Finite Group Databasebeta

← #13358918

Subgroups — #13358918

Order 1792.

1518 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90239C211yesmaximal, characteristic
2C2×#89477C211yesmaximal, characteristic, complemented
3C4×D7×C4⋊C4C211yesmaximal, characteristic, complemented
4C7×#1762C211yesmaximal, characteristic, complemented
5#11729363C211yesmaximal, characteristic, complemented
6#11727954C212yesmaximal, complemented
7#11727915C212yesmaximal, complemented
8#11727566C211yesmaximal, characteristic, complemented
9#11727228C211yesmaximal, characteristic, complemented
10#11726634C212yesmaximal, complemented
11#11726553C212yesmaximal, complemented
12C22×#1157—21no—
13C2×C4×C7×D4C2211yescharacteristic
14C2×#6503C2211yescharacteristic
15C2×#6501C2211yescharacteristic
16C2×C4×C22⋊D7—21no—
17#90239—24no—
18#90239C418yescomplemented
19C2×#6458C2211yescharacteristic
20C2×D7×C4⋊C4C2212yes—
21C2×#6451C2211yescharacteristic
22C2×#6448—21no—
23C2×#6447—21no—
24C2×C4×D28C2211yescharacteristic
25C2×C42×D7—21no—
26C2×C42×D7C2211yescharacteristic
27C2×C7×#102C2212yes—
28C2×C7×#97C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C2×#6406C2212yes—
31C2×#6395C2212yes—
32C2×(C2×D7)⋊C8C2212yes—
33C2×Dic7⋊C8C2212yes—
34C4⋊C4×Dic7C2212yes—
35C2×#6383C2212yes—
36C4×C4⋊Dic7C2211yescharacteristic
37C4×Dic7⋊C4—21no—
38C2×#6377C2211yescharacteristic
39#89477—24no—
40D7×#95—21no—
41#89421—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21395—71nomaximal, Sylow
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C22×C22⋊D7—21no—
50C22×C22⋊D7—21no—
51C22×C22⋊D7—21no—
52C2×#1157—28no—
53C2×#1157—48no—
54C22×D28—21no—
55C22×D28C2311yescharacteristic
56C22×C4×D7—22no—
57C22×C4×D7—21no—
58C22×C4×D7D411yescharacteristic
59C22×C4×D7D411yescharacteristic
60C22×C4×D7C2311yescharacteristic
61C22×Dic14—21no—
62C22×C7×C8C2312yes—
63C4×C7×D4—22no—
64C4×C7×D4C2×C414yescomplemented
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2312yes—
67C2×C7×C22⋊C4—21no—
68C2×C42×C7—21no—
69C2×C42×C7C2311yescharacteristic
70C2×C22⋊Dic7—21no—
71#6503—22no—
72#6503C2×C414yescomplemented
73#6501—22no—
74#6501C2×C414yes—
75C4×C22⋊D7—28no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2312yes—
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—22no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7D411yescharacteristic
85C2×C4×Dic7D411yescharacteristic
86C2×C4×Dic7C2311yescharacteristic
87C22×C7⋊C8C2312yes—
88#6458—22no—
89#6458C2×C414yes—
90D7×C4⋊C4—24no—
91D7×C4⋊C4—24no—
92#6451—22no—
93#6451C2×C414yescomplemented
94#6448—28no—
95#6447—28no—
96C4×D28—22no—
97C4×D28C2×C414yescomplemented
98C42×D7—44no—
99C42×D7C2×C414yes—
100C42×D7D414yes—
101C7×#102—24no—
102C7×#97—22no—
103C7×#95—21no—
104#6406—24no—
105C7⋊#95—21no—
106C7⋊#95—22no—
107#6395—24no—
108(C2×D7)⋊C8—24no—
109Dic7⋊C8—24no—
110#6383—24no—
111#6377—22no—
112C2×#532—71no—
113C2×#432—71no—
114C2×C4×C4⋊C4—71no—
115#1987—72no—
116#1982—72no—
117#1874—72no—
118#1874—72no—
119#1762—71no—
120#1762—71no—
121#1762—71no—
122#1762D711yescharacteristic, complemented
123C24×C7—21no—
124C23×D7—21no—
125C23×D7C2×D411yescharacteristic
126C2×C7×D4—21no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4C22×C412yes—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—24no—
133C22×C4×C7—44no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2411yescharacteristic
137C2×C22⋊D7—24no—
138C2×C22⋊D7—24no—
139C2×C22⋊D7—24no—
140C2×C22⋊D7—44no—
141C2×C22⋊D7—44no—
142C2×C22⋊D7—42no—
143C2×C22⋊D7—42no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7C2×D411yescharacteristic
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149#1157—432no—
150C2×D28—22no—
151C2×D28—22no—
152C2×D28—24no—
153C2×D28—21no—
154C2×D28—44no—
155C2×D28C22×C412yes—
156C2×C4×D7—24no—
157C2×C4×D7—22no—
158C2×C4×D7—22no—
159C2×C4×D7—22no—
160C2×C4×D7—22no—
161C2×C4×D7—28no—
162C2×C4×D7—24no—
163C2×C4×D7—44no—
164C2×C4×D7—44no—
165C2×C4×D7—44no—
166C2×C4×D7C22⋊C414yes—
167C2×C4×D7C22⋊C414yes—
168C2×C4×D7C22×C412yes—
169C2×C4×D7C2×D412yes—
170C2×Dic14—24no—
171C2×Dic14—44no—
172C2×C7×C8—22no—
173C2×C7×C8—22no—
174C2×C7×C8—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4C22×C412yes—
178C7×C4⋊C4#4114yescomplemented
179C7×C22⋊C4—24no—
180C42×C7—42no—
181C42×C7C22×C412yes—
182C42×C7C2×D412yes—
183C22⋊Dic7—24no—
184(C2×D7)⋊C4—24no—
185(C2×D7)⋊C4—24no—
186C4⋊Dic7—21no—
187C4⋊Dic7—22no—
188C4⋊Dic7C22×C412yes—
189C4⋊Dic7#4114yescomplemented
190Dic7⋊C4—28no—
191Dic7⋊C4—24no—
192C4×Dic7—42no—
193C4×Dic7—44no—
194C4×Dic7C22⋊C414yes—
195C4×Dic7C22⋊C414yes—
196C4×Dic7C22×C412yes—
197C4×Dic7C2×D412yes—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—22no—
201C22×#41—141no—
202#532—78no—
203#532—144no—
204C2×C4⋊Q8—71no—
205C2×#127—71no—
206C2×#124—141no—
207C2×#121—71no—
208C2×#121—71no—
209C2×#121—141no—
210C2×C4×D4—71no—
211C2×C4×D4—141no—
212C2×C4×D4C2×D711yescharacteristic
213C22×C4⋊C4—72no—
214C22×C42—71no—
215C22×C42—141no—
216#432—144no—
217C2×C4⋊C8—72no—
218C2×#102—72no—
219C2×#102—72no—
220C2×#102—72no—
221C2×#102C2×D712yes—
222C2×C22⋊C8—72no—
223C2×#97—71no—
224C2×#97C2×D711yescharacteristic
225C4×C4⋊C4—72no—
226C4×C4⋊C4—71no—
227C4×C4⋊C4—141no—
228C4×C4⋊C4—141no—
229C4×C4⋊C4C2×D711yescharacteristic
230C2×#95—141no—
231#340—72no—
232#340—72no—
233#340—72no—
234#340C2×D712yes—
235C23×C7—22no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7C22×D411yescharacteristic
241C22×D7—22no—
242C22×D7—21no—
243C22×D7—21no—
244C22×D7—22no—
245C22×D7—42no—
246C22×D7—42no—
247C22×D7—41no—
248C22×D7—41no—
249C22×D7C2×C22⋊C412yes—
250C22×D7AGL(1,ℤ/8)14yes—
251C7×D4—22no—
252C7×D4—22no—
253C7×D4—42no—
254C2×C4×C7—21no—
255C2×C4×C7—22no—
256C2×C4×C7—21no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—48no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7C2×C22⋊C412yes—
264C2×C4×C7C2×C22⋊C412yes—
265C2×C4×C7C23×C411yescharacteristic
266C2×C4×C7C22×D411yescharacteristic
267C2×C4×C7C2×#4112yes—
268C2×C4×C7C2×#4112yes—
269C22⋊D7—48no—
270C22⋊D7—48no—
271C22⋊D7—48no—
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—24no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7C2×C22⋊C412yes—
280C2×Dic7C2×C22⋊C412yes—
281C2×Dic7C2×C22⋊C412yes—
282C2×Dic7AGL(1,ℤ/8)12yes—
283C2×Dic7#13712yes—
284D28—22no—
285D28—22no—
286D28—48no—
287D28—42no—
288C4×D7—28no—
289C4×D7—22no—
290C4×D7—22no—
291C4×D7—24no—
292C4×D7—28no—
293C4×D7—48no—
294C4×D7—416no—
295Dic14—48no—
296C7×C8—42no—
297C7⋊C8—42no—
298C2×#41—148no—
299C2×#41—288no—
300C22×Q8—141no—
301C22×D4—71no—
302C22×D4—141no—
303C22×D4—141no—
304C22×D4—141no—
305C22×D4—141no—
306C22×D4C22×D711yescharacteristic
307C23×C4—21no—
308C23×C4—71no—
309C23×C4—71no—
310C23×C4—71no—
311C23×C4—142no—
312C23×C4—141no—
313C22×C8—72no—
314C22×C8C22×D712yes—
315C4⋊Q8—74no—
316C4⋊Q8—142no—
317#127—74no—
318#127—142no—
319#124—148no—
320#121—74no—
321#121—74no—
322#121—148no—
323#121—142no—
324#121—142no—
325C4×D4—22no—
326C4×D4—74no—
327C4×D4—148no—
328C4×D4—142no—
329C4×D4C4×D714yescomplemented
330C2×C4⋊C4—72no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—144no—
333C2×C4⋊C4—142no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—144no—
336C2×C4⋊C4C22×D711yescharacteristic
337C2×C4⋊C4C22×D712yes—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—141no—
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C42—21no—
343C2×C42—71no—
344C2×C42—71no—
345C2×C42—71no—
346C2×C42—74no—
347C2×C42—74no—
348C2×C42—142no—
349C2×C42—141no—
350C2×C42—284no—
351C2×C42C22×D711yescharacteristic
352C4⋊C8—144no—
353#102—24no—
354#102—144no—
355#102—144no—
356#102—144no—
357C22⋊C8—144no—
358#97—22no—
359#97—142no—
360#95—21no—
361#95—142no—
362#95—141no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—82no—
367C22×C7—82no—
368C22×C7C22×C22⋊C411yescharacteristic
369C22×C7#52512yes—
370C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
371C22×C7C2×AGL(1,ℤ/8)12yes—
372C22×C7C2×#13711yescharacteristic
373C2×D7—24no—
374C2×D7—24no—
375C2×D7—22no—
376C2×D7—22no—
377C2×D7—42no—
378C2×D7—42no—
379C2×D7—42no—
380C2×D7—82no—
381C2×D7—82no—
382C2×D7#41814yes—
383C4×C7—22no—
384C4×C7—22no—
385C4×C7—44no—
386C4×C7—42no—
387C4×C7C4×#4111yescharacteristic
388C4×C7C4×#4111yescharacteristic
389C4×C7#53212yes—
390Dic7—22no—
391Dic7—22no—
392Dic7—44no—
393Dic7—42no—
394Dic7#41714yes—
395C24—21no—
396C24—71no—
397C24—141no—
398#41—2832no—
399C2×Q8—144no—
400C2×Q8—284no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—72no—
405C2×D4—144no—
406C2×D4—144no—
407C2×D4—144no—
408C2×D4—144no—
409C2×D4—142no—
410C2×D4—141no—
411C2×D4—142no—
412C2×D4—284no—
413C2×D4—282no—
414C2×D4—282no—
415C2×D4—284no—
416C2×D4—284no—
417C2×D4C2×C4×D712yes—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—71no—
423C22×C4—71no—
424C22×C4—71no—
425C22×C4—74no—
426C22×C4—74no—
427C22×C4—72no—
428C22×C4—72no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—144no—
432C22×C4—148no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—141no—
437C22×C4—144no—
438C22×C4—284no—
439C22×C4—284no—
440C22×C4—284no—
441C22×C4D4×D711yescharacteristic
442C22×C4D4×D711yescharacteristic
443C22×C4C23×D711yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—142no—
448C2×C8—142no—
449C2×C8—142no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—72no—
453C4⋊C4—74no—
454C4⋊C4—144no—
455C4⋊C4—148no—
456C4⋊C4—142no—
457C4⋊C4—141no—
458C4⋊C4C2×C4×D712yes—
459C4⋊C4#115714yescomplemented
460C22⋊C4—24no—
461C22⋊C4—144no—
462C22⋊C4—144no—
463C22⋊C4—144no—
464C42—42no—
465C42—74no—
466C42—74no—
467C42—72no—
468C42—72no—
469C42—282no—
470C42—284no—
471C42C2×C4×D712yes—
472C42D4×D712yes—
473C2×C7—82no—
474C2×C7#229711yescharacteristic
475C2×C7C2×#41811yescharacteristic
476C2×C7C2×#41711yescharacteristic
477C2×C7#310912yes—
478C2×C7#311212yes—
479D7—24no—
480D7—82no—
481C23—22no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—72no—
487C23—74no—
488C23—141no—
489C23—141no—
490C23—142no—
491C23—142no—
492C23—282no—
493C23—281no—
494C23—281no—
495C23—282no—
496C23C2×D4×D711yescharacteristic
497Q8—288no—
498D4—22no—
499D4—22no—
500D4—42no—
501D4—142no—
502D4—142no—
503D4—288no—
504D4—288no—
505D4—288no—
506D4—288no—
507D4—282no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—72no—
518C2×C4—72no—
519C2×C4—72no—
520C2×C4—72no—
521C2×C4—72no—
522C2×C4—148no—
523C2×C4—148no—
524C2×C4—142no—
525C2×C4—142no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—144no—
529C2×C4—142no—
530C2×C4—144no—
531C2×C4—288no—
532C2×C4—2816no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4D7×C22⋊C412yes—
537C2×C4D7×C22⋊C412yes—
538C2×C4C22×C4×D711yescharacteristic
539C2×C4C2×#115712yes—
540C2×C4C2×D4×D711yescharacteristic
541C2×C4D7×#4112yes—
542C8—42no—
543C8—282no—
544C7#2139511yescharacteristic, complemented, Sylow
545C22—42no—
546C22—42no—
547C22—42no—
548C22—74no—
549C22—82no—
550C22—82no—
551C22—142no—
552C22—142no—
553C22—144no—
554C22—144no—
555C22—282no—
556C22—282no—
557C22—282no—
558C22—562no—
559C22—562no—
560C22C2×D7×C22⋊C411yescharacteristic
561C22#9023812yes—
562C22D7×AGL(1,ℤ/8)11yescharacteristic
563C22D7×AGL(1,ℤ/8)12yes—
564C22D7×#13711yescharacteristic
565C4—22no—
566C4—22no—
567C4—42no—
568C4—44no—
569C4—74no—
570C4—142no—
571C4—142no—
572C4—282no—
573C4—284no—
574C4C4×#115711yescharacteristic
575C4#9021311yescharacteristic
576C4#9023912yes—
577C2—82no—
578C2—144no—
579C2—562no—
580C2#1173100811yescharacteristic
581C2D7×#41711yescharacteristic
582C2D7×#41811yescharacteristic
583C2#1173722612yes—
584C2#1173731412yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1762C2
socleC23×C7C22×D4
prime radicalC2×#90239C2
cosocleC22×C4×C7C24
hypercenter#1762D7
hypocenterC7#21395
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#89477C2semidirect
3. C4×D7×C4⋊C4C2semidirect
4. C7×#1762C2semidirect
5. #11729363C2semidirect
6. #11727954C2semidirect
7. #11727915C2semidirect
8. #11727566C2semidirect
9. #11727228C2semidirect
10. #11726634C2semidirect
11. #11726553C2semidirect
18. #90239C4semidirect
64. C4×C7×D4C2×C4semidirect
72. #6503C2×C4semidirect
93. #6451C2×C4semidirect
97. C4×D28C2×C4semidirect
122. #1762D7semidirect
178. C7×C4⋊C4#41semidirect
189. C4⋊Dic7#41semidirect
329. C4×D4C4×D7semidirect
459. C4⋊C4#1157semidirect
544. C7#21395semidirect

Maximal subgroups

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

subgroupindex
C2×#902392
C2×#894772
C4×D7×C4⋊C42
C7×#17622
#117293632
#117279542
#117279152
#117275662
#117272282
#117266342
#117265532
#213957