Finite Group Databasebeta

← #13370341

Subgroups — #13370341

Order 1792.

1354 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic14C211yesmaximal, characteristic
2#11731321C212yesmaximal, complemented
3C7×#1799C211yesmaximal, characteristic
4C4×#6408C211yesmaximal, characteristic
5#11727782C212yesmaximal
6Dic7×#99C212yesmaximal
7#11727283C212yesmaximal, complemented
8#11726709C214yesmaximal, complemented
9C23×Dic14C411yescharacteristic, complemented
10C23×Dic14C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C2×D4×Dic7C2212yes—
13C2×#6487C411yescharacteristic, complemented
14C2×#6487C2211yescharacteristic
15D4×Dic14—22no—
16D4×Dic14—24no—
17D4×Dic14—22no—
18C2×#6451C412yescomplemented
19C2×#6450—22no—
20C2×#6441C412yescomplemented
21C2×#6441C2212yes—
22C2×#6432C411yescharacteristic, complemented
23C2×C4×Dic14C411yescharacteristic, complemented
24C2×C7×#99C2212yes—
25C7×#380C2212yes—
26C2×#6408C2211yescharacteristic
27C2×#6408C2211yescharacteristic
28C4×C22⋊Dic7C2212yes—
29#89746—22no—
30#89714C2214yes—
31C22⋊C4×Dic7C2214yes—
32#89418—24no—
33C7×#342C2212yes—
34#89322C2214yes—
35#89322C2214yes—
36#89321C2212yes—
37Q8×#99—71nomaximal, Sylow
38C22×C7×D4C2311yescharacteristic
39C23×C4×C7C2×C411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7C2×C412yes—
42C23×Dic7C2312yes—
43C22×Dic14—22no—
44C22×Dic14—21no—
45C22×Dic14—21no—
46C22×Dic14—44no—
47C22×Dic14—44no—
48C22×Dic14C2×C411yescharacteristic
49C22×Dic14C2×C411yescharacteristic
50C22×Dic14C2×C412yes—
51C22×Dic14C2×C411yescharacteristic
52C22×Dic14C2×C412yes—
53C22×Dic14D412yes—
54C22×Dic14D414yes—
55C22×Dic14D412yes—
56C4×C7×D4—22no—
57C4×C7×D4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4C2×C411yescharacteristic
60C2×C7×C22⋊C4C2×C411yescharacteristic
61C2×C7×C22⋊C4C2314yes—
62C2×C7×C22⋊C4C2311yescharacteristic
63C2×C42×C7C2×C411yescharacteristic
64C2×C22⋊Dic7C2×C412yes—
65C2×C22⋊Dic7C2312yes—
66C2×C22⋊Dic7C2312yes—
67C2×C22⋊Dic7C2312yes—
68D4×Dic7—24no—
69D4×Dic7—24no—
70C2×C4⋊Dic7C2×C411yescharacteristic
71C2×C4⋊Dic7C2×C412yes—
72C2×C4⋊Dic7C2×C412yes—
73#6487—22no—
74#6487—22no—
75#6487—22no—
76#6487—22no—
77C2×Dic7⋊C4—24no—
78C2×Dic7⋊C4C2×C412yes—
79C2×Dic7⋊C4C2×C412yes—
80C2×Dic7⋊C4C2×C412yes—
81C2×C4×Dic7—22no—
82C2×C4×Dic7—24no—
83C2×C4×Dic7C2×C412yes—
84#6451—24no—
85#6451—24no—
86#6450—48no—
87#6441—24no—
88#6441—24no—
89#6441—24no—
90#6441—24no—
91#6432—22no—
92#6432—22no—
93C4×Dic14—21no—
94C4×Dic14—22no—
95C4×Dic14—21no—
96C7×#99—24no—
97C7×#95—22no—
98#6408—22no—
99#6408Q814yescomplemented
100C7⋊#95—22no—
101C7⋊#95—24no—
102C7⋊#95—24no—
103C2×D4×Q8—71no—
104Q8×C22⋊C4—72no—
105#1996—72no—
106#1996—74no—
107#1799—72no—
108#1799D711yescharacteristic
109C4×#99—72no—
110C4×#99—71no—
111C24×C7C22×C411yescharacteristic
112C24×C7C2411yescharacteristic
113C2×C7×D4—22no—
114C2×C7×D4—22no—
115C2×C7×D4C2×Q812yes—
116C2×C7×D4#4112yes—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—24no—
120C22×C4×C7—42no—
121C22×C4×C7—42no—
122C22×C4×C7C22×C411yescharacteristic
123C22×C4×C7C22×C411yescharacteristic
124C22×C4×C7C22×C411yescharacteristic
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D412yes—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7—22no—
133C22×Dic7—44no—
134C22×Dic7—44no—
135C22×Dic7C22×C412yes—
136C22×Dic7C22×C412yes—
137C22×Dic7C22×C412yes—
138C22×Dic7C22×C412yes—
139C22×Dic7C2×D414yes—
140C22×Dic7C2×D414yes—
141C2×Dic14—28no—
142C2×Dic14—24no—
143C2×Dic14—21no—
144C2×Dic14—21no—
145C2×Dic14—22no—
146C2×Dic14—22no—
147C2×Dic14—44no—
148C2×Dic14—48no—
149C2×Dic14—42no—
150C2×Dic14—44no—
151C2×Dic14—44no—
152C2×Dic14—48no—
153C2×Dic14—44no—
154C2×Dic14—44no—
155C2×Dic14—42no—
156C2×Dic14C22⋊C411yescharacteristic
157C2×Dic14C22⋊C412yes—
158C2×Dic14C22⋊C412yes—
159C2×Dic14C22⋊C414yes—
160C2×Dic14C22⋊C412yes—
161C2×Dic14C22⋊C414yes—
162C2×Dic14C22⋊C411yescharacteristic
163C7×C4⋊C4—22no—
164C7×C4⋊C4—44no—
165C7×C22⋊C4—22no—
166C7×C22⋊C4—22no—
167C7×C22⋊C4—24no—
168C7×C22⋊C4—24no—
169C42×C7—21no—
170C42×C7—21no—
171C22⋊Dic7—24no—
172C22⋊Dic7—22no—
173C22⋊Dic7—24no—
174C22⋊Dic7—22no—
175C22⋊Dic7—22no—
176C22⋊Dic7C2×Q814yescomplemented
177C4⋊Dic7—21no—
178C4⋊Dic7—21no—
179C4⋊Dic7—24no—
180C4⋊Dic7—24no—
181Dic7⋊C4—24no—
182Dic7⋊C4—22no—
183Dic7⋊C4—22no—
184Dic7⋊C4—24no—
185Dic7⋊C4—48no—
186C4×Dic7—22no—
187C4×Dic7—22no—
188C4×Dic7—48no—
189C4×Dic7—44no—
190C23×Q8—71no—
191C23×Q8—71no—
192D4×Q8—142no—
193D4×Q8—142no—
194D4×Q8—144no—
195C2×C4⋊Q8—72no—
196C2×C4⋊Q8—71no—
197C2×C22⋊Q8—71no—
198C2×C22⋊Q8—71no—
199C2×C22⋊Q8—72no—
200C2×C22⋊Q8—72no—
201C2×C4×Q8—71no—
202C2×C4×Q8—142no—
203C2×C4×D4—72no—
204C2×C4×D4C2×D711yescharacteristic
205C2×#99—71no—
206C2×#99—71no—
207C2×#99C2×D712yes—
208#391—144no—
209#391—142no—
210#380—74no—
211#380C2×D712yes—
212C4×C22⋊C4—72no—
213C4×C22⋊C4—74no—
214#342—74no—
215#342—72no—
216#342—74no—
217#342C2×D712yes—
218C23×C7—21no—
219C23×C7—21no—
220C23×C7—41no—
221C23×C7—41no—
222C23×C7—41no—
223C23×C7—41no—
224C23×C7C4×Q812yescomplemented
225C23×C7C23×C411yescharacteristic
226C23×C7C22×D412yes—
227C23×C7C22×Q811yescharacteristic
228C23×C7C2×#4111yescharacteristic
229C23×C72−1+412yes—
230C7×D4—24no—
231C7×D4—24no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—24no—
235C2×C4×C7—22no—
236C2×C4×C7—21no—
237C2×C4×C7—21no—
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×C7—44no—
245C2×C4×C7—44no—
246C2×C4×C7—44no—
247C2×C4×C7C2×C22⋊C412yes—
248C2×C4×C7C2×C22⋊C411yescharacteristic
249C2×C4×C7C2×C22⋊C411yescharacteristic
250C2×C4×C7C4×Q812yescomplemented
251C2×Dic7—28no—
252C2×Dic7—24no—
253C2×Dic7—22no—
254C2×Dic7—22no—
255C2×Dic7—24no—
256C2×Dic7—42no—
257C2×Dic7—44no—
258C2×Dic7—44no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—44no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7C2×C22⋊C412yes—
268C2×Dic7C2×C22⋊C414yes—
269C2×Dic7C2×C22⋊C412yes—
270Dic14—22no—
271Dic14—24no—
272Dic14—24no—
273Dic14—28no—
274Dic14—44no—
275Dic14—42no—
276Dic14—44no—
277Dic14—42no—
278Dic14—42no—
279Dic14—44no—
280Dic14#9914yescomplemented
281C22×Q8—71no—
282C22×Q8—72no—
283C22×Q8—71no—
284C22×Q8—72no—
285C22×Q8—71no—
286C22×Q8—72no—
287C22×Q8—74no—
288C22×Q8—72no—
289C22×Q8—141no—
290C22×Q8—142no—
291C22×Q8—141no—
292C22×Q8—284no—
293C22×Q8—284no—
294C22×D4C22×D711yescharacteristic
295C23×C4—72no—
296C23×C4—72no—
297C23×C4C4×D711yescharacteristic
298C23×C4C22×D711yescharacteristic
299C4⋊Q8—142no—
300C4⋊Q8—142no—
301C4⋊Q8—144no—
302C4⋊Q8—144no—
303C22⋊Q8—144no—
304C22⋊Q8—144no—
305C22⋊Q8—144no—
306C22⋊Q8—144no—
307C22⋊Q8—142no—
308C22⋊Q8—142no—
309C22⋊Q8—142no—
310C22⋊Q8—142no—
311C4×Q8—141no—
312C4×Q8—142no—
313C4×Q8—141no—
314C4×Q8—288no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—144no—
318C4×D4—144no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—72no—
321C2×C4⋊C4—72no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—72no—
324C2×C4⋊C4—72no—
325C2×C4⋊C4—72no—
326C2×C4⋊C4—144no—
327C2×C4⋊C4C4×D711yescharacteristic
328C2×C22⋊C4—72no—
329C2×C22⋊C4—72no—
330C2×C22⋊C4—72no—
331C2×C22⋊C4—72no—
332C2×C22⋊C4C4×D711yescharacteristic
333C2×C22⋊C4C22×D714yes—
334C2×C22⋊C4C22×D711yescharacteristic
335C2×C42—72no—
336C2×C42—144no—
337C2×C42—142no—
338C2×C42C4×D711yescharacteristic
339#99—24no—
340#99—74no—
341#99—142no—
342#95—22no—
343#95—144no—
344#95—144no—
345#95—142no—
346C22×C7—22no—
347C22×C7—21no—
348C22×C7—22no—
349C22×C7—21no—
350C22×C7—22no—
351C22×C7—41no—
352C22×C7—41no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—41no—
357C22×C7—41no—
358C22×C7C22×C22⋊C411yescharacteristic
359C22×C7C2×C4×Q811yescharacteristic
360C22×C7#51911yescharacteristic
361C22×C7#54812yes—
362C22×C7D4×Q812yes—
363C4×C7—22no—
364C4×C7—21no—
365C4×C7—22no—
366C4×C7—41no—
367C4×C7—41no—
368C4×C7—84no—
369C4×C7C2×#9912yes—
370Dic7—22no—
371Dic7—24no—
372Dic7—44no—
373Dic7—42no—
374Dic7—42no—
375Dic7—82no—
376Dic7C2×#9914yes—
377C24C2×C4×D711yescharacteristic
378C24C23×D711yescharacteristic
379C2×Q8—71no—
380C2×Q8—72no—
381C2×Q8—71no—
382C2×Q8—74no—
383C2×Q8—72no—
384C2×Q8—74no—
385C2×Q8—72no—
386C2×Q8—142no—
387C2×Q8—142no—
388C2×Q8—148no—
389C2×Q8—144no—
390C2×Q8—141no—
391C2×Q8—141no—
392C2×Q8—284no—
393C2×Q8—282no—
394C2×Q8—282no—
395C2×Q8—284no—
396C2×Q8—288no—
397C2×Q8—284no—
398C2×Q8—284no—
399C2×Q8—288no—
400C2×Q8—284no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4#115912yes—
404C2×D4Q8×D712yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—72no—
411C22×C4—74no—
412C22×C4—72no—
413C22×C4—74no—
414C22×C4—72no—
415C22×C4—72no—
416C22×C4—142no—
417C22×C4—142no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—284no—
421C22×C4—284no—
422C22×C4C2×C4×D711yescharacteristic
423C22×C4C2×C4×D711yescharacteristic
424C22×C4C2×C4×D711yescharacteristic
425C22×C4C2×C4×D711yescharacteristic
426C22×C4C2×C4×D711yescharacteristic
427C22×C4D4×D712yes—
428C22×C4D4×D712yes—
429C4⋊C4—22no—
430C4⋊C4—44no—
431C4⋊C4—142no—
432C4⋊C4—142no—
433C4⋊C4—144no—
434C4⋊C4—144no—
435C4⋊C4—144no—
436C4⋊C4—144no—
437C4⋊C4—141no—
438C4⋊C4—141no—
439C4⋊C4—288no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—74no—
445C22⋊C4—144no—
446C22⋊C4—144no—
447C22⋊C4—142no—
448C22⋊C4—142no—
449C22⋊C4—142no—
450C42—21no—
451C42—21no—
452C42—142no—
453C42—142no—
454C42—284no—
455C42—288no—
456C2×C7—22no—
457C2×C7—22no—
458C2×C7—21no—
459C2×C7—21no—
460C2×C7—42no—
461C2×C7Q8×C22⋊C411yescharacteristic
462C2×C7C22×#9911yescharacteristic
463C2×C7#288411yescharacteristic
464C23—21no—
465C23—21no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23#645012yescomplemented
471C23C22×C4×D711yescharacteristic
472C23C2×D4×D712yes—
473C23C2×#115911yescharacteristic
474C23C2×Q8×D711yescharacteristic
475C23#655412yes—
476Q8—74no—
477Q8—148no—
478Q8—144no—
479Q8—144no—
480Q8—142no—
481Q8—282no—
482Q8—284no—
483Q8—282no—
484Q8—284no—
485Q8—284no—
486Q8—282no—
487D4—24no—
488D4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—72no—
505C2×C4—72no—
506C2×C4—74no—
507C2×C4—148no—
508C2×C4—144no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—144no—
512C2×C4—284no—
513C2×C4—284no—
514C2×C4—282no—
515C2×C4—284no—
516C2×C4—282no—
517C2×C4—282no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4D7×C22⋊C411yescharacteristic
524C2×C4D7×C22⋊C412yes—
525C2×C4D7×C22⋊C411yescharacteristic
526C2×C4#645012yescomplemented
527C7Q8×#9911yescharacteristic, complemented, Sylow
528C22—22no—
529C22—21no—
530C22—22no—
531C22—21no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22C2×D7×C22⋊C411yescharacteristic
541C22C2×#645011yescharacteristic
542C22#9020711yescharacteristic
543C22#9028912yes—
544C22#9032312yes—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—74no—
551C4—84no—
552C4—142no—
553C4—144no—
554C4—284no—
555C4—282no—
556C4—282no—
557C4—562no—
558C4D7×#9912yes—
559C2—22no—
560C2—22no—
561C2—21no—
562C2—21no—
563C2—42no—
564C2#1173132111yescharacteristic
565C2C2×D7×#9911yescharacteristic
566C2#1173613411yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC24C23×D7
FittingC7×#1799C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C7×D4C23
cosocleC24×C7C24
hypercenter#1799D7
hypocenterC7Q8×#99
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #11731321C2semidirect
7. #11727283C2semidirect
8. #11726709C2semidirect
9. C23×Dic14C4semidirect
13. C2×#6487C4semidirect
18. C2×#6451C4semidirect
20. C2×#6441C4semidirect
22. C2×#6432C4semidirect
23. C2×C4×Dic14C4semidirect
99. #6408Q8semidirect
176. C22⋊Dic7C2×Q8semidirect
224. C23×C7C4×Q8semidirect
250. C2×C4×C7C4×Q8semidirect
280. Dic14#99semidirect
470. C23#6450semidirect
526. C2×C4#6450semidirect
527. C7Q8×#99semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic142
#117313212
C7×#17992
C4×#64082
#117277822
Dic7×#992
#117272832
#117267092
Q8×#997