Finite Group Databasebeta

← #11737981

Subgroups — #11737981

Order 896.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6477C211yesmaximal, characteristic, complemented
2C2×D7×D8C211yesmaximal, characteristic, complemented
3#90365C211yesmaximal, characteristic, complemented
4#90236C211yesmaximal, characteristic, complemented
5C7×#463C211yesmaximal, characteristic, complemented
6#89937C211yesmaximal, characteristic, complemented
7#89922C211yesmaximal, characteristic, complemented
8#89829C211yesmaximal, characteristic, complemented
9#89787C211yesmaximal, characteristic, complemented
10#89615C211yesmaximal, characteristic, complemented
11#89611C211yesmaximal, characteristic, complemented
12#89591C211yesmaximal, characteristic, complemented
13#89576C211yesmaximal, characteristic, complemented
14#89558C211yesmaximal, characteristic, complemented
15#89542C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×D4×D7C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic, complemented
20C2×C7×D8C2211yescharacteristic
21C7×#124C2211yescharacteristic, complemented
22C4×C7×D4C2211yescharacteristic, complemented
23#6510C2211yescharacteristic, complemented
24C2×#1143C2211yescharacteristic, complemented
25#6503C2211yescharacteristic, complemented
26D4×Dic7C2211yescharacteristic, complemented
27C2×#1141C2211yescharacteristic
28C2×#1141C2211yescharacteristic
29C2×#1141C2211yescharacteristic, complemented
30C4×C22⋊D7—21no—
31C2×(C2×D7)⋊C4—21no—
32#6477—24no—
33D7×D8—24no—
34C2×D56C2211yescharacteristic
35C2×D56C2211yescharacteristic, complemented
36C2×#1131C2211yescharacteristic
37C2×C8×D7C2211yescharacteristic
38#6456C2211yescharacteristic, complemented
39#6455C2211yescharacteristic
40#6447—21no—
41#6447—21no—
42#6445—21no—
43#6445—21no—
44(C22⋊D7)⋊C4—21no—
45D7×C22⋊C4—21no—
46#6437C2211yescharacteristic, complemented
47C4×D28C2211yescharacteristic, complemented
48#6435C2211yescharacteristic
49C7×C4⋊C8C2211yescharacteristic, complemented
50C7×#103C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic, complemented
52#6409C2211yescharacteristic, complemented
53#6406C2211yescharacteristic, complemented
54#6395C2211yescharacteristic, complemented
55#6395C2211yescharacteristic, complemented
56Dic7⋊C8C2211yescharacteristic, complemented
57Dic7⋊C8C2211yescharacteristic, complemented
58#6383C2211yescharacteristic, complemented
59#6383C2211yescharacteristic, complemented
60C4⋊(C7⋊C8)C2211yescharacteristic, complemented
61#3041—71nomaximal, Sylow
62C23×D7—21no—
63C23×D7—21no—
64C2×C7×Q8C2311yescharacteristic
65C2×C7×D4C2311yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7—21no—
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72C22×Dic7—21no—
73#1161—22no—
74#1161—22no—
75D4×D7—22no—
76D4×D7—22no—
77D4×D7—22no—
78D4×D7D414yescomplemented
79C2×D28—21no—
80C2×D28—21no—
81C2×D28—21no—
82C2×D28C2311yescharacteristic
83C2×D28C2311yescharacteristic
84C2×D28C2311yescharacteristic
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×C4×D7C2311yescharacteristic
89C7×SD16—22no—
90C7×D8—22no—
91C2×C7×C8C2311yescharacteristic
92C2×C7×C8C2311yescharacteristic
93C7×C4⋊C4C2311yescharacteristic
94C7×C22⋊C4—21no—
95C7×C22⋊C4—21no—
96C7×C22⋊C4—21no—
97C42×C7C2311yescharacteristic
98C22⋊Dic7—21no—
99#1143—22no—
100#1141—22no—
101#1141—22no—
102#1141—22no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107(C2×D7)⋊C4—21no—
108(C2×D7)⋊C4D411yescharacteristic
109(C2×D7)⋊C4D411yescharacteristic
110C4⋊Dic7C2311yescharacteristic
111Dic7⋊C4D411yescharacteristic
112Dic7⋊C4D411yescharacteristic
113C4×Dic7—21no—
114C4×Dic7C2311yescharacteristic
115C2×C7⋊C8C2311yescharacteristic
116C2×C7⋊C8C2311yescharacteristic
117D56—22no—
118D56—22no—
119#1131—22no—
120C8×D7—22no—
121C2×AGL(1,ℤ/8)—71no—
122C22×D8—71no—
123#535—71no—
124#518—71no—
125#463—71no—
126#463—71no—
127#463—71no—
128#463D711yescharacteristic, complemented
129#459—71no—
130#459—71no—
131#459—71no—
132#459—71no—
133#424—71no—
134#418—71no—
135#416—71no—
136C23×C7—21no—
137C23×C7—21no—
138C22×D7—21no—
139C22×D7—24no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—42no—
143C22×D7—81no—
144C22×D7—81no—
145C22×D7—81no—
146C22×D7—81no—
147C22×D7C2×D411yescharacteristic
148C7×Q8—21no—
149C7×Q8—21no—
150C7×D4—21no—
151C7×D4—21no—
152C7×D4—21no—
153C7×D4C2×D412yes—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—81no—
157C2×C4×C7C2×D411yescharacteristic
158C2×C4×C7C2×D411yescharacteristic
159C2×C4×C7C2411yescharacteristic
160C22⋊D7—24no—
161C22⋊D7—42no—
162C22⋊D7—81no—
163C22⋊D7—81no—
164C2×Dic7—21no—
165C2×Dic7—81no—
166C2×Dic7C2×D411yescharacteristic
167C2×Dic7C2×D411yescharacteristic
168C2×Dic7C2×D411yescharacteristic
169D28—21no—
170D28—21no—
171D28—21no—
172D28—21no—
173D28—21no—
174D28—42no—
175D28—81no—
176D28—81no—
177D28C2×D412yes—
178C4×D7—42no—
179C4×D7—81no—
180C4×D7—81no—
181C4×D7C2×D412yes—
182C4×D7#4112yes—
183C7×C8—21no—
184C7×C8—21no—
185C7⋊C8—21no—
186C7⋊C8—21no—
187C2×#41—71no—
188C22×D4—71no—
189C22×D4—71no—
190AGL(1,ℤ/8)—144no—
191C2×SD16—71no—
192C2×SD16C2×D711yescharacteristic, complemented
193C2×D8—71no—
194C2×D8—71no—
195C2×D8—71no—
196C2×D8—71no—
197C2×D8—71no—
198C2×D8—144no—
199C2×D8C2×D711yescharacteristic
200C2×M16—71no—
201C22×C8—71no—
202#127—71no—
203#127—71no—
204#124—71no—
205#124C2×D711yescharacteristic, complemented
206#121—141no—
207#121—141no—
208#120—141no—
209#120—141no—
210C4×D4—71no—
211C4×D4—71no—
212C4×D4—71no—
213C4×D4—141no—
214C4×D4—141no—
215C4×D4C2×D711yescharacteristic, complemented
216#117—71no—
217#117—71no—
218C2×C22⋊C4—141no—
219C2×C22⋊C4—141no—
220C4⋊C8—71no—
221C4⋊C8—71no—
222C4⋊C8—71no—
223C4⋊C8C2×D711yescharacteristic, complemented
224#103—71no—
225#103C2×D711yescharacteristic, complemented
226#102—71no—
227#102—71no—
228#102—71no—
229#102—71no—
230#102—71no—
231#102C2×D711yescharacteristic, complemented
232C22×C7—22no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—41no—
238C22×C7C22×D411yescharacteristic
239C2×D7—21no—
240C2×D7—21no—
241C2×D7—22no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—41no—
250C2×D7—84no—
251C2×D7—84no—
252C2×D7#12112yes—
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—41no—
256C4×C7—41no—
257C4×C7C22×D411yescharacteristic
258C4×C7C2×#4111yescharacteristic
259Dic7—21no—
260Dic7—21no—
261Dic7—41no—
262Dic7#12112yes—
263C24—141no—
264C24—141no—
265#41—142no—
266#41—142no—
267C2×Q8C22×D711yescharacteristic
268C2×D4—74no—
269C2×D4—71no—
270C2×D4—71no—
271C2×D4—71no—
272C2×D4—141no—
273C2×D4—142no—
274C2×D4—142no—
275C2×D4—142no—
276C2×D4—141no—
277C2×D4—141no—
278C2×D4—141no—
279C2×D4—141no—
280C2×D4—141no—
281C2×D4—141no—
282C2×D4C22×D711yescharacteristic
283C2×D4C22×D711yescharacteristic
284C22×C4—21no—
285C22×C4—71no—
286C22×C4—141no—
287C22×C4—141no—
288C22×C4—141no—
289C22×C4—141no—
290SD16—22no—
291SD16—142no—
292D8—22no—
293D8—142no—
294D8—142no—
295D8—142no—
296D8—142no—
297D8—142no—
298M16—142no—
299C2×C8—71no—
300C2×C8—71no—
301C2×C8—142no—
302C2×C8C22×D711yescharacteristic
303C2×C8C22×D711yescharacteristic
304C4⋊C4—71no—
305C4⋊C4—71no—
306C4⋊C4—71no—
307C4⋊C4C22×D711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—71no—
312C22⋊C4—71no—
313C22⋊C4—141no—
314C22⋊C4—141no—
315C22⋊C4—141no—
316C22⋊C4—141no—
317C22⋊C4—141no—
318C22⋊C4—141no—
319C42—71no—
320C42—141no—
321C42C22×D711yescharacteristic
322C2×C7—42no—
323C2×C7—81no—
324C2×C7C2×#12111yescharacteristic
325C2×C7#57611yescharacteristic
326C2×C7#57611yescharacteristic
327D7—22no—
328D7—42no—
329D7—81no—
330D7—81no—
331C23—21no—
332C23—21no—
333C23—71no—
334C23—144no—
335C23—141no—
336C23—141no—
337C23—141no—
338C23—282no—
339C23—561no—
340C23—561no—
341C23—561no—
342C23—561no—
343Q8—21no—
344Q8—21no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—72no—
349D4—144no—
350D4—141no—
351D4—141no—
352D4—141no—
353D4—141no—
354D4—141no—
355D4—282no—
356D4—282no—
357D4—561no—
358D4—561no—
359D4—561no—
360D4—561no—
361D4D4×D712yes—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—71no—
365C2×C4—72no—
366C2×C4—72no—
367C2×C4—71no—
368C2×C4—71no—
369C2×C4—81no—
370C2×C4—141no—
371C2×C4—282no—
372C2×C4—561no—
373C2×C4—561no—
374C2×C4—561no—
375C2×C4D4×D711yescharacteristic
376C2×C4D4×D711yescharacteristic
377C2×C4C23×D711yescharacteristic
378C8—21no—
379C8—21no—
380C8—141no—
381C8—141no—
382C7#304111yescharacteristic, complemented, Sylow
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—72no—
390C22—141no—
391C22—141no—
392C22—142no—
393C22—281no—
394C22—281no—
395C22—281no—
396C22—281no—
397C22—281no—
398C22—281no—
399C22—281no—
400C22—281no—
401C22—564no—
402C22—564no—
403C22C2×D4×D711yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—72no—
409C4—141no—
410C4—141no—
411C4—281no—
412C4C2×D4×D711yescharacteristic
413C4D7×#4111yescharacteristic
414C2—42no—
415C2—81no—
416C2—142no—
417C2—282no—
418C2—561no—
419C2—561no—
420C2D7×#12111yescharacteristic
421C2#9046011yescharacteristic
422C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#463C2
socleC22×C7C22×D4
hypercenter#463D7
hypocenterC7#3041
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#6477C2semidirect
2. C2×D7×D8C2semidirect
3. #90365C2semidirect
4. #90236C2semidirect
5. C7×#463C2semidirect
6. #89937C2semidirect
7. #89922C2semidirect
8. #89829C2semidirect
9. #89787C2semidirect
10. #89615C2semidirect
11. #89611C2semidirect
12. #89591C2semidirect
13. #89576C2semidirect
14. #89558C2semidirect
15. #89542C2semidirect
19. C2×C7×SD16C22semidirect
21. C7×#124C22semidirect
22. C4×C7×D4C22semidirect
23. #6510C22semidirect
24. C2×#1143C22semidirect
25. #6503C22semidirect
26. D4×Dic7C22semidirect
29. C2×#1141C22semidirect
35. C2×D56C22semidirect
38. #6456C22semidirect
46. #6437C22semidirect
47. C4×D28C22semidirect
49. C7×C4⋊C8C22semidirect
50. C7×#103C22semidirect
51. C7×#102C22semidirect
52. #6409C22semidirect
53. #6406C22semidirect
54. #6395C22semidirect
55. #6395C22semidirect
56. Dic7⋊C8C22semidirect
57. Dic7⋊C8C22semidirect
58. #6383C22semidirect
59. #6383C22semidirect
60. C4⋊(C7⋊C8)C22semidirect
78. D4×D7D4semidirect
128. #463D7semidirect
192. C2×SD16C2×D7semidirect
205. #124C2×D7semidirect
215. C4×D4C2×D7semidirect
223. C4⋊C8C2×D7semidirect
225. #103C2×D7semidirect
231. #102C2×D7semidirect
382. C7#3041semidirect

Maximal subgroups

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

subgroupindex
C2×#64772
C2×D7×D82
#903652
#902362
C7×#4632
#899372
#899222
#898292
#897872
#896152
#896112
#895912
#895762
#895582
#895422
#30417