Finite Group Databasebeta

← #11736684

Subgroups — #11736684

Order 896.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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#90208C211yesmaximal, characteristic, complemented
2C2×#6456C211yesmaximal, characteristic
3C7×#428C211yesmaximal, characteristic, complemented
4#89903C211yesmaximal, characteristic
5#89890C211yesmaximal, characteristic, complemented
6#89773C211yesmaximal, characteristic
7#89736C211yesmaximal, characteristic, complemented
8#89663C212yesmaximal, complemented
9#89662C212yesmaximal, complemented
10#89643C212yesmaximal, complemented
11#89642C212yesmaximal, complemented
12C22×D28C2211yescharacteristic
13C22×C4×D7—21no—
14C2×C7×M16C2211yescharacteristic
15C7×#117C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4—21no—
18#6489C2211yescharacteristic
19C2×C4×Dic7C2211yescharacteristic
20C2×#1136C2211yescharacteristic
21#6456—21no—
22#6456—21no—
23#6456C2212yes—
24#6456C2212yes—
25#6453C2211yescharacteristic, complemented
26#6453C2211yescharacteristic, complemented
27#6451C2211yescharacteristic, complemented
28#6451C2211yescharacteristic, complemented
29#6441—21no—
30#6441—21no—
31C4×Dic14—21no—
32C4×Dic14—21no—
33C7×#107C2212yescomplemented
34C7×#106C2212yescomplemented
35#6395C2212yes—
36#6395C2212yes—
37Dic7⋊C8C2212yescomplemented
38Dic7⋊C8C2212yescomplemented
39#6383C2212yes—
40#6383C2212yes—
41#6382C2212yescomplemented
42#6381C2212yescomplemented
43#3077—71nomaximal, Sylow
44C23×D7—21no—
45C22×C4×C7—21no—
46C22×C4×C7C2311yescharacteristic
47C22×Dic7D411yescharacteristic
48C22×Dic7D411yescharacteristic
49C2×D28—21no—
50C2×D28—22no—
51C2×D28Q814yescomplemented
52C2×D28C2312yes—
53C2×C4×D7—22no—
54C2×C4×D7—22no—
55C2×C4×D7—44no—
56C2×Dic14—21no—
57C2×Dic14—21no—
58C7×M16—22no—
59C2×C7×C8C2312yes—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2311yescharacteristic
62C7×C4⋊C4C2312yes—
63C7×C4⋊C4C2311yescharacteristic
64C7×C22⋊C4—21no—
65C42×C7—21no—
66C22⋊Dic7—21no—
67(C2×D7)⋊C4—22no—
68(C2×D7)⋊C4—22no—
69C4⋊Dic7—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7C2311yescharacteristic
72C4⋊Dic7C2311yescharacteristic
73Dic7⋊C4—21no—
74Dic7⋊C4—21no—
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77C4×Dic7—21no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C4×Dic7C2311yescharacteristic
82#1136—22no—
83C2×C7⋊C8C2312yes—
84#533—71no—
85C2×C4×D4—71no—
86#478—72no—
87#478—72no—
88#476—72no—
89#474—72no—
90#428—71no—
91#428D711yescharacteristic, complemented
92#423—71no—
93#418—71no—
94#418—71no—
95C23×C7C2×D411yescharacteristic
96C22×D7—24no—
97C22×D7—22no—
98C22×D7—42no—
99C22×D7—42no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—42no—
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2×Q812yes—
106C2×C4×C7#4112yes—
107C2×C4×C7C2411yescharacteristic
108C2×Dic7—22no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—41no—
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115C2×Dic7C2×D411yescharacteristic
116D28—24no—
117D28—22no—
118D28C2×Q814yes—
119C4×D7—44no—
120C4×D7—44no—
121Dic14—42no—
122Dic14—42no—
123C7×C8—22no—
124C7⋊C8—22no—
125C22×D4—71no—
126C23×C4—141no—
127C2×M16—71no—
128C2×M16C2×D711yescharacteristic
129C4⋊Q8—71no—
130C4⋊Q8—71no—
131#125—71no—
132#125—71no—
133C22⋊Q8—141no—
134C22⋊Q8—141no—
135C4×Q8—141no—
136C4×Q8—141no—
137C4×D4—72no—
138C4×D4—72no—
139C4×D4—141no—
140C4×D4—141no—
141#117—71no—
142#117C2×D711yescharacteristic
143C2×C4⋊C4C2×D711yescharacteristic
144C2×C22⋊C4—141no—
145C2×C42—71no—
146#107—72no—
147#107C2×D712yescomplemented
148#106—72no—
149#106C2×D712yescomplemented
150C4⋊C8—72no—
151C4⋊C8—72no—
152#102—72no—
153#102—72no—
154#102—72no—
155#102—72no—
156C22×C7—21no—
157C22×C7—21no—
158C22×C7C22⋊Q812yes—
159C22×C7C22×D411yescharacteristic
160C2×D7—24no—
161C2×D7—48no—
162C2×D7—42no—
163C2×D7—42no—
164C4×C7—41no—
165C4×C7—42no—
166C4×C7—41no—
167C4×C7C22⋊Q812yes—
168C4×C7C22×Q811yescharacteristic
169C4×C7C2×#4111yescharacteristic
170Dic7—21no—
171Dic7—21no—
172Dic7—21no—
173Dic7—41no—
174Dic7—41no—
175Dic7AGL(1,ℤ/8)12yes—
176C24—141no—
177C2×Q8—141no—
178C2×Q8—141no—
179C2×D4—74no—
180C2×D4—72no—
181C2×D4—142no—
182C2×D4—141no—
183C22×C4—21no—
184C22×C4—71no—
185C22×C4—71no—
186C22×C4—142no—
187C22×C4—142no—
188C22×C4—284no—
189C22×C4C22×D711yescharacteristic
190M16—22no—
191M16—142no—
192C2×C8—72no—
193C2×C8C22×D712yes—
194C4⋊C4—21no—
195C4⋊C4—71no—
196C4⋊C4—71no—
197C4⋊C4—141no—
198C4⋊C4—141no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4C22×D711yescharacteristic
204C4⋊C4C22×D711yescharacteristic
205C4⋊C4C22×D712yes—
206C22⋊C4—21no—
207C22⋊C4—142no—
208C22⋊C4—142no—
209C22⋊C4—141no—
210C42—21no—
211C42—71no—
212C42—71no—
213C42—71no—
214C42—71no—
215C42—141no—
216C2×C7—22no—
217C2×C7C2×C22⋊Q811yescharacteristic
218C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C7#57511yescharacteristic
220D7—44no—
221C23—144no—
222C23—142no—
223C23—282no—
224C23—282no—
225C23D4×D711yescharacteristic
226Q8—282no—
227Q8—282no—
228D4—74no—
229D4—144no—
230D4—142no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—71no—
236C2×C4—71no—
237C2×C4—71no—
238C2×C4—71no—
239C2×C4—142no—
240C2×C4—141no—
241C2×C4—141no—
242C2×C4—284no—
243C2×C4—284no—
244C2×C4—281no—
245C2×C4#115712yes—
246C2×C4D4×D711yescharacteristic
247C2×C4Q8×D712yes—
248C2×C4C23×D711yescharacteristic
249C8—22no—
250C8—142no—
251C7#307711yescharacteristic, complemented, Sylow
252C22—21no—
253C22—21no—
254C22—144no—
255C22—288no—
256C22—282no—
257C22—282no—
258C22#645912yes—
259C22C2×D4×D711yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—72no—
264C4—141no—
265C4—141no—
266C4—141no—
267C4—281no—
268C4—281no—
269C4#645912yes—
270C4C2×#115711yescharacteristic
271C4C2×Q8×D711yescharacteristic
272C2—22no—
273C2—284no—
274C2C2×#645911yescharacteristic
275C2D7×AGL(1,ℤ/8)11yescharacteristic
276C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radicalC22×D28C22
hypercenter#428D7
hypocenterC7#3077
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #90208C2semidirect
3. C7×#428C2semidirect
5. #89890C2semidirect
7. #89736C2semidirect
8. #89663C2semidirect
9. #89662C2semidirect
10. #89643C2semidirect
11. #89642C2semidirect
25. #6453C22semidirect
26. #6453C22semidirect
27. #6451C22semidirect
28. #6451C22semidirect
33. C7×#107C22semidirect
34. C7×#106C22semidirect
37. Dic7⋊C8C22semidirect
38. Dic7⋊C8C22semidirect
41. #6382C22semidirect
42. #6381C22semidirect
51. C2×D28Q8semidirect
91. #428D7semidirect
147. #107C2×D7semidirect
149. #106C2×D7semidirect
251. C7#3077semidirect

Maximal subgroups

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

subgroupindex
#902082
C2×#64562
C7×#4282
#899032
#898902
#897732
#897362
#896632
#896622
#896432
#896422
#30777