Dic7C2×D412yes—
168C22×Dic7C2×Q814yes—
169C22×Dic7C2×Q814yes—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—48no—
172C4⋊Dic7—44no—
173C4⋊Dic7—44no—
174C4⋊Dic7—48no—
175C4×Dic7—44no—
176C4×Dic7—48no—
177C23×Q8C2×D711yescharacteristic
178C22×C4⋊C4—72no—
179C22×C4⋊C4—71no—
180C22×C4⋊C4—71no—
181C22×C4⋊C4C2×D711yescharacteristic
182C22×C4⋊C4C2×D712yes—
183C22×C42—72no—
184C22×C42—71no—
185#395Dic718yescomplemented
186#391—144no—
187#387—144no—
188C2×#95—72no—
189C2×#95—71no—
190C2×#95—71no—
191C2×#95—72no—
192C2×#95—72no—
193C2×#95—71no—
194C2×#95—72no—
195C2×#95—71no—
196C2×#95—71no—
197C2×#95—72no—
198C2×#95—72no—
199C2×#95—72no—
200C2×#95—72no—
201C2×#95C2×D711yescharacteristic
202C2×#95C2×D712yes—
203C23×C7C23×C411yescharacteristic
204C23×C7C22×D411yescharacteristic
205C23×C7C22×Q812yes—
206C23×C7C2×#4111yescharacteristic
207C23×C7C2×#4112yes—
208C23×C7H(5,𝔽2)12yes—
209C23×C7H(5,𝔽2)12yes—
210C23×C7H(5,𝔽2)11yescharacteristic
211C23×C72−1+411yescharacteristic
212C23×C72−1+411yescharacteristic
213C23×C72−1+411yescharacteristic
214C7×Q8—416no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—24no—
221C2×C4×C7—24no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—24no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—24no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—41no—
234C2×C4×C7—42no—
235C2×C4×C7—41no—
236C2×C4×C7—42no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—42no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7C4×D414yes—
245C2×C4×C7C4×Q818yes—
246C2×Dic7—22no—
247C2×Dic7—22no—
248C2×Dic7—24no—
249C2×Dic7—24no—
250C2×Dic7—22no—
251C2×Dic7—24no—
252C2×Dic7—22no—
253C2×Dic7—22no—
254C2×Dic7—24no—
255C2×Dic7—24no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—41no—
263C2×Dic7—42no—
264C2×Dic7—41no—
265C2×Dic7—41no—
266C2×Dic7—42no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7C4⋊Q814yes—
280C22×Q8—22no—
281C22×Q8—22no—
282C22×Q8—42no—
283C22×Q8—42no—
284C22×Q8C2×Dic714yes—
285C23×C4—71no—
286C23×C4—71no—
287C23×C4—72no—
288C23×C4—71no—
289C23×C4—72no—
290C23×C4—71no—
291C23×C4C22×D711yescharacteristic
292C23×C4C22×D712yes—
293C23×C4C22×D711yescharacteristic
294C23×C4C22×D711yescharacteristic
295C23×C4C22×D712yes—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—142no—
301C2×C4⋊C4—142no—
302C2×C4⋊C4—144no—
303C2×C4⋊C4—282no—
304C2×C4⋊C4—282no—
305C2×C4⋊C4—282no—
306C2×C4⋊C4—282no—
307C2×C4⋊C4C2×Dic714yes—
308C2×C4⋊C4C2×Dic718yes—
309C2×C42—74no—
310C2×C42—78no—
311C2×C42—282no—
312C2×C42—282no—
313C2×C42—282no—
314#95—142no—
315#95—142no—
316#95—142no—
317#95—144no—
318#95—144no—
319#95—142no—
320#95—142no—
321#95—144no—
322#95—144no—
323#95—282no—
324#95—282no—
325#95—282no—
326#95—282no—
327#95C2×Dic718yes—
328#95C2×Dic714yes—
329C22×C7C2×C4×D411yescharacteristic
330C22×C7C2×C4×Q812yes—
331C22×C7#51812yes—
332C22×C7#51911yescharacteristic
333C22×C7#52011yescharacteristic
334C22×C7C2×C4⋊Q811yescharacteristic
335C22×C7#53211yescharacteristic
336C22×C7#53312yes—
337C22×C7#53411yescharacteristic
338C22×C7#53511yescharacteristic
339C22×C7#53611yescharacteristic
340C22×C7#53711yescharacteristic
341C22×C7#53812yes—
342C22×C7#53812yes—
343C22×C7#53912yes—
344C22×C7#54111yescharacteristic
345C22×C7#54212yes—
346C22×C7#54212yes—
347C22×C7#54211yescharacteristic
348C22×C7#54312yes—
349C22×C7#54412yes—
350C22×C7#54412yes—
351C22×C7#54412yes—
352C4×C7—24no—
353C4×C7—28no—
354C4×C7—42no—
355C4×C7—44no—
356C4×C7—42no—
357Dic7—24no—
358Dic7—42no—
359Dic7—44no—
360Dic7—82no—
361Dic7—81no—
362Dic7—81no—
363C24C23×D711yescharacteristic
364C2×Q8—28no—
365C2×Q8—48no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—48no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—72no—
392C22×C4—74no—
393C22×C4—72no—
394C22×C4—74no—
395C22×C4—142no—
396C22×C4—141no—
397C22×C4—141no—
398C22×C4—142no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4—142no—
402C22×C4—142no—
403C22×C4—142no—
404C22×C4—142no—
405C22×C4—141no—
406C22×C4—142no—
407C22×C4—141no—
408C22×C4—141no—
409C22×C4—141no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—142no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—142no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4D4×D712yes—
426C22×C4#115912yes—
427C22×C4Q8×D714yes—
428C22×C4#116114yes—
429C22×C4C22×Dic712yes—
430C22×C4C22×Dic714yes—
431C22×C4C22×Dic712yes—
432C22×C4C22×Dic712yes—
433C22×C4C22×Dic714yes—
434C4⋊C4—28no—
435C4⋊C4—48no—
436C4⋊C4—284no—
437C4⋊C4—288no—
438C4⋊C4—284no—
439C42—284no—
440C42—288no—
441C2×C7C4×C4⋊Q811yescharacteristic
442C2×C7#232111yescharacteristic
443C2×C7#232311yescharacteristic
444C2×C7#232912yes—
445C2×C7#233112yes—
446C2×C7#265911yescharacteristic
447C2×C7#266011yescharacteristic
448C2×C7#266211yescharacteristic
449C2×C7#266311yescharacteristic
450C2×C7#266512yes—
451C2×C7#266612yes—
452C23C2×D4×D711yescharacteristic
453C23C2×#115911yescharacteristic
454C23#654812yes—
455C23#654812yes—
456C23C2×Q8×D712yes—
457C23C2×#116112yes—
458C23#655111yescharacteristic
459C23#655111yescharacteristic
460C23#655311yescharacteristic
461C23#655411yescharacteristic
462C23C23×Dic711yescharacteristic
463Q8—416no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—74no—
494C2×C4—144no—
495C2×C4—142no—
496C2×C4—142no—
497C2×C4—144no—
498C2×C4—142no—
499C2×C4—144no—
500C2×C4—144no—
501C2×C4—142no—
502C2×C4—142no—
503C2×C4—144no—
504C2×C4—282no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—282no—
510C2×C4—281no—
511C2×C4—282no—
512C2×C4—281no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4—282no—
517C2×C4—281no—
518C2×C4—282no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—282no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—282no—
527C2×C4D4×Dic714yes—
528C2×C4Q8×Dic718yes—
529C7#1432711yescharacteristic, complemented, Sylow
530C22#9028011yescharacteristic
531C22#9028311yescharacteristic
532C22#9028412yes—
533C22#9031412yes—
534C22#9031611yescharacteristic
535C22#9031911yescharacteristic
536C22#9032011yescharacteristic
537C22#9032512yes—
538C22#9033411yescharacteristic
539C22#9033512yes—
540C22#9033612yes—
541C22D7×C4⋊Q811yescharacteristic
542C22#9041611yescharacteristic
543C22#9041912yes—
544C22#9042012yes—
545C22#9042012yes—
546C22#9042412yes—
547C22#9042412yes—
548C22C2×D4×Dic711yescharacteristic
549C22#9048912yes—
550C22C2×Q8×Dic712yes—
551C22#9050311yescharacteristic
552C22#9051811yescharacteristic
553C4—28no—
554C4—24no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—144no—
559C4—282no—
560C4—284no—
561C4—562no—
562C4—561no—
563C4—561no—
564C2#1173465311yescharacteristic
565C2#1173465411yescharacteristic
566C2#1173465511yescharacteristic
567C2#1173465612yes—
568C2#1173465812yes—
569C2#1173470311yescharacteristic
570C2#1174002311yescharacteristic
571C2#1174018911yescharacteristic
572C2#1174019012yes—
573C2Dic7×C4⋊Q811yescharacteristic
574C2#1174067312yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#395C2
socle, prime radical, cosocleC24×C7C24
hypercenterC2×#395D7
hypocenterC7#14327
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
20. C7×#395C4semidirect
72. C2×C4×Dic7Q8semidirect
185. #395Dic7semidirect
529. C7#14327semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3952
C2×#900022
C2×#897532
#117298882
#117293192
#117293172
#117271732
#117271662
#117271552
Dic7×#952
#143277