Finite Group Databasebeta

← #11728821

Subgroups — #11728821

Order 896.

194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 57 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#90261C211yesmaximal, characteristic
2C7×#462C211yesmaximal, characteristic
3#89866C211yesmaximal, characteristic
4#89851C211yesmaximal, characteristic
5Q8⋊(C7⋊C8)C211yesmaximal, characteristic
6#89281C211yesmaximal, characteristic
7#89279C211yesmaximal, characteristic
8C2×C7×Q16—21no—
9C7×C4⋊Q8C2211yescharacteristic
10C4×C7×Q8C2211yescharacteristic
11Q8×Dic7—21no—
12#6453—21no—
13#6453—22no—
14#6432—21no—
15#6432C2211yescharacteristic
16C4×Dic14—21no—
17C4×Dic14C2211yescharacteristic
18C7×C4⋊C8C2211yescharacteristic
19C7×#103—21no—
20#6384—21no—
21#6382—21no—
22#6382—22no—
23C4⋊(C7⋊C8)C2211yescharacteristic
24C4×C7⋊C8C2211yescharacteristic
25#1653—71nomaximal, Sylow
26C2×C7×Q8—21no—
27C2×C7×Q8D411yescharacteristic
28C2×Dic14—21no—
29C2×Dic14D411yescharacteristic
30C7×Q16—42no—
31C2×C7×C8—21no—
32C7×C4⋊C4—21no—
33C7×C4⋊C4—21no—
34C7×C4⋊C4—21no—
35C7×C4⋊C4D411yescharacteristic
36C42×C7—21no—
37C42×C7C2311yescharacteristic
38C4⋊Dic7—22no—
39C4⋊Dic7—22no—
40C4⋊Dic7—21no—
41C4⋊Dic7D411yescharacteristic
42C4⋊Dic7D412yes—
43Dic7⋊C4—22no—
44Dic7⋊C4—21no—
45C4×Dic7—22no—
46C4×Dic7—21no—
47C2×C7⋊C8—21no—
48C2×C7⋊C8—21no—
49#557—71no—
50#498—71no—
51#475—71no—
52#462D711yescharacteristic
53#332—71no—
54Q8⋊C8—71no—
55Q8⋊C8—71no—
56C7×Q8—21no—
57C7×Q8—41no—
58C7×Q8—41no—
59C7×Q8SD1612yes—
60C2×C4×C7—21no—
61C2×C4×C7—21no—
62C2×C4×C7—41no—
63C2×C4×C7C2×D411yescharacteristic
64C2×C4×C7C2×D411yescharacteristic
65C2×C4×C7C2×D411yescharacteristic
66C2×Dic7—21no—
67C2×Dic7—21no—
68C2×Dic7—22no—
69Dic14—21no—
70Dic14—21no—
71Dic14—41no—
72Dic14—41no—
73C7×C8—41no—
74C7⋊C8—22no—
75C7⋊C8—41no—
76C2×Q16—21no—
77C4⋊Q8—71no—
78C4⋊Q8—141no—
79C4⋊Q8C2×D711yescharacteristic
80#125—142no—
81#125—141no—
82C4×Q8—71no—
83C4×Q8—141no—
84C4×Q8—141no—
85C4×Q8C2×D711yescharacteristic
86#106—141no—
87#106—142no—
88C4⋊C8—71no—
89C4⋊C8C2×D711yescharacteristic
90#103—21no—
91#103—141no—
92C4×C8—71no—
93C22×C7#12011yescharacteristic
94C4×C7—22no—
95C4×C7—21no—
96C4×C7—41no—
97C4×C7—81no—
98C4×C7C2×SD1611yescharacteristic
99C4×C7#13511yescharacteristic
100C4×C7#13711yescharacteristic
101C4×C7#13711yescharacteristic
102Dic7—41no—
103Dic7—42no—
104Dic7—41no—
105C2×Q8—21no—
106C2×Q8—71no—
107C2×Q8—141no—
108C2×Q8C22⋊D711yescharacteristic
109Q16—42no—
110C2×C8—21no—
111C2×C8—141no—
112C2×C8—141no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—71no—
117C4⋊C4—72no—
118C4⋊C4—142no—
119C4⋊C4—142no—
120C4⋊C4—142no—
121C4⋊C4—141no—
122C4⋊C4—141no—
123C4⋊C4C22⋊D711yescharacteristic
124C42—21no—
125C42—142no—
126C42—141no—
127C42C22×D711yescharacteristic
128C2×C7#44811yescharacteristic
129C2×C7#45211yescharacteristic
130C2×C7#45611yescharacteristic
131Q8—21no—
132Q8—41no—
133Q8—41no—
134Q8—141no—
135Q8—141no—
136Q8—281no—
137Q8—281no—
138Q8#114212yes—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—41no—
142C2×C4—141no—
143C2×C4—142no—
144C2×C4—141no—
145C2×C4D4×D711yescharacteristic
146C2×C4D4×D711yescharacteristic
147C2×C4C2×C22⋊D711yescharacteristic
148C8—41no—
149C8—142no—
150C8—281no—
151C7#165311yescharacteristic, complemented, Sylow
152C22#650011yescharacteristic
153C4—21no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—281no—
158C4—282no—
159C4—281no—
160C4#648111yescharacteristic
161C4#648211yescharacteristic
162C4C2×#114211yescharacteristic
163C4#651411yescharacteristic
164C2#8981211yescharacteristic
165C2#8995911yescharacteristic
166C2#8997711yescharacteristic
subgroupG/H
centerC22#6500
commutator, cosocleC42×C7C23
FrattiniC42C22×D7
FittingC7×#462C2
socle, prime radicalC22×C7#120
hypercenter#462D7
hypocenterC7#1653
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
151. C7#1653semidirect

Maximal subgroups

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

subgroupindex
#902612
C7×#4622
#898662
#898512
Q8⋊(C7⋊C8)2
#892812
#892792
#16537