Finite Group Databasebeta

← #13331031

Subgroups — #13331031

Order 1792.

1262 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3498C211yesmaximal, characteristic
2#11740762C211yesmaximal, characteristic, complemented
3C2×#89842C211yesmaximal, characteristic, complemented
4#11739953C211yesmaximal, characteristic, complemented
5#11729675C214yesmaximal, complemented
6#11729662C214yesmaximal, complemented
7#11729556C211yesmaximal, characteristic, complemented
8#11729513C211yesmaximal, characteristic, complemented
9#11729392C211yesmaximal, characteristic, complemented
10C2×C7×H(5,𝔽2)—21no—
11C7×#550—21no—
12C7×#550C2212yes—
13C7×D42—22no—
14C7×D42—22no—
15C7×D42—21no—
16C7×D42C2212yes—
17C7×#535—22no—
18C7×#535—21no—
19C7×#532—21no—
20C7×#532C2211yescharacteristic
21C2×C7×#127—21no—
22C2×C7×#127C2211yescharacteristic
23C4×C7×#41C2211yescharacteristic
24C2×#6432C2211yescharacteristic, complemented
25#90004—21no—
26C2×#6406—22no—
27#89865—21no—
28#89848—21no—
29#89842—22no—
30#89842C2214yescomplemented
31#89812—22no—
32#89793—21no—
33#89792C2212yescomplemented
34#89791C2214yescomplemented
35#89780—21no—
36#89779C2212yescomplemented
37#89767—21no—
38#89697C2211yescharacteristic, complemented
39#89696C2211yescharacteristic, complemented
40C2×C4×C7⋊C8C2211yescharacteristic, complemented
41#89339C2214yescomplemented
42Q8⋊(C7⋊C8)C2212yescomplemented
43D4⋊(C7⋊C8)C2214yescomplemented
44D4⋊(C7⋊C8)C2212yescomplemented
45#18631—71nomaximal, Sylow
46C7×H(5,𝔽2)—44no—
47C7×H(5,𝔽2)—44no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—21no—
51C2×C7×#41D411yescharacteristic
52C22×C7×D4—22no—
53C22×C7×D4—24no—
54C22×C7×D4—22no—
55C22×C7×D4—22no—
56C22×C7×D4—22no—
57C22×C7×D4—21no—
58C22×C7×D4—21no—
59C22×C7×D4D412yes—
60C22×Dic14—21no—
61C7×C4⋊Q8D411yescharacteristic
62C7×#127—22no—
63C7×#127—22no—
64C7×#127—22no—
65C7×#127—21no—
66C7×#127—21no—
67C7×#127D411yescharacteristic
68C7×#127D412yes—
69C7×#127C2312yes—
70C7×#124—22no—
71C7×#124—21no—
72C7×#121—22no—
73C7×#121—24no—
74C7×#121—24no—
75C7×#121—21no—
76C7×#121—22no—
77C7×#121—22no—
78C7×#121—22no—
79C7×#121D412yes—
80C7×#120—24no—
81C7×#120—24no—
82C7×#120—24no—
83C4×C7×Q8D411yescharacteristic
84C4×C7×Q8C2311yescharacteristic
85C4×C7×D4—22no—
86C4×C7×D4—21no—
87C4×C7×D4—21no—
88C4×C7×D4—22no—
89C4×C7×D4—21no—
90C4×C7×D4D411yescharacteristic
91C4×C7×D4C2311yescharacteristic
92C4×C7×D4C2312yes—
93C7×#117—21no—
94C7×#117D411yescharacteristic
95C2×C42×C7—21no—
96C2×C42×C7C2311yescharacteristic
97C2×#1142—24no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7—21no—
100C2×#1136—21no—
101C22×C7⋊C8—21no—
102#6432—21no—
103#6432—21no—
104#6432—21no—
105#6432C2312yes—
106#6409—22no—
107#6406—28no—
108#6406—22no—
109C22⋊(C7⋊C8)—21no—
110#6384—24no—
111#6382—22no—
112#6381—22no—
113C4⋊(C7⋊C8)—21no—
114C4⋊(C7⋊C8)C2312yes—
115C4⋊(C7⋊C8)C2312yes—
116C4×C7⋊C8—21no—
117C4×C7⋊C8—21no—
118C4×C7⋊C8C2312yes—
119#3498D711yescharacteristic
120#3206—71no—
121C2×#486—71no—
122#3079—71no—
123#1642—74no—
124#1639—74no—
125#1545—71no—
126#1492—71no—
127#1476—71no—
128C24×C7—24no—
129C24×C7—22no—
130C7×#41—22no—
131C7×#41—48no—
132C7×#41—44no—
133C7×#41—44no—
134C7×#41—42no—
135C7×#41—42no—
136C7×#41SD1614yescomplemented
137C2×C7×Q8—21no—
138C2×C7×Q8C2×D411yescharacteristic
139C2×C7×D4—21no—
140C2×C7×D4—24no—
141C2×C7×D4—24no—
142C2×C7×D4—22no—
143C2×C7×D4—24no—
144C2×C7×D4—24no—
145C2×C7×D4—22no—
146C2×C7×D4—21no—
147C2×C7×D4—22no—
148C2×C7×D4—21no—
149C2×C7×D4—22no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—48no—
153C2×C7×D4—44no—
154C2×C7×D4—416no—
155C2×C7×D4—44no—
156C2×C7×D4—44no—
157C2×C7×D4—42no—
158C2×C7×D4—42no—
159C2×C7×D4—82no—
160C2×C7×D4—84no—
161C2×C7×D4—84no—
162C2×C7×D4C2×D412yes—
163C2×C7×D4C2×D411yescharacteristic
164C2×C7×D4C2×D414yes—
165C22×C4×C7—21no—
166C22×C4×C7—24no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—22no—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×Dic7—41no—
174C2×Dic14—22no—
175C2×Dic14—41no—
176C2×Dic14—82no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4C2×D411yescharacteristic
181C7×C4⋊C4C2×D411yescharacteristic
182C7×C4⋊C4C2×D412yes—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—21no—
186C7×C22⋊C4—21no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—21no—
189C7×C22⋊C4—22no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7C2×D411yescharacteristic
194C42×C7C2×D411yescharacteristic
195C42×C7C2×D411yescharacteristic
196C42×C7C2411yescharacteristic
197#1142—84no—
198C4⋊Dic7—22no—
199C4⋊Dic7—22no—
200C4⋊Dic7—41no—
201C4⋊Dic7—41no—
202#1136—42no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—42no—
207C2×H(5,𝔽2)—21no—
208#550—21no—
209#550C2×D712yes—
210D42—22no—
211D42—22no—
212D42—21no—
213D42C2×D712yes—
214#535—22no—
215#535—21no—
216#532—21no—
217#532C2×D711yescharacteristic
218C2×C4⋊Q8—71no—
219C2×#127—21no—
220C2×#127C2×D711yescharacteristic
221C4×#41C2×D711yescharacteristic
222#492—141no—
223#487—141no—
224#486—74no—
225#486—142no—
226#477—141no—
227#476—141no—
228#475—72no—
229#474—72no—
230#461—74no—
231#448—142no—
232#431—71no—
233#428—141no—
234#423—71no—
235#417—141no—
236C2×#102—142no—
237C2×C4×C8—71no—
238#331—74no—
239Q8⋊C8—72no—
240D4⋊C8—72no—
241D4⋊C8—74no—
242C23×C7—21no—
243C23×C7—22no—
244C23×C7—24no—
245C23×C7—48no—
246C23×C7—44no—
247C23×C7—82no—
248C23×C7—82no—
249C23×C7—84no—
250C23×C7—84no—
251C23×C7#12011yescharacteristic
252C7×Q8—21no—
253C7×Q8—42no—
254C7×Q8C2×SD1612yes—
255C7×D4—24no—
256C7×D4—21no—
257C7×D4—24no—
258C7×D4—44no—
259C7×D4—48no—
260C7×D4—48no—
261C7×D4—44no—
262C7×D4—42no—
263C7×D4—42no—
264C7×D4—44no—
265C7×D4—42no—
266C7×D4—44no—
267C7×D4—44no—
268C7×D4—42no—
269C7×D4—82no—
270C7×D4—84no—
271C7×D4—84no—
272C7×D4C2×SD1612yes—
273C2×C4×C7—21no—
274C2×C4×C7—22no—
275C2×C4×C7—24no—
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—48no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7#12011yescharacteristic
286C2×C4×C7#12011yescharacteristic
287C2×C4×C7#12011yescharacteristic
288C2×C4×C7C2×SD1612yes—
289C2×C4×C7AGL(1,ℤ/8)12yes—
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×Dic7—42no—
294C2×Dic7—82no—
295Dic14—44no—
296Dic14—82no—
297C7⋊C8—42no—
298C7⋊C8—42no—
299C7⋊C8—42no—
300H(5,𝔽2)—44no—
301H(5,𝔽2)—44no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—21no—
305C2×#41C22⋊D711yescharacteristic
306C22×Q8—141no—
307C22×D4—22no—
308C22×D4—24no—
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—22no—
312C22×D4—21no—
313C22×D4—21no—
314C22×D4C22⋊D712yes—
315C2×SD16—144no—
316C2×M16—141no—
317C22×C8—141no—
318C4⋊Q8—72no—
319C4⋊Q8—141no—
320C4⋊Q8—141no—
321C4⋊Q8—141no—
322C4⋊Q8C22⋊D711yescharacteristic
323#127—22no—
324#127—22no—
325#127—22no—
326#127—21no—
327#127—21no—
328#127C22⋊D711yescharacteristic
329#127C22⋊D712yes—
330#127C22×D712yes—
331#124—22no—
332#124—21no—
333#121—22no—
334#121—24no—
335#121—24no—
336#121—21no—
337#121—22no—
338#121—22no—
339#121—22no—
340#121C22⋊D712yes—
341#120—24no—
342#120—24no—
343#120—24no—
344C4×Q8C22⋊D711yescharacteristic
345C4×Q8C22×D711yescharacteristic
346C4×D4—22no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—22no—
350C4×D4—21no—
351C4×D4C22⋊D711yescharacteristic
352C4×D4C22×D711yescharacteristic
353C4×D4C22×D712yes—
354#117—21no—
355#117C22⋊D711yescharacteristic
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C42—21no—
359C2×C42C22×D711yescharacteristic
360#107—142no—
361#106—142no—
362C4⋊C8—72no—
363C4⋊C8—72no—
364C4⋊C8—141no—
365#103—142no—
366#103—144no—
367#102—148no—
368#102—142no—
369C22⋊C8—141no—
370C4×C8—72no—
371C4×C8—141no—
372C4×C8—141no—
373C22×C7—21no—
374C22×C7—21no—
375C22×C7—22no—
376C22×C7—41no—
377C22×C7—41no—
378C22×C7—44no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—44no—
382C22×C7—44no—
383C22×C7—42no—
384C22×C7—164no—
385C22×C7—168no—
386C22×C7#45012yes—
387C22×C7C2×#12011yescharacteristic
388C4×C7—22no—
389C4×C7—21no—
390C4×C7—21no—
391C4×C7—42no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7#45012yes—
395C4×C7#45512yes—
396C4×C7C22×SD1611yescharacteristic
397C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
398C4×C7#57612yes—
399Dic7—82no—
400C24—24no—
401C24—22no—
402#41—22no—
403#41—48no—
404#41—44no—
405#41—44no—
406#41—42no—
407#41—42no—
408#41#114214yescomplemented
409C2×Q8—21no—
410C2×Q8—142no—
411C2×Q8—281no—
412C2×Q8—562no—
413C2×Q8C2×C22⋊D711yescharacteristic
414C2×D4—21no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—48no—
428C2×D4—44no—
429C2×D4—416no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—82no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4C2×C22⋊D712yes—
438C2×D4C2×C22⋊D711yescharacteristic
439C2×D4C2×C22⋊D714yes—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—281no—
446C22×C4C2×C22⋊D711yescharacteristic
447C22×C4C2×C22⋊D711yescharacteristic
448C22×C4C2×C22⋊D711yescharacteristic
449SD16—564no—
450M16—282no—
451C2×C8—142no—
452C2×C8—142no—
453C2×C8—142no—
454C2×C8—282no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—142no—
459C4⋊C4—142no—
460C4⋊C4—281no—
461C4⋊C4—281no—
462C4⋊C4C2×C22⋊D712yes—
463C4⋊C4C2×C22⋊D711yescharacteristic
464C4⋊C4C2×C22⋊D711yescharacteristic
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—22no—
470C22⋊C4—21no—
471C22⋊C4—22no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42C2×C22⋊D711yescharacteristic
476C42C2×C22⋊D711yescharacteristic
477C42C2×C22⋊D711yescharacteristic
478C42C23×D711yescharacteristic
479C2×C7—22no—
480C2×C7—42no—
481C2×C7—82no—
482C2×C7—84no—
483C2×C7C2×#45011yescharacteristic
484C2×C7#300811yescharacteristic
485C2×C7C2×#45511yescharacteristic
486C23—21no—
487C23—22no—
488C23—24no—
489C23—48no—
490C23—44no—
491C23—82no—
492C23—82no—
493C23—84no—
494C23—84no—
495C23#651611yescharacteristic
496Q8—21no—
497Q8—42no—
498Q8—284no—
499Q8—562no—
500Q8C2×#114212yes—
501D4—24no—
502D4—21no—
503D4—24no—
504D4—44no—
505D4—48no—
506D4—48no—
507D4—44no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—44no—
512D4—44no—
513D4—42no—
514D4—44no—
515D4—82no—
516D4—84no—
517D4—84no—
518D4C2×#114212yes—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—48no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—282no—
532C2×C4—562no—
533C2×C4#649512yes—
534C2×C4C2×#114212yes—
535C2×C4#651611yescharacteristic
536C2×C4#651611yescharacteristic
537C2×C4#651611yescharacteristic
538C2×C4C22×C22⋊D711yescharacteristic
539C2×C4C22×C22⋊D711yescharacteristic
540C2×C4C22×C22⋊D711yescharacteristic
541C8—282no—
542C8—282no—
543C8—282no—
544C7#1863111yescharacteristic, complemented, Sylow
545C22—21no—
546C22—21no—
547C22—22no—
548C22—41no—
549C22—41no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—164no—
557C22—168no—
558C22#8999012yes—
559C22C2×#651611yescharacteristic
560C4—22no—
561C4—21no—
562C4—21no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—562no—
567C4#8999012yes—
568C4#9001412yes—
569C4C2×#649511yescharacteristic
570C4C22×#114211yescharacteristic
571C4#9052612yes—
572C2—22no—
573C2—42no—
574C2—82no—
575C2—84no—
576C2C2×#8999011yescharacteristic
577C2#1174154611yescharacteristic
578C2C2×#9001411yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3498C2
socleC22×C7C2×#120
hypercenter#3498D7
hypocenterC7#18631
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3498

Complements

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

NH ≅ G/Ntype
2. #11740762C2semidirect
3. C2×#89842C2semidirect
4. #11739953C2semidirect
5. #11729675C2semidirect
6. #11729662C2semidirect
7. #11729556C2semidirect
8. #11729513C2semidirect
9. #11729392C2semidirect
24. C2×#6432C22semidirect
30. #89842C22semidirect
33. #89792C22semidirect
34. #89791C22semidirect
36. #89779C22semidirect
38. #89697C22semidirect
39. #89696C22semidirect
40. C2×C4×C7⋊C8C22semidirect
41. #89339C22semidirect
42. Q8⋊(C7⋊C8)C22semidirect
43. D4⋊(C7⋊C8)C22semidirect
44. D4⋊(C7⋊C8)C22semidirect
136. C7×#41SD16semidirect
408. #41#1142semidirect
544. C7#18631semidirect

Maximal subgroups

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

subgroupindex
C7×#34982
#117407622
C2×#898422
#117399532
#117296752
#117296622
#117295562
#117295132
#117293922
#186317