Finite Group Databasebeta

← #11740968

Subgroups — #11740968

Order 896.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C7×#547C211yesmaximal, characteristic, complemented
2C2×#6512C212yesmaximal, complemented
3D4×D28C211yesmaximal, characteristic, complemented
4#90010C212yesmaximal, complemented
5#89857C211yesmaximal, characteristic, complemented
6#89855C211yesmaximal, characteristic, complemented
7#89821C212yesmaximal, complemented
8#89818C212yesmaximal, complemented
9#89787C211yesmaximal, characteristic, complemented
10C4×#1141C211yesmaximal, characteristic, complemented
11#89784C211yesmaximal, characteristic, complemented
12C2×C7×#41C2212yescomplemented
13C2×D4×D7—21no—
14C22×D28C2212yes—
15C7×C4⋊Q8C2211yescharacteristic, complemented
16C7×#123—22no—
17C7×C22⋊Q8C2212yescomplemented
18C7×#121—21no—
19C4×C7×D4C2211yescharacteristic
20C4×C7×D4C2211yescharacteristic, complemented
21C2×C7×C4⋊C4—21no—
22#6512—28no—
23C2×#1143C2212yes—
24C2×#1143C2212yescomplemented
25C2×#1143C2212yescomplemented
26C2×#1141C2211yescharacteristic
27C2×#1141C2212yescomplemented
28#6493C2212yes—
29C2×#1136C2212yescomplemented
30#6458—21no—
31#6445—22no—
32#6437C2211yescharacteristic
33C4×D28C2211yescharacteristic
34#6409C2212yescomplemented
35#6406C2211yescharacteristic, complemented
36C22⋊(C7⋊C8)C2212yescomplemented
37#6383C2211yescharacteristic
38#6383C2212yescomplemented
39#6381C2211yescharacteristic, complemented
40C4⋊(C7⋊C8)C2211yescharacteristic, complemented
41C4×C7⋊C8C2211yescharacteristic, complemented
42#3272—71nomaximal, Sylow
43C23×D7—22no—
44C7×#41—24no—
45C7×#41—42no—
46C2×C7×Q8C2312yes—
47C2×C7×D4—22no—
48C2×C7×D4D411yescharacteristic
49C2×C7×D4C2311yescharacteristic
50C22×C4×C7—21no—
51C22×C4×C7—22no—
52C22×C4×C7C2312yes—
53C2×C22⋊D7—22no—
54D4×D7—44no—
55C2×D28—24no—
56C2×D28—21no—
57C2×D28—22no—
58C2×D28—21no—
59C2×D28—42no—
60C2×D28C2312yes—
61C2×D28C2311yescharacteristic
62C2×C4×D7—21no—
63C7×C4⋊C4—21no—
64C7×C4⋊C4—22no—
65C7×C4⋊C4—21no—
66C7×C4⋊C4D411yescharacteristic
67C7×C4⋊C4C2311yescharacteristic
68C7×C22⋊C4—22no—
69C7×C22⋊C4—21no—
70C7×C22⋊C4D412yes—
71C42×C7C2311yescharacteristic
72#1143—24no—
73#1143—24no—
74#1143—42no—
75#1141—24no—
76#1141D414yescomplemented
77(C2×D7)⋊C4—21no—
78(C2×D7)⋊C4—22no—
79C4⋊Dic7C2311yescharacteristic
80#1136—24no—
81C2×C7⋊C8C2311yescharacteristic
82C2×C7⋊C8C2312yes—
83C2×C7⋊C8C2311yescharacteristic
84C2×AGL(1,ℤ/8)—72no—
85#547D711yescharacteristic, complemented
86D42—71no—
87#492—71no—
88#468—72no—
89#461—71no—
90#459—71no—
91#450—72no—
92#449—72no—
93C4×D8—71no—
94#436—71no—
95C23×C7—21no—
96C23×C7C2×D412yes—
97C22×D7—21no—
98C22×D7—22no—
99C22×D7—44no—
100C22×D7—44no—
101C22×D7—82no—
102C7×Q8—22no—
103C7×Q8—22no—
104C7×D4—22no—
105C7×D4—21no—
106C7×D4—44no—
107C7×D4C2×D412yescomplemented
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—21no—
111C2×C4×C7—22no—
112C2×C4×C7—44no—
113C2×C4×C7—42no—
114C2×C4×C7C2×D411yescharacteristic
115C2×C4×C7C2×D412yes—
116C2×C4×C7C2×D411yescharacteristic
117C2×C4×C7C2411yescharacteristic
118C22⋊D7—44no—
119C2×Dic7—21no—
120D28—22no—
121D28—21no—
122D28—22no—
123D28—44no—
124D28—42no—
125D28—81no—
126D28C2×D412yes—
127C4×D7—42no—
128C7⋊C8—22no—
129C7⋊C8—21no—
130C7⋊C8C2×D412yescomplemented
131C2×#41C2×D712yescomplemented
132C22×D4—72no—
133C22×D4—141no—
134AGL(1,ℤ/8)—148no—
135C2×SD16—72no—
136C2×SD16—72no—
137C2×SD16—72no—
138C2×D8—71no—
139C2×D8—72no—
140C2×M16—72no—
141C4⋊Q8C2×D711yescharacteristic, complemented
142#127—71no—
143#123—22no—
144C22⋊Q8C2×D712yescomplemented
145#121—21no—
146#121—72no—
147#121—141no—
148#120—142no—
149C4×D4—71no—
150C4×D4C2×D711yescharacteristic
151C4×D4C2×D711yescharacteristic, complemented
152C2×C4⋊C4—21no—
153#107—71no—
154C4⋊C8—71no—
155#103—72no—
156#102—71no—
157#102—71no—
158#102—72no—
159C22⋊C8—72no—
160C4×C8—71no—
161C22×C7—22no—
162C22×C7—22no—
163C22×C7—22no—
164C22×C7—22no—
165C22×C7—41no—
166C22×C7—41no—
167C22×C7C22×D411yescharacteristic
168C2×D7—22no—
169C2×D7—41no—
170C2×D7—41no—
171C2×D7—42no—
172C2×D7—42no—
173C2×D7—42no—
174C2×D7—88no—
175C4×C7—22no—
176C4×C7—21no—
177C4×C7—42no—
178C4×C7—41no—
179C4×C7AGL(1,ℤ/8)12yescomplemented
180C4×C7C22×D411yescharacteristic
181C4×C7H(5,𝔽2)11yescharacteristic
182Dic7—41no—
183C24—142no—
184#41—24no—
185#41—42no—
186C2×Q8C22×D712yes—
187C2×D4—22no—
188C2×D4—71no—
189C2×D4—72no—
190C2×D4—144no—
191C2×D4—142no—
192C2×D4—142no—
193C2×D4—141no—
194C2×D4—141no—
195C2×D4—282no—
196C2×D4—284no—
197C2×D4C22⋊D711yescharacteristic
198C2×D4C22×D711yescharacteristic
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—141no—
202C22×C4C22×D712yes—
203SD16—144no—
204SD16—144no—
205SD16—282no—
206D8—74no—
207D8—144no—
208M16—144no—
209C2×C8—71no—
210C2×C8—71no—
211C2×C8—72no—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—21no—
215C4⋊C4—71no—
216C4⋊C4C22⋊D711yescharacteristic
217C4⋊C4C22×D711yescharacteristic
218C22⋊C4—22no—
219C22⋊C4—21no—
220C22⋊C4—141no—
221C22⋊C4—142no—
222C22⋊C4C22⋊D712yes—
223C42C22×D711yescharacteristic
224C2×C7—42no—
225C2×C7—42no—
226C2×C7D4211yescharacteristic
227C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
228C2×C7#57711yescharacteristic
229D7—42no—
230D7—82no—
231C23—21no—
232C23—141no—
233C23—142no—
234C23—284no—
235C23—284no—
236C23—562no—
237C23C2×C22⋊D712yes—
238Q8—22no—
239Q8—22no—
240D4—22no—
241D4—21no—
242D4—44no—
243D4—72no—
244D4—142no—
245D4—141no—
246D4—142no—
247D4—284no—
248D4—284no—
249D4—282no—
250D4—561no—
251D4D4×D712yescomplemented
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—141no—
259C2×C4—282no—
260C2×C4C2×C22⋊D711yescharacteristic
261C2×C4C2×C22⋊D712yes—
262C2×C4C2×C22⋊D711yescharacteristic
263C2×C4C23×D711yescharacteristic
264C8—72no—
265C8—142no—
266C8—141no—
267C7#327211yescharacteristic, complemented, Sylow
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—142no—
275C22—281no—
276C22—281no—
277C22—282no—
278C22—282no—
279C22—282no—
280C22—568no—
281C22C22×C22⋊D711yescharacteristic
282C4—22no—
283C4—21no—
284C4—42no—
285C4—41no—
286C4—281no—
287C4#651212yescomplemented
288C4C2×D4×D711yescharacteristic
289C4#654811yescharacteristic
290C2—42no—
291C2—42no—
292C2—282no—
293C2—562no—
294C2D4×C22⋊D711yescharacteristic
295C2C2×#651211yescharacteristic
296C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#547C2
socleC22×C7C22×D4
hypercenter#547D7
hypocenterC7#3272
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C7×#547C2semidirect
2. C2×#6512C2semidirect
3. D4×D28C2semidirect
4. #90010C2semidirect
5. #89857C2semidirect
6. #89855C2semidirect
7. #89821C2semidirect
8. #89818C2semidirect
9. #89787C2semidirect
10. C4×#1141C2semidirect
11. #89784C2semidirect
12. C2×C7×#41C22semidirect
15. C7×C4⋊Q8C22semidirect
17. C7×C22⋊Q8C22semidirect
20. C4×C7×D4C22semidirect
24. C2×#1143C22semidirect
25. C2×#1143C22semidirect
27. C2×#1141C22semidirect
29. C2×#1136C22semidirect
34. #6409C22semidirect
35. #6406C22semidirect
36. C22⋊(C7⋊C8)C22semidirect
38. #6383C22semidirect
39. #6381C22semidirect
40. C4⋊(C7⋊C8)C22semidirect
41. C4×C7⋊C8C22semidirect
76. #1141D4semidirect
85. #547D7semidirect
107. C7×D4C2×D4semidirect
130. C7⋊C8C2×D4semidirect
131. C2×#41C2×D7semidirect
141. C4⋊Q8C2×D7semidirect
144. C22⋊Q8C2×D7semidirect
151. C4×D4C2×D7semidirect
179. C4×C7AGL(1,ℤ/8)semidirect
251. D4D4×D7semidirect
267. C7#3272semidirect
287. C4#6512semidirect

Maximal subgroups

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

subgroupindex
C7×#5472
C2×#65122
D4×D282
#900102
#898572
#898552
#898212
#898182
#897872
C4×#11412
#897842
#32727