Finite Group Databasebeta

← #11739085

Subgroups — #11739085

Order 896.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#90379C211yesmaximal, characteristic, complemented
2D7×#124C211yesmaximal, characteristic
3C7×#488C211yesmaximal, characteristic, complemented
4#89840C211yesmaximal, characteristic, complemented
5#89828C211yesmaximal, characteristic, complemented
6#89574C212yesmaximal, complemented
7#89567C212yesmaximal, complemented
8#89542C212yesmaximal
9#89537C212yesmaximal, complemented
10#89469C211yesmaximal, characteristic, complemented
11C4×#1131C211yesmaximal, characteristic, complemented
12C2×Q8×D7C2211yescharacteristic
13C2×D4×D7C2211yescharacteristic
14C7×#125C2211yescharacteristic, complemented
15C7×#124C2211yescharacteristic
16#6510C2211yescharacteristic
17#6499C2211yescharacteristic
18C2×#1131C2212yes—
19#6459—22no—
20#6455C2212yes—
21#6451C2212yescomplemented
22#6450—22no—
23#6448—22no—
24D7×C22⋊C4—22no—
25#6438C2211yescharacteristic
26C42×D7C2211yescharacteristic
27#6433C2211yescharacteristic, complemented
28C7×#103C2212yescomplemented
29C7×#102C2212yes—
30C4×C7×C8C2211yescharacteristic, complemented
31#6409C2212yescomplemented
32#6406C2212yes—
33#6395C2212yes—
34#6390C2212yescomplemented
35#6389C2212yescomplemented
36#6384C2212yescomplemented
37#6383C2212yes—
38C4×C7⋊C8C2211yescharacteristic, complemented
39#3140—71nomaximal, Sylow
40C23×D7—21no—
41C2×C7×Q8C2311yescharacteristic
42C2×C7×D4C2311yescharacteristic
43C2×C22⋊D7—21no—
44Q8×D7—24no—
45D4×D7—24no—
46C2×D28C2311yescharacteristic
47C2×C4×D7—21no—
48C2×C4×D7D412yes—
49C2×C4×D7C2311yescharacteristic
50C2×Dic14—21no—
51C2×Dic14—22no—
52C2×Dic14C2311yescharacteristic
53C2×C7×C8C2312yes—
54C7×C4⋊C4—22no—
55C7×C4⋊C4C2312yes—
56C7×C22⋊C4—22no—
57C42×C7C2311yescharacteristic
58C22⋊Dic7—22no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—22no—
62C4⋊Dic7C2312yes—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65C4×Dic7—22no—
66C4×Dic7D412yes—
67C4×Dic7C2311yescharacteristic
68C2×C7⋊C8C2312yes—
69#1131—24no—
70#533—71no—
71C2×#124—71no—
72#489—72no—
73#489—72no—
74#488—71no—
75#488—71no—
76#488—71no—
77#488D711yescharacteristic, complemented
78#419—72no—
79#418—72no—
80C4×M16—71no—
81C23×C7—21no—
82C22×D7—21no—
83C22×D7—44no—
84C22×D7—42no—
85C22×D7C2×D411yescharacteristic
86C7×Q8—22no—
87C7×D4—22no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7C2×D412yes—
91C2×C4×C7C2411yescharacteristic
92C22⋊D7—42no—
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7C2×D412yes—
96C2×Dic7C2×D411yescharacteristic
97D28—22no—
98C4×D7—42no—
99C4×D7—42no—
100C4×D7#4114yes—
101Dic14—22no—
102Dic14—44no—
103Dic14—42no—
104C7×C8—22no—
105C7⋊C8—22no—
106C22×Q8—71no—
107C22×D4—71no—
108C2×M16—72no—
109C4⋊Q8—72no—
110#125—71no—
111#125C2×D711yescharacteristic, complemented
112#124—71no—
113#124—71no—
114#124—71no—
115#124—142no—
116#124C2×D711yescharacteristic
117C22⋊Q8—142no—
118C4×Q8—142no—
119#117—72no—
120C2×C22⋊C4—142no—
121C2×C42—71no—
122#103—72no—
123#103—72no—
124#103—72no—
125#103C2×D712yescomplemented
126#102—72no—
127#102—72no—
128#102—72no—
129#102C2×D712yes—
130#97—72no—
131C4×C8—71no—
132C4×C8C2×D711yescharacteristic, complemented
133C22×C7—41no—
134C22×C7—42no—
135C22×C7C22×D411yescharacteristic
136C2×D7—22no—
137C2×D7—41no—
138C2×D7—42no—
139C2×D7—84no—
140C2×D7#12412yes—
141C4×C7—22no—
142C4×C7—41no—
143C4×C7—42no—
144C4×C7C2×#4112yes—
145Dic7—22no—
146Dic7—41no—
147Dic7—42no—
148Dic7#12412yes—
149C24—141no—
150C2×Q8—71no—
151C2×Q8—142no—
152C2×Q8—141no—
153C2×Q8—144no—
154C2×Q8C22×D711yescharacteristic
155C2×D4—71no—
156C2×D4—144no—
157C2×D4—141no—
158C2×D4C22×D711yescharacteristic
159C22×C4—71no—
160C22×C4—72no—
161C22×C4—141no—
162M16—144no—
163C2×C8—72no—
164C2×C8C22×D712yes—
165C4⋊C4—22no—
166C4⋊C4—72no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C4⋊C4C22×D712yes—
170C22⋊C4—22no—
171C22⋊C4—142no—
172C22⋊C4—142no—
173C22⋊C4—142no—
174C22⋊C4—142no—
175C42—72no—
176C42—71no—
177C42—142no—
178C42C22×D711yescharacteristic
179C2×C7—81no—
180C2×C7C2×#12411yescharacteristic
181C2×C7#57512yes—
182D7—22no—
183D7—81no—
184C23—21no—
185C23—71no—
186C23—141no—
187C23—282no—
188C23—284no—
189Q8—22no—
190Q8—142no—
191Q8—282no—
192Q8—284no—
193D4—22no—
194D4—142no—
195D4—282no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—71no—
199C2×C4—74no—
200C2×C4—72no—
201C2×C4—142no—
202C2×C4—141no—
203C2×C4—282no—
204C2×C4—282no—
205C2×C4D4×D712yes—
206C2×C4C23×D711yescharacteristic
207C8—22no—
208C8—142no—
209C7#314011yescharacteristic, complemented, Sylow
210C22—42no—
211C22—41no—
212C22—72no—
213C22—142no—
214C22—282no—
215C22—281no—
216C22—564no—
217C22C2×D4×D711yescharacteristic
218C4—22no—
219C4—42no—
220C4—41no—
221C4—72no—
222C4—142no—
223C4—282no—
224C4—281no—
225C4D7×#4112yes—
226C2—81no—
227C2—142no—
228C2—561no—
229C2D7×#12411yescharacteristic
230C2#9045912yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#488C2
socleC22×C7C22×D4
prime radicalC2×D4×D7C22
hypercenter#488D7
hypocenterC7#3140
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #90379C2semidirect
3. C7×#488C2semidirect
4. #89840C2semidirect
5. #89828C2semidirect
6. #89574C2semidirect
7. #89567C2semidirect
9. #89537C2semidirect
10. #89469C2semidirect
11. C4×#1131C2semidirect
14. C7×#125C22semidirect
21. #6451C22semidirect
27. #6433C22semidirect
28. C7×#103C22semidirect
30. C4×C7×C8C22semidirect
31. #6409C22semidirect
34. #6390C22semidirect
35. #6389C22semidirect
36. #6384C22semidirect
38. C4×C7⋊C8C22semidirect
77. #488D7semidirect
111. #125C2×D7semidirect
125. #103C2×D7semidirect
132. C4×C8C2×D7semidirect
209. C7#3140semidirect

Maximal subgroups

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

subgroupindex
#903792
D7×#1242
C7×#4882
#898402
#898282
#895742
#895672
#895422
#895372
#894692
C4×#11312
#31407