Finite Group Databasebeta

← #11737583

Subgroups — #11737583

Order 896.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6503C211yesmaximal, characteristic
2#90313C211yesmaximal, characteristic, complemented
3C7×#450C211yesmaximal, characteristic, complemented
4#89990C211yesmaximal, characteristic, complemented
5#89947C212yesmaximal, complemented
6#89935C212yesmaximal, complemented
7#89818C211yesmaximal, characteristic, complemented
8#89534C212yesmaximal, complemented
9#89532C212yesmaximal, complemented
10#89503C211yesmaximal, characteristic, complemented
11#89490C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7—21no—
14C22×C22⋊D7—21no—
15C22×D28C2211yescharacteristic
16C2×C7×SD16C2212yescomplemented
17C7×C22⋊Q8C2211yescharacteristic
18Q8×Dic7C2211yescharacteristic, complemented
19#6507C2211yescharacteristic, complemented
20C2×#1143C2212yescomplemented
21#6503—21no—
22#6503—21no—
23#6503C2212yes—
24#6503C2212yes—
25C2×#1142C2212yescomplemented
26#6487C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28C2×#1132C2212yescomplemented
29#6452—21no—
30#6451C2211yescharacteristic, complemented
31#6450—21no—
32#6450C2211yescharacteristic, complemented
33#6441—21no—
34#6440—21no—
35C7×#102C2212yescomplemented
36C7×C22⋊C8C2211yescharacteristic
37#6406C2212yescomplemented
38C22⋊(C7⋊C8)C2211yescharacteristic
39#6395C2212yescomplemented
40Dic7⋊C8C2212yescomplemented
41C8×Dic7C2212yescomplemented
42#6383C2212yescomplemented
43#3208—71nomaximal, Sylow
44C24×C7—21no—
45C23×D7—21no—
46C2×C7×Q8C2311yescharacteristic
47C2×C7×D4—21no—
48C2×C7×D4—42no—
49C2×C7×D4C2312yes—
50C22×C4×C7C2311yescharacteristic
51C2×C22⋊D7—22no—
52C2×C22⋊D7—22no—
53C2×C22⋊D7—22no—
54C2×C22⋊D7—22no—
55C2×C22⋊D7—44no—
56C2×C22⋊D7—44no—
57C22×Dic7D411yescharacteristic
58C22×Dic7D411yescharacteristic
59C2×D28—21no—
60C2×D28—42no—
61C2×D28C2312yes—
62C2×Dic14—21no—
63C2×Dic14C2311yescharacteristic
64C7×SD16—42no—
65C2×C7×C8C2312yes—
66C7×C4⋊C4—21no—
67C7×C4⋊C4C2311yescharacteristic
68C7×C22⋊C4—21no—
69C22⋊Dic7—21no—
70#1143—42no—
71#1142—42no—
72C4⋊Dic7—21no—
73C4⋊Dic7C2311yescharacteristic
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79C4×Dic7—21no—
80C4×Dic7—21no—
81C4×Dic7D411yescharacteristic
82C4×Dic7D411yescharacteristic
83C4×Dic7C2311yescharacteristic
84C4×Dic7C2311yescharacteristic
85C2×C7⋊C8C2312yes—
86#1132—42no—
87#533—71no—
88C2×#127—71no—
89#492—72no—
90#486—72no—
91#460—72no—
92#460—72no—
93#450—71no—
94#450—71no—
95#450—71no—
96#450D711yescharacteristic, complemented
97#431—71no—
98C23×C7—22no—
99C23×C7—42no—
100C23×C7—42no—
101C23×C7—42no—
102C23×C7C2×D411yescharacteristic
103C22×D7—22no—
104C22×D7—42no—
105C22×D7—42no—
106C22×D7—42no—
107C7×Q8—22no—
108C7×D4—22no—
109C7×D4—22no—
110C7×D4—42no—
111C2×C4×C7—22no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7C2×D411yescharacteristic
115C2×C4×C7C2411yescharacteristic
116C22⋊D7—44no—
117C22⋊D7—44no—
118C22⋊D7—44no—
119C22⋊D7—44no—
120C2×Dic7—22no—
121C2×Dic7—21no—
122C2×Dic7—21no—
123C2×Dic7SD1614yescomplemented
124C2×Dic7C2×D411yescharacteristic
125C2×Dic7C2×D411yescharacteristic
126C2×Dic7C2×D411yescharacteristic
127C2×Dic7C2×D411yescharacteristic
128D28—22no—
129D28—22no—
130D28—42no—
131Dic14—22no—
132Dic14—42no—
133C7×C8—22no—
134C7⋊C8—22no—
135C22×D4—71no—
136C22×D4—141no—
137C22×D4—141no—
138C22×D4C2×D711yescharacteristic
139C2×SD16—72no—
140C2×SD16—72no—
141C2×SD16—72no—
142C2×SD16C2×D712yescomplemented
143C4⋊Q8—71no—
144C4⋊Q8—71no—
145#127—72no—
146#127—72no—
147#127—141no—
148#127—141no—
149#125—141no—
150C22⋊Q8—71no—
151C22⋊Q8—141no—
152C22⋊Q8C2×D711yescharacteristic
153C4×Q8—71no—
154C4×Q8—71no—
155C4×Q8—141no—
156#117—141no—
157C2×C42—71no—
158C4⋊C8—72no—
159#102—72no—
160#102—72no—
161#102—72no—
162#102C2×D712yescomplemented
163C22⋊C8—71no—
164C22⋊C8C2×D711yescharacteristic
165C4×C8—72no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—42no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7—84no—
172C22×C7C2×SD1612yes—
173C22×C7C22×D411yescharacteristic
174C2×D7—42no—
175C2×D7—42no—
176C2×D7—42no—
177C2×D7—84no—
178C4×C7—21no—
179C4×C7—41no—
180C4×C7—41no—
181C4×C7H(5,𝔽2)12yes—
182Dic7—21no—
183Dic7—41no—
184Dic7—41no—
185Dic7C2×SD1612yes—
186Dic7C2×SD1612yes—
187Dic7AGL(1,ℤ/8)12yes—
188C24—21no—
189C24—141no—
190C2×Q8—71no—
191C2×Q8—141no—
192C2×Q8C22×D711yescharacteristic
193C2×D4—21no—
194C2×D4—42no—
195C2×D4—72no—
196C2×D4—142no—
197C2×D4—142no—
198C2×D4—142no—
199C2×D4—142no—
200C2×D4—141no—
201C2×D4—282no—
202C2×D4—284no—
203C2×D4—284no—
204C2×D4C22×D712yes—
205C22×C4—71no—
206C22×C4—71no—
207C22×C4C22×D711yescharacteristic
208SD16—42no—
209SD16—282no—
210SD16—282no—
211SD16—282no—
212C2×C8—72no—
213C2×C8C22×D712yes—
214C4⋊C4—21no—
215C4⋊C4—71no—
216C4⋊C4—141no—
217C4⋊C4—141no—
218C4⋊C4—141no—
219C4⋊C4—141no—
220C4⋊C4—141no—
221C4⋊C4—141no—
222C4⋊C4C22×D711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—141no—
225C42—71no—
226C42—71no—
227C42—71no—
228C42—71no—
229C42—141no—
230C42—141no—
231C2×C7—22no—
232C2×C7—82no—
233C2×C7#53511yescharacteristic
234C2×C7C22×SD1611yescharacteristic
235C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
236D7—82no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—142no—
242C23—282no—
243C23—282no—
244C23—282no—
245C23D4×D711yescharacteristic
246Q8—22no—
247Q8—142no—
248Q8—282no—
249D4—22no—
250D4—22no—
251D4—42no—
252D4—142no—
253D4—142no—
254D4—282no—
255D4—284no—
256D4—284no—
257D4—284no—
258D4—284no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—74no—
263C2×C4—71no—
264C2×C4—71no—
265C2×C4—71no—
266C2×C4—71no—
267C2×C4—142no—
268C2×C4—141no—
269C2×C4—141no—
270C2×C4D4×D711yescharacteristic
271C2×C4C23×D711yescharacteristic
272C8—22no—
273C8—142no—
274C7#320811yescharacteristic, complemented, Sylow
275C22—21no—
276C22—21no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—84no—
281C22—282no—
282C22—282no—
283C22—282no—
284C22—564no—
285C22D7×SD1612yes—
286C22C2×D4×D711yescharacteristic
287C4—21no—
288C4—41no—
289C4—41no—
290C4—72no—
291C4—72no—
292C4—72no—
293C4—141no—
294C4—281no—
295C4—281no—
296C4#654812yes—
297C2—22no—
298C2—82no—
299C2—562no—
300C2#9028311yescharacteristic
301C2C2×D7×SD1611yescharacteristic
302C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#450C2
socleC22×C7C22×D4
prime radicalC2×#6503C2
hypercenter#450D7
hypocenterC7#3208
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #90313C2semidirect
3. C7×#450C2semidirect
4. #89990C2semidirect
5. #89947C2semidirect
6. #89935C2semidirect
7. #89818C2semidirect
8. #89534C2semidirect
9. #89532C2semidirect
10. #89503C2semidirect
11. #89490C2semidirect
16. C2×C7×SD16C22semidirect
18. Q8×Dic7C22semidirect
19. #6507C22semidirect
20. C2×#1143C22semidirect
25. C2×#1142C22semidirect
28. C2×#1132C22semidirect
30. #6451C22semidirect
32. #6450C22semidirect
35. C7×#102C22semidirect
37. #6406C22semidirect
39. #6395C22semidirect
40. Dic7⋊C8C22semidirect
41. C8×Dic7C22semidirect
42. #6383C22semidirect
96. #450D7semidirect
123. C2×Dic7SD16semidirect
142. C2×SD16C2×D7semidirect
162. #102C2×D7semidirect
274. C7#3208semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
#903132
C7×#4502
#899902
#899472
#899352
#898182
#895342
#895322
#895032
#894902
#32087