Finite Group Databasebeta

← #11728883

Subgroups — #11728883

Order 896.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×Dic14C211yesmaximal, characteristic
2C7×#475C211yesmaximal, characteristic
3#89865C211yesmaximal, characteristic
4#89862C211yesmaximal, characteristic
5Q8⋊(C7⋊C8)C211yesmaximal, characteristic
6#89279C211yesmaximal, characteristic
7#89276C211yesmaximal, characteristic
8C7×C4⋊Q8C2211yescharacteristic
9C4×C7×Q8C2211yescharacteristic
10Q8×Dic7—21no—
11C2×#1144—21no—
12#6451—22no—
13#6451—21no—
14#6432—21no—
15#6432C2211yescharacteristic
16C4×Dic14—21no—
17C4×Dic14C2211yescharacteristic
18C7×#106—21no—
19C7×C4⋊C8C2211yescharacteristic
20C7×#103—21no—
21#6409—21no—
22#6384—22no—
23C4⋊(C7⋊C8)C2211yescharacteristic
24C4×C7⋊C8C2211yescharacteristic
25#1645—71nomaximal, Sylow
26C2×C7×Q8—21no—
27C2×C7×Q8D411yescharacteristic
28C2×Dic14—22no—
29C2×Dic14D411yescharacteristic
30C2×Dic14D412yes—
31C2×C7×C8—21no—
32C7×C4⋊C4—21no—
33C7×C4⋊C4—21no—
34C7×C4⋊C4—21no—
35C7×C4⋊C4D411yescharacteristic
36C42×C7—21no—
37C42×C7C2311yescharacteristic
38#1144—42no—
39C4⋊Dic7—22no—
40C4⋊Dic7—21no—
41C4⋊Dic7—21no—
42C4⋊Dic7D411yescharacteristic
43Dic7⋊C4—22no—
44Dic7⋊C4—21no—
45C4×Dic7—22no—
46C4×Dic7—21no—
47C2×C7⋊C8—21no—
48C2×C7⋊C8—21no—
49Q82—71no—
50#487—71no—
51#475D711yescharacteristic
52#462—71no—
53#333—71no—
54Q8⋊C8—71no—
55Q8⋊C8—71no—
56C7×Q8—21no—
57C7×Q8—41no—
58C7×Q8—41no—
59C7×Q8Q1612yescomplemented
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—22no—
68C2×Dic7—21no—
69Dic14—21no—
70Dic14—42no—
71Dic14—82no—
72Dic14SD1612yes—
73C7×C8—41no—
74C7⋊C8—41no—
75C7⋊C8—41no—
76C2×Q16—141no—
77C4⋊Q8—71no—
78C4⋊Q8—141no—
79C4⋊Q8—142no—
80C4⋊Q8—141no—
81C4⋊Q8C2×D711yescharacteristic
82C4×Q8—71no—
83C4×Q8—141no—
84C4×Q8—141no—
85C4×Q8C2×D711yescharacteristic
86#106—21no—
87C4⋊C8—71no—
88C4⋊C8C2×D711yescharacteristic
89#103—21no—
90#103—142no—
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×C7C2×Q1611yescharacteristic
100C4×C7#13711yescharacteristic
101C4×C7#13711yescharacteristic
102Dic7—22no—
103Dic7—42no—
104Dic7—41no—
105C2×Q8—21no—
106C2×Q8—71no—
107C2×Q8—72no—
108C2×Q8—142no—
109C2×Q8C22⋊D711yescharacteristic
110Q16—282no—
111C2×C8—21no—
112C2×C8—141no—
113C2×C8—141no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—71no—
118C4⋊C4—142no—
119C4⋊C4—142no—
120C4⋊C4—141no—
121C4⋊C4—141no—
122C4⋊C4—141no—
123C4⋊C4C22⋊D711yescharacteristic
124C42—21no—
125C42—142no—
126C42—141no—
127C42C22×D711yescharacteristic
128C2×C7#44811yescharacteristic
129C2×C7#45111yescharacteristic
130C2×C7#45511yescharacteristic
131Q8—21no—
132Q8—41no—
133Q8—41no—
134Q8—72no—
135Q8—141no—
136Q8—282no—
137Q8—562no—
138Q8#114412yescomplemented
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—41no—
142C2×C4—141no—
143C2×C4—141no—
144C2×C4—142no—
145C2×C4D4×D711yescharacteristic
146C2×C4D4×D711yescharacteristic
147C2×C4C2×C22⋊D711yescharacteristic
148C8—41no—
149C8—281no—
150C8—281no—
151C7#164511yescharacteristic, complemented, Sylow
152C22#650011yescharacteristic
153C4—21no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—142no—
158C4—282no—
159C4—281no—
160C4D7×SD1611yescharacteristic
161C4#647811yescharacteristic
162C4C2×#114411yescharacteristic
163C4#651411yescharacteristic
164C2#8982211yescharacteristic
165C2#8994211yescharacteristic
166C2#8997411yescharacteristic
subgroupG/H
centerC22#6500
commutator, cosocleC42×C7C23
FrattiniC42C22×D7
FittingC7×#475C2
socle, prime radicalC22×C7#120
hypercenter#475D7
hypocenterC7#1645
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
59. C7×Q8Q16semidirect
138. Q8#1144semidirect
151. C7#1645semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic142
C7×#4752
#898652
#898622
Q8⋊(C7⋊C8)2
#892792
#892762
#16457