Finite Group Databasebeta

← #11737793

Subgroups — #11737793

Order 896.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6475C211yesmaximal, characteristic, complemented
2C2×#6474C211yesmaximal, characteristic, complemented
3#90409C211yesmaximal, characteristic, complemented
4#90245C211yesmaximal, characteristic, complemented
5C7×#459C211yesmaximal, characteristic, complemented
6#89930C211yesmaximal, characteristic, complemented
7#89928C211yesmaximal, characteristic, complemented
8#89846C211yesmaximal, characteristic, complemented
9#89790C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic, complemented
11#89608C211yesmaximal, characteristic, complemented
12#89555C211yesmaximal, characteristic, complemented
13#89553C211yesmaximal, characteristic, complemented
14#89552C211yesmaximal, characteristic, complemented
15#89547C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×#1159C2211yescharacteristic, complemented
19C2×D4×D7C2211yescharacteristic
20C2×C7×D8C2211yescharacteristic
21C2×C7×D8C2211yescharacteristic
22C7×#127C2211yescharacteristic, complemented
23C4×C7×D4C2211yescharacteristic
24#6503—21no—
25#6502—21no—
26#6502—21no—
27#6501C2211yescharacteristic
28#6501C2211yescharacteristic, complemented
29D4×Dic7—21no—
30C2×#1142C2211yescharacteristic
31C2×#1142C2211yescharacteristic, complemented
32C2×#1142C2211yescharacteristic, complemented
33C2×#1141C2211yescharacteristic
34C2×#1141C2211yescharacteristic
35#6493—21no—
36C2×(C2×D7)⋊C4—21no—
37#6475—24no—
38#6474—24no—
39C2×Dic28C2211yescharacteristic, complemented
40C2×#1132C2211yescharacteristic
41C2×#1131C2211yescharacteristic
42C2×C8×D7C2211yescharacteristic, complemented
43#6460C2211yescharacteristic
44#6458C2211yescharacteristic
45#6449—21no—
46#6445—21no—
47#6438C2211yescharacteristic
48C4×D28C2211yescharacteristic
49C4×Dic14C2211yescharacteristic, complemented
50C7×C4⋊C8C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic, complemented
52C7×#102C2211yescharacteristic, complemented
53#6406C2211yescharacteristic
54#6395C2211yescharacteristic
55(C2×D7)⋊C8C2211yescharacteristic
56(C2×D7)⋊C8C2211yescharacteristic, complemented
57#6390C2211yescharacteristic, complemented
58#6384C2211yescharacteristic, complemented
59#6381C2211yescharacteristic, complemented
60#6377C2211yescharacteristic, complemented
61#3268—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×D4—21no—
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C2×C7×D4C2311yescharacteristic
68C22×C4×C7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72C2×C22⋊D7—21no—
73C2×C22⋊D7—21no—
74C2×C22⋊D7—21no—
75C22×Dic7—21no—
76C22×Dic7—21no—
77C22×Dic7—21no—
78#1159—22no—
79#1159—22no—
80#1159—22no—
81#1159—41no—
82#1159—41no—
83#1159—41no—
84D4×D7—22no—
85D4×D7—41no—
86C2×D28—21no—
87C2×D28D411yescharacteristic
88C2×D28C2311yescharacteristic
89C2×C4×D7C2311yescharacteristic
90C2×C4×D7C2311yescharacteristic
91C2×Dic14C2311yescharacteristic
92C2×Dic14C2311yescharacteristic
93C7×D8—22no—
94C7×D8—22no—
95C2×C7×C8C2311yescharacteristic
96C2×C7×C8C2311yescharacteristic
97C7×C4⋊C4C2311yescharacteristic
98C7×C22⋊C4—21no—
99C42×C7C2311yescharacteristic
100C22⋊Dic7—21no—
101C22⋊Dic7—21no—
102#1142—22no—
103#1142—41no—
104#1142D414yescomplemented
105#1141—22no—
106#1141—41no—
107(C2×D7)⋊C4—21no—
108(C2×D7)⋊C4—21no—
109(C2×D7)⋊C4—21no—
110(C2×D7)⋊C4—21no—
111(C2×D7)⋊C4D411yescharacteristic
112(C2×D7)⋊C4D411yescharacteristic
113C4⋊Dic7—21no—
114C4⋊Dic7D411yescharacteristic
115C4⋊Dic7C2311yescharacteristic
116Dic7⋊C4—21no—
117C4×Dic7—21no—
118C2×C7⋊C8C2311yescharacteristic
119C2×C7⋊C8C2311yescharacteristic
120Dic28—22no—
121#1132—22no—
122#1131—22no—
123C8×D7—22no—
124C2×AGL(1,ℤ/8)—71no—
125C2×#135—71no—
126#550—71no—
127#546—71no—
128#496—71no—
129#468—71no—
130#466—71no—
131#464—71no—
132#459D711yescharacteristic, complemented
133#452—71no—
134#449—71no—
135#449—71no—
136#447—71no—
137#440—71no—
138#435—71no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C22×D7—21no—
143C22×D7—42no—
144C22×D7—42no—
145C22×D7—81no—
146C22×D7C2×D411yescharacteristic
147C22×D7C2×D411yescharacteristic
148C7×D4—21no—
149C7×D4—21no—
150C7×D4—21no—
151C7×D4—21no—
152C7×D4—21no—
153C7×D4—42no—
154C7×D4—81no—
155C7×D4C2×D412yes—
156C2×C4×C7—21no—
157C2×C4×C7—81no—
158C2×C4×C7C2×D411yescharacteristic
159C2×C4×C7C2×D411yescharacteristic
160C2×C4×C7C2411yescharacteristic
161C22⋊D7—42no—
162C22⋊D7—42no—
163C22⋊D7—42no—
164C22⋊D7—42no—
165C22⋊D7—81no—
166C22⋊D7—81no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—42no—
171C2×Dic7—42no—
172C2×Dic7—42no—
173C2×Dic7C2×D411yescharacteristic
174C2×Dic7C2×D411yescharacteristic
175D28—21no—
176D28—21no—
177D28—41no—
178D28—81no—
179C4×D7—21no—
180C4×D7—21no—
181C4×D7—21no—
182C4×D7—21no—
183Dic14—21no—
184Dic14—21no—
185Dic14—21no—
186Dic14C2×D412yes—
187C7×C8—21no—
188C7×C8—21no—
189C7⋊C8—21no—
190C7⋊C8C2×D412yes—
191C2×#41—71no—
192C2×#41—71no—
193C2×#41—71no—
194C22×D4—71no—
195AGL(1,ℤ/8)—144no—
196#135—144no—
197C2×Q16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16—71no—
201C2×SD16—71no—
202C2×D8—71no—
203C2×D8—71no—
204C2×D8C2×D711yescharacteristic
205C2×D8C2×D711yescharacteristic
206C2×M16—71no—
207C22×C8—71no—
208#127—141no—
209#127C2×D711yescharacteristic, complemented
210#124—71no—
211#123—141no—
212C22⋊Q8—71no—
213#121—71no—
214#121—71no—
215#121—71no—
216#121—141no—
217#121—141no—
218#121—141no—
219#120—141no—
220C4×Q8—71no—
221C4×D4—71no—
222C4×D4—141no—
223C4×D4C2×D711yescharacteristic
224C2×C22⋊C4—141no—
225#107—71no—
226C4⋊C8C2×D711yescharacteristic, complemented
227#103—71no—
228#103—71no—
229#102—71no—
230#102—71no—
231#102C2×D711yescharacteristic, complemented
232#102C2×D711yescharacteristic, complemented
233C22⋊C8—71no—
234C22⋊C8—71no—
235#97—71no—
236C22×C7—22no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7C22×D411yescharacteristic
246C2×D7—21no—
247C2×D7—21no—
248C2×D7—21no—
249C2×D7—21no—
250C2×D7—21no—
251C2×D7—21no—
252C2×D7—41no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—84no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—41no—
259C4×C7C22×D411yescharacteristic
260C4×C7H(5,𝔽2)11yescharacteristic
261Dic7—22no—
262Dic7—21no—
263Dic7—21no—
264Dic7—41no—
265Dic7—41no—
266C24—141no—
267#41—142no—
268#41—142no—
269#41—142no—
270#41—281no—
271#41—281no—
272#41—281no—
273C2×Q8—71no—
274C2×Q8—71no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—71no—
278C2×D4—71no—
279C2×D4—141no—
280C2×D4—141no—
281C2×D4—142no—
282C2×D4—141no—
283C2×D4—141no—
284C2×D4—141no—
285C2×D4—141no—
286C2×D4—141no—
287C2×D4—281no—
288C2×D4C22×D711yescharacteristic
289C2×D4C22×D711yescharacteristic
290C2×D4C22×D711yescharacteristic
291C22×C4—21no—
292C22×C4—71no—
293C22×C4—71no—
294C22×C4—141no—
295C22×C4—141no—
296C22×C4—141no—
297Q16—142no—
298SD16—74no—
299SD16—142no—
300SD16—142no—
301SD16—281no—
302D8—22no—
303D8—22no—
304D8—142no—
305D8—281no—
306M16—142no—
307C2×C8—71no—
308C2×C8—71no—
309C2×C8—142no—
310C2×C8C22×D711yescharacteristic
311C2×C8C22×D711yescharacteristic
312C4⋊C4—71no—
313C4⋊C4—71no—
314C4⋊C4—141no—
315C4⋊C4—141no—
316C4⋊C4C22×D711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—71no—
319C22⋊C4—71no—
320C22⋊C4—141no—
321C22⋊C4—141no—
322C22⋊C4—141no—
323C22⋊C4—141no—
324C22⋊C4—141no—
325C22⋊C4—141no—
326C42—141no—
327C42C22×D711yescharacteristic
328C2×C7—42no—
329C2×C7—81no—
330C2×C7—81no—
331C2×C7D4211yescharacteristic
332C2×C7#57511yescharacteristic
333C2×C7#57611yescharacteristic
334D7—41no—
335D7—41no—
336D7—81no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—71no—
341C23—71no—
342C23—141no—
343C23—282no—
344C23—282no—
345C23—561no—
346Q8—72no—
347Q8—141no—
348Q8—141no—
349Q8—141no—
350D4—21no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—81no—
357D4—141no—
358D4—141no—
359D4—281no—
360D4—282no—
361D4—282no—
362D4—282no—
363D4—282no—
364D4—561no—
365D4—561no—
366D4—561no—
367D4D4×D712yes—
368C2×C4—21no—
369C2×C4—71no—
370C2×C4—71no—
371C2×C4—81no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—141no—
376C2×C4—141no—
377C2×C4—141no—
378C2×C4—141no—
379C2×C4—282no—
380C2×C4—282no—
381C2×C4—282no—
382C2×C4D4×D711yescharacteristic
383C2×C4D4×D711yescharacteristic
384C2×C4C23×D711yescharacteristic
385C8—21no—
386C8—21no—
387C8—72no—
388C8—141no—
389C7#326811yescharacteristic, complemented, Sylow
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—141no—
400C22—141no—
401C22—141no—
402C22—141no—
403C22—141no—
404C22—141no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22—564no—
409C22C2×D4×D711yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—141no—
414C4—141no—
415C4—142no—
416C4—281no—
417C4—281no—
418C4C2×D4×D711yescharacteristic
419C4#655311yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—281no—
424C2—281no—
425C2—561no—
426C2#9030011yescharacteristic
427C2#9044811yescharacteristic
428C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#459C2
socleC22×C7C22×D4
hypercenter#459D7
hypocenterC7#3268
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#6475C2semidirect
2. C2×#6474C2semidirect
3. #90409C2semidirect
4. #90245C2semidirect
5. C7×#459C2semidirect
6. #89930C2semidirect
7. #89928C2semidirect
8. #89846C2semidirect
9. #89790C2semidirect
10. #89617C2semidirect
11. #89608C2semidirect
12. #89555C2semidirect
13. #89553C2semidirect
14. #89552C2semidirect
15. #89547C2semidirect
18. C2×#1159C22semidirect
22. C7×#127C22semidirect
28. #6501C22semidirect
31. C2×#1142C22semidirect
32. C2×#1142C22semidirect
39. C2×Dic28C22semidirect
42. C2×C8×D7C22semidirect
49. C4×Dic14C22semidirect
50. C7×C4⋊C8C22semidirect
51. C7×#102C22semidirect
52. C7×#102C22semidirect
56. (C2×D7)⋊C8C22semidirect
57. #6390C22semidirect
58. #6384C22semidirect
59. #6381C22semidirect
60. #6377C22semidirect
104. #1142D4semidirect
132. #459D7semidirect
209. #127C2×D7semidirect
226. C4⋊C8C2×D7semidirect
231. #102C2×D7semidirect
232. #102C2×D7semidirect
389. C7#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#64752
C2×#64742
#904092
#902452
C7×#4592
#899302
#899282
#898462
#897902
#896172
#896082
#895552
#895532
#895522
#895472
#32687