Finite Group Databasebeta

← #13355213

Subgroups — #13355213

Order 1792.

1766 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6505C211yesmaximal, characteristic
2C2×#90214C211yesmaximal, characteristic
3C7×#2310C211yesmaximal, characteristic, complemented
4C22×#6409C211yesmaximal, characteristic, complemented
5C2×#89774C211yesmaximal, characteristic
6#11729335C211yesmaximal, characteristic, complemented
7#11729333C211yesmaximal, characteristic, complemented
8#11727444C214yesmaximal, complemented
9#11727441C214yesmaximal, complemented
10C23×C7×Q8C2211yescharacteristic
11C22×#1157C2211yescharacteristic
12C2×C7×#117C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×#1144C2212yes—
15C2×#6505—24no—
16C2×#6505—24no—
17C2×#6505C418yescomplemented
18C22×#1143C2212yes—
19C22×C4⋊Dic7C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C23×C7⋊C8C2211yescharacteristic
22#90214—24no—
23#90214—24no—
24C2×#6455—21no—
25C2×D7×C4⋊C4—21no—
26C2×(C22⋊D7)⋊C4—22no—
27C2×C4×D28C2212yes—
28C2×C4×Dic14C2212yes—
29C7×#391—21no—
30C7×#391—21no—
31C7×#391C2214yes—
32C7×#381—22no—
33C2×#6409—24no—
34C2×#6409C2214yes—
35#89774—24no—
36C2×#6384C2212yes—
37C2×#6383C2212yes—
38#89324C2214yes—
39#89319C2214yes—
40#20931—71nomaximal, Sylow
41C22×C7×Q8—21no—
42C22×C7×Q8—22no—
43C22×C7×Q8—22no—
44C22×C7×Q8—22no—
45C22×C7×Q8—22no—
46C22×C7×Q8—22no—
47C22×C7×Q8C2×C414yescomplemented
48C22×C7×Q8C2312yes—
49C23×C4×C7—21no—
50C23×C4×C7C2311yescharacteristic
51C22×C22⋊D7—21no—
52C23×Dic7—21no—
53C2×#1157—22no—
54C2×#1157—22no—
55C2×#1157—22no—
56C2×#1157C2×C414yes—
57C2×#1157D414yes—
58C2×#1157D414yes—
59C22×D28C2311yescharacteristic
60C22×C4×D7—21no—
61C22×C4×D7—22no—
62C22×Dic14C2311yescharacteristic
63C7×#117—22no—
64C7×#117—22no—
65C7×#117—22no—
66C7×#117—22no—
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4D412yes—
72C2×C42×C7—21no—
73C2×C42×C7C2312yes—
74C2×#1144—24no—
75C2×#1144—24no—
76C2×#1144C2×C418yescomplemented
77#6505—232no—
78C2×#1143—24no—
79C2×#1143—24no—
80C2×#1143C2×C418yescomplemented
81C2×(C2×D7)⋊C4—22no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—22no—
84C2×C4⋊Dic7D414yes—
85C2×C4⋊Dic7C2312yes—
86C2×Dic7⋊C4—22no—
87C2×C4×Dic7—22no—
88C2×#1136—22no—
89C2×#1136—22no—
90C2×#1136C2×C414yescomplemented
91C22×C7⋊C8—22no—
92C22×C7⋊C8—22no—
93C22×C7⋊C8—22no—
94C22×C7⋊C8C2312yes—
95C22×C7⋊C8C2312yes—
96#6455—28no—
97D7×C4⋊C4—28no—
98(C22⋊D7)⋊C4—216no—
99C4×D28—24no—
100C4×D28—24no—
101C4×Dic14—24no—
102C4×Dic14—24no—
103C7×#95—24no—
104C7×#95—24no—
105#6409—28no—
106#6384—24no—
107#6383—24no—
108C22×#137—71no—
109C2×#520—71no—
110C2×#416—71no—
111C22×#103—71no—
112#2310D711yescharacteristic, complemented
113#1878—74no—
114#1877—74no—
115#1736—71no—
116#1729—71no—
117C24×C7C2×D411yescharacteristic
118C23×D7—21no—
119C2×C7×Q8—24no—
120C2×C7×Q8—22no—
121C2×C7×Q8—28no—
122C2×C7×Q8—28no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8—22no—
127C2×C7×Q8—42no—
128C2×C7×Q8—42no—
129C2×C7×Q8—44no—
130C2×C7×Q8—44no—
131C2×C7×Q8C22×C414yes—
132C22×C4×C7—24no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—42no—
136C22×C4×C7—42no—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2×D412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7C2×D412yes—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D412yes—
144C22×C4×C7#4112yes—
145C22×C4×C7#4112yes—
146C22×C4×C7C2411yescharacteristic
147C2×C22⋊D7—24no—
148C2×C22⋊D7—24no—
149C2×C22⋊D7—24no—
150C22×Dic7—21no—
151C22×Dic7—22no—
152C22×Dic7—24no—
153C22×Dic7—82no—
154#1157—28no—
155#1157—28no—
156#1157—28no—
157#1157C22⋊C4116yes—
158C2×D28—21no—
159C2×D28—21no—
160C2×D28—21no—
161C2×D28C22×C412yes—
162C2×D28C2×D412yes—
163C2×D28C2×D412yes—
164C2×C4×D7—24no—
165C2×C4×D7—24no—
166C2×C4×D7—24no—
167C2×C4×D7—28no—
168C2×C4×D7—84no—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14C22×C412yes—
173C2×Dic14C2×D412yes—
174C2×Dic14C2×D412yes—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—42no—
181C42×C7—22no—
182C42×C7—22no—
183C42×C7—42no—
184#1144—216no—
185#1143—216no—
186(C2×D7)⋊C4—28no—
187C4⋊Dic7—24no—
188C4⋊Dic7—22no—
189C4⋊Dic7C2×D414yes—
190Dic7⋊C4—28no—
191C4×Dic7—28no—
192#1136—28no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—44no—
199C2×C7⋊C8C22×C414yes—
200C22×#41—71no—
201C23×Q8C2×D711yescharacteristic
202C2×#137—78no—
203C2×#137—144no—
204C2×#137—144no—
205C22×Q16—72no—
206C22×SD16—72no—
207C22×M16—71no—
208C23×C8—71no—
209#520—144no—
210#520—144no—
211C2×C4×Q8—72no—
212C2×C4×D4—72no—
213C2×C4×D4—142no—
214C2×#117—141no—
215C2×#117C2×D711yescharacteristic
216C2×#117C2×D711yescharacteristic
217C22×C4⋊C4—71no—
218C22×C4⋊C4—141no—
219#416—144no—
220C2×#103—74no—
221C2×#103—72no—
222C2×#103—144no—
223C2×#102—72no—
224#391—21no—
225#391—21no—
226#391C2×D714yes—
227#381—22no—
228#340—74no—
229#336—74no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7C4×D412yes—
233C23×C7#12012yes—
234C23×C7#12112yes—
235C23×C7C22×D411yescharacteristic
236C22×D7—22no—
237C22×D7—22no—
238C22×D7—22no—
239C22×D7—42no—
240C22×D7—42no—
241C7×Q8—24no—
242C7×Q8—24no—
243C7×Q8—42no—
244C7×Q8—42no—
245C7×Q8—48no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—48no—
254C2×C4×C7—42no—
255C2×C4×C7—42no—
256C2×C4×C7—42no—
257C2×C4×C7C2×C22⋊C414yes—
258C2×C4×C7C4×D414yes—
259C2×C4×C7C4×D412yes—
260C2×C4×C7#12012yes—
261C2×C4×C7#12114yes—
262C2×C4×C7#12112yes—
263C2×C4×C7#12314yes—
264C2×C4×C7C23×C411yescharacteristic
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×D411yescharacteristic
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C2×#4111yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C22⋊D7—216no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—24no—
275C2×Dic7—44no—
276C2×Dic7—84no—
277D28—22no—
278D28—22no—
279D28—22no—
280D28C2×C22⋊C414yes—
281C4×D7—216no—
282C4×D7—88no—
283Dic14—22no—
284Dic14—22no—
285Dic14—22no—
286Dic14C2×C22⋊C414yes—
287C7⋊C8—24no—
288C7⋊C8—42no—
289C2×#41—74no—
290C2×#41—74no—
291C2×#41—74no—
292C2×#41—142no—
293C2×#41—142no—
294C2×#41—142no—
295C22×Q8—22no—
296C22×Q8—21no—
297C22×Q8—22no—
298C22×Q8—22no—
299C22×Q8—22no—
300C22×Q8—22no—
301C22×Q8—71no—
302C22×Q8C4×D714yescomplemented
303C22×Q8C22×D712yes—
304C22×D4—71no—
305C22×D4—141no—
306C23×C4—21no—
307C23×C4—141no—
308C23×C4—141no—
309C23×C4—142no—
310C23×C4C22×D711yescharacteristic
311#137—1432no—
312C2×Q16—78no—
313C2×Q16—144no—
314C2×Q16—144no—
315C2×SD16—78no—
316C2×SD16—144no—
317C2×SD16—144no—
318C2×M16—74no—
319C2×M16—142no—
320C2×M16—142no—
321C22×C8—72no—
322C22×C8—72no—
323C22×C8—142no—
324C22×C8—142no—
325C22×C8—142no—
326C4×Q8—144no—
327C4×Q8—144no—
328C4×D4—144no—
329C4×D4—144no—
330C4×D4—1416no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—148no—
336C2×C4⋊C4—74no—
337C2×C4⋊C4—72no—
338C2×C4⋊C4—148no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—142no—
342C2×C4⋊C4C22⋊D711yescharacteristic
343C2×C4⋊C4C22⋊D711yescharacteristic
344C2×C4⋊C4C22×D712yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—142no—
347C2×C22⋊C4C22⋊D712yes—
348C2×C42—21no—
349C2×C42—142no—
350C2×C42C22×D712yes—
351#103—144no—
352#103—148no—
353#102—144no—
354#95—24no—
355#95—24no—
356C22×C7—24no—
357C22×C7—22no—
358C22×C7—22no—
359C22×C7—24no—
360C22×C7#38614yes—
361C22×C7C2×C4×D411yescharacteristic
362C22×C7C2×#12011yescharacteristic
363C22×C7C2×#12111yescharacteristic
364C22×C7#57712yes—
365C22×C7#57812yes—
366C2×D7—24no—
367C2×D7—44no—
368C2×D7—42no—
369C2×D7—42no—
370C2×D7—44no—
371C4×C7—42no—
372C4×C7—42no—
373C4×C7—44no—
374C4×C7—82no—
375C4×C7#38614yes—
376C4×C7C22×C22⋊C411yescharacteristic
377C4×C7C2×C4×D411yescharacteristic
378C4×C7C2×#12111yescharacteristic
379C4×C7C2×#12311yescharacteristic
380Dic7—24no—
381Dic7—44no—
382C24—141no—
383C24C2×C22⋊D711yescharacteristic
384#41—716no—
385#41—148no—
386#41—148no—
387#41—148no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—72no—
401C2×Q8—72no—
402C2×Q8—72no—
403C2×Q8—141no—
404C2×Q8—141no—
405C2×Q8—141no—
406C2×Q8C2×C4×D714yes—
407C2×D4—72no—
408C2×D4—72no—
409C2×D4—72no—
410C2×D4—144no—
411C2×D4—144no—
412C2×D4—144no—
413C2×D4—141no—
414C2×D4—141no—
415C2×D4—141no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—148no—
422C22×C4—141no—
423C22×C4—144no—
424C22×C4—144no—
425C22×C4—144no—
426C22×C4—142no—
427C22×C4—144no—
428C22×C4—564no—
429C22×C4—562no—
430C22×C4C2×C4×D712yes—
431C22×C4#115712yes—
432C22×C4D4×D712yes—
433C22×C4D4×D712yes—
434C22×C4D4×D712yes—
435C22×C4#115912yes—
436C22×C4C2×C22⋊D712yes—
437C22×C4C2×C22⋊D712yes—
438C22×C4C2×C22⋊D711yescharacteristic
439C22×C4C23×D711yescharacteristic
440Q16—1416no—
441SD16—1416no—
442M16—148no—
443C2×C8—74no—
444C2×C8—142no—
445C2×C8—142no—
446C2×C8—142no—
447C2×C8—142no—
448C2×C8—142no—
449C2×C8—284no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—74no—
455C4⋊C4—148no—
456C4⋊C4—144no—
457C4⋊C4—142no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—148no—
461C42—22no—
462C42—22no—
463C42—42no—
464C42—148no—
465C2×C7—24no—
466C2×C7C2×#38611yescharacteristic
467C2×C7#294712yes—
468C2×C7#300911yescharacteristic
469C2×C7#301111yescharacteristic
470C2×C7#304612yes—
471D7—44no—
472C23—22no—
473C23—22no—
474C23—142no—
475C23—142no—
476C23—142no—
477C23—282no—
478C23—282no—
479C23C4×C22⋊D712yes—
480C23#650012yes—
481C23#650112yes—
482C23C22×C22⋊D711yescharacteristic
483Q8—24no—
484Q8—24no—
485Q8—42no—
486Q8—42no—
487Q8—48no—
488Q8—74no—
489Q8—142no—
490Q8—142no—
491Q8—142no—
492D4—74no—
493D4—1416no—
494D4—142no—
495D4—142no—
496D4—142no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—142no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—1416no—
512C2×C4—144no—
513C2×C4—284no—
514C2×C4—568no—
515C2×C4—564no—
516C2×C4D7×C22⋊C414yes—
517C2×C4(C22⋊D7)⋊C414yes—
518C2×C4#644614yes—
519C2×C4#644714yes—
520C2×C4C4×C22⋊D712yes—
521C2×C4#650012yes—
522C2×C4#650112yes—
523C2×C4C22×C4×D711yescharacteristic
524C2×C4C2×#115711yescharacteristic
525C2×C4C2×D4×D711yescharacteristic
526C2×C4C2×D4×D711yescharacteristic
527C2×C4C2×D4×D711yescharacteristic
528C2×C4C2×#115911yescharacteristic
529C8—144no—
530C8—282no—
531C7#2093111yescharacteristic, complemented, Sylow
532C22—24no—
533C22—22no—
534C22—24no—
535C22—22no—
536C22—144no—
537C22—284no—
538C22—282no—
539C22—284no—
540C22—282no—
541C22#8972814yes—
542C22C2×C4×C22⋊D711yescharacteristic
543C22C2×#650011yescharacteristic
544C22C2×#650111yescharacteristic
545C22#9052812yes—
546C22#9052912yes—
547C4—44no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—144no—
552C4—284no—
553C4#8972814yes—
554C4C2×D7×C22⋊C411yescharacteristic
555C4C2×(C22⋊D7)⋊C411yescharacteristic
556C4C2×#644611yescharacteristic
557C4C2×#644711yescharacteristic
558C2—24no—
559C2—284no—
560C2C2×#8972811yescharacteristic
561C2#1173999212yes—
562C2#1174084911yescharacteristic
563C2#1174085112yes—
564C2#1174085611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2310C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#2310D7
hypocenterC7#20931
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. C7×#2310C2semidirect
4. C22×#6409C2semidirect
6. #11729335C2semidirect
7. #11729333C2semidirect
8. #11727444C2semidirect
9. #11727441C2semidirect
17. C2×#6505C4semidirect
47. C22×C7×Q8C2×C4semidirect
76. C2×#1144C2×C4semidirect
80. C2×#1143C2×C4semidirect
90. C2×#1136C2×C4semidirect
112. #2310D7semidirect
302. C22×Q8C4×D7semidirect
531. C7#20931semidirect

Maximal subgroups

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

subgroupindex
C22×#65052
C2×#902142
C7×#23102
C22×#64092
C2×#897742
#117293352
#117293332
#117274442
#117274412
#209317