Finite Group Databasebeta

← #14240576

Subgroups — #14240576

Order 1792.

2136 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
1C7×#3499C211yesmaximal, characteristic, complemented
2#11743320C211yesmaximal, characteristic, complemented
3#11741558C214yesmaximal, complemented
4D4×#1143C214yesmaximal, complemented
5#11741000C214yesmaximal, complemented
6#11740968C214yesmaximal, complemented
7#11740956C214yesmaximal, complemented
8#11740852C214yesmaximal, complemented
9#11740723C211yesmaximal, characteristic, complemented
10#11740658C212yesmaximal, complemented
11#11739974C212yesmaximal, complemented
12C2×C7×2−1+4C2212yescomplemented
13C7×D4×Q8C2214yescomplemented
14C7×#547—24no—
15C7×#547C2214yescomplemented
16C7×#536—22no—
17C7×#536C2212yescomplemented
18C7×#532—21no—
19C7×#532C2211yescharacteristic
20C2×C7×C4⋊Q8—21no—
21C2×C7×C4⋊Q8C2211yescharacteristic, complemented
22C4×C7×#41C2211yescharacteristic, complemented
23#90521C2214yes—
24C2×#6512C2214yescomplemented
25C22×#1143C2214yescomplemented
26#90425—21no—
27#90408—21no—
28#90374—22no—
29#90298—24no—
30#90254C2214yes—
31D4×D28C2214yes—
32#90168C2212yes—
33#90010C2218yescomplemented
34#90010C2214yescomplemented
35#90006C2214yescomplemented
36#90005C2214yescomplemented
37#89999C2218yescomplemented
38#89999C2218yescomplemented
39#89997C2214yescomplemented
40#89857C2214yes—
41#89855C2214yescomplemented
42#89855C2214yescomplemented
43#89853C2214yescomplemented
44#89821C2218yescomplemented
45#89820C2218yescomplemented
46#89818C2218yescomplemented
47#89817C2218yescomplemented
48#89799C2212yes—
49C4×#1143C2214yescomplemented
50#89793C2212yescomplemented
51#89787C2212yes—
52C4×#1141C2214yescomplemented
53#89784C2214yescomplemented
54D4×C7⋊C8C2214yescomplemented
55#89780C2212yescomplemented
56#89775C2214yescomplemented
57#89697C2212yescomplemented
58#61664—71nomaximal, Sylow
59C7×2−1+4—44no—
60C7×2−1+4—44no—
61C2×C7×#41—22no—
62C2×C7×#41—24no—
63C2×C7×#41C2311yescharacteristic
64C2×C7×#41C2314yes—
65C2×C7×#41C2312yes—
66C22×C7×Q8—22no—
67C22×C7×Q8C2314yes—
68C22×C7×Q8C2312yes—
69C2×D4×D7—24no—
70C2×#1157—22no—
71C22×D28C2314yes—
72C7×C4⋊Q8—22no—
73C7×C4⋊Q8—24no—
74C7×C4⋊Q8—21no—
75C7×C4⋊Q8—21no—
76C7×C4⋊Q8D411yescharacteristic
77C7×C4⋊Q8C2314yes—
78C7×#127D411yescharacteristic
79C7×#124—22no—
80C7×#124D412yes—
81C7×#123—24no—
82C7×#123—28no—
83C7×C22⋊Q8—28no—
84C7×C22⋊Q8C2318yes—
85C7×#121—24no—
86C7×#121D414yes—
87C4×C7×Q8C2312yes—
88C4×C7×D4—24no—
89C4×C7×D4—22no—
90C4×C7×D4C2312yes—
91C4×C7×D4C2314yes—
92C7×#117—21no—
93C7×#117C2311yescharacteristic
94C2×C7×C4⋊C4—22no—
95C2×C7×C4⋊C4—24no—
96C2×C42×C7—21no—
97C2×C42×C7C2311yescharacteristic
98#6512—48no—
99#6510—24no—
100C2×#1143—48no—
101C2×#1143C2314yes—
102C2×#1143C2318yes—
103C2×#1143C2318yes—
104#6503—24no—
105#6501—24no—
106#6500—24no—
107#6500—24no—
108C2×#1141C2314yes—
109#6493—22no—
110#6493C2314yes—
111#6493C2318yes—
112#6489C2314yes—
113C2×#1136C2314yes—
114C22×C7⋊C8C2314yes—
115#6458—24no—
116#6455—24no—
117#6448—28no—
118#6445—28no—
119#6438—22no—
120#6437C2312yes—
121C4×D28—22no—
122C4×D28C2314yes—
123#6432C2312yes—
124#6409C2314yes—
125#6409C2318yes—
126#6409C2318yes—
127#6406C2314yes—
128C22⋊(C7⋊C8)C2314yes—
129C22⋊(C7⋊C8)C2318yes—
130#6383C2314yes—
131#6383C2314yes—
132#6382C2314yes—
133#6381C2314yes—
134C4⋊(C7⋊C8)C2312yes—
135C4⋊(C7⋊C8)C2312yes—
136C4×C7⋊C8C2314yes—
137#3549—71no—
138#3499D711yescharacteristic, complemented
139#3329—74no—
140#3303—74no—
141D4×SD16—74no—
142#3272—74no—
143#3206—72no—
144#3200—74no—
145#3106—72no—
146#3010—74no—
147#2979—71no—
148C23×D7—24no—
149C7×#41—22no—
150C7×#41—22no—
151C7×#41—28no—
152C7×#41—42no—
153C7×#41—42no—
154C7×#41—44no—
155C7×#41—48no—
156C7×#41—416no—
157C2×C7×Q8—24no—
158C2×C7×Q8—22no—
159C2×C7×Q8—28no—
160C2×C7×Q8—44no—
161C2×C7×Q8—42no—
162C2×C7×Q8—42no—
163C2×C7×Q8—48no—
164C2×C7×Q8C2×D412yes—
165C2×C7×Q8C2414yes—
166C2×C7×Q8C2411yescharacteristic
167C2×C7×D4—21no—
168C2×C7×D4—24no—
169C2×C7×D4C2×D414yes—
170C2×C7×D4C2×D412yes—
171C2×C7×D4C2×D414yes—
172C2×C7×D4C2411yescharacteristic
173C22×C4×C7—21no—
174C22×C4×C7—22no—
175C22×C4×C7—24no—
176C22×C4×C7—22no—
177C22×C4×C7—24no—
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2411yescharacteristic
180C22×C4×C7C2414yes—
181C2×C22⋊D7—28no—
182C2×C22⋊D7—24no—
183D4×D7—88no—
184#1157—84no—
185C2×D28—22no—
186C2×D28—22no—
187C2×D28—24no—
188C2×D28—44no—
189C2×D28—44no—
190C2×D28C2414yes—
191C2×C4×D7—24no—
192C2×Dic14—22no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—28no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4C2×D414yes—
198C7×C4⋊C4C2412yes—
199C7×C22⋊C4—21no—
200C7×C22⋊C4—28no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4C2×D418yes—
203C42×C7—21no—
204C42×C7—22no—
205C42×C7C2×D411yescharacteristic
206C42×C7C2×D411yescharacteristic
207C42×C7C2412yes—
208C22⋊Dic7—24no—
209#1143—28no—
210#1143—48no—
211#1143—48no—
212#1141—28no—
213(C2×D7)⋊C4—24no—
214(C2×D7)⋊C4—24no—
215(C2×D7)⋊C4—28no—
216C4⋊Dic7—22no—
217C4⋊Dic7C2414yes—
218C4×Dic7—24no—
219#1136—44no—
220C2×C7⋊C8—44no—
221C2×C7⋊C8C2414yes—
222C2×C7⋊C8C2414yes—
223C2×2−1+4C2×D712yescomplemented
224C2×AGL(1,ℤ/8)—74no—
225C22×SD16—74no—
226H(3,𝔽4)—142no—
227#555—74no—
228#555—141no—
229D4×Q8C2×D714yescomplemented
230#547—24no—
231#547C2×D714yescomplemented
232D42—74no—
233D42—141no—
234#536—22no—
235#536C2×D712yescomplemented
236#535—74no—
237#535—144no—
238#532—21no—
239#532—72no—
240#532C2×D711yescharacteristic
241C2×C4⋊Q8—21no—
242C2×C4⋊Q8C2×D711yescharacteristic, complemented
243C4×#41C2×D711yescharacteristic, complemented
244#492—74no—
245#487—74no—
246#485—78no—
247#482—74no—
248#477—72no—
249#476—72no—
250#468—78no—
251#465—78no—
252#461—74no—
253#461—74no—
254#460—72no—
255#459—72no—
256#450—78no—
257#449—74no—
258#449—78no—
259#448—78no—
260#448—78no—
261C4×SD16—74no—
262C4×D8—74no—
263#436—74no—
264C8×D4—74no—
265#431—72no—
266#419—74no—
267#416—74no—
268#408—74no—
269C23×C7—21no—
270C23×C7C22×D414yes—
271C23×C7C22×D411yescharacteristic
272C22×D7—24no—
273C22×D7—84no—
274C22×D7—88no—
275C7×Q8—21no—
276C7×Q8—24no—
277C7×Q8—24no—
278C7×Q8—42no—
279C7×Q8—48no—
280C7×Q8—44no—
281C7×Q8H(5,𝔽2)12yes—
282C7×D4—24no—
283C7×D4—28no—
284C7×D4—21no—
285C7×D4—44no—
286C7×D4—42no—
287C7×D4—48no—
288C7×D4H(5,𝔽2)12yes—
289C2×C4×C7—24no—
290C2×C4×C7—21no—
291C2×C4×C7—24no—
292C2×C4×C7—24no—
293C2×C4×C7—24no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—24no—
298C2×C4×C7—48no—
299C2×C4×C7—48no—
300C2×C4×C7—44no—
301C2×C4×C7—44no—
302C2×C4×C7C22×D414yes—
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×D412yes—
305C2×C4×C7C22×D412yes—
306C2×C4×C7C2511yescharacteristic
307C22⋊D7—84no—
308C22⋊D7—88no—
309C2×Dic7—24no—
310D28—24no—
311D28—24no—
312D28—44no—
313D28—84no—
314D28—82no—
315C4×D7—48no—
316Dic14—82no—
317C7⋊C8—24no—
318C7⋊C8—24no—
3192−1+4—44no—
3202−1+4—44no—
321C2×#41—22no—
322C2×#41—24no—
323C2×#41—142no—
324C2×#41C22×D711yescharacteristic
325C2×#41C22×D714yes—
326C2×#41C22×D712yes—
327C22×Q8—22no—
328C22×Q8C22×D714yes—
329C22×Q8C22×D712yes—
330C22×D4—74no—
331C22×D4—144no—
332AGL(1,ℤ/8)—288no—
333C2×SD16—74no—
334C2×SD16—78no—
335C2×SD16—78no—
336C2×SD16—288no—
337C2×D8—74no—
338C2×M16—74no—
339C22×C8—74no—
340C4⋊Q8—22no—
341C4⋊Q8—24no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8—72no—
345C4⋊Q8C22⋊D711yescharacteristic
346C4⋊Q8C22×D714yes—
347#127—72no—
348#127—144no—
349#127C22⋊D711yescharacteristic
350#124—22no—
351#124—144no—
352#124—148no—
353#124—142no—
354#124C22⋊D712yes—
355#123—24no—
356#123—28no—
357C22⋊Q8—28no—
358C22⋊Q8C22×D718yes—
359#121—24no—
360#121—74no—
361#121—78no—
362#121—144no—
363#121—142no—
364#121—144no—
365#121C22⋊D714yes—
366#120—144no—
367#120—144no—
368#120—148no—
369C4×Q8C22×D712yes—
370C4×D4—24no—
371C4×D4—22no—
372C4×D4—74no—
373C4×D4—142no—
374C4×D4C22×D712yes—
375C4×D4C22×D714yes—
376#117—21no—
377#117—74no—
378#117—144no—
379#117C22×D711yescharacteristic
380C2×C4⋊C4—22no—
381C2×C4⋊C4—24no—
382C2×C42—21no—
383C2×C42C22×D711yescharacteristic
384#107—74no—
385#106—74no—
386C4⋊C8—72no—
387C4⋊C8—72no—
388#103—78no—
389#103—74no—
390#103—78no—
391#102—74no—
392#102—74no—
393#102—74no—
394C22⋊C8—74no—
395C22⋊C8—78no—
396C4×C8—74no—
397C22×C7—21no—
398C22×C7—22no—
399C22×C7—24no—
400C22×C7—22no—
401C22×C7—28no—
402C22×C7—42no—
403C22×C7C23×D411yescharacteristic
404C2×D7—44no—
405C2×D7—44no—
406C2×D7—44no—
407C2×D7—168no—
408C4×C7—24no—
409C4×C7—21no—
410C4×C7—22no—
411C4×C7—22no—
412C4×C7—44no—
413C4×C7—42no—
414C4×C7#57714yes—
415C4×C7C2×H(5,𝔽2)11yescharacteristic
416C4×C7C2×H(5,𝔽2)11yescharacteristic
417Dic7—44no—
418C24—144no—
419#41—22no—
420#41—22no—
421#41—28no—
422#41—42no—
423#41—42no—
424#41—44no—
425#41—48no—
426#41—416no—
427#41—564no—
428C2×Q8—24no—
429C2×Q8—22no—
430C2×Q8—28no—
431C2×Q8—44no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—48no—
435C2×Q8—142no—
436C2×Q8C2×C22⋊D712yes—
437C2×Q8C23×D714yes—
438C2×Q8C23×D711yescharacteristic
439C2×D4—21no—
440C2×D4—24no—
441C2×D4—74no—
442C2×D4—148no—
443C2×D4—144no—
444C2×D4—142no—
445C2×D4—142no—
446C2×D4—144no—
447C2×D4—284no—
448C2×D4—284no—
449C2×D4—568no—
450C2×D4C2×C22⋊D714yes—
451C2×D4C2×C22⋊D712yes—
452C2×D4C2×C22⋊D714yes—
453C2×D4C23×D711yescharacteristic
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—144no—
460C22×C4C2×C22⋊D712yes—
461C22×C4C23×D711yescharacteristic
462C22×C4C23×D714yes—
463SD16—148no—
464SD16—288no—
465SD16—288no—
466D8—148no—
467M16—284no—
468C2×C8—74no—
469C2×C8—74no—
470C2×C8—284no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—74no—
476C4⋊C4—142no—
477C4⋊C4C2×C22⋊D714yes—
478C4⋊C4C23×D712yes—
479C22⋊C4—21no—
480C22⋊C4—28no—
481C22⋊C4—22no—
482C22⋊C4—144no—
483C22⋊C4—148no—
484C22⋊C4—144no—
485C22⋊C4—144no—
486C22⋊C4C2×C22⋊D718yes—
487C42—21no—
488C42—22no—
489C42—144no—
490C42C2×C22⋊D711yescharacteristic
491C42C2×C22⋊D711yescharacteristic
492C42C23×D712yes—
493C2×C7—44no—
494C2×C7—41no—
495C2×C7—42no—
496C2×C7#348411yescharacteristic
497C2×C7C2×#57712yes—
498D7—84no—
499C23—21no—
500C23—144no—
501C23—564no—
502C23—568no—
503C23C22×C22⋊D714yes—
504C23C22×C22⋊D711yescharacteristic
505Q8—21no—
506Q8—24no—
507Q8—24no—
508Q8—42no—
509Q8—48no—
510Q8—44no—
511Q8—562no—
512Q8#654812yes—
513D4—24no—
514D4—28no—
515D4—21no—
516D4—42no—
517D4—44no—
518D4—48no—
519D4—144no—
520D4—144no—
521D4—284no—
522D4—564no—
523D4—562no—
524D4—568no—
525D4—564no—
526D4#654812yes—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—144no—
541C2×C4—288no—
542C2×C4C22×C22⋊D711yescharacteristic
543C2×C4C22×C22⋊D714yes—
544C2×C4C22×C22⋊D712yes—
545C2×C4C22×C22⋊D712yes—
546C2×C4C24×D711yescharacteristic
547C8—144no—
548C8—144no—
549C7#6166411yescharacteristic, complemented, Sylow
550C22—21no—
551C22—22no—
552C22—24no—
553C22—22no—
554C22—28no—
555C22—42no—
556C22—284no—
557C22—284no—
558C22—284no—
559C22—1128no—
560C22C23×C22⋊D711yescharacteristic
561C4—21no—
562C4—22no—
563C4—24no—
564C4—22no—
565C4—44no—
566C4—42no—
567C4—284no—
568C4#9052814yes—
569C4C2×#654811yescharacteristic
570C4C2×#654811yescharacteristic
571C2—44no—
572C2—41no—
573C2—42no—
574C2—564no—
575C2#1174430711yescharacteristic
576C2C2×#9052812yes—
subgroupG/H
centerC22C23×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC7×#3499C2
socleC22×C7C23×D4
hypercenter#3499D7
hypocenterC7#61664
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #532 ◁ #3499

Complements

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

NH ≅ G/Ntype
1. C7×#3499C2semidirect
2. #11743320C2semidirect
3. #11741558C2semidirect
4. D4×#1143C2semidirect
5. #11741000C2semidirect
6. #11740968C2semidirect
7. #11740956C2semidirect
8. #11740852C2semidirect
9. #11740723C2semidirect
10. #11740658C2semidirect
11. #11739974C2semidirect
12. C2×C7×2−1+4C22semidirect
13. C7×D4×Q8C22semidirect
15. C7×#547C22semidirect
17. C7×#536C22semidirect
21. C2×C7×C4⋊Q8C22semidirect
22. C4×C7×#41C22semidirect
24. C2×#6512C22semidirect
25. C22×#1143C22semidirect
33. #90010C22semidirect
34. #90010C22semidirect
35. #90006C22semidirect
36. #90005C22semidirect
37. #89999C22semidirect
38. #89999C22semidirect
39. #89997C22semidirect
41. #89855C22semidirect
42. #89855C22semidirect
43. #89853C22semidirect
44. #89821C22semidirect
45. #89820C22semidirect
46. #89818C22semidirect
47. #89817C22semidirect
49. C4×#1143C22semidirect
50. #89793C22semidirect
52. C4×#1141C22semidirect
53. #89784C22semidirect
54. D4×C7⋊C8C22semidirect
55. #89780C22semidirect
56. #89775C22semidirect
57. #89697C22semidirect
138. #3499D7semidirect
223. C2×2−1+4C2×D7semidirect
229. D4×Q8C2×D7semidirect
231. #547C2×D7semidirect
235. #536C2×D7semidirect
242. C2×C4⋊Q8C2×D7semidirect
243. C4×#41C2×D7semidirect
549. C7#61664semidirect

Maximal subgroups

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

subgroupindex
C7×#34992
#117433202
#117415582
D4×#11432
#117410002
#117409682
#117409562
#117408522
#117407232
#117406582
#117399742
#616647