Finite Group Databasebeta

← #13359477

Subgroups — #13359477

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×#6450C211yesmaximal, characteristic
2Dic7×C22⋊Q8C211yesmaximal, characteristic
3C2×#89822C211yesmaximal, characteristic
4C2×#89561C212yesmaximal, complemented
5C7×#1872C211yesmaximal, characteristic
6#11729310C211yesmaximal, characteristic
7#11728035C212yesmaximal, complemented
8#11727717C211yesmaximal, characteristic
9#11727660C211yesmaximal, characteristic
10#11726661C212yesmaximal, complemented
11#11726537C212yesmaximal, complemented
12C23×Dic14C2211yescharacteristic
13C2×C7×C22⋊Q8C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×Q8×Dic7C2211yescharacteristic
16C22×#1144C2212yes—
17C22×Dic7⋊C4—21no—
18C22×C4×Dic7—21no—
19C22×C4×Dic7C2211yescharacteristic
20C2×#6450—21no—
21C2×#6450—21no—
22C2×#6450—28no—
23C2×#6450C2212yes—
24C2×#6450C2212yes—
25C2×C7×#103C2212yes—
26C2×C7×C22⋊C8C2211yescharacteristic
27#90002C2211yescharacteristic
28#89984C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×#6390C2212yes—
31C2×#6390C2212yes—
32C2×#6389C2212yes—
33#89822C418yescomplemented
34#89757C2211yescharacteristic
35#89753—21no—
36C4⋊C4×Dic7—21no—
37C4⋊C4×Dic7C2211yescharacteristic
38C2×#6384C2212yes—
39C2×#6382C2212yes—
40#89722—21no—
41C22⋊C4×Dic7—21no—
42#89561—28no—
43C7×#340C2212yes—
44#89345C2212yes—
45#89324C2212yes—
46#21455—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C23×C4×C7—21no—
49C23×C4×C7C2311yescharacteristic
50C23×Dic7—21no—
51C23×Dic7D411yescharacteristic
52C23×Dic7D411yescharacteristic
53C22×Dic14—21no—
54C22×Dic14—22no—
55C22×Dic14—24no—
56C22×Dic14—24no—
57C22×Dic14C2×C414yes—
58C22×Dic14C2312yes—
59C22×C7×C8C2312yes—
60C7×C22⋊Q8—22no—
61C7×C22⋊Q8C2×C414yes—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2312yes—
67C2×C7×C22⋊C4—21no—
68C2×C22⋊Dic7—21no—
69Q8×Dic7—24no—
70C2×#1144—24no—
71C2×#1144—24no—
72C2×#1144C2×C418yescomplemented
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7—21no—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×C4⋊Dic7C2311yescharacteristic
77C2×Dic7⋊C4—28no—
78C2×Dic7⋊C4—22no—
79C2×Dic7⋊C4—22no—
80C2×C4×Dic7—28no—
81C2×C4×Dic7—22no—
82C2×C4×Dic7—22no—
83C2×C4×Dic7—24no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7D411yescharacteristic
87C2×C4×Dic7D411yescharacteristic
88C2×C4×Dic7C2311yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8C2312yes—
91#6450—28no—
92#6450—28no—
93#6450—24no—
94#6450—24no—
95#6450—24no—
96#6450D418yes—
97C7×#103—24no—
98C7×C22⋊C8—22no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C22⋊(C7⋊C8)C2×C414yes—
105#6390—24no—
106#6390—24no—
107#6389—24no—
108#6384C2×C418yescomplemented
109#6382—24no—
110C22×C4×Q8—71no—
111C2×#451—71no—
112C2×#441—72no—
113C4×C22⋊Q8—71no—
114#1878—72no—
115#1872D711yescharacteristic
116#1865—72no—
117#1853—71no—
118#1787—71no—
119#1763—72no—
120#1749—71no—
121C24×C7C2×D411yescharacteristic
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8C22×C412yes—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—24no—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7C22×C412yes—
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—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—41no—
150C22×Dic7C2×D411yescharacteristic
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C2×Dic14—24no—
155C2×Dic14—24no—
156C2×Dic14—216no—
157C2×Dic14—24no—
158C2×Dic14—22no—
159C2×Dic14—22no—
160C2×Dic14—22no—
161C2×Dic14—216no—
162C2×Dic14C22×C414yes—
163C2×Dic14C2×D414yes—
164C2×Dic14#4114yes—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4C22×C412yes—
173C7×C4⋊C4C2×D414yes—
174C7×C22⋊C4—24no—
175C22⋊Dic7—42no—
176#1144—216no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—42no—
180C4⋊Dic7—42no—
181Dic7⋊C4—28no—
182Dic7⋊C4—28no—
183C4×Dic7—28no—
184C4×Dic7—22no—
185C4×Dic7—22no—
186C4×Dic7—22no—
187C4×Dic7—28no—
188C4×Dic7—42no—
189C4×Dic7—42no—
190C4×Dic7C2×D414yes—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8C22×C414yes—
194C23×Q8—71no—
195C22×Q16—72no—
196C2×C22⋊Q8C2×D711yescharacteristic
197C2×C4×Q8—71no—
198C2×C4×Q8—72no—
199C2×C4×Q8—72no—
200C2×C4×Q8—141no—
201C2×C4×Q8—141no—
202C2×C4×Q8—148no—
203C22×C4⋊C4—141no—
204C22×C4⋊C4C2×D711yescharacteristic
205C22×C42—71no—
206C22×C42—141no—
207#451—78no—
208#441—148no—
209C2×#106—72no—
210C2×#103—72no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#103C2×D712yes—
214C2×C22⋊C8—71no—
215C2×C22⋊C8C2×D711yescharacteristic
216C2×#97—72no—
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#340C2×D712yes—
227#336—72no—
228C23×C7—21no—
229C23×C7—21no—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7C4×D412yes—
233C23×C7#13712yes—
234C23×C7#13712yes—
235C23×C7C22×D411yescharacteristic
236C7×Q8—24no—
237C7×Q8—42no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—24no—
241C2×C4×C7—24no—
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—24no—
245C2×C4×C7—48no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—41no—
251C2×C4×C7—42no—
252C2×C4×C7C4×D412yes—
253C2×C4×C7C23×C411yescharacteristic
254C2×C4×C7C22×D412yes—
255C2×C4×C7C2×#4111yescharacteristic
256C2×C4×C7C2×#4112yes—
257C2×Dic7—216no—
258C2×Dic7—28no—
259C2×Dic7—24no—
260C2×Dic7—21no—
261C2×Dic7—24no—
262C2×Dic7—24no—
263C2×Dic7—21no—
264C2×Dic7—21no—
265C2×Dic7—21no—
266C2×Dic7—21no—
267C2×Dic7—21no—
268C2×Dic7—21no—
269C2×Dic7—21no—
270C2×Dic7—21no—
271C2×Dic7—44no—
272C2×Dic7—41no—
273C2×Dic7—41no—
274C2×Dic7—41no—
275C2×Dic7—41no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7#12012yes—
279C2×Dic7#12012yes—
280C2×Dic7#13712yes—
281Dic14—216no—
282Dic14—24no—
283Dic14—24no—
284Dic14—24no—
285Dic14C4×D418yes—
286C7×C8—42no—
287C7⋊C8—24no—
288C22×Q8—74no—
289C22×Q8—72no—
290C22×Q8—144no—
291C22×Q8—144no—
292C22×Q8—141no—
293C22×Q8—142no—
294C22×Q8C22×D711yescharacteristic
295C23×C4—21no—
296C23×C4—71no—
297C23×C4—71no—
298C23×C4—141no—
299C23×C4C22×D711yescharacteristic
300C2×Q16—78no—
301C2×Q16—144no—
302C2×Q16—144no—
303C22×C8—72no—
304C22×C8C22×D712yes—
305C22⋊Q8—22no—
306C22⋊Q8C4×D714yes—
307C4×Q8—78no—
308C4×Q8—144no—
309C4×Q8—144no—
310C4×Q8—144no—
311C4×Q8—148no—
312C4×Q8—148no—
313C4×Q8—144no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—148no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4—141no—
324C2×C4⋊C4C22×D711yescharacteristic
325C2×C4⋊C4C22×D712yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—141no—
328C2×C42—71no—
329C2×C42—71no—
330C2×C42—71no—
331C2×C42—71no—
332C2×C42—144no—
333C2×C42—142no—
334C2×C42—142no—
335C2×C42—148no—
336C2×C42—141no—
337C2×C42—141no—
338#106—144no—
339#103—24no—
340#103—78no—
341#103—144no—
342#103—144no—
343C22⋊C8—22no—
344C22⋊C8—74no—
345#97—144no—
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—42no—
367C4×C7—44no—
368C4×C7—42no—
369C4×C7C2×C4×D412yes—
370C4×C7#52512yes—
371Dic7—22no—
372Dic7—22no—
373Dic7—22no—
374Dic7—28no—
375Dic7—41no—
376Dic7—81no—
377Dic7—81no—
378C24D4×D711yescharacteristic
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—74no—
383C2×Q8—74no—
384C2×Q8—74no—
385C2×Q8—1416no—
386C2×Q8—144no—
387C2×Q8—144no—
388C2×Q8—1416no—
389C2×Q8—144no—
390C2×Q8—142no—
391C2×Q8—142no—
392C2×Q8—142no—
393C2×Q8C2×C4×D712yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—71no—
403C22×C4—71no—
404C22×C4—71no—
405C22×C4—71no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—144no—
410C22×C4—144no—
411C22×C4—144no—
412C22×C4—142no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—281no—
418C22×C4C2×C4×D712yes—
419C22×C4D4×D711yescharacteristic
420C22×C4#115912yes—
421C22×C4C23×D711yescharacteristic
422Q16—1416no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—74no—
427C2×C8—142no—
428C2×C8—142no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—148no—
434C4⋊C4—148no—
435C4⋊C4—142no—
436C4⋊C4—142no—
437C4⋊C4—282no—
438C4⋊C4—282no—
439C4⋊C4C2×C4×D712yes—
440C4⋊C4D4×D714yes—
441C22⋊C4—24no—
442C22⋊C4—282no—
443C42—74no—
444C42—142no—
445C42—142no—
446C42—148no—
447C42—148no—
448C42—142no—
449C42—282no—
450C42—282no—
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—
463C23(C22⋊D7)⋊C412yes—
464C23#647812yes—
465C23#648112yes—
466C23C2×D4×D711yescharacteristic
467Q8—24no—
468Q8—42no—
469Q8—78no—
470Q8—1416no—
471Q8—144no—
472Q8—144no—
473Q8—144no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—48no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—72no—
489C2×C4—72no—
490C2×C4—72no—
491C2×C4—148no—
492C2×C4—1416no—
493C2×C4—144no—
494C2×C4—141no—
495C2×C4—141no—
496C2×C4—141no—
497C2×C4—141no—
498C2×C4—141no—
499C2×C4—141no—
500C2×C4—144no—
501C2×C4—144no—
502C2×C4—141no—
503C2×C4—141no—
504C2×C4—141no—
505C2×C4—284no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4(C22⋊D7)⋊C412yes—
513C2×C4C22×C4×D711yescharacteristic
514C2×C4C2×D4×D712yes—
515C2×C4C2×#115911yescharacteristic
516C2×C4D7×#4112yes—
517C8—42no—
518C8—144no—
519C7#2145511yescharacteristic, complemented, Sylow
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22#8956114yescomplemented
527C22C2×(C22⋊D7)⋊C411yescharacteristic
528C22D7×#12011yescharacteristic
529C22#9034711yescharacteristic
530C22C2×#647811yescharacteristic
531C22C2×#648111yescharacteristic
532C22#9046411yescharacteristic
533C22D7×#13711yescharacteristic
534C4—22no—
535C4—42no—
536C4—42no—
537C4—44no—
538C4—142no—
539C4—142no—
540C4—148no—
541C4—142no—
542C4—281no—
543C4—561no—
544C4—561no—
545C4C4×D4×D712yes—
546C4#9029712yes—
547C2—24no—
548C2#1173210411yescharacteristic
549C2C2×#8956111yescharacteristic
550C2#1173631211yescharacteristic
551C2#1173759211yescharacteristic
552C2#1173762811yescharacteristic
553C2#1173885911yescharacteristic
554C2#1173890011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1872C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1872D7
hypocenterC7#21455
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. C2×#89561C2semidirect
7. #11728035C2semidirect
10. #11726661C2semidirect
11. #11726537C2semidirect
33. #89822C4semidirect
72. C2×#1144C2×C4semidirect
108. #6384C2×C4semidirect
357. C22×C7#441semidirect
519. C7#21455semidirect
526. C22#89561semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
Dic7×C22⋊Q82
C2×#898222
C2×#895612
C7×#18722
#117293102
#117280352
#117277172
#117276602
#117266612
#117265372
#214557