Finite Group Databasebeta

← #13370010

Subgroups — #13370010

Order 1792.

1474 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2D4×C4⋊Dic7C211yesmaximal, characteristic, complemented
3C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
4#11729541C211yesmaximal, characteristic, complemented
5#11729534C211yesmaximal, characteristic, complemented
6#11729435C212yesmaximal, complemented
7#11729431C212yesmaximal, complemented
8C4×#6406C211yesmaximal, characteristic, complemented
9#11729358C211yesmaximal, characteristic, complemented
10C2×#89324C212yesmaximal, complemented
11#11729307C212yesmaximal, complemented
12C22×C7×#41—21no—
13C7×#547—24no—
14C7×#547C418yescomplemented
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#123—22no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×#121C2212yes—
19C2×C4×C7×D4C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C22×C7×C4⋊C4C2212yes—
22C2×D4×Dic7—21no—
23C22×C4⋊Dic7C2212yes—
24C23×C7⋊C8C2212yes—
25C2×#6406C2211yescharacteristic
26C2×#6406C2211yescharacteristic
27C2×#6406C2212yes—
28#89984C2212yes—
29C2×C22⋊(C7⋊C8)C2212yes—
30D4×C7⋊C8—24no—
31#89757—21no—
32#89722—22no—
33#89707C2211yescharacteristic
34C4×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊(C7⋊C8)C2211yescharacteristic
36C2×C4×C7⋊C8C2211yescharacteristic
37#89324C2212yes—
38#89324C2212yes—
39#89324C2212yes—
40#89324C2212yes—
41#89324C2211yescharacteristic
42#89324C2212yes—
43#89324C2211yescharacteristic
44#22485—71nomaximal, Sylow
45C2×C7×#41—28no—
46C2×C7×#41—44no—
47C2×C7×#41—44no—
48C22×C7×Q8—21no—
49C22×C7×D4—22no—
50C22×C7×D4D411yescharacteristic
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7—22no—
53C23×C4×C7—21no—
54C23×C4×C7C2312yes—
55C23×Dic7—22no—
56C7×C4⋊Q8—22no—
57C7×C4⋊Q8C2×C414yescomplemented
58C7×#123—216no—
59C7×C22⋊Q8—28no—
60C7×#121—24no—
61C7×#121C2×C418yescomplemented
62C4×C7×D4—22no—
63C4×C7×D4C2×C414yes—
64C4×C7×D4C2×C414yescomplemented
65C4×C7×D4D414yes—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2×C418yescomplemented
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—22no—
75C2×C7×C22⋊C4D412yes—
76C2×C42×C7C2311yescharacteristic
77C2×C22⋊Dic7—22no—
78D4×Dic7—44no—
79C2×C4⋊Dic7—24no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—42no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2312yes—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4×Dic7—21no—
88C22×C7⋊C8—24no—
89C22×C7⋊C8—24no—
90C22×C7⋊C8—42no—
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2311yescharacteristic
94#6406—22no—
95#6406—22no—
96#6406—42no—
97C7⋊#95—21no—
98C7⋊#95—22no—
99C22⋊(C7⋊C8)—24no—
100C4⋊(C7⋊C8)—22no—
101C4×C7⋊C8—22no—
102C2×#547D711yescharacteristic
103C2×C8×D4—71no—
104D4×C4⋊C4—71no—
105#1990—71no—
106#1988—71no—
107#1893—72no—
108#1874—72no—
109#1823—72no—
110#1802—71no—
111C4×#102—71no—
112C2×#340—72no—
113C24×C7—21no—
114C24×C7C2×D412yes—
115C7×#41—432no—
116C2×C7×Q8—24no—
117C2×C7×Q8—42no—
118C2×C7×Q8—42no—
119C2×C7×D4—28no—
120C2×C7×D4—21no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—44no—
126C2×C7×D4—44no—
127C2×C7×D4C22⋊C414yes—
128C2×C7×D4D814yes—
129C2×C7×D4SD1614yes—
130C2×C7×D4C22×C412yes—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—28no—
136C22×C4×C7—21no—
137C22×C4×C7—22no—
138C22×C4×C7—24no—
139C22×C4×C7—44no—
140C22×C4×C7—44no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7C22×C414yes—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D412yes—
146C22×C4×C7C2×D414yes—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—22no—
152C22×Dic7—44no—
153C22×Dic7—44no—
154C22×Dic7—82no—
155C7×C4⋊C4—28no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4C22⋊C414yes—
160C7×C4⋊C4C22×C412yes—
161C7×C4⋊C4C22×C414yes—
162C7×C22⋊C4—24no—
163C7×C22⋊C4—28no—
164C7×C22⋊C4C22⋊C418yes—
165C42×C7C22×C412yes—
166C42×C7C2×D412yes—
167C22⋊Dic7—44no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170C4⋊Dic7—22no—
171C4⋊Dic7—22no—
172C4⋊Dic7—21no—
173C4⋊Dic7—44no—
174C4⋊Dic7—42no—
175C4⋊Dic7—81no—
176C4⋊Dic7#4112yescomplemented
177C4×Dic7—42no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8—22no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—22no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—21no—
186C2×C7⋊C8—48no—
187C2×C7⋊C8#4112yescomplemented
188C22×#41—21no—
189C23×C8—72no—
190#547—24no—
191#547Dic718yescomplemented
192C2×C4⋊Q8C2×D711yescharacteristic
193C2×#123—22no—
194C2×C22⋊Q8—21no—
195C2×#121C2×D712yes—
196C2×C4×D4—141no—
197C2×C4×D4C2×D711yescharacteristic
198C2×C4×D4C2×D711yescharacteristic
199C22×C4⋊C4—72no—
200C22×C4⋊C4C2×D712yes—
201C8×D4—144no—
202C2×C4⋊C8—71no—
203C2×#102—71no—
204C2×#102—71no—
205C2×#102—72no—
206C2×C22⋊C8—72no—
207C2×C4×C8—71no—
208#389—141no—
209#385—142no—
210#384—71no—
211#380—72no—
212C4×C4⋊C4—71no—
213#340—72no—
214#340—71no—
215#340—72no—
216#340—72no—
217#340—72no—
218#340—72no—
219#340—71no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7C2×C22⋊C414yes—
232C23×C7C2×D814yes—
233C23×C7C2×SD1614yes—
234C23×C7C22×D411yescharacteristic
235C7×Q8—48no—
236C7×D4—22no—
237C7×D4—22no—
238C7×D4—416no—
239C7×D4—41no—
240C7×D4—41no—
241C7×D4#102116yescomplemented
242C2×C4×C7—22no—
243C2×C4×C7—22no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—24no—
252C2×C4×C7—41no—
253C2×C4×C7—42no—
254C2×C4×C7—41no—
255C2×C4×C7—42no—
256C2×C4×C7—416no—
257C2×C4×C7—48no—
258C2×C4×C7C2×C22⋊C412yes—
259C2×C4×C7C2×C22⋊C414yes—
260C2×C4×C7C2×C22⋊C412yes—
261C2×C4×C7C2×D812yes—
262C2×C4×C7C2×SD1612yes—
263C2×C4×C7C23×C411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C2×#4111yescharacteristic
266C2×C4×C7C2×#4111yescharacteristic
267C2×C4×C7H(5,𝔽2)11yescharacteristic
268C2×C4×C72−1+411yescharacteristic
269C2×Dic7—22no—
270C2×Dic7—41no—
271C2×Dic7—41no—
272C2×Dic7—42no—
273C2×Dic7—41no—
274C2×Dic7—41no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—41no—
278C2×Dic7—88no—
279C7⋊C8—22no—
280C7⋊C8—42no—
281C7⋊C8—41no—
282C2×#41—28no—
283C2×#41—44no—
284C2×#41—44no—
285C22×Q8—21no—
286C22×D4—22no—
287C22×D4C22⋊D711yescharacteristic
288C22×D4C22×D711yescharacteristic
289C23×C4—22no—
290C23×C4—21no—
291C23×C4—142no—
292C23×C4C22×D712yes—
293C22×C8—71no—
294C22×C8—72no—
295C22×C8—71no—
296C22×C8—144no—
297C22×C8—144no—
298C22×C8—282no—
299C4⋊Q8—22no—
300C4⋊Q8C2×Dic714yescomplemented
301#123—216no—
302C22⋊Q8—28no—
303#121—24no—
304#121C2×Dic718yescomplemented
305C4×D4—22no—
306C4×D4—284no—
307C4×D4C2×Dic714yes—
308C4×D4C2×Dic714yescomplemented
309C4×D4C22⋊D714yes—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—71no—
314C2×C4⋊C4—72no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—144no—
317C2×C4⋊C4—142no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4—141no—
320C2×C4⋊C4—282no—
321C2×C4⋊C4C2×Dic718yescomplemented
322C2×C4⋊C4C22⋊D711yescharacteristic
323C2×C4⋊C4C22×D711yescharacteristic
324C2×C4⋊C4C22×D712yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—142no—
328C2×C22⋊C4C22⋊D712yes—
329C2×C42—141no—
330C2×C42C22×D711yescharacteristic
331C4⋊C8—142no—
332#102—142no—
333#102—142no—
334#102—282no—
335C22⋊C8—144no—
336C4×C8—142no—
337#95—141no—
338#95—142no—
339C22×C7—24no—
340C22×C7—24no—
341C22×C7—24no—
342C22×C7—24no—
343C22×C7—24no—
344C22×C7—24no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7C2×#10218yes—
353C22×C7C22×C22⋊C411yescharacteristic
354C22×C7#54611yescharacteristic
355C22×C7#54711yescharacteristic
356C22×C7C22×D811yescharacteristic
357C22×C7C22×SD1611yescharacteristic
358C22×C7#57711yescharacteristic
359C22×C7#57811yescharacteristic
360C4×C7—24no—
361C4×C7—22no—
362C4×C7—42no—
363C4×C7—44no—
364C4×C7C2×#10214yes—
365C4×C7C4×#4111yescharacteristic
366C4×C7#52011yescharacteristic
367C4×C7#54611yescharacteristic
368C4×C7#54711yescharacteristic
369Dic7—42no—
370Dic7—81no—
371Dic7—82no—
372C24—21no—
373C24C2×C22⋊D712yes—
374#41—432no—
375C2×Q8—24no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×D4—28no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4#114114yes—
387C2×D4#114214yes—
388C2×D4C22⋊Dic714yes—
389C2×D4C22×Dic712yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—141no—
403C22×C4—141no—
404C22×C4—142no—
405C22×C4—284no—
406C22×C4—284no—
407C22×C4—562no—
408C22×C4C22×Dic714yes—
409C22×C4C2×C22⋊D711yescharacteristic
410C22×C4C2×C22⋊D712yes—
411C22×C4C2×C22⋊D714yes—
412C22×C4C2×C22⋊D711yescharacteristic
413C22×C4C23×D711yescharacteristic
414C2×C8—72no—
415C2×C8—141no—
416C2×C8—141no—
417C2×C8—141no—
418C2×C8—142no—
419C2×C8—142no—
420C2×C8—141no—
421C2×C8—141no—
422C2×C8—142no—
423C2×C8—288no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—72no—
429C4⋊C4—141no—
430C4⋊C4—141no—
431C4⋊C4—142no—
432C4⋊C4—142no—
433C4⋊C4—141no—
434C4⋊C4—284no—
435C4⋊C4—282no—
436C4⋊C4—561no—
437C4⋊C4C22⋊Dic714yes—
438C4⋊C4C22×Dic712yes—
439C4⋊C4C22×Dic714yes—
440C22⋊C4—24no—
441C22⋊C4—28no—
442C22⋊C4—284no—
443C22⋊C4C22⋊Dic718yes—
444C42—282no—
445C42C22×Dic712yes—
446C42C2×C22⋊D712yes—
447C2×C7—28no—
448C2×C7—82no—
449C2×C7#233911yescharacteristic
450C2×C7C22×#10211yescharacteristic
451C2×C7#289811yescharacteristic
452C2×C7#329411yescharacteristic
453C2×C7#329811yescharacteristic
454C2×C7#333411yescharacteristic
455C2×C7#333511yescharacteristic
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23C2×#114114yes—
468C23C2×#114214yes—
469C23C2×C22⋊Dic714yes—
470C23C22×C22⋊D711yescharacteristic
471Q8—48no—
472D4—22no—
473D4—22no—
474D4—416no—
475D4—41no—
476D4—41no—
477D4#6406116yescomplemented
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—416no—
493C2×C4—48no—
494C2×C4—142no—
495C2×C4—281no—
496C2×C4—281no—
497C2×C4—281no—
498C2×C4—282no—
499C2×C4—281no—
500C2×C4—282no—
501C2×C4—282no—
502C2×C4—281no—
503C2×C4—568no—
504C2×C4C2×#114112yes—
505C2×C4C2×#114212yes—
506C2×C4C2×C22⋊Dic712yes—
507C2×C4C2×C22⋊Dic714yes—
508C2×C4C2×C22⋊Dic712yes—
509C2×C4D7×#4111yescharacteristic
510C2×C4D7×#4111yescharacteristic
511C2×C4#655311yescharacteristic
512C2×C4#655411yescharacteristic
513C2×C4C23×Dic711yescharacteristic
514C2×C4C22×C22⋊D711yescharacteristic
515C8—142no—
516C8—281no—
517C8—282no—
518C7#2248511yescharacteristic, complemented, Sylow
519C22—24no—
520C22—24no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22C2×#640618yes—
533C22C22×#114111yescharacteristic
534C22C22×#114211yescharacteristic
535C22#9051511yescharacteristic
536C22#9052011yescharacteristic
537C22#9052811yescharacteristic
538C22#9052911yescharacteristic
539C22C22×C22⋊Dic711yescharacteristic
540C4—24no—
541C4—22no—
542C4—42no—
543C4—44no—
544C4—282no—
545C4—561no—
546C4—562no—
547C4C2×#640614yes—
548C4#9051511yescharacteristic
549C4#41×Dic711yescharacteristic
550C4#9051811yescharacteristic
551C4#9052011yescharacteristic
552C2—28no—
553C2—82no—
554C2#1174096111yescharacteristic
555C2#1174096211yescharacteristic
556C2#1174097011yescharacteristic
557C2#1174097611yescharacteristic
558C2C22×#640611yescharacteristic
559C2#1174153311yescharacteristic
560C2#1174155611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#547C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#547D7
hypocenterC7#22485
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊Dic7C2semidirect
3. C2×D4×C7⋊C8C2semidirect
4. #11729541C2semidirect
5. #11729534C2semidirect
6. #11729435C2semidirect
7. #11729431C2semidirect
8. C4×#6406C2semidirect
9. #11729358C2semidirect
10. C2×#89324C2semidirect
11. #11729307C2semidirect
14. C7×#547C4semidirect
57. C7×C4⋊Q8C2×C4semidirect
61. C7×#121C2×C4semidirect
64. C4×C7×D4C2×C4semidirect
69. C2×C7×C4⋊C4C2×C4semidirect
176. C4⋊Dic7#41semidirect
187. C2×C7⋊C8#41semidirect
191. #547Dic7semidirect
241. C7×D4#102semidirect
300. C4⋊Q8C2×Dic7semidirect
304. #121C2×Dic7semidirect
308. C4×D4C2×Dic7semidirect
321. C2×C4⋊C4C2×Dic7semidirect
477. D4#6406semidirect
518. C7#22485semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5472
D4×C4⋊Dic72
C2×D4×C7⋊C82
#117295412
#117295342
#117294352
#117294312
C4×#64062
#117293582
C2×#893242
#117293072
#224857