Finite Group Databasebeta

← #13355532

Subgroups — #13355532

Order 1792.

1946 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#535C211yesmaximal, characteristic
2C22×#6511C211yesmaximal, characteristic, complemented
3#11741532C211yesmaximal, characteristic, complemented
4C2×#89987C211yesmaximal, characteristic, complemented
5C22×#6406C211yesmaximal, characteristic, complemented
6#11729436C214yesmaximal, complemented
7#11729435C214yesmaximal, complemented
8#11729335C211yesmaximal, characteristic, complemented
9#11729334C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C7×D4C2211yescharacteristic
12C7×#535—24no—
13C7×#535C418yescomplemented
14C2×C7×#127—21no—
15C2×C7×#124—21no—
16C2×C7×#121C2214yes—
17C2×C7×#120—22no—
18C2×C7×#117C2211yescharacteristic
19C2×#6511—24no—
20C2×#6511—24no—
21C2×#6511—24no—
22C2×#6489C2211yescharacteristic
23C22×C4⋊Dic7C2211yescharacteristic
24C22×#1136C2211yescharacteristic
25C22×#1136C2212yescomplemented
26C23×C7⋊C8C2211yescharacteristic
27C23×C7⋊C8C2212yescomplemented
28#90002—21no—
29#89993—21no—
30#89992—22no—
31#89987—24no—
32C2×#6406—24no—
33C2×#6406C2212yes—
34C2×#6406C2214yes—
35C2×#6406C2212yes—
36#89984C2214yescomplemented
37#89324C2212yescomplemented
38#89324C2214yescomplemented
39#89324C2212yescomplemented
40#20975—71nomaximal, Sylow
41C25×C7—21no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41C2×C414yes—
47C2×C7×#41D414yes—
48C22×C7×Q8D411yescharacteristic
49C22×C7×D4—24no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4—21no—
53C22×C7×D4—22no—
54C22×C7×D4—22no—
55C22×C7×D4—22no—
56C22×C7×D4C2×C414yes—
57C22×C7×D4D412yes—
58C22×C7×D4D411yescharacteristic
59C22×C7×D4C2312yes—
60C23×C4×C7C2312yes—
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—21no—
63C7×#127—28no—
64C7×#124—28no—
65C7×#121—28no—
66C7×#121C2×C4116yescomplemented
67C7×#120—216no—
68C7×#117—22no—
69C7×#117C2×C414yes—
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—24no—
73C2×C42×C7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—24no—
76#6511—232no—
77#6489—22no—
78#6489—22no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2312yes—
83C2×C4⋊Dic7C2312yes—
84C2×C4×Dic7—21no—
85C2×#1136—22no—
86C2×#1136—22no—
87C2×#1136—24no—
88C2×#1136—24no—
89C2×#1136—24no—
90C2×#1136D414yescomplemented
91C22×C7⋊C8—22no—
92C22×C7⋊C8—22no—
93C22×C7⋊C8—24no—
94C22×C7⋊C8—24no—
95C22×C7⋊C8—24no—
96C22×C7⋊C8D414yescomplemented
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2312yes—
100#6406—24no—
101#6406—24no—
102#6406—28no—
103C7⋊#95—24no—
104C22×#131—71no—
105C2×#535D711yescharacteristic
106C2×#418—71no—
107C22×#102—71no—
108#2311—71no—
109#1895—74no—
110#1893—74no—
111#1736—71no—
112#1735—71no—
113C24×C7—22no—
114C24×C7—24no—
115C24×C7—42no—
116C24×C7—42no—
117C24×C7—82no—
118C24×C7—82no—
119C24×C7C2×D412yes—
120C24×C7C2×D411yescharacteristic
121C7×#41—28no—
122C7×#41—28no—
123C7×#41—28no—
124C7×#41—28no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8C22⋊C414yes—
128C2×C7×D4—24no—
129C2×C7×D4—24no—
130C2×C7×D4—216no—
131C2×C7×D4—24no—
132C2×C7×D4—28no—
133C2×C7×D4—28no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—44no—
142C2×C7×D4—42no—
143C2×C7×D4—42no—
144C2×C7×D4—44no—
145C2×C7×D4—84no—
146C2×C7×D4—84no—
147C2×C7×D4C22⋊C418yes—
148C2×C7×D4C22⋊C414yes—
149C2×C7×D4C22×C414yes—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7C22×C414yes—
156C22×C4×C7C22×C412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2×D414yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7#4112yes—
164C22×C4×C7#4112yes—
165C22×C4×C7C2411yescharacteristic
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—42no—
169C22×Dic7—82no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4C22×C414yes—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—216no—
174C42×C7—24no—
175C22⋊Dic7—42no—
176C22⋊Dic7—84no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—22no—
181C4⋊Dic7—42no—
182C4×Dic7—42no—
183#1136—216no—
184#1136—28no—
185C2×C7⋊C8—216no—
186C2×C7⋊C8—28no—
187C2×C7⋊C8—21no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8C2×D412yes—
194C2×C7⋊C8C2×D414yes—
195C2×C7⋊C8C2×D412yes—
196C22×#41C2×D711yescharacteristic
197C23×D4C2×D711yescharacteristic
198C2×#131—144no—
199C2×#131—144no—
200C2×#131—144no—
201C22×M16—71no—
202C22×M16—72no—
203C23×C8—71no—
204C23×C8—72no—
205#535—24no—
206#535Dic718yescomplemented
207C2×#127—21no—
208C2×#124—21no—
209C2×#121C2×D714yes—
210C2×#120—22no—
211C2×#117—71no—
212C2×#117C2×D711yescharacteristic
213C22×C4⋊C4—71no—
214#418—144no—
215C2×#102—72no—
216C2×#102—72no—
217C2×#102—74no—
218C2×#102—144no—
219#391—141no—
220#390—141no—
221#386—142no—
222#380—74no—
223#340—72no—
224#340—74no—
225#340—72no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—24no—
231C23×C7—24no—
232C23×C7—24no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—44no—
236C23×C7—44no—
237C23×C7—48no—
238C23×C7—88no—
239C23×C7—84no—
240C23×C7—84no—
241C23×C7—88no—
242C23×C7C2×C22⋊C414yes—
243C23×C7C2×C22⋊C412yes—
244C23×C7#12112yes—
245C23×C7#12112yes—
246C23×C7C22×D411yescharacteristic
247C7×Q8—28no—
248C7×D4—216no—
249C7×D4—24no—
250C7×D4—24no—
251C7×D4—28no—
252C7×D4—42no—
253C7×D4—42no—
254C7×D4—48no—
255C7×D4—816no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7C2×C22⋊C414yes—
266C2×C4×C7C2×C22⋊C412yes—
267C2×C4×C7C4×D414yes—
268C2×C4×C7C4×D414yes—
269C2×C4×C7#12014yes—
270C2×C4×C7#12112yes—
271C2×C4×C7#12112yes—
272C2×C4×C7#12314yes—
273C2×C4×C7C23×C411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C22×D411yescharacteristic
276C2×C4×C7C22×D411yescharacteristic
277C2×C4×C7C2×#4111yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—44no—
283C2×Dic7—42no—
284C2×Dic7—84no—
285C7⋊C8—22no—
286C7⋊C8—24no—
287C7⋊C8—22no—
288C25—21no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41C2×Dic714yes—
294C2×#41C22⋊D714yes—
295C22×Q8C22⋊D711yescharacteristic
296C22×D4—24no—
297C22×D4—22no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—21no—
302C22×D4—22no—
303C22×D4C2×Dic714yes—
304C22×D4C22⋊D712yes—
305C22×D4C22⋊D711yescharacteristic
306C22×D4C22×D712yes—
307C23×C4—141no—
308C23×C4C22×D712yes—
309C23×C4C22×D711yescharacteristic
310#131—1432no—
311C2×M16—74no—
312C2×M16—144no—
313C2×M16—144no—
314C2×M16—142no—
315C2×M16—144no—
316C2×M16—142no—
317C22×C8—74no—
318C22×C8—71no—
319C22×C8—71no—
320C22×C8—72no—
321C22×C8—144no—
322C22×C8—144no—
323C22×C8—142no—
324C22×C8—144no—
325C22×C8—142no—
326#127—28no—
327#124—28no—
328#121—28no—
329#121C2×Dic7116yescomplemented
330#120—216no—
331#117—22no—
332#117—142no—
333#117—142no—
334#117C2×Dic714yes—
335C2×C4⋊C4—72no—
336C2×C4⋊C4—72no—
337C2×C4⋊C4—142no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4C22×D712yes—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—144no—
345C2×C42—21no—
346C2×C42—141no—
347#102—144no—
348#102—144no—
349#102—148no—
350#95—144no—
351C22×C7—24no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—24no—
355C22×C7—28no—
356C22×C7—24no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—44no—
360C22×C7—48no—
361C22×C7—44no—
362C22×C7—816no—
363C22×C7—82no—
364C22×C7—82no—
365C22×C7—84no—
366C22×C7#38614yes—
367C22×C7C22×C22⋊C411yescharacteristic
368C22×C7C2×#12111yescharacteristic
369C22×C7C2×#12111yescharacteristic
370C22×C7#57612yes—
371C22×C7#57712yes—
372C4×C7—24no—
373C4×C7—44no—
374C4×C7#38614yes—
375C4×C7C2×C4×D411yescharacteristic
376C4×C7C2×C4×D411yescharacteristic
377C4×C7C2×#12011yescharacteristic
378C4×C7C2×#12311yescharacteristic
379Dic7—82no—
380Dic7—82no—
381C24—22no—
382C24—24no—
383C24—42no—
384C24—42no—
385C24—82no—
386C24—82no—
387C24C2×C22⋊D712yes—
388C24C2×C22⋊D711yescharacteristic
389#41—28no—
390#41—28no—
391#41—28no—
392#41—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C22⋊Dic714yes—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—216no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4C22⋊Dic718yes—
416C2×D4C22⋊Dic714yes—
417C2×D4C22×Dic714yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—282no—
426C22×C4—562no—
427C22×C4D4×D712yes—
428C22×C4D4×D712yes—
429C22×C4#115912yes—
430C22×C4#115912yes—
431C22×C4C22×Dic714yes—
432C22×C4C22×Dic712yes—
433C22×C4C2×C22⋊D712yes—
434C22×C4C2×C22⋊D714yes—
435C22×C4C2×C22⋊D712yes—
436C22×C4C2×C22⋊D711yescharacteristic
437C22×C4C23×D711yescharacteristic
438M16—1416no—
439M16—148no—
440C2×C8—72no—
441C2×C8—72no—
442C2×C8—74no—
443C2×C8—1416no—
444C2×C8—148no—
445C2×C8—141no—
446C2×C8—142no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—142no—
451C4⋊C4—22no—
452C4⋊C4—142no—
453C4⋊C4—142no—
454C4⋊C4—142no—
455C4⋊C4—142no—
456C4⋊C4—282no—
457C4⋊C4C22×Dic714yes—
458C22⋊C4—24no—
459C22⋊C4—216no—
460C22⋊C4—282no—
461C22⋊C4—564no—
462C42—24no—
463C42—282no—
464C2×C7—24no—
465C2×C7—44no—
466C2×C7—84no—
467C2×C7C2×#38611yescharacteristic
468C2×C7#289712yes—
469C2×C7#305711yescharacteristic
470C2×C7#305811yescharacteristic
471C2×C7#306212yes—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—42no—
480C23—42no—
481C23—48no—
482C23—44no—
483C23—44no—
484C23—88no—
485C23—84no—
486C23—84no—
487C23—88no—
488C23#650212yes—
489C23#650212yes—
490C23C2×C22⋊Dic714yes—
491C23C2×C22⋊Dic712yes—
492C23C22×C22⋊D711yescharacteristic
493Q8—28no—
494D4—216no—
495D4—24no—
496D4—24no—
497D4—28no—
498D4—42no—
499D4—42no—
500D4—48no—
501D4—816no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4—282no—
514C2×C4—284no—
515C2×C4—282no—
516C2×C4—564no—
517C2×C4D4×Dic714yes—
518C2×C4D4×Dic714yes—
519C2×C4#649814yes—
520C2×C4#650014yes—
521C2×C4#650212yes—
522C2×C4#650212yes—
523C2×C4C2×C22⋊Dic714yes—
524C2×C4C2×C22⋊Dic712yes—
525C2×C4C2×D4×D711yescharacteristic
526C2×C4C2×D4×D711yescharacteristic
527C2×C4C2×#115911yescharacteristic
528C2×C4C2×#115911yescharacteristic
529C2×C4C23×Dic711yescharacteristic
530C2×C4C22×C22⋊D711yescharacteristic
531C8—142no—
532C8—142no—
533C8—144no—
534C7#2097511yescharacteristic, complemented, Sylow
535C22—24no—
536C22—22no—
537C22—22no—
538C22—24no—
539C22—24no—
540C22—28no—
541C22—42no—
542C22—42no—
543C22—44no—
544C22—44no—
545C22—48no—
546C22—816no—
547C22—82no—
548C22—82no—
549C22—84no—
550C22#8999214yes—
551C22C2×#650211yescharacteristic
552C22C2×#650211yescharacteristic
553C22#9052612yes—
554C22#9052712yes—
555C22C22×C22⋊Dic711yescharacteristic
556C4—24no—
557C4—44no—
558C4—562no—
559C4—562no—
560C4#8999214yes—
561C4C2×D4×Dic711yescharacteristic
562C4C2×D4×Dic711yescharacteristic
563C4C2×#649811yescharacteristic
564C4C2×#650011yescharacteristic
565C2—24no—
566C2—44no—
567C2—84no—
568C2#1174082211yescharacteristic
569C2#1174082311yescharacteristic
570C2#1174082412yes—
571C2C2×#8999211yescharacteristic
572C2#1174154412yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#535C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#535D7
hypocenterC7#20975
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. C22×#6511C2semidirect
3. #11741532C2semidirect
4. C2×#89987C2semidirect
5. C22×#6406C2semidirect
6. #11729436C2semidirect
7. #11729435C2semidirect
8. #11729335C2semidirect
9. #11729334C2semidirect
13. C7×#535C4semidirect
25. C22×#1136C22semidirect
27. C23×C7⋊C8C22semidirect
36. #89984C22semidirect
37. #89324C22semidirect
38. #89324C22semidirect
39. #89324C22semidirect
66. C7×#121C2×C4semidirect
90. C2×#1136D4semidirect
96. C22×C7⋊C8D4semidirect
206. #535Dic7semidirect
329. #121C2×Dic7semidirect
534. C7#20975semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5352
C22×#65112
#117415322
C2×#899872
C22×#64062
#117294362
#117294352
#117293352
#117293342
#209757