Finite Group Databasebeta

← #11738327

Subgroups — #11738327

Order 896.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6474C211yesmaximal, characteristic, complemented
2C2×#6471C211yesmaximal, characteristic, complemented
3#90300C211yesmaximal, characteristic, complemented
4#90290C211yesmaximal, characteristic, complemented
5C7×#469C211yesmaximal, characteristic, complemented
6#89923C211yesmaximal, characteristic, complemented
7#89900C211yesmaximal, characteristic, complemented
8#89809C211yesmaximal, characteristic, complemented
9#89808C211yesmaximal, characteristic, complemented
10#89637C211yesmaximal, characteristic, complemented
11#89624C211yesmaximal, characteristic, complemented
12#89558C211yesmaximal, characteristic, complemented
13#89553C211yesmaximal, characteristic, complemented
14#89545C211yesmaximal, characteristic, complemented
15#89534C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic, complemented
17C2×D4×D7—21no—
18C2×D4×D7C2211yescharacteristic
19C2×#1157C2211yescharacteristic, complemented
20C22×D28C2211yescharacteristic
21C2×C7×D8C2211yescharacteristic, complemented
22C2×C7×M16C2211yescharacteristic, complemented
23C7×#121C2211yescharacteristic
24C7×#121C2211yescharacteristic, complemented
25#6503—21no—
26#6503C2211yescharacteristic
27#6503C2211yescharacteristic, complemented
28#6502—21no—
29#6500—21no—
30C2×#1142C2211yescharacteristic, complemented
31C2×#1141C2211yescharacteristic
32C2×#1141C2211yescharacteristic, complemented
33C2×#1141C2211yescharacteristic, complemented
34C4×C22⋊D7C2211yescharacteristic
35#6474—24no—
36#6471—24no—
37C2×D56C2211yescharacteristic
38C2×D56C2211yescharacteristic, complemented
39C2×#1132C2211yescharacteristic
40C2×#1132C2211yescharacteristic, complemented
41C2×#1131C2211yescharacteristic, complemented
42#6458C2211yescharacteristic
43#6458C2211yescharacteristic, complemented
44#6456C2211yescharacteristic
45#6450C2211yescharacteristic, complemented
46#6447—21no—
47#6447—21no—
48#6445—21no—
49(C22⋊D7)⋊C4—21no—
50C7×#106C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic, complemented
52C7×#102C2211yescharacteristic, complemented
53C22⋊(C7⋊C8)C2211yescharacteristic, complemented
54(C2×D7)⋊C8C2211yescharacteristic, complemented
55Dic7⋊C8C2211yescharacteristic, complemented
56C8×Dic7C2211yescharacteristic, complemented
57#6384C2211yescharacteristic, complemented
58#6383C2211yescharacteristic
59#6383C2211yescharacteristic, complemented
60#6383C2211yescharacteristic, complemented
61#3274—71nomaximal, Sylow
62C23×D7—21no—
63C23×D7—21no—
64C2×C7×D4—21no—
65C2×C7×D4—21no—
66C2×C7×D4C2311yescharacteristic
67C2×C7×D4C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72C2×C22⋊D7—21no—
73C2×C22⋊D7—21no—
74C2×C22⋊D7—21no—
75C2×C22⋊D7—21no—
76C2×C22⋊D7D411yescharacteristic
77C22×Dic7—21no—
78#1159—22no—
79#1159—41no—
80D4×D7—22no—
81D4×D7—41no—
82D4×D7—44no—
83#1157—22no—
84#1157—41no—
85C2×D28—22no—
86C2×D28—21no—
87C2×D28—21no—
88C2×D28—21no—
89C2×D28—41no—
90C2×D28C2311yescharacteristic
91C2×D28C2311yescharacteristic
92C2×D28C2311yescharacteristic
93C2×C4×D7—21no—
94C2×C4×D7—21no—
95C2×C4×D7C2311yescharacteristic
96C2×Dic14C2311yescharacteristic
97C7×D8—22no—
98C7×M16—22no—
99C2×C7×C8C2311yescharacteristic
100C2×C7×C8C2311yescharacteristic
101C7×C4⋊C4C2311yescharacteristic
102C7×C4⋊C4C2311yescharacteristic
103C7×C22⋊C4—21no—
104C7×C22⋊C4—21no—
105C22⋊Dic7D411yescharacteristic
106#1142—22no—
107#1141—22no—
108#1141—41no—
109#1141—41no—
110(C2×D7)⋊C4—21no—
111(C2×D7)⋊C4—21no—
112(C2×D7)⋊C4—21no—
113(C2×D7)⋊C4D411yescharacteristic
114Dic7⋊C4—21no—
115Dic7⋊C4D411yescharacteristic
116C4×Dic7—21no—
117C4×Dic7C2311yescharacteristic
118C2×C7⋊C8C2311yescharacteristic
119C2×C7⋊C8C2311yescharacteristic
120D56—22no—
121D56—22no—
122#1132—22no—
123#1132D414yescomplemented
124#1131—22no—
125C2×AGL(1,ℤ/8)—71no—
126C2×AGL(1,ℤ/8)—71no—
127#550—71no—
128D42—71no—
129#493—71no—
130#469—71no—
131#469D711yescharacteristic, complemented
132#460—71no—
133#459—71no—
134#449—71no—
135#449—71no—
136#447—71no—
137#447—71no—
138C4×SD16—71no—
139#436—71no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7C2×D411yescharacteristic
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—42no—
147C22×D7—42no—
148C22×D7—42no—
149C22×D7—42no—
150C22×D7—81no—
151C22×D7—81no—
152C22×D7C2×D411yescharacteristic
153C7×D4—21no—
154C7×D4—21no—
155C7×D4—21no—
156C7×D4—21no—
157C7×D4—42no—
158C7×D4—81no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7C2×D411yescharacteristic
164C2×C4×C7C2411yescharacteristic
165C22⋊D7—22no—
166C22⋊D7—42no—
167C22⋊D7—42no—
168C22⋊D7—42no—
169C22⋊D7—42no—
170C22⋊D7—42no—
171C22⋊D7—81no—
172C22⋊D7—81no—
173C2×Dic7—21no—
174C2×Dic7—42no—
175C2×Dic7C2×D411yescharacteristic
176C2×Dic7C2×D411yescharacteristic
177C2×Dic7C2×D411yescharacteristic
178D28—21no—
179D28—21no—
180D28—21no—
181D28—21no—
182D28—21no—
183D28—42no—
184D28—42no—
185D28—81no—
186D28C2×D412yes—
187C4×D7—21no—
188C4×D7—21no—
189C4×D7—42no—
190C4×D7—42no—
191Dic14—21no—
192Dic14C2×D412yescomplemented
193C7×C8—21no—
194C7×C8C2×D412yescomplemented
195C7⋊C8—21no—
196C7⋊C8—21no—
197C2×#41—71no—
198C2×#41—71no—
199C22×D4—71no—
200C22×D4—71no—
201C22×D4—141no—
202AGL(1,ℤ/8)—144no—
203AGL(1,ℤ/8)—144no—
204C2×SD16—71no—
205C2×SD16—71no—
206C2×SD16—71no—
207C2×D8—71no—
208C2×D8—71no—
209C2×D8—71no—
210C2×D8—71no—
211C2×D8—71no—
212C2×D8C2×D711yescharacteristic, complemented
213C2×M16—71no—
214C2×M16C2×D711yescharacteristic, complemented
215#127—71no—
216#127—71no—
217#127—141no—
218#121—71no—
219#121—71no—
220#121—141no—
221#121—141no—
222#121—141no—
223#121C2×D711yescharacteristic
224#121C2×D711yescharacteristic, complemented
225#120—141no—
226#120—141no—
227C4×Q8—71no—
228C4×D4—71no—
229C4×D4—71no—
230C4×D4—141no—
231#106C2×D711yescharacteristic, complemented
232C4⋊C8—71no—
233#103—71no—
234#102—71no—
235#102—71no—
236#102—71no—
237#102C2×D711yescharacteristic, complemented
238#102C2×D711yescharacteristic, complemented
239C22⋊C8—71no—
240C22⋊C8—71no—
241C4×C8—71no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7C22×D411yescharacteristic
252C2×D7—21no—
253C2×D7—22no—
254C2×D7—21no—
255C2×D7—21no—
256C2×D7—41no—
257C2×D7—41no—
258C2×D7—41no—
259C2×D7—41no—
260C2×D7—41no—
261C2×D7—41no—
262C2×D7—41no—
263C2×D7—41no—
264C2×D7—84no—
265C2×D7—84no—
266C4×C7—21no—
267C4×C7—41no—
268C4×C7—41no—
269C4×C7C22×D411yescharacteristic
270C4×C7H(5,𝔽2)11yescharacteristic
271Dic7—22no—
272Dic7—21no—
273Dic7—21no—
274Dic7AGL(1,ℤ/8)12yescomplemented
275C24—141no—
276C24—141no—
277#41—142no—
278#41—142no—
279#41—281no—
280#41—281no—
281C2×Q8—71no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—71no—
285C2×D4—71no—
286C2×D4—71no—
287C2×D4—71no—
288C2×D4—142no—
289C2×D4—141no—
290C2×D4—141no—
291C2×D4—141no—
292C2×D4—141no—
293C2×D4—141no—
294C2×D4—141no—
295C2×D4—141no—
296C2×D4—142no—
297C2×D4—141no—
298C2×D4—141no—
299C2×D4—141no—
300C2×D4—281no—
301C2×D4—281no—
302C2×D4—284no—
303C2×D4C22×D711yescharacteristic
304C2×D4C22×D711yescharacteristic
305C22×C4—71no—
306C22×C4—141no—
307C22×C4—141no—
308C22×C4—141no—
309C22×C4C22×D711yescharacteristic
310SD16—74no—
311SD16—142no—
312SD16—142no—
313D8—22no—
314D8—142no—
315D8—142no—
316D8—142no—
317D8—281no—
318D8—281no—
319M16—22no—
320M16—142no—
321C2×C8—71no—
322C2×C8—71no—
323C2×C8C22×D711yescharacteristic
324C2×C8C22×D711yescharacteristic
325C4⋊C4—71no—
326C4⋊C4—141no—
327C4⋊C4C22×D711yescharacteristic
328C4⋊C4C22×D711yescharacteristic
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—71no—
332C22⋊C4—71no—
333C22⋊C4—141no—
334C22⋊C4—141no—
335C22⋊C4—141no—
336C42—71no—
337C42—141no—
338C2×C7—41no—
339C2×C7—81no—
340C2×C7—81no—
341C2×C7D4211yescharacteristic
342C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
343C2×C7#57611yescharacteristic
344D7—41no—
345D7—42no—
346D7—81no—
347D7—81no—
348C23—21no—
349C23—21no—
350C23—71no—
351C23—141no—
352C23—141no—
353C23—141no—
354C23—282no—
355C23—282no—
356C23—282no—
357C23—282no—
358C23—561no—
359C23—561no—
360C23D4×D711yescharacteristic
361Q8—72no—
362Q8—141no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—21no—
367D4—42no—
368D4—72no—
369D4—81no—
370D4—142no—
371D4—141no—
372D4—141no—
373D4—141no—
374D4—141no—
375D4—141no—
376D4—282no—
377D4—282no—
378D4—282no—
379D4—282no—
380D4—282no—
381D4—282no—
382D4—282no—
383D4—561no—
384D4—561no—
385D4—561no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—71no—
391C2×C4—71no—
392C2×C4—71no—
393C2×C4—141no—
394C2×C4—141no—
395C2×C4—141no—
396C2×C4—282no—
397C2×C4—282no—
398C2×C4—282no—
399C2×C4D4×D711yescharacteristic
400C2×C4C23×D711yescharacteristic
401C8—21no—
402C8—141no—
403C8—141no—
404C8D4×D712yescomplemented
405C7#327411yescharacteristic, complemented, Sylow
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—141no—
416C22—141no—
417C22—141no—
418C22—142no—
419C22—281no—
420C22—281no—
421C22—281no—
422C22—281no—
423C22—281no—
424C22—281no—
425C22—281no—
426C22—281no—
427C22—564no—
428C22—564no—
429C22C2×D4×D711yescharacteristic
430C4—21no—
431C4—41no—
432C4—41no—
433C4—72no—
434C4—141no—
435C4—141no—
436C4—142no—
437C4C2×D4×D711yescharacteristic
438C4#655311yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—281no—
443C2—282no—
444C2—561no—
445C2—561no—
446C2#9030011yescharacteristic
447C2D7×AGL(1,ℤ/8)11yescharacteristic
448C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#469C2
socleC22×C7C22×D4
hypercenter#469D7
hypocenterC7#3274
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#6474C2semidirect
2. C2×#6471C2semidirect
3. #90300C2semidirect
4. #90290C2semidirect
5. C7×#469C2semidirect
6. #89923C2semidirect
7. #89900C2semidirect
8. #89809C2semidirect
9. #89808C2semidirect
10. #89637C2semidirect
11. #89624C2semidirect
12. #89558C2semidirect
13. #89553C2semidirect
14. #89545C2semidirect
15. #89534C2semidirect
16. C2×#1159C22semidirect
19. C2×#1157C22semidirect
21. C2×C7×D8C22semidirect
22. C2×C7×M16C22semidirect
24. C7×#121C22semidirect
27. #6503C22semidirect
30. C2×#1142C22semidirect
32. C2×#1141C22semidirect
33. C2×#1141C22semidirect
38. C2×D56C22semidirect
40. C2×#1132C22semidirect
41. C2×#1131C22semidirect
43. #6458C22semidirect
45. #6450C22semidirect
50. C7×#106C22semidirect
51. C7×#102C22semidirect
52. C7×#102C22semidirect
53. C22⋊(C7⋊C8)C22semidirect
54. (C2×D7)⋊C8C22semidirect
55. Dic7⋊C8C22semidirect
56. C8×Dic7C22semidirect
57. #6384C22semidirect
59. #6383C22semidirect
60. #6383C22semidirect
123. #1132D4semidirect
131. #469D7semidirect
192. Dic14C2×D4semidirect
194. C7×C8C2×D4semidirect
212. C2×D8C2×D7semidirect
214. C2×M16C2×D7semidirect
224. #121C2×D7semidirect
231. #106C2×D7semidirect
237. #102C2×D7semidirect
238. #102C2×D7semidirect
274. Dic7AGL(1,ℤ/8)semidirect
404. C8D4×D7semidirect
405. C7#3274semidirect

Maximal subgroups

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

subgroupindex
C2×#64742
C2×#64712
#903002
#902902
C7×#4692
#899232
#899002
#898092
#898082
#896372
#896242
#895582
#895532
#895452
#895342
#32747