Finite Group Databasebeta

← #11736069

Subgroups — #11736069

Order 896.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#6493C211yesmaximal, characteristic
2C2×#6489C211yesmaximal, characteristic, complemented
3C7×#407C211yesmaximal, characteristic, complemented
4#89903C212yesmaximal, complemented
5#89893C212yesmaximal, complemented
6#89509C214yesmaximal, complemented
7#89508C212yesmaximal, complemented
8#89507C212yesmaximal, complemented
9C23×C4×C7C2211yescharacteristic
10C22×C22⋊D7—21no—
11C22×D28C2211yescharacteristic
12C2×C7×M16C2212yes—
13C2×C22⋊Dic7—21no—
14#6493—21no—
15#6493—21no—
16#6493C2212yes—
17#6493C2212yes—
18C2×(C2×D7)⋊C4—21no—
19#6489—21no—
20#6489—21no—
21#6489C2212yes—
22#6489C2212yescomplemented
23C2×C4⋊Dic7C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic, complemented
25C2×C4⋊Dic7C2211yescharacteristic, complemented
26C2×C4×Dic7—21no—
27C7×C22⋊C8C2212yescomplemented
28C7×C22⋊C8C2212yescomplemented
29#6395C2214yescomplemented
30#6395C2214yescomplemented
31#6392C2214yescomplemented
32#6391C2214yescomplemented
33#3094—71nomaximal, Sylow
34C24×C7D411yescharacteristic
35C23×D7—21no—
36C22×C4×C7—24no—
37C22×C4×C7D411yescharacteristic
38C22×C4×C7D411yescharacteristic
39C22×C4×C7D411yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C2×C22⋊D7—22no—
44C2×C22⋊D7—22no—
45C2×C22⋊D7—44no—
46C22×Dic7—21no—
47C22×Dic7—21no—
48C22×Dic7—21no—
49C2×D28—24no—
50C2×D28—21no—
51C2×D28C2312yes—
52C7×M16—24no—
53C2×C7×C8C2314yes—
54C22⋊Dic7—22no—
55C22⋊Dic7—22no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58C4⋊Dic7—21no—
59C4⋊Dic7—21no—
60C4⋊Dic7—21no—
61C4⋊Dic7C2312yes—
62C4⋊Dic7C2312yes—
63C4⋊Dic7C2312yes—
64C4×Dic7—22no—
65C4×Dic7—22no—
66C2×#121—71no—
67C2×#117—71no—
68#482—74no—
69#481—72no—
70#480—72no—
71#428—72no—
72#418—72no—
73#407D711yescharacteristic, complemented
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—41no—
77C23×C7C2×D411yescharacteristic
78C23×C7C2×D411yescharacteristic
79C23×C7C2×D411yescharacteristic
80C22×D7—22no—
81C22×D7—42no—
82C22×D7—42no—
83C22×D7—42no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7C2×D411yescharacteristic
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7#4114yes—
93C2×C4×C7C2411yescharacteristic
94C22⋊D7—44no—
95C22⋊D7—44no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—42no—
100C2×Dic7—42no—
101C2×Dic7—42no—
102D28—24no—
103D28—42no—
104C7×C8—24no—
105C22×D4—71no—
106C22×D4—141no—
107C23×C4C2×D711yescharacteristic
108C2×M16C2×D712yes—
109#121—72no—
110#121—72no—
111#121—141no—
112#121—141no—
113#117—72no—
114#117—72no—
115#117—141no—
116#117—141no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—71no—
119C2×C4⋊C4—71no—
120C2×C22⋊C4—141no—
121C2×C22⋊C4—141no—
122C2×C42—141no—
123#107—74no—
124#106—74no—
125#102—74no—
126#102—74no—
127C22⋊C8C2×D712yescomplemented
128C22⋊C8C2×D712yescomplemented
129C22×C7—21no—
130C22×C7—21no—
131C22×C7—21no—
132C22×C7—21no—
133C22×C7—21no—
134C22×C7—21no—
135C22×C7—21no—
136C22×C7—44no—
137C22×C7#12312yes—
138C22×C7AGL(1,ℤ/8)12yes—
139C22×C7C22×D411yescharacteristic
140C2×D7—42no—
141C2×D7—42no—
142C2×D7—42no—
143C2×D7—84no—
144C4×C7—21no—
145C4×C7—21no—
146C4×C7#12312yes—
147C4×C7C2×#4112yes—
148Dic7—42no—
149Dic7—42no—
150Dic7—42no—
151C24—141no—
152C24D2811yescharacteristic
153C2×D4—72no—
154C2×D4—142no—
155C2×D4—142no—
156C2×D4—141no—
157C2×D4—144no—
158C2×D4—284no—
159C22×C4—24no—
160C22×C4—141no—
161C22×C4—141no—
162C22×C4—141no—
163C22×C4D2811yescharacteristic
164C22×C4D2811yescharacteristic
165C22×C4D2811yescharacteristic
166C22×C4C22×D711yescharacteristic
167C22×C4C22×D711yescharacteristic
168C22×C4C22×D711yescharacteristic
169M16—24no—
170C2×C8C22×D714yes—
171C4⋊C4—72no—
172C4⋊C4—72no—
173C4⋊C4—72no—
174C4⋊C4—141no—
175C4⋊C4—141no—
176C4⋊C4—141no—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C22⋊C4—142no—
180C22⋊C4—142no—
181C42—142no—
182C42—142no—
183C2×C7—22no—
184C2×C7—22no—
185C2×C7—41no—
186C2×C7C2×#12311yescharacteristic
187C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C7#57511yescharacteristic
189D7—82no—
190C23—22no—
191C23—22no—
192C23—41no—
193C23—142no—
194C23—282no—
195C23—282no—
196C23—282no—
197C23C2×D2811yescharacteristic
198C23C2×D2811yescharacteristic
199C23C2×D2811yescharacteristic
200D4—144no—
201D4—284no—
202D4—284no—
203D4—282no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—142no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—282no—
213C2×C4—282no—
214C2×C4—282no—
215C2×C4C2×D2811yescharacteristic
216C2×C4C2×D2811yescharacteristic
217C2×C4C2×D2811yescharacteristic
218C2×C4#115914yes—
219C2×C4C23×D711yescharacteristic
220C8—24no—
221C7#309411yescharacteristic, complemented, Sylow
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—282no—
231C22—282no—
232C22—282no—
233C22—564no—
234C22#644912yes—
235C22#647112yes—
236C22C22×D2811yescharacteristic
237C4—21no—
238C4—21no—
239C4—282no—
240C4—282no—
241C4—282no—
242C4#644912yes—
243C4C2×#115912yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—562no—
248C2C2×#644911yescharacteristic
249C2C2×#647111yescharacteristic
250C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#407C2
socleC22×C7C22×D4
prime radicalC2×#6493C2
hypercenter#407D7
hypocenterC7#3094
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#6489C2semidirect
3. C7×#407C2semidirect
4. #89903C2semidirect
5. #89893C2semidirect
6. #89509C2semidirect
7. #89508C2semidirect
8. #89507C2semidirect
22. #6489C22semidirect
24. C2×C4⋊Dic7C22semidirect
25. C2×C4⋊Dic7C22semidirect
27. C7×C22⋊C8C22semidirect
28. C7×C22⋊C8C22semidirect
29. #6395C22semidirect
30. #6395C22semidirect
31. #6392C22semidirect
32. #6391C22semidirect
73. #407D7semidirect
127. C22⋊C8C2×D7semidirect
128. C22⋊C8C2×D7semidirect
221. C7#3094semidirect

Maximal subgroups

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

subgroupindex
C2×#64932
C2×#64892
C7×#4072
#899032
#898932
#895092
#895082
#895072
#30947