Finite Group Databasebeta

← #11736745

Subgroups — #11736745

Order 896.

454 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#90170C211yesmaximal, characteristic, complemented
2C2×C4×D28C211yesmaximal, characteristic, complemented
3C7×#431C211yesmaximal, characteristic, complemented
4#89619C212yesmaximal, complemented
5#89616C212yesmaximal, complemented
6#89507C211yesmaximal, characteristic, complemented
7#89504C211yesmaximal, characteristic
8#89503C211yesmaximal, characteristic, complemented
9#89494C211yesmaximal, characteristic, complemented
10C4×D56C212yesmaximal, complemented
11C4×#1132C212yesmaximal, complemented
12C22×D28C2211yescharacteristic
13C22×C4×D7—21no—
14C2×C42×C7C2211yescharacteristic
15#6493C2211yescharacteristic
16#6492—21no—
17C4×C22⋊D7—21no—
18C2×(C2×D7)⋊C4—21no—
19C2×C4⋊Dic7C2211yescharacteristic
20#6487C2211yescharacteristic
21C2×D56C2212yes—
22C2×#1132C2212yes—
23#6439—21no—
24#6438C2211yescharacteristic, complemented
25C4×D28—21no—
26C4×D28—21no—
27C4×D28C2212yes—
28C4×D28C2212yes—
29C4×D28C2211yescharacteristic, complemented
30#6435—21no—
31#6433C2211yescharacteristic, complemented
32C4×Dic14C2211yescharacteristic, complemented
33C7×C4⋊C8C2212yescomplemented
34C7×C22⋊C8C2211yescharacteristic
35C7×C22⋊C8C2211yescharacteristic
36C4×C7×C8C2212yescomplemented
37#6395C2212yes—
38#6395C2212yes—
39#6395C2212yescomplemented
40#6392C2212yescomplemented
41#6391C2212yescomplemented
42#6390C2212yescomplemented
43#3105—71nomaximal, Sylow
44C23×D7—21no—
45C22×C4×C7D411yescharacteristic
46C22×C4×C7D411yescharacteristic
47C22×C4×C7C2311yescharacteristic
48C2×C22⋊D7—21no—
49C22×Dic7—21no—
50C2×D28—21no—
51C2×D28—24no—
52C2×D28C2311yescharacteristic
53C2×D28C2312yes—
54C2×C4×D7—21no—
55C2×C4×D7—22no—
56C2×C4×D7—22no—
57C2×C4×D7—44no—
58C2×Dic14C2311yescharacteristic
59C2×C7×C8C2312yes—
60C2×C7×C8C2312yes—
61C42×C7D411yescharacteristic
62C42×C7D411yescharacteristic
63C42×C7C2311yescharacteristic
64C42×C7C2311yescharacteristic
65C22⋊Dic7—21no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—21no—
71(C2×D7)⋊C4—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7C2311yescharacteristic
74C4⋊Dic7C2311yescharacteristic
75C4⋊Dic7C2312yes—
76Dic7⋊C4—21no—
77Dic7⋊C4—21no—
78Dic7⋊C4—21no—
79Dic7⋊C4—21no—
80C4×Dic7—21no—
81D56—24no—
82#1132—24no—
83#529—71no—
84C2×C4×D4—71no—
85#484—71no—
86#481—71no—
87#478—72no—
88#463—72no—
89#450—71no—
90#447—71no—
91C4×SD16—72no—
92C4×D8—72no—
93#431D711yescharacteristic, complemented
94C23×C7C2×D411yescharacteristic
95C22×D7—24no—
96C22×D7—22no—
97C22×D7—21no—
98C22×D7—42no—
99C22×D7—42no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2411yescharacteristic
109C22⋊D7—42no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—42no—
115D28—24no—
116D28—22no—
117D28—22no—
118D28#4114yes—
119C4×D7—44no—
120C4×D7—42no—
121C4×D7—44no—
122Dic14—22no—
123C7×C8—22no—
124C7×C8—22no—
125C22×D4—71no—
126C23×C4—141no—
127C2×SD16—72no—
128C2×D8—72no—
129#126—141no—
130#125—71no—
131#124—71no—
132#123—141no—
133C22⋊Q8—71no—
134#121—71no—
135C4×Q8—71no—
136C4×D4—72no—
137C4×D4—72no—
138C4×D4—71no—
139C4×D4—141no—
140C4×D4—141no—
141C4×D4—141no—
142#117—141no—
143C2×C4⋊C4—71no—
144C2×C22⋊C4—141no—
145C2×C42C2×D711yescharacteristic
146#107—72no—
147#106—72no—
148C4⋊C8C2×D712yescomplemented
149#103—72no—
150#102—72no—
151#102—72no—
152#102—72no—
153C22⋊C8C2×D711yescharacteristic
154C22⋊C8C2×D711yescharacteristic
155C4×C8C2×D712yescomplemented
156C22×C7—21no—
157C22×C7—21no—
158C22×C7#13512yes—
159C22×C7C22×D411yescharacteristic
160C2×D7—24no—
161C2×D7—48no—
162C2×D7—42no—
163C2×D7—42no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—41no—
167C4×C7—21no—
168C4×C7—21no—
169C4×C7—21no—
170C4×C7—21no—
171C4×C7AGL(1,ℤ/8)12yes—
172C4×C7C2×#4112yes—
173Dic7—41no—
174Dic7—41no—
175Dic7—42no—
176Dic7—41no—
177C24—141no—
178C2×Q8—71no—
179C2×D4—72no—
180C2×D4—71no—
181C2×D4—144no—
182C2×D4—141no—
183C2×D4—141no—
184C22×C4—141no—
185C22×C4—142no—
186C22×C4—141no—
187C22×C4—142no—
188C22×C4—284no—
189C22×C4D2811yescharacteristic
190C22×C4D2811yescharacteristic
191C22×C4C22×D711yescharacteristic
192SD16—144no—
193D8—144no—
194C2×C8C22×D712yes—
195C2×C8C22×D712yes—
196C4⋊C4—71no—
197C4⋊C4—71no—
198C4⋊C4—72no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C22⋊C4—141no—
205C22⋊C4—142no—
206C22⋊C4—142no—
207C22⋊C4—141no—
208C22⋊C4—141no—
209C22⋊C4—141no—
210C22⋊C4—141no—
211C42—141no—
212C42D2811yescharacteristic
213C42D2811yescharacteristic
214C42C22×D711yescharacteristic
215C42C22×D711yescharacteristic
216C2×C7—22no—
217C2×C7#52511yescharacteristic
218C2×C7C2×#13511yescharacteristic
219C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
220D7—44no—
221D7—81no—
222C23—144no—
223C23—142no—
224C23—141no—
225C23—282no—
226C23—282no—
227C23C2×D2811yescharacteristic
228Q8—142no—
229D4—74no—
230D4—144no—
231D4—142no—
232D4—142no—
233D4—282no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—141no—
238C2×C4—141no—
239C2×C4—141no—
240C2×C4—142no—
241C2×C4—284no—
242C2×C4—282no—
243C2×C4—282no—
244C2×C4—284no—
245C2×C4C2×D2811yescharacteristic
246C2×C4C2×D2811yescharacteristic
247C2×C4C2×D2811yescharacteristic
248C2×C4C2×D2811yescharacteristic
249C2×C4C2×D2811yescharacteristic
250C2×C4C23×D711yescharacteristic
251C8—22no—
252C8—22no—
253C7#310511yescharacteristic, complemented, Sylow
254C22—21no—
255C22—21no—
256C22—144no—
257C22—288no—
258C22—282no—
259C22—282no—
260C22—281no—
261C22—281no—
262C22—281no—
263C22#646712yes—
264C22C22×D2811yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—281no—
270C4—282no—
271C4—281no—
272C4—281no—
273C4#647112yes—
274C4D7×#4112yes—
275C2—22no—
276C2—284no—
277C2—561no—
278C2#9021511yescharacteristic
279C2C2×#646711yescharacteristic
280C2C2×#647111yescharacteristic
subgroupG/H
centerC2×C4C2×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#431C2
socleC22×C7C22×D4
prime radical#89504C2
hypercenter#431D7
hypocenterC7#3105
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #90170C2semidirect
2. C2×C4×D28C2semidirect
3. C7×#431C2semidirect
4. #89619C2semidirect
5. #89616C2semidirect
6. #89507C2semidirect
8. #89503C2semidirect
9. #89494C2semidirect
10. C4×D56C2semidirect
11. C4×#1132C2semidirect
24. #6438C22semidirect
29. C4×D28C22semidirect
31. #6433C22semidirect
32. C4×Dic14C22semidirect
33. C7×C4⋊C8C22semidirect
36. C4×C7×C8C22semidirect
39. #6395C22semidirect
40. #6392C22semidirect
41. #6391C22semidirect
42. #6390C22semidirect
93. #431D7semidirect
148. C4⋊C8C2×D7semidirect
155. C4×C8C2×D7semidirect
253. C7#3105semidirect

Maximal subgroups

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

subgroupindex
#901702
C2×C4×D282
C7×#4312
#896192
#896162
#895072
#895042
#895032
#894942
C4×D562
C4×#11322
#31057