Finite Group Databasebeta

← #13359476

Subgroups — #13359476

Order 1792.

1436 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic14C211yesmaximal, characteristic
2C4×#6487C211yesmaximal, characteristic
3C2×#89511C211yesmaximal, characteristic
4C2×#89488C212yesmaximal, complemented
5C7×#1749C211yesmaximal, characteristic
6#11727717C211yesmaximal, characteristic
7#11727684C211yesmaximal, characteristic
8#11727682C211yesmaximal, characteristic
9#11727567C212yesmaximal, complemented
10#11726439C212yesmaximal, complemented
11#11726438C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C2×#6487C2211yescharacteristic
16C22×Dic7⋊C4—21no—
17C22×C4×Dic7—21no—
18C22×Dic28C2212yes—
19C2×C4×Dic14—21no—
20C2×C4×Dic14—21no—
21C2×C4×Dic14—28no—
22C2×C4×Dic14C2212yes—
23C2×C4×Dic14C2211yescharacteristic
24C2×C4×Dic14C2212yes—
25C2×C7×C22⋊C8C2211yescharacteristic
26C2×C7×C22⋊C8C2211yescharacteristic
27C2×C7×#97C2212yes—
28#89984C2211yescharacteristic
29#89982—21no—
30C4×C22⋊Dic7—21no—
31C2×#6391C2212yes—
32C2×#6390C2212yes—
33C2×#6390C2212yes—
34C2×#6390C2212yes—
35C2×#6390C2212yes—
36C4×C4⋊Dic7C2211yescharacteristic
37#89705—21no—
38C4×Dic7⋊C4—21no—
39#89701C2211yescharacteristic
40#89700C2211yescharacteristic
41#89511C418yescomplemented
42#89488—28no—
43C7×#336C2212yes—
44#89346C2212yes—
45#89346C2212yes—
46#21455—71nomaximal, Sylow
47C23×C4×C7D411yescharacteristic
48C23×C4×C7D411yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×Dic7—21no—
51C23×Dic7—21no—
52C22×Dic14—21no—
53C22×Dic14—22no—
54C22×Dic14—24no—
55C22×Dic14—24no—
56C22×Dic14C2×C414yes—
57C22×Dic14C2311yescharacteristic
58C22×Dic14C2312yes—
59C22×C7×C8C2312yes—
60C22×C7×C8C2312yes—
61C2×C42×C7—24no—
62C2×C42×C7D411yescharacteristic
63C2×C42×C7D411yescharacteristic
64C2×C42×C7C2311yescharacteristic
65C2×C42×C7C2311yescharacteristic
66C2×C22⋊Dic7—21no—
67C2×C22⋊Dic7—21no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7—24no—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×C4⋊Dic7C2311yescharacteristic
72C2×C4⋊Dic7C2311yescharacteristic
73C2×C4⋊Dic7C2312yes—
74#6487—22no—
75#6487C2×C414yes—
76C2×Dic7⋊C4—28no—
77C2×Dic7⋊C4—22no—
78C2×Dic7⋊C4—22no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×C4×Dic7—28no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7—22no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×Dic28—24no—
88C2×Dic28—24no—
89C2×Dic28C2×C418yescomplemented
90C4×Dic14—28no—
91C4×Dic14—28no—
92C4×Dic14—24no—
93C4×Dic14—24no—
94C4×Dic14—24no—
95C4×Dic14—24no—
96C4×Dic14D418yes—
97C7×C22⋊C8—22no—
98C7×C22⋊C8C2×C414yes—
99C7×#97—24no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105#6391—24no—
106#6390—24no—
107#6390—24no—
108#6390—24no—
109#6390C2×C418yescomplemented
110C22×C4×Q8—71no—
111C2×#451—71no—
112C2×#441—72no—
113C4×C22⋊Q8—71no—
114#1878—72no—
115#1872—71no—
116#1865—72no—
117#1853—71no—
118#1787—71no—
119#1763—72no—
120#1749D711yescharacteristic
121C24×C7C2×D411yescharacteristic
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—22no—
129C22×C4×C7—41no—
130C22×C4×C7C22×C412yes—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7#4112yes—
137C22×C4×C7C2411yescharacteristic
138C22×Dic7—24no—
139C22×Dic7—24no—
140C22×Dic7—24no—
141C22×Dic7—22no—
142C22×Dic7—24no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—22no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—42no—
149C22×Dic7—42no—
150C2×Dic14—24no—
151C2×Dic14—24no—
152C2×Dic14—216no—
153C2×Dic14—24no—
154C2×Dic14—22no—
155C2×Dic14—22no—
156C2×Dic14—22no—
157C2×Dic14—216no—
158C2×Dic14—22no—
159C2×Dic14—22no—
160C2×Dic14—21no—
161C2×Dic14C22×C412yes—
162C2×Dic14C22×C414yes—
163C2×Dic14C2×D414yes—
164C2×Dic14#4114yes—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8C22×C414yes—
171C42×C7—22no—
172C42×C7—22no—
173C42×C7—22no—
174C42×C7C2×D414yes—
175C22⋊Dic7—24no—
176C22⋊Dic7—42no—
177C4⋊Dic7—24no—
178C4⋊Dic7—22no—
179C4⋊Dic7—21no—
180C4⋊Dic7—22no—
181C4⋊Dic7—22no—
182C4⋊Dic7C22×C412yes—
183C4⋊Dic7C2×D414yes—
184Dic7⋊C4—28no—
185Dic7⋊C4—24no—
186Dic7⋊C4—28no—
187Dic7⋊C4—42no—
188Dic7⋊C4—42no—
189C4×Dic7—28no—
190C4×Dic7—28no—
191C4×Dic7—42no—
192C4×Dic7—42no—
193Dic28—216no—
194C23×Q8—71no—
195C22×Q16—72no—
196C2×C22⋊Q8—71no—
197C2×C4×Q8—72no—
198C2×C4×Q8—72no—
199C2×C4×Q8—71no—
200C2×C4×Q8—148no—
201C2×C4×Q8—141no—
202C2×C4×Q8—141no—
203C22×C4⋊C4—71no—
204C22×C4⋊C4—141no—
205C22×C42—141no—
206C22×C42C2×D711yescharacteristic
207#451—78no—
208#441—148no—
209C2×#106—72no—
210C2×#103—72no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#103—72no—
214C2×C22⋊C8C2×D711yescharacteristic
215C2×C22⋊C8C2×D711yescharacteristic
216C2×#97C2×D712yes—
217#391—71no—
218#389—71no—
219#387—141no—
220#385—141no—
221#380—71no—
222C4×C4⋊C4—71no—
223C4×C4⋊C4—141no—
224C4×C22⋊C4—141no—
225#340—72no—
226#340—72no—
227#336C2×D712yes—
228C23×C7—21no—
229C23×C7—21no—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7C4×D412yes—
233C23×C7#13712yes—
234C23×C7#13712yes—
235C23×C7C22×D411yescharacteristic
236C2×C4×C7—28no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—24no—
246C2×C4×C7—24no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—44no—
251C2×C4×C7C4×D412yes—
252C2×C4×C7#12012yes—
253C2×C4×C7#12012yes—
254C2×C4×C7#13712yes—
255C2×C4×C7C23×C411yescharacteristic
256C2×C4×C7C22×D412yes—
257C2×C4×C7C2×#4111yescharacteristic
258C2×C4×C7C2×#4112yes—
259C2×Dic7—216no—
260C2×Dic7—24no—
261C2×Dic7—24no—
262C2×Dic7—24no—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—24no—
266C2×Dic7—48no—
267C2×Dic7—41no—
268C2×Dic7—41no—
269C2×Dic7—41no—
270C2×Dic7—42no—
271C2×Dic7—41no—
272C2×Dic7—42no—
273C2×Dic7—41no—
274C2×Dic7—41no—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—41no—
279Dic14—216no—
280Dic14—24no—
281Dic14—24no—
282Dic14—24no—
283Dic14—24no—
284Dic14—42no—
285Dic14C4×D418yes—
286C7×C8—24no—
287C7×C8—42no—
288C22×Q8—74no—
289C22×Q8—72no—
290C22×Q8—71no—
291C22×Q8—144no—
292C22×Q8—144no—
293C22×Q8—141no—
294C22×Q8—142no—
295C23×C4—141no—
296C23×C4—141no—
297C23×C4D2811yescharacteristic
298C23×C4D2811yescharacteristic
299C23×C4C22×D711yescharacteristic
300C2×Q16—78no—
301C2×Q16—144no—
302C2×Q16—144no—
303C22×C8C22×D712yes—
304C22×C8C22×D712yes—
305C22⋊Q8—74no—
306C22⋊Q8—142no—
307C4×Q8—78no—
308C4×Q8—144no—
309C4×Q8—144no—
310C4×Q8—148no—
311C4×Q8—148no—
312C4×Q8—144no—
313C4×Q8—144no—
314C2×C4⋊C4—72no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—71no—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—144no—
319C2×C4⋊C4—148no—
320C2×C4⋊C4—141no—
321C2×C4⋊C4—142no—
322C2×C4⋊C4—142no—
323C2×C4⋊C4—141no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4—141no—
326C2×C22⋊C4—141no—
327C2×C22⋊C4—141no—
328C2×C42—24no—
329C2×C42—148no—
330C2×C42—142no—
331C2×C42—142no—
332C2×C42—141no—
333C2×C42—141no—
334C2×C42D2811yescharacteristic
335C2×C42D2811yescharacteristic
336C2×C42C22×D711yescharacteristic
337C2×C42C22×D711yescharacteristic
338#106—144no—
339#103—78no—
340#103—144no—
341#103—144no—
342#103—144no—
343C22⋊C8—22no—
344C22⋊C8C4×D714yes—
345#97—24no—
346#95—141no—
347#95—141no—
348#95—141no—
349#95—141no—
350#95—141no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—22no—
356C22×C7—22no—
357C22×C7#44114yescomplemented
358C22×C7C2×C4×D411yescharacteristic
359C22×C7C2×#12011yescharacteristic
360C22×C7#52511yescharacteristic
361C22×C7C2×#13711yescharacteristic
362C22×C7C2×#13711yescharacteristic
363C22×C7C2×#13711yescharacteristic
364C22×C7#57511yescharacteristic
365C4×C7—22no—
366C4×C7—22no—
367C4×C7—22no—
368C4×C7—22no—
369C4×C7—41no—
370C4×C7C2×C4×D412yes—
371C4×C7#52512yes—
372Dic7—28no—
373Dic7—42no—
374Dic7—44no—
375Dic7—42no—
376Dic7—81no—
377Dic7—81no—
378C24C2×D2811yescharacteristic
379C2×Q8—74no—
380C2×Q8—72no—
381C2×Q8—74no—
382C2×Q8—74no—
383C2×Q8—1416no—
384C2×Q8—144no—
385C2×Q8—144no—
386C2×Q8—1416no—
387C2×Q8—144no—
388C2×Q8—142no—
389C2×Q8—141no—
390C2×Q8—142no—
391C2×Q8—142no—
392C2×Q8—142no—
393C2×Q8—142no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—144no—
403C22×C4—144no—
404C22×C4—144no—
405C22×C4—142no—
406C22×C4—144no—
407C22×C4—141no—
408C22×C4—141no—
409C22×C4—142no—
410C22×C4—141no—
411C22×C4—141no—
412C22×C4—282no—
413C22×C4—282no—
414C22×C4C2×C4×D712yes—
415C22×C4C2×D2811yescharacteristic
416C22×C4C2×D2811yescharacteristic
417C22×C4C2×D2811yescharacteristic
418C22×C4C2×D2811yescharacteristic
419C22×C4C2×D2811yescharacteristic
420C22×C4#115712yes—
421C22×C4C23×D711yescharacteristic
422Q16—1416no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8C2×C4×D714yes—
429C4⋊C4—72no—
430C4⋊C4—74no—
431C4⋊C4—148no—
432C4⋊C4—144no—
433C4⋊C4—144no—
434C4⋊C4—148no—
435C4⋊C4—141no—
436C4⋊C4—142no—
437C4⋊C4—142no—
438C4⋊C4—142no—
439C4⋊C4—282no—
440C4⋊C4—282no—
441C22⋊C4—144no—
442C22⋊C4—282no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—148no—
447C42—148no—
448C42—282no—
449C42—282no—
450C42D4×D714yes—
451C2×C7—24no—
452C2×C7C4×#12011yescharacteristic
453C2×C7C2×#44111yescharacteristic
454C2×C7C4×#13711yescharacteristic
455C2×C7#300411yescharacteristic
456C2×C7#300711yescharacteristic
457C2×C7#311311yescharacteristic
458C2×C7#311711yescharacteristic
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23C4×D2812yes—
464C23#647212yes—
465C23#647212yes—
466C23C22×D2811yescharacteristic
467Q8—78no—
468Q8—1416no—
469Q8—144no—
470Q8—144no—
471Q8—144no—
472Q8—144no—
473Q8—282no—
474C2×C4—28no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—44no—
489C2×C4—1416no—
490C2×C4—144no—
491C2×C4—144no—
492C2×C4—144no—
493C2×C4—142no—
494C2×C4—142no—
495C2×C4—144no—
496C2×C4—288no—
497C2×C4—281no—
498C2×C4—281no—
499C2×C4—282no—
500C2×C4—281no—
501C2×C4—282no—
502C2×C4—281no—
503C2×C4—281no—
504C2×C4—281no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4C4×D2812yes—
510C2×C4#644512yes—
511C2×C4#644512yes—
512C2×C4#647212yes—
513C2×C4C22×C4×D711yescharacteristic
514C2×C4C2×#115711yescharacteristic
515C2×C4C2×D4×D712yes—
516C2×C4D7×#4112yes—
517C8—24no—
518C8—42no—
519C7#2145511yescharacteristic, complemented, Sylow
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22#8948814yescomplemented
527C22C2×C4×D2811yescharacteristic
528C22C2×#644511yescharacteristic
529C22#9021511yescharacteristic
530C22C2×#647211yescharacteristic
531C22C2×#647211yescharacteristic
532C22C2×#647211yescharacteristic
533C22#9043911yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—148no—
540C4—284no—
541C4—282no—
542C4—282no—
543C4—561no—
544C4—561no—
545C4C4×D4×D712yes—
546C4#9023812yes—
547C2—24no—
548C2C4×#644511yescharacteristic
549C2C2×#8948811yescharacteristic
550C2C4×#647211yescharacteristic
551C2#1173608511yescharacteristic
552C2#1173609011yescharacteristic
553C2#1173680111yescharacteristic
554C2#1173680211yescharacteristic
subgroupG/H
centerC23C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1749C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1749D7
hypocenterC7#21455
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
4. C2×#89488C2semidirect
9. #11727567C2semidirect
10. #11726439C2semidirect
11. #11726438C2semidirect
41. #89511C4semidirect
89. C2×Dic28C2×C4semidirect
109. #6390C2×C4semidirect
357. C22×C7#441semidirect
519. C7#21455semidirect
526. C22#89488semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic142
C4×#64872
C2×#895112
C2×#894882
C7×#17492
#117277172
#117276842
#117276822
#117275672
#117264392
#117264382
#214557