Finite Group Databasebeta

← #13357501

Subgroups — #13357501

Order 1792.

1526 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C2×#90004C211yesmaximal, characteristic
3C2×#90005C211yesmaximal, characteristic
4C2×#89997C211yesmaximal, characteristic, complemented
5C22×#6409C211yesmaximal, characteristic, complemented
6#11739678C211yesmaximal, characteristic, complemented
7C2×#89769C211yesmaximal, characteristic, complemented
8#11729540C212yesmaximal, complemented
9#11729539C212yesmaximal, complemented
10#11729480C212yesmaximal, complemented
11#11729479C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic
14C7×#536—24no—
15C7×#536C418yescomplemented
16C2×C7×C4⋊Q8C2212yes—
17C2×C7×#124C2212yes—
18C2×C7×#123—22no—
19C2×C7×C22⋊Q8—22no—
20C2×C7×#117C2211yescharacteristic
21C2×#6489C2211yescharacteristic
22C22×C4⋊Dic7C2211yescharacteristic
23C22×#1136C2211yescharacteristic
24C23×C7⋊C8C2211yescharacteristic
25#90005—24no—
26#90004—24no—
27#89997—24no—
28C2×#6409—24no—
29C2×#6409C2212yes—
30C2×#6409C2212yes—
31C2×#6409C2212yes—
32C2×#6409C2212yes—
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2212yes—
36C2×#6406C2211yescharacteristic
37C2×#6406C2212yes—
38C2×#6406C2211yescharacteristic
39#89769—24no—
40#89757—22no—
41#89722—22no—
42#89708C2212yes—
43#89707C2212yes—
44C2×#6377C2212yes—
45C2×C4×C7⋊C8C2212yes—
46#21205—71nomaximal, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41C2×C414yes—
53C22×C7×Q8—21no—
54C22×C7×Q8—22no—
55C22×C7×Q8—22no—
56C22×C7×Q8—22no—
57C22×C7×Q8—22no—
58C22×C7×Q8—22no—
59C22×C7×Q8C2×C414yes—
60C22×C7×Q8C2312yes—
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4—21no—
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C23×C4×C7C2311yescharacteristic
67C23×Dic7—21no—
68C7×C4⋊Q8—24no—
69C7×C4⋊Q8C2×C418yescomplemented
70C7×#124—24no—
71C7×#124C2×C418yescomplemented
72C7×#123—216no—
73C7×C22⋊Q8—216no—
74C7×#117C2×C414yes—
75C7×#117D414yes—
76C2×C7×C4⋊C4—24no—
77C2×C7×C4⋊C4D412yes—
78C2×C7×C22⋊C4—24no—
79C2×C7×C22⋊C4D412yes—
80C2×C42×C7C2312yes—
81C2×C22⋊Dic7—21no—
82#6489—22no—
83#6489—22no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—24no—
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2312yes—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4×Dic7—21no—
92C2×#1136—22no—
93C2×#1136—22no—
94C2×#1136D414yes—
95C22×C7⋊C8—22no—
96C22×C7⋊C8—22no—
97C22×C7⋊C8D414yes—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2312yes—
100C22×C7⋊C8C2311yescharacteristic
101#6409—24no—
102#6409—24no—
103#6409—24no—
104#6409—24no—
105#6409—24no—
106#6409—22no—
107#6409—22no—
108#6406—24no—
109#6406—22no—
110#6406—22no—
111C7⋊#95—24no—
112#6377—24no—
113C4×C7⋊C8—24no—
114C2×#536D711yescharacteristic
115C2×#417—71no—
116C2×#419—71no—
117C2×#416—71no—
118C22×#103—71no—
119C2×#405—71no—
120#2317—71no—
121#1960—72no—
122#1959—72no—
123#1958—72no—
124#1956—72no—
125C24×C7—21no—
126C24×C7C2×D411yescharacteristic
127C7×#41—28no—
128C7×#41—28no—
129C7×#41—44no—
130C7×#41—44no—
131C2×C7×Q8—24no—
132C2×C7×Q8—22no—
133C2×C7×Q8—21no—
134C2×C7×Q8—28no—
135C2×C7×Q8—28no—
136C2×C7×Q8—22no—
137C2×C7×Q8—22no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—22no—
141C2×C7×Q8—22no—
142C2×C7×Q8—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—44no—
145C2×C7×Q8—44no—
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8C22×C414yes—
149C2×C7×Q8C22×C412yes—
150C2×C7×D4—24no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4—21no—
156C2×C7×D4—42no—
157C2×C7×D4—42no—
158C2×C7×D4C22×C412yes—
159C22×C4×C7—24no—
160C22×C4×C7—22no—
161C22×C4×C7—21no—
162C22×C4×C7—24no—
163C22×C4×C7—42no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7—42no—
167C22×C4×C7C22×C412yes—
168C22×C4×C7C2×D414yes—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7#4112yes—
174C22×C4×C7#4112yes—
175C22×C4×C7C2411yescharacteristic
176C22×Dic7—21no—
177C22×Dic7—22no—
178C22×Dic7—21no—
179C22×Dic7—42no—
180C22×Dic7—82no—
181C7×C4⋊C4—216no—
182C7×C4⋊C4C22⋊C418yes—
183C7×C22⋊C4—216no—
184C7×C22⋊C4C22⋊C418yes—
185C42×C7C22×C414yes—
186C42×C7C2×D414yes—
187C22⋊Dic7—42no—
188C4⋊Dic7—22no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—22no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—42no—
195C4⋊Dic7—84no—
196C4×Dic7—42no—
197#1136—28no—
198C2×C7⋊C8—28no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8C2×D412yes—
206C2×C7⋊C8C2×D412yes—
207C2×C7⋊C8C2×D414yes—
208C22×#41C2×D711yescharacteristic
209C23×Q8C2×D711yescharacteristic
210C22×M16—71no—
211C23×C8—71no—
212#536—24no—
213#536Dic718yescomplemented
214C2×C4⋊Q8C2×D712yes—
215C2×#124C2×D712yes—
216C2×#123—22no—
217C2×C22⋊Q8—22no—
218C2×#117—71no—
219C2×#117C2×D711yescharacteristic
220C22×C4⋊C4—71no—
221#419—144no—
222#417—144no—
223#416—144no—
224C2×#103—72no—
225C2×#103—72no—
226C2×#103—71no—
227C2×#103—71no—
228C2×#103—72no—
229C2×#103—72no—
230C2×#103—72no—
231C2×#103—144no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—72no—
235#405—144no—
236C2×#97—72no—
237C2×C4×C8—72no—
238#389—142no—
239#385—142no—
240#384—72no—
241#382—72no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7C2×C22⋊C412yes—
250C23×C7#12412yes—
251C23×C7#12712yes—
252C23×C7C22×D411yescharacteristic
253C7×Q8—24no—
254C7×Q8—24no—
255C7×Q8—22no—
256C7×Q8—22no—
257C7×Q8—42no—
258C7×Q8—42no—
259C7×Q8—41no—
260C7×Q8—41no—
261C7×Q8—48no—
262C7×D4—22no—
263C7×D4—22no—
264C7×D4—48no—
265C7×D4—41no—
266C7×D4—41no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—48no—
276C2×C4×C7—48no—
277C2×C4×C7C2×C22⋊C418yes—
278C2×C4×C7C2×C22⋊C412yes—
279C2×C4×C7C4×D414yes—
280C2×C4×C7C4×D414yes—
281C2×C4×C7#12114yes—
282C2×C4×C7#12114yes—
283C2×C4×C7#12412yes—
284C2×C4×C7#12712yes—
285C2×C4×C7C23×C411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—42no—
294C2×Dic7—42no—
295C2×Dic7—44no—
296C2×Dic7—42no—
297C2×Dic7—84no—
298C7⋊C8—22no—
299C7⋊C8—22no—
300C7⋊C8—24no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41C2×Dic714yes—
307C22×Q8—21no—
308C22×Q8—22no—
309C22×Q8—22no—
310C22×Q8—22no—
311C22×Q8—22no—
312C22×Q8—22no—
313C22×Q8C2×Dic714yes—
314C22×Q8C22×D712yes—
315C22×Q8C22×D711yescharacteristic
316C22×D4—21no—
317C22×D4C22×D711yescharacteristic
318C23×C4—21no—
319C23×C4—21no—
320C23×C4—141no—
321C23×C4C22×D711yescharacteristic
322C2×M16—74no—
323C2×M16—142no—
324C2×M16—142no—
325C22×C8—74no—
326C22×C8—71no—
327C22×C8—71no—
328C22×C8—72no—
329C22×C8—142no—
330C22×C8—142no—
331C4⋊Q8—24no—
332C4⋊Q8C2×Dic718yescomplemented
333#124—24no—
334#124C2×Dic718yescomplemented
335#123—216no—
336C22⋊Q8—216no—
337#117—142no—
338#117—142no—
339#117C2×Dic714yes—
340#117C22⋊D714yes—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—72no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4C22⋊D712yes—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4C22⋊D712yes—
353C2×C42—141no—
354C2×C42C22×D712yes—
355#103—144no—
356#103—144no—
357#103—142no—
358#103—142no—
359#103—144no—
360#103—144no—
361#103—144no—
362#102—142no—
363#102—142no—
364#102—144no—
365#97—144no—
366C4×C8—144no—
367#95—144no—
368C22×C7—24no—
369C22×C7—24no—
370C22×C7—22no—
371C22×C7—22no—
372C22×C7—44no—
373C22×C7—42no—
374C22×C7—82no—
375C22×C7—81no—
376C22×C7—81no—
377C22×C7#39014yes—
378C22×C7C22×C22⋊C411yescharacteristic
379C22×C7C2×#12411yescharacteristic
380C22×C7C2×#12711yescharacteristic
381C22×C7#57712yes—
382C22×C7#57812yes—
383C4×C7—28no—
384C4×C7—42no—
385C4×C7—44no—
386C4×C7#39014yes—
387C4×C7C2×C4×D411yescharacteristic
388C4×C7C2×C4×D411yescharacteristic
389C4×C7C2×#12111yescharacteristic
390C4×C7C2×#12111yescharacteristic
391Dic7—81no—
392Dic7—82no—
393Dic7—81no—
394C24—21no—
395C24C2×C22⋊D711yescharacteristic
396#41—28no—
397#41—28no—
398#41—44no—
399#41—44no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—42no—
417C2×Q8C22×Dic714yes—
418C2×Q8C22×Dic712yes—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C22×Dic712yes—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—141no—
437C22×C4—142no—
438C22×C4—141no—
439C22×C4—282no—
440C22×C4—562no—
441C22×C4D4×D712yes—
442C22×C4D4×D712yes—
443C22×C4#115912yes—
444C22×C4#115912yes—
445C22×C4C22×Dic712yes—
446C22×C4C2×C22⋊D714yes—
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D712yes—
449C22×C4C23×D711yescharacteristic
450M16—148no—
451C2×C8—72no—
452C2×C8—72no—
453C2×C8—74no—
454C2×C8—148no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—142no—
458C2×C8—141no—
459C2×C8—142no—
460C2×C8—141no—
461C4⋊C4—216no—
462C4⋊C4—142no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—142no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—282no—
469C4⋊C4—564no—
470C4⋊C4C22⋊Dic718yes—
471C22⋊C4—216no—
472C22⋊C4—282no—
473C22⋊C4C22⋊Dic718yes—
474C42—282no—
475C42C22×Dic714yes—
476C42C2×C22⋊D714yes—
477C2×C7—24no—
478C2×C7—82no—
479C2×C7C2×#39011yescharacteristic
480C2×C7#289812yes—
481C2×C7#313712yes—
482C2×C7#315411yescharacteristic
483C2×C7#315611yescharacteristic
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23#649912yes—
492C23#650312yes—
493C23C2×C22⋊Dic712yes—
494C23C22×C22⋊D711yescharacteristic
495Q8—24no—
496Q8—24no—
497Q8—22no—
498Q8—22no—
499Q8—42no—
500Q8—42no—
501Q8—41no—
502Q8—41no—
503Q8—48no—
504D4—22no—
505D4—22no—
506D4—48no—
507D4—41no—
508D4—41no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4—284no—
524C2×C4—282no—
525C2×C4—564no—
526C2×C4D4×Dic714yes—
527C2×C4D4×Dic714yes—
528C2×C4#649912yes—
529C2×C4#650114yes—
530C2×C4#650114yes—
531C2×C4#650312yes—
532C2×C4C2×C22⋊Dic718yes—
533C2×C4C2×C22⋊Dic712yes—
534C2×C4C2×D4×D711yescharacteristic
535C2×C4C2×D4×D711yescharacteristic
536C2×C4C2×#115911yescharacteristic
537C2×C4C2×#115911yescharacteristic
538C2×C4C23×Dic711yescharacteristic
539C2×C4C22×C22⋊D711yescharacteristic
540C8—142no—
541C8—142no—
542C8—144no—
543C7#2120511yescharacteristic, complemented, Sylow
544C22—24no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—42no—
549C22—44no—
550C22—82no—
551C22—81no—
552C22—81no—
553C22#8999314yes—
554C22C2×#649911yescharacteristic
555C22C2×#650311yescharacteristic
556C22#9052812yes—
557C22#9052912yes—
558C22C22×C22⋊Dic711yescharacteristic
559C4—28no—
560C4—42no—
561C4—44no—
562C4—562no—
563C4—561no—
564C4—561no—
565C4#8999314yes—
566C4C2×D4×Dic711yescharacteristic
567C4C2×D4×Dic711yescharacteristic
568C4C2×#650111yescharacteristic
569C4C2×#650111yescharacteristic
570C2—24no—
571C2—82no—
572C2#1174084612yes—
573C2#1174084711yescharacteristic
574C2#1174084811yescharacteristic
575C2C2×#8999311yescharacteristic
576C2#1174155612yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#536C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#536D7
hypocenterC7#21205
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
4. C2×#89997C2semidirect
5. C22×#6409C2semidirect
6. #11739678C2semidirect
7. C2×#89769C2semidirect
8. #11729540C2semidirect
9. #11729539C2semidirect
10. #11729480C2semidirect
11. #11729479C2semidirect
15. C7×#536C4semidirect
69. C7×C4⋊Q8C2×C4semidirect
71. C7×#124C2×C4semidirect
213. #536Dic7semidirect
332. C4⋊Q8C2×Dic7semidirect
334. #124C2×Dic7semidirect
543. C7#21205semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C2×#900042
C2×#900052
C2×#899972
C22×#64092
#117396782
C2×#897692
#117295402
#117295392
#117294802
#117294792
#212057