Finite Group Databasebeta

← #11728992

Subgroups — #11728992

Order 896.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#90421C211yesmaximal, characteristic
2C7×#487C211yesmaximal, characteristic, complemented
3#89862C211yesmaximal, characteristic, complemented
4#89859C211yesmaximal, characteristic
5#89341C211yesmaximal, characteristic, complemented
6#89254C211yesmaximal, characteristic
7#89251C211yesmaximal, characteristic, complemented
8C7×C4⋊Q8C2211yescharacteristic
9C7×C4⋊Q8C2211yescharacteristic
10#6509—22no—
11C2×#1144—21no—
12#6460—21no—
13#6456—21no—
14D7×C4⋊C4—21no—
15#6453—21no—
16C4×D28C2211yescharacteristic
17C4×Dic14C2211yescharacteristic
18C7×#103—22no—
19C4×C7×C8C2211yescharacteristic
20#6409—21no—
21#6383—21no—
22#6381—21no—
23C4⋊(C7⋊C8)C2211yescharacteristic
24C4⋊(C7⋊C8)C2211yescharacteristic
25#1644—71nomaximal, Sylow
26C2×C7×Q8—21no—
27C2×C7×Q8D412yes—
28C2×D28D411yescharacteristic
29C2×C4×D7—21no—
30C2×C4×D7—22no—
31C2×Dic14D411yescharacteristic
32C2×C7×C8—21no—
33C7×C4⋊C4—21no—
34C7×C4⋊C4—21no—
35C7×C4⋊C4—21no—
36C7×C4⋊C4—21no—
37C42×C7C2311yescharacteristic
38#1144—42no—
39(C2×D7)⋊C4—22no—
40(C2×D7)⋊C4—21no—
41C4⋊Dic7—22no—
42C4⋊Dic7D411yescharacteristic
43C4⋊Dic7D411yescharacteristic
44Dic7⋊C4—22no—
45Dic7⋊C4—21no—
46C4×Dic7—21no—
47C2×C7⋊C8—21no—
48C2×C7⋊C8—21no—
49#554—71no—
50#487D711yescharacteristic, complemented
51#474—71no—
52#462—71no—
53#333—71no—
54Q8⋊C8—71no—
55D4⋊C8—71no—
56C22×D7—21no—
57C7×Q8—42no—
58C7×Q8—41no—
59C7×Q8—41no—
60C2×C4×C7—22no—
61C2×C4×C7—41no—
62C2×C4×C7C2×D411yescharacteristic
63C2×C4×C7C2×D411yescharacteristic
64C2×C4×C7C2×D411yescharacteristic
65C2×Dic7—21no—
66C2×Dic7—21no—
67C2×Dic7—21no—
68D28—21no—
69D28Q1612yescomplemented
70C4×D7—41no—
71C4×D7—41no—
72C4×D7—82no—
73Dic14—21no—
74Dic14—21no—
75C7×C8—41no—
76C7⋊C8—41no—
77C7⋊C8—41no—
78C2×Q16—141no—
79C4⋊Q8C2×D711yescharacteristic
80C4⋊Q8C2×D711yescharacteristic
81#125—141no—
82C22⋊Q8—141no—
83C22⋊Q8—142no—
84C4×Q8—71no—
85C4×D4—71no—
86C4×D4—141no—
87C2×C4⋊C4—141no—
88#107—141no—
89C4⋊C8—71no—
90C4⋊C8—71no—
91#103—22no—
92#103—141no—
93#102—141no—
94C4×C8C2×D711yescharacteristic
95C22×C7#12011yescharacteristic
96C2×D7—22no—
97C2×D7—41no—
98C2×D7—41no—
99C4×C7—21no—
100C4×C7—42no—
101C4×C7—81no—
102C4×C7C2×Q1611yescharacteristic
103C4×C7#13511yescharacteristic
104C4×C7AGL(1,ℤ/8)11yescharacteristic
105C4×C7#13711yescharacteristic
106Dic7—41no—
107Dic7—41no—
108Dic7—41no—
109C2×Q8—21no—
110C2×Q8—71no—
111C2×Q8C22⋊D712yes—
112C2×D4—71no—
113C22×C4—142no—
114C22×C4—141no—
115Q16—282no—
116C2×C8—21no—
117C2×C8—141no—
118C2×C8—141no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—71no—
124C4⋊C4—71no—
125C4⋊C4—142no—
126C4⋊C4—142no—
127C4⋊C4—141no—
128C22⋊C4—142no—
129C22⋊C4—141no—
130C42—141no—
131C42C22×D711yescharacteristic
132C2×C7#44911yescharacteristic
133C2×C7#45111yescharacteristic
134C2×C7#45511yescharacteristic
135D7—42no—
136C23—141no—
137Q8—42no—
138Q8—41no—
139Q8—41no—
140Q8—141no—
141Q8—141no—
142D4—72no—
143D4—141no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—141no—
147C2×C4—141no—
148C2×C4—141no—
149C2×C4—281no—
150C2×C4—281no—
151C2×C4—562no—
152C2×C4D4×D711yescharacteristic
153C2×C4D4×D711yescharacteristic
154C2×C4C2×C22⋊D711yescharacteristic
155C8—41no—
156C8—281no—
157C8—281no—
158C7#164411yescharacteristic, complemented, Sylow
159C22—142no—
160C22—281no—
161C22—281no—
162C22#650011yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—281no—
167C4—281no—
168C4—281no—
169C4#647711yescharacteristic
170C4#647911yescharacteristic
171C4D7×Q1611yescharacteristic
172C4#648111yescharacteristic
173C2—282no—
174C2#8983311yescharacteristic
175C2#8994411yescharacteristic
176C2#8995811yescharacteristic
subgroupG/H
centerC22#6500
commutator, cosocleC42×C7C23
FrattiniC42C22×D7
FittingC7×#487C2
socleC22×C7#120
prime radicalC2×D28D4
hypercenter#487D7
hypocenterC7#1644
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C7×#487C2semidirect
3. #89862C2semidirect
5. #89341C2semidirect
7. #89251C2semidirect
50. #487D7semidirect
69. D28Q16semidirect
158. C7#1644semidirect

Maximal subgroups

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

subgroupindex
#904212
C7×#4872
#898622
#898592
#893412
#892542
#892512
#16447