Finite Group Databasebeta

← #13357497

Subgroups — #13357497

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×#90516C211yesmaximal, characteristic
2C2×C7×#417C211yesmaximal, characteristic
3C2×#90005C211yesmaximal, characteristic
4C2×#89892C211yesmaximal, characteristic, complemented
5C2×#89894C211yesmaximal, characteristic, complemented
6#11739602C211yesmaximal, characteristic, complemented
7C22×#6384C211yesmaximal, characteristic, complemented
8#11727982C212yesmaximal, complemented
9#11727979C212yesmaximal, complemented
10#11727913C212yesmaximal, complemented
11#11727912C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×Dic14C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C22×C7×C4⋊C4C2211yescharacteristic
16#90516—24no—
17#90516C418yescomplemented
18C2×#6507C2212yes—
19C2×#6499C2212yes—
20C2×#6498—22no—
21C2×#6489C2211yescharacteristic
22C2×#6489C2211yescharacteristic
23C2×#6487—22no—
24C23×C7⋊C8C2211yescharacteristic
25C7×#417—24no—
26C2×C7×#103C2212yes—
27C2×C7×#102C2212yes—
28#90005—24no—
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33#89982—22no—
34#89894—24no—
35#89892—24no—
36C2×#6390C2212yes—
37C2×#6390C2212yes—
38C2×#6389C2212yes—
39C2×C8×Dic7C2212yes—
40#89749C2212yes—
41#89748C2212yes—
42#89745—22no—
43C2×#6384—24no—
44C2×#6384C2212yes—
45C2×#6384C2212yes—
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×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2311yescharacteristic
59C23×Dic7—21no—
60C22×Dic14—22no—
61C22×Dic14—22no—
62C22×Dic14—22no—
63C22×Dic14—22no—
64C22×Dic14—21no—
65C22×Dic14—22no—
66C22×Dic14C2×C414yes—
67C22×Dic14C2312yes—
68C2×C7×M16—22no—
69C2×C7×M16—22no—
70C2×C7×M16D414yes—
71C22×C7×C8C2312yes—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2312yes—
76C2×C22⋊Dic7—24no—
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7D412yes—
79#6507—24no—
80#6507C2×C418yescomplemented
81#6499—24no—
82#6499C2×C418yescomplemented
83#6498—216no—
84#6489—22no—
85#6489—22no—
86#6489C2×C414yes—
87#6489D414yes—
88C2×C4⋊Dic7D412yes—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91#6487—216no—
92C2×Dic7⋊C4—24no—
93C2×Dic7⋊C4—24no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7C2312yes—
96C22×C7⋊C8—22no—
97C22×C7⋊C8—22no—
98C22×C7⋊C8D414yes—
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C7×#103—24no—
102C7×#102—24no—
103#6409—22no—
104#6409—22no—
105#6406—22no—
106#6406—22no—
107C7⋊#95—24no—
108#6390—24no—
109#6390—24no—
110#6389—24no—
111C8×Dic7—24no—
112#6384—24no—
113#6384—24no—
114C2×#536—71no—
115C2×#417D711yescharacteristic
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—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—21no—
135C2×C7×Q8—21no—
136C2×C7×Q8C22×C412yes—
137C2×C7×D4—24no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—42no—
144C2×C7×D4—42no—
145C2×C7×D4C22×C412yes—
146C22×C4×C7—21no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7—82no—
153C22×C4×C7C22×C412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D412yes—
158C22×C4×C7#4112yes—
159C22×C4×C7#4112yes—
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—24no—
162C22×Dic7—22no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—42no—
166C22×Dic7—42no—
167C22×Dic7C2×D414yes—
168C2×Dic14—28no—
169C2×Dic14—28no—
170C2×Dic14—22no—
171C2×Dic14—22no—
172C2×Dic14—22no—
173C2×Dic14—22no—
174C2×Dic14—24no—
175C2×Dic14—22no—
176C2×Dic14—44no—
177C2×Dic14—44no—
178C2×Dic14—42no—
179C2×Dic14—42no—
180C2×Dic14C22×C414yes—
181C7×M16—28no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8C2×D414yes—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—42no—
188C22⋊Dic7—216no—
189C22⋊Dic7—42no—
190C22⋊Dic7C22⋊C418yes—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7C22⋊C418yes—
196Dic7⋊C4—216no—
197Dic7⋊C4—84no—
198C4×Dic7—42no—
199C4×Dic7C22×C414yes—
200C4×Dic7C2×D414yes—
201C2×C7⋊C8—28no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8C2×D412yes—
207C2×C7⋊C8C2×D412yes—
208C22×#41C2×D711yescharacteristic
209C23×Q8—71no—
210C22×M16C2×D711yescharacteristic
211C23×C8—71no—
212#536—78no—
213#536—144no—
214C2×C4⋊Q8—72no—
215C2×#124—72no—
216C2×#123—142no—
217C2×C22⋊Q8—142no—
218C2×#117—71no—
219C2×#117—71no—
220C22×C4⋊C4C2×D711yescharacteristic
221#419—144no—
222#417—24no—
223#416—144no—
224C2×#103—71no—
225C2×#103—71no—
226C2×#103—72no—
227C2×#103—72no—
228C2×#103—72no—
229C2×#103—72no—
230C2×#103—144no—
231C2×#103C2×D712yes—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102C2×D712yes—
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—22no—
254C7×Q8—22no—
255C7×Q8—41no—
256C7×Q8—41no—
257C7×D4—22no—
258C7×D4—22no—
259C7×D4—48no—
260C7×D4—41no—
261C7×D4—41no—
262C2×C4×C7—22no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—48no—
266C2×C4×C7—42no—
267C2×C4×C7—44no—
268C2×C4×C7—84no—
269C2×C4×C7C2×C22⋊C412yes—
270C2×C4×C7C4×D414yes—
271C2×C4×C7C4×D414yes—
272C2×C4×C7#12114yes—
273C2×C4×C7#12114yes—
274C2×C4×C7#12412yes—
275C2×C4×C7#12712yes—
276C2×C4×C7C23×C411yescharacteristic
277C2×C4×C7C22×D411yescharacteristic
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—48no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—42no—
291C2×Dic7—42no—
292C2×Dic7C2×C22⋊C418yes—
293Dic14—24no—
294Dic14—24no—
295Dic14—42no—
296Dic14—42no—
297Dic14—48no—
298C7×C8—24no—
299C7⋊C8—22no—
300C7⋊C8—22no—
301C2×#41—22no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41C4×D714yes—
307C22×Q8—74no—
308C22×Q8—72no—
309C22×Q8—141no—
310C22×Q8—142no—
311C22×Q8—142no—
312C22×Q8—142no—
313C22×Q8—142no—
314C22×Q8—142no—
315C22×Q8C22×D711yescharacteristic
316C22×D4—21no—
317C22×D4C22×D711yescharacteristic
318C23×C4—21no—
319C23×C4—21no—
320C23×C4—141no—
321C23×C4C22×D711yescharacteristic
322C2×M16—22no—
323C2×M16—22no—
324C2×M16C22⋊D714yes—
325C22×C8—74no—
326C22×C8—71no—
327C22×C8—71no—
328C22×C8—142no—
329C22×C8—142no—
330C22×C8C22×D712yes—
331C4⋊Q8—78no—
332C4⋊Q8—144no—
333#124—78no—
334#124—144no—
335#123—1416no—
336C22⋊Q8—1416no—
337#117—74no—
338#117—74no—
339#117—142no—
340#117—142no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—72no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—144no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4C22×D712yes—
350C2×C22⋊C4—72no—
351C2×C22⋊C4—144no—
352C2×C22⋊C4—141no—
353C2×C42—72no—
354C2×C42—141no—
355#103—24no—
356#103—144no—
357#103—144no—
358#103—142no—
359#103—142no—
360#103—144no—
361#103—144no—
362#102—24no—
363#102—142no—
364#102—142no—
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—42no—
384C4×C7—82no—
385C4×C7#39014yes—
386C4×C7C2×C4×D411yescharacteristic
387C4×C7C2×C4×D411yescharacteristic
388C4×C7C2×#12111yescharacteristic
389C4×C7C2×#12111yescharacteristic
390Dic7—28no—
391Dic7—44no—
392Dic7—81no—
393Dic7—81no—
394C24—21no—
395C24D4×D711yescharacteristic
396#41—28no—
397#41—28no—
398#41—44no—
399#41—44no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—74no—
406C2×Q8—144no—
407C2×Q8—142no—
408C2×Q8—148no—
409C2×Q8—148no—
410C2×Q8—142no—
411C2×Q8—142no—
412C2×Q8—142no—
413C2×Q8—142no—
414C2×Q8—284no—
415C2×Q8—282no—
416C2×Q8—284no—
417C2×Q8—282no—
418C2×Q8C2×C4×D712yes—
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×D4C2×C4×D712yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—74no—
435C22×C4—82no—
436C22×C4—144no—
437C22×C4—142no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—282no—
441C22×C4—282no—
442C22×C4C2×C4×D712yes—
443C22×C4#115712yes—
444C22×C4D4×D711yescharacteristic
445C22×C4D4×D712yes—
446C22×C4D4×D712yes—
447C22×C4#115912yes—
448C22×C4C2×C22⋊D712yes—
449C22×C4C23×D711yescharacteristic
450M16—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—72no—
454C2×C8—72no—
455C2×C8—148no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8C2×C22⋊D714yes—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—78no—
465C4⋊C4—1416no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—564no—
471C22⋊C4—78no—
472C22⋊C4—1416no—
473C22⋊C4—282no—
474C42—74no—
475C42—74no—
476C42—282no—
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—42no—
489C23—42no—
490C23—41no—
491C23D7×C22⋊C412yes—
492C23#644812yes—
493C23#650312yes—
494C23C2×D4×D711yescharacteristic
495Q8—22no—
496Q8—22no—
497Q8—41no—
498Q8—41no—
499Q8—144no—
500Q8—144no—
501Q8—282no—
502Q8—282no—
503Q8—288no—
504D4—22no—
505D4—22no—
506D4—48no—
507D4—41no—
508D4—41no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—78no—
516C2×C4—84no—
517C2×C4—144no—
518C2×C4—144no—
519C2×C4—144no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—288no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4D7×C22⋊C412yes—
528C2×C4(C22⋊D7)⋊C414yes—
529C2×C4#644714yes—
530C2×C4#644812yes—
531C2×C4C4×C22⋊D714yes—
532C2×C4#650214yes—
533C2×C4#650312yes—
534C2×C4C22×C4×D711yescharacteristic
535C2×C4C2×#115711yescharacteristic
536C2×C4C2×D4×D711yescharacteristic
537C2×C4C2×D4×D711yescharacteristic
538C2×C4C2×#115911yescharacteristic
539C2×C4C22×C22⋊D711yescharacteristic
540C8—24no—
541C8—142no—
542C8—142no—
543C7#2120511yescharacteristic, complemented, Sylow
544C22—24no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—44no—
549C22—42no—
550C22—82no—
551C22—81no—
552C22—81no—
553C22#8972914yes—
554C22C2×D7×C22⋊C411yescharacteristic
555C22C2×#644811yescharacteristic
556C22#9046612yes—
557C22#9047012yes—
558C22C2×#650311yescharacteristic
559C4—42no—
560C4—82no—
561C4—148no—
562C4—284no—
563C4—561no—
564C4—561no—
565C4#8972914yes—
566C4C2×(C22⋊D7)⋊C411yescharacteristic
567C4C2×#644711yescharacteristic
568C4C2×C4×C22⋊D711yescharacteristic
569C4C2×#650211yescharacteristic
570C2—24no—
571C2—82no—
572C2#1173632212yes—
573C2#1173632812yes—
574C2C2×#8972911yescharacteristic
575C2#1174138411yescharacteristic
576C2#1174142011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#21205
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
4. C2×#89892C2semidirect
5. C2×#89894C2semidirect
6. #11739602C2semidirect
7. C22×#6384C2semidirect
8. #11727982C2semidirect
9. #11727979C2semidirect
10. #11727913C2semidirect
11. #11727912C2semidirect
17. #90516C4semidirect
80. #6507C2×C4semidirect
82. #6499C2×C4semidirect
543. C7#21205semidirect

Maximal subgroups

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

subgroupindex
C2×#905162
C2×C7×#4172
C2×#900052
C2×#898922
C2×#898942
#117396022
C22×#63842
#117279822
#117279792
#117279132
#117279122
#212057