Finite Group Databasebeta

← #11737980

Subgroups — #11737980

Order 896.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6478C211yesmaximal, characteristic, complemented
2C2×#6474C211yesmaximal, characteristic
3D7×#124C211yesmaximal, characteristic, complemented
4C4×#1159C211yesmaximal, characteristic, complemented
5C7×#463C211yesmaximal, characteristic, complemented
6#89937C211yesmaximal, characteristic, complemented
7#89925C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89788C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic, complemented
11#89609C211yesmaximal, characteristic, complemented
12#89574C211yesmaximal, characteristic, complemented
13#89570C211yesmaximal, characteristic, complemented
14#89542C211yesmaximal, characteristic, complemented
15#89539C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×D4×D7C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic, complemented
20C2×C7×D8C2211yescharacteristic
21C7×#124C2211yescharacteristic
22C4×C7×D4C2211yescharacteristic, complemented
23#6510C2211yescharacteristic
24C2×#1144C2211yescharacteristic, complemented
25#6499C2211yescharacteristic
26D4×Dic7C2211yescharacteristic, complemented
27C2×#1142C2211yescharacteristic
28C2×#1142C2211yescharacteristic, complemented
29C2×#1141C2211yescharacteristic
30C4×C22⋊D7—21no—
31C2×C4×Dic7—21no—
32#6478—24no—
33#6474—24no—
34C2×Dic28C2211yescharacteristic, complemented
35C2×#1132C2211yescharacteristic
36C2×#1131C2211yescharacteristic
37C2×#1131C2211yescharacteristic
38#6455C2211yescharacteristic
39#6450C2211yescharacteristic, complemented
40#6448—21no—
41#6448—21no—
42(C22⋊D7)⋊C4—21no—
43D7×C22⋊C4—21no—
44D7×C22⋊C4—21no—
45#6440—21no—
46#6438C2211yescharacteristic
47C42×D7C2211yescharacteristic
48C4×Dic14C2211yescharacteristic, complemented
49C7×C4⋊C8C2211yescharacteristic, complemented
50C7×#103C2211yescharacteristic, complemented
51C7×#102C2211yescharacteristic
52#6409C2211yescharacteristic, complemented
53#6406C2211yescharacteristic
54#6395C2211yescharacteristic
55#6390C2211yescharacteristic, complemented
56Dic7⋊C8C2211yescharacteristic, complemented
57Dic7⋊C8C2211yescharacteristic, complemented
58#6384C2211yescharacteristic, complemented
59#6383C2211yescharacteristic
60C4⋊(C7⋊C8)C2211yescharacteristic, complemented
61#3040—71nomaximal, Sylow
62C23×D7—21no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C22×Dic7—21no—
70C22×Dic7—21no—
71Q8×D7—22no—
72Q8×D7—22no—
73#1159—22no—
74#1159D414yes—
75D4×D7—22no—
76D4×D7—22no—
77C2×D28C2311yescharacteristic
78C2×C4×D7—21no—
79C2×C4×D7D411yescharacteristic
80C2×C4×D7D411yescharacteristic
81C2×C4×D7C2311yescharacteristic
82C2×Dic14—21no—
83C2×Dic14C2311yescharacteristic
84C2×Dic14C2311yescharacteristic
85C7×SD16—22no—
86C7×D8—22no—
87C2×C7×C8C2311yescharacteristic
88C2×C7×C8C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C22⋊C4—21no—
91C7×C22⋊C4—21no—
92C7×C22⋊C4—21no—
93C42×C7C2311yescharacteristic
94C22⋊Dic7—21no—
95C22⋊Dic7—21no—
96C22⋊Dic7—21no—
97#1144—22no—
98#1142—22no—
99#1142—22no—
100#1141—22no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103(C2×D7)⋊C4—21no—
104(C2×D7)⋊C4—21no—
105(C2×D7)⋊C4—21no—
106C4⋊Dic7C2311yescharacteristic
107Dic7⋊C4—21no—
108Dic7⋊C4—21no—
109C4×Dic7—21no—
110C4×Dic7—21no—
111C4×Dic7—21no—
112C4×Dic7D411yescharacteristic
113C4×Dic7D411yescharacteristic
114C4×Dic7C2311yescharacteristic
115C2×C7⋊C8C2311yescharacteristic
116C2×C7⋊C8C2311yescharacteristic
117Dic28—22no—
118#1132—22no—
119#1131—22no—
120#1131—22no—
121C2×#137—71no—
122C2×AGL(1,ℤ/8)—71no—
123C2×#124—71no—
124C4×#41—71no—
125#464—71no—
126#464—71no—
127#464—71no—
128#464—71no—
129#463—71no—
130#463—71no—
131#463—71no—
132#463D711yescharacteristic, complemented
133#423—71no—
134#419—71no—
135#418—71no—
136C23×C7—21no—
137C23×C7—21no—
138C22×D7—21no—
139C22×D7—42no—
140C22×D7—42no—
141C22×D7—42no—
142C22×D7C2×D411yescharacteristic
143C7×Q8—21no—
144C7×Q8—21no—
145C7×D4—21no—
146C7×D4—21no—
147C7×D4—21no—
148C7×D4C2×D412yes—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—42no—
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2×D411yescharacteristic
154C2×C4×C7C2411yescharacteristic
155C22⋊D7—24no—
156C22⋊D7—42no—
157C2×Dic7—21no—
158C2×Dic7—24no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—42no—
162C2×Dic7C2×D411yescharacteristic
163C2×Dic7C2×D411yescharacteristic
164C2×Dic7C2×D411yescharacteristic
165D28—21no—
166D28—21no—
167C4×D7—41no—
168C4×D7—41no—
169C4×D7—42no—
170C4×D7C2×D412yes—
171C4×D7#4112yes—
172Dic14—21no—
173Dic14—21no—
174Dic14—21no—
175Dic14—42no—
176Dic14C2×D412yes—
177C7×C8—21no—
178C7×C8—21no—
179C7⋊C8—21no—
180C7⋊C8—21no—
181C2×#41—71no—
182C22×Q8—71no—
183C22×D4—71no—
184#137—144no—
185AGL(1,ℤ/8)—144no—
186C2×Q16—71no—
187C2×Q16—71no—
188C2×SD16—71no—
189C2×SD16—71no—
190C2×SD16—71no—
191C2×SD16C2×D711yescharacteristic, complemented
192C2×D8—71no—
193C2×D8C2×D711yescharacteristic
194C2×M16—71no—
195C2×M16—71no—
196#124—71no—
197#124—71no—
198#124—71no—
199#124—141no—
200#124—141no—
201#124C2×D711yescharacteristic
202C4×Q8—71no—
203C4×Q8—71no—
204C4×D4—71no—
205C4×D4—141no—
206C4×D4—141no—
207C4×D4C2×D711yescharacteristic, complemented
208#117—71no—
209#117—141no—
210C2×C22⋊C4—141no—
211C2×C22⋊C4—141no—
212C2×C42—71no—
213C2×C42—141no—
214C4⋊C8—71no—
215C4⋊C8—71no—
216C4⋊C8—71no—
217C4⋊C8C2×D711yescharacteristic, complemented
218#103—71no—
219#103—71no—
220#103—71no—
221#103C2×D711yescharacteristic, complemented
222#102—71no—
223#102—71no—
224#102—71no—
225#102C2×D711yescharacteristic
226C22×C7—22no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7C22×D411yescharacteristic
233C2×D7—21no—
234C2×D7—21no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—84no—
239C2×D7#12112yes—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7C22×D411yescharacteristic
245C4×C7C2×#4111yescharacteristic
246Dic7—22no—
247Dic7—21no—
248Dic7—21no—
249Dic7—41no—
250Dic7—41no—
251Dic7#12112yes—
252C24—141no—
253#41—74no—
254#41—142no—
255C2×Q8—71no—
256C2×Q8—71no—
257C2×Q8—141no—
258C2×Q8—142no—
259C2×Q8—142no—
260C2×Q8C22×D711yescharacteristic
261C2×D4—71no—
262C2×D4—142no—
263C2×D4—142no—
264C2×D4—141no—
265C2×D4—141no—
266C2×D4C22×D711yescharacteristic
267C2×D4C22×D711yescharacteristic
268C22×C4—21no—
269C22×C4—71no—
270C22×C4—71no—
271C22×C4—71no—
272C22×C4—141no—
273C22×C4—141no—
274C22×C4—141no—
275Q16—142no—
276Q16—142no—
277SD16—22no—
278SD16—142no—
279SD16—142no—
280SD16—142no—
281D8—22no—
282D8—142no—
283M16—142no—
284M16—142no—
285C2×C8—71no—
286C2×C8—71no—
287C2×C8C22×D711yescharacteristic
288C2×C8C22×D711yescharacteristic
289C4⋊C4—71no—
290C4⋊C4—141no—
291C4⋊C4—141no—
292C4⋊C4C22×D711yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—141no—
297C22⋊C4—141no—
298C22⋊C4—141no—
299C22⋊C4—141no—
300C22⋊C4—141no—
301C22⋊C4—141no—
302C22⋊C4—141no—
303C22⋊C4—141no—
304C42—71no—
305C42—71no—
306C42—71no—
307C42—141no—
308C42—141no—
309C42—141no—
310C42C22×D711yescharacteristic
311C2×C7—42no—
312C2×C7—81no—
313C2×C7C2×#12111yescharacteristic
314C2×C7#57511yescharacteristic
315C2×C7#57511yescharacteristic
316D7—22no—
317D7—81no—
318C23—21no—
319C23—21no—
320C23—71no—
321C23—141no—
322C23—282no—
323C23—282no—
324C23—282no—
325Q8—21no—
326Q8—21no—
327Q8—72no—
328Q8—141no—
329Q8—141no—
330Q8—141no—
331Q8—282no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—144no—
336D4—141no—
337D4—141no—
338D4—282no—
339D4D4×D712yes—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—71no—
344C2×C4—72no—
345C2×C4—72no—
346C2×C4—71no—
347C2×C4—71no—
348C2×C4—144no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—281no—
353C2×C4—281no—
354C2×C4—282no—
355C2×C4—282no—
356C2×C4D4×D711yescharacteristic
357C2×C4D4×D711yescharacteristic
358C2×C4C23×D711yescharacteristic
359C8—21no—
360C8—21no—
361C8—141no—
362C8—141no—
363C7#304011yescharacteristic, complemented, Sylow
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—72no—
371C22—141no—
372C22—141no—
373C22—281no—
374C22—281no—
375C22—281no—
376C22—564no—
377C22C2×D4×D711yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—72no—
383C4—141no—
384C4—141no—
385C4—142no—
386C4—281no—
387C4—281no—
388C4C2×D4×D711yescharacteristic
389C4D7×#4111yescharacteristic
390C2—42no—
391C2—81no—
392C2—142no—
393C2—561no—
394C2D7×#12111yescharacteristic
395C2#9045911yescharacteristic
396C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#463C2
socleC22×C7C22×D4
prime radicalC2×#6474C2
hypercenter#463D7
hypocenterC7#3040
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×#6478C2semidirect
3. D7×#124C2semidirect
4. C4×#1159C2semidirect
5. C7×#463C2semidirect
6. #89937C2semidirect
7. #89925C2semidirect
8. #89826C2semidirect
9. #89788C2semidirect
10. #89617C2semidirect
11. #89609C2semidirect
12. #89574C2semidirect
13. #89570C2semidirect
14. #89542C2semidirect
15. #89539C2semidirect
19. C2×C7×SD16C22semidirect
22. C4×C7×D4C22semidirect
24. C2×#1144C22semidirect
26. D4×Dic7C22semidirect
28. C2×#1142C22semidirect
34. C2×Dic28C22semidirect
39. #6450C22semidirect
48. C4×Dic14C22semidirect
49. C7×C4⋊C8C22semidirect
50. C7×#103C22semidirect
52. #6409C22semidirect
55. #6390C22semidirect
56. Dic7⋊C8C22semidirect
57. Dic7⋊C8C22semidirect
58. #6384C22semidirect
60. C4⋊(C7⋊C8)C22semidirect
132. #463D7semidirect
191. C2×SD16C2×D7semidirect
207. C4×D4C2×D7semidirect
217. C4⋊C8C2×D7semidirect
221. #103C2×D7semidirect
363. C7#3040semidirect

Maximal subgroups

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

subgroupindex
C2×#64782
C2×#64742
D7×#1242
C4×#11592
C7×#4632
#899372
#899252
#898262
#897882
#896172
#896092
#895742
#895702
#895422
#895392
#30407