Finite Group Databasebeta

← #13054053

Subgroups — #13054053

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×C3×Dic14C211yesmaximal, characteristic
2C2×C7×Dic3⋊C4C211yesmaximal, characteristic
3C2×#66577C212yesmaximal
4#10620670C211yesmaximal, characteristic
5#10620529C212yesmaximal
6#89751—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×Dic21C2212yes—
9C22×C7×Dic3C2211yescharacteristic
10C22×C7×Dic3C2211yescharacteristic
11C22×C3×Dic7C2212yes—
12C2×C3×Dic14—22no—
13C2×C3×Dic14—24no—
14C7×Dic3⋊C4—22no—
15#66577—24no—
16C22×Dic14D311yescharacteristic
17C2×C7×C4⋊C4—31no—
18C2×Dic7⋊C4—32no—
19C7⋊#95—31no—
20C7⋊#95—32no—
21#5230—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—21no—
29C2×C7×Dic3—21no—
30C2×C7×Dic3—22no—
31C2×C7×Dic3—21no—
32C2×C7×Dic3Q812yescomplemented
33C2×C3×Dic7—22no—
34C2×C3×Dic7—22no—
35C2×C3×Dic7D414yes—
36C3×Dic14—28no—
37C22×C4×C7—31no—
38C22×C4×C7—31no—
39C22×C4×C7C2×D311yescharacteristic
40C22×Dic7—32no—
41C22×Dic7C2×D312yes—
42C2×Dic14—22no—
43C2×Dic14—24no—
44C7×C4⋊C4—62no—
45Dic7⋊C4—64no—
46C22×C3×Q8—71no—
47C2×Dic3⋊C4—72no—
48C2×Dic3⋊C4D711yescharacteristic
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—24no—
59C7×Dic3—22no—
60C7×Dic3—41no—
61#395—211noSylow
62C23×C7C22×D311yescharacteristic
63C2×C4×C7—22no—
64C2×C4×C7—32no—
65C2×C4×C7—61no—
66C2×C4×C7—61no—
67C2×C4×C7—62no—
68C2×C4×C7—61no—
69C2×C4×C7#6712yes—
70C2×Dic7—22no—
71C2×Dic7—22no—
72C2×Dic7—34no—
73C2×Dic7—62no—
74C2×Dic7—62no—
75C2×Dic7#6714yes—
76Dic14—28no—
77C2×C3×Q8—144no—
78C2×C3×Q8—142no—
79C22×C3×C4—72no—
80C22×C3×C4C2×D711yescharacteristic
81C22×Dic3—72no—
82C22×Dic3C2×D711yescharacteristic
83C22×Dic3C2×D711yescharacteristic
84Dic3⋊C4—22no—
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—71no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—212no—
94#95—212no—
95#95—211no—
96C22×C7D3×Q811yescharacteristic
97C22×C7D3×Q812yes—
98C22×C7#23911yescharacteristic
99C22×C7C2×#6711yescharacteristic
100C22×C7C2×#6712yes—
101C4×C7—22no—
102C4×C7—62no—
103C4×C7—121no—
104Dic7—24no—
105Dic7—64no—
106C23×C3C22×D711yescharacteristic
107C3×Q8—148no—
108C2×C3×C4—22no—
109C2×C3×C4—74no—
110C2×C3×C4—142no—
111C2×C3×C4—142no—
112C2×C3×C4C22⋊D712yes—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—22no—
116C2×Dic3—21no—
117C2×Dic3—74no—
118C2×Dic3—142no—
119C2×Dic3—142no—
120C2×Dic3Dic1412yescomplemented
121C3×C7#39511yescharacteristic, complemented, Hall
122C2×Q8—144no—
123C2×Q8—142no—
124C22×C4—31no—
125C22×C4—31no—
126C22×C4—72no—
127C22×C4—212no—
128C22×C4D3×D711yescharacteristic
129C4⋊C4—62no—
130C4⋊C4—424no—
131C2×C7#94411yescharacteristic
132C2×C7#94412yes—
133C2×C7#94611yescharacteristic
134C2×C7#94612yes—
135C2×C7#95311yescharacteristic
136C22×C3C2×Dic1411yescharacteristic
137C22×C3#115711yescharacteristic
138C22×C3D4×D712yes—
139C22×C3Q8×D712yes—
140C22×C3C2×C22⋊D711yescharacteristic
141C3×C4—22no—
142C3×C4—144no—
143Dic3—22no—
144Dic3—41no—
145Dic3—144no—
146C23C2×D3×D711yescharacteristic
147Q8—148no—
148C2×C4—22no—
149C2×C4—32no—
150C2×C4—61no—
151C2×C4—61no—
152C2×C4—61no—
153C2×C4—62no—
154C2×C4—74no—
155C2×C4—142no—
156C2×C4—142no—
157C2×C4—214no—
158C2×C4—422no—
159C2×C4—422no—
160C2×C4#424512yes—
161C7#523011yescharacteristic, complemented, Sylow
162C2×C3#645112yes—
163C2×C3#645912yes—
164C2×C3#648711yescharacteristic
165C2×C3#650011yescharacteristic
166C2×C3#650711yescharacteristic
167C22D3×Dic1411yescharacteristic
168C22#6667311yescharacteristic
169C22D21⋊Q812yes—
170C22C2×#424511yescharacteristic
171C22D7×#6712yes—
172C4—22no—
173C4—62no—
174C4—121no—
175C4—144no—
176C4—424no—
177C3#8975111yescharacteristic, complemented, Sylow
178C2#1062088912yes—
179C2#1062093111yescharacteristic
180C2#1062094711yescharacteristic
181C2#1062097312yes—
182C2#1062112511yescharacteristic
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
32. C2×C7×Dic3Q8semidirect
120. C2×Dic3Dic14semidirect
121. C3×C7#395semidirect
161. C7#5230semidirect
177. C3#89751semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic142
C2×C7×Dic3⋊C42
C2×#665772
#106206702
#106205292
#897513
#52307