Finite Group Databasebeta

← #13054085

Subgroups — #13054085

Order 1344.

364 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C7×Dic6C211yesmaximal, characteristic
2C2×C3×Dic7⋊C4C211yesmaximal, characteristic
3C2×#66577C212yesmaximal
4#10620670C211yesmaximal, characteristic
5#10620529C212yesmaximal
6#90001—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×Dic21C2212yes—
9C22×C7×Dic3C2212yes—
10C2×C7×Dic6—22no—
11C2×C7×Dic6—24no—
12C22×C3×Dic7C2211yescharacteristic
13C22×C3×Dic7C2211yescharacteristic
14C3×Dic7⋊C4—22no—
15#66577—24no—
16C22×C7×Q8—31no—
17C2×Dic7⋊C4—32no—
18C2×Dic7⋊C4D311yescharacteristic
19C7⋊#95—31no—
20C7⋊#95—32no—
21#4979—71nomaximal, Hall
22C23×C3×C7C2311yescharacteristic
23C2×C3×C4×C7—22no—
24C2×C3×C4×C7D412yes—
25C2×Dic21—22no—
26C2×Dic21—22no—
27C2×Dic21Q814yescomplemented
28C2×C7×Dic3—22no—
29C2×C7×Dic3—22no—
30C2×C7×Dic3D414yes—
31C7×Dic6—28no—
32C2×C3×Dic7—21no—
33C2×C3×Dic7—21no—
34C2×C3×Dic7—22no—
35C2×C3×Dic7—21no—
36C2×C3×Dic7Q812yescomplemented
37C2×C7×Q8—64no—
38C2×C7×Q8—62no—
39C22×C4×C7—32no—
40C22×C4×C7C2×D311yescharacteristic
41C22×Dic7—32no—
42C22×Dic7C2×D311yescharacteristic
43C22×Dic7C2×D311yescharacteristic
44Dic7⋊C4—22no—
45Dic7⋊C4—64no—
46C22×Dic6D711yescharacteristic
47C2×C3×C4⋊C4—71no—
48C2×Dic3⋊C4—72no—
49C3⋊#95—71no—
50C3⋊#95—72no—
51C22×C3×C7C2×D412yes—
52C22×C3×C7C2×D411yescharacteristic
53C22×C3×C7C2×Q811yescharacteristic
54C22×C3×C7C2×Q812yes—
55C22×C3×C7#4111yescharacteristic
56C3×C4×C7—22no—
57Dic21—24no—
58C3×Dic7—22no—
59C3×Dic7—41no—
60C7×Dic3—24no—
61#395—211noSylow
62C23×C7C22×D311yescharacteristic
63C7×Q8—68no—
64C2×C4×C7—22no—
65C2×C4×C7—34no—
66C2×C4×C7—62no—
67C2×C4×C7—62no—
68C2×C4×C7#6712yes—
69C2×Dic7—21no—
70C2×Dic7—21no—
71C2×Dic7—22no—
72C2×Dic7—21no—
73C2×Dic7—34no—
74C2×Dic7—62no—
75C2×Dic7—62no—
76C2×Dic7Dic612yescomplemented
77C22×C3×C4—71no—
78C22×C3×C4—71no—
79C22×C3×C4C2×D711yescharacteristic
80C22×Dic3—72no—
81C22×Dic3C2×D712yes—
82C2×Dic6—22no—
83C2×Dic6—24no—
84C3×C4⋊C4—142no—
85Dic3⋊C4—144no—
86C2×C3×C7#12011yescharacteristic
87C2×C3×C7C22⋊Q812yes—
88C2×C3×C7C22⋊Q811yescharacteristic
89C2×C3×C7C4⋊Q812yes—
90C2×C3×C7C4⋊Q811yescharacteristic
91C22×Q8—31no—
92C2×C4⋊C4—71no—
93C2×C4⋊C4—212no—
94#95—212no—
95#95—211no—
96C22×C7C2×Dic611yescharacteristic
97C22×C7#23511yescharacteristic
98C22×C7D3×D412yes—
99C22×C7D3×Q812yes—
100C22×C7C2×#6711yescharacteristic
101C4×C7—22no—
102C4×C7—64no—
103Dic7—22no—
104Dic7—41no—
105Dic7—64no—
106C23×C3C22×D711yescharacteristic
107C2×C3×C4—22no—
108C2×C3×C4—72no—
109C2×C3×C4—141no—
110C2×C3×C4—141no—
111C2×C3×C4—142no—
112C2×C3×C4—141no—
113C2×C3×C4C22⋊D712yes—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—74no—
117C2×Dic3—142no—
118C2×Dic3—142no—
119C2×Dic3C22⋊D714yes—
120Dic6—28no—
121C3×C7#39511yescharacteristic, complemented, Hall
122C2×Q8—64no—
123C2×Q8—62no—
124C22×C4—32no—
125C22×C4—71no—
126C22×C4—71no—
127C22×C4—212no—
128C22×C4D3×D711yescharacteristic
129C4⋊C4—142no—
130C4⋊C4—424no—
131C2×C7#88812yes—
132C2×C7#89612yes—
133C2×C7#92411yescharacteristic
134C2×C7#93711yescharacteristic
135C2×C7#94411yescharacteristic
136C22×C3Q8×D711yescharacteristic
137C22×C3Q8×D712yes—
138C22×C3#116111yescharacteristic
139C22×C3C2×C22⋊D711yescharacteristic
140C22×C3C2×C22⋊D712yes—
141C3×C4—22no—
142C3×C4—142no—
143C3×C4—281no—
144Dic3—24no—
145Dic3—144no—
146C23C2×D3×D711yescharacteristic
147Q8—68no—
148C2×C4—22no—
149C2×C4—34no—
150C2×C4—62no—
151C2×C4—62no—
152C2×C4—72no—
153C2×C4—141no—
154C2×C4—141no—
155C2×C4—141no—
156C2×C4—142no—
157C2×C4—214no—
158C2×C4—422no—
159C2×C4—422no—
160C2×C4#424512yes—
161C7#497911yescharacteristic, complemented, Sylow
162C2×C3#650711yescharacteristic
163C2×C3#650712yes—
164C2×C3#650912yes—
165C2×C3#650911yescharacteristic
166C2×C3#651611yescharacteristic
167C22D7×Dic611yescharacteristic
168C22D21⋊Q812yes—
169C22#6667711yescharacteristic
170C22C2×#424511yescharacteristic
171C22D3×C22⋊D712yes—
172C4—22no—
173C4—64no—
174C4—142no—
175C4—281no—
176C4—424no—
177C3#9000111yescharacteristic, complemented, Sylow
178C2#1062089112yes—
179C2#1062092111yescharacteristic
180C2#1062094711yescharacteristic
181C2#1062098612yes—
182C2#1062113011yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, prime radical, cosocleC23×C3×C7C23
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#395
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
27. C2×Dic21Q8semidirect
36. C2×C3×Dic7Q8semidirect
76. C2×Dic7Dic6semidirect
121. C3×C7#395semidirect
161. C7#4979semidirect
177. C3#90001semidirect

Maximal subgroups

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

subgroupindex
C22×C7×Dic62
C2×C3×Dic7⋊C42
C2×#665772
#106206702
#106205292
#900013
#49797