Finite Group Databasebeta

← #13056615

Subgroups — #13056615

Order 1344.

300 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×C3×C7×C4⋊C4C211yesmaximal, characteristic
2C2×#66629C211yesmaximal, characteristic
3C2×#66629C212yesmaximal
4#10620670C212yesmaximal
5#10620670C211yesmaximal, characteristic
6#89752—31nomaximal, Hall
7C22×C3×C4×C7C2212yes—
8C22×C3×C4×C7C2211yescharacteristic
9C22×Dic21C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2212yes—
12C3×C7×C4⋊C4—22no—
13#66629—22no—
14#66629—24no—
15C2×C7×C4⋊C4D311yescharacteristic
16C2×Dic7⋊C4—32no—
17C2×Dic7⋊C4—31no—
18C7⋊#95—31no—
19C7⋊#95—32no—
20#4983—71nomaximal, Hall
21C23×C3×C7C2311yescharacteristic
22C2×C3×C4×C7—22no—
23C2×C3×C4×C7—22no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—22no—
26C2×C3×C4×C7D412yes—
27C2×Dic21—22no—
28C2×Dic21—21no—
29C2×Dic21—22no—
30C2×Dic21—21no—
31C2×Dic21—21no—
32C2×Dic21—22no—
33C2×Dic21D412yes—
34C2×Dic21Q814yes—
35C22×C4×C7C2×D312yes—
36C22×C4×C7C2×D311yescharacteristic
37C22×Dic7—32no—
38C22×Dic7—31no—
39C22×Dic7—31no—
40C7×C4⋊C4—22no—
41Dic7⋊C4—64no—
42Dic7⋊C4—62no—
43C2×C3×C4⋊C4D711yescharacteristic
44C2×Dic3⋊C4—72no—
45C2×Dic3⋊C4—71no—
46C3⋊#95—71no—
47C3⋊#95—72no—
48C22×C3×C7C2×D411yescharacteristic
49C22×C3×C7C2×D411yescharacteristic
50C22×C3×C7C2×Q812yes—
51C22×C3×C7#4112yes—
52C22×C3×C7#4111yescharacteristic
53C3×C4×C7—22no—
54C3×C4×C7—42no—
55Dic21—22no—
56Dic21—24no—
57Dic21—41no—
58#398—211noSylow
59C23×C7C22×D311yescharacteristic
60C2×C4×C7—22no—
61C2×C4×C7—22no—
62C2×C4×C7—22no—
63C2×C4×C7—22no—
64C2×C4×C7#6712yes—
65C2×Dic7—32no—
66C2×Dic7—34no—
67C2×Dic7—62no—
68C2×Dic7—62no—
69C2×Dic7—61no—
70C2×Dic7—61no—
71C2×Dic7—62no—
72C2×Dic7—61no—
73C22×C3×C4C2×D712yes—
74C22×C3×C4C2×D711yescharacteristic
75C22×Dic3—72no—
76C22×Dic3—71no—
77C22×Dic3—71no—
78C3×C4⋊C4—22no—
79Dic3⋊C4—144no—
80Dic3⋊C4—142no—
81C2×C3×C7#12111yescharacteristic
82C2×C3×C7C22⋊Q812yes—
83C2×C3×C7#12311yescharacteristic
84C2×C3×C7#12512yes—
85C2×C3×C7C4⋊Q811yescharacteristic
86C2×C4⋊C4—212no—
87C2×C4⋊C4—211no—
88C2×C4⋊C4D2111yescharacteristic
89#95—211no—
90#95—212no—
91C22×C7#23512yes—
92C22×C7D3×D411yescharacteristic
93C22×C7#23711yescharacteristic
94C22×C7D3×Q812yes—
95C22×C7C2×#6711yescharacteristic
96C4×C7—22no—
97C4×C7—42no—
98Dic7—64no—
99Dic7—62no—
100Dic7—121no—
101C23×C3C22×D711yescharacteristic
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4C22⋊D712yes—
107C2×Dic3—72no—
108C2×Dic3—74no—
109C2×Dic3—142no—
110C2×Dic3—142no—
111C2×Dic3—141no—
112C2×Dic3—141no—
113C2×Dic3—142no—
114C2×Dic3—141no—
115C3×C7#39811yescharacteristic, complemented, Hall
116C22×C4—212no—
117C22×C4—211no—
118C22×C4—211no—
119C22×C4C2×D2112yes—
120C22×C4C2×D2111yescharacteristic
121C4⋊C4—22no—
122C4⋊C4—424no—
123C4⋊C4—422no—
124C2×C7#88912yes—
125C2×C7#89612yes—
126C2×C7#92911yescharacteristic
127C2×C7#93811yescharacteristic
128C2×C7#94411yescharacteristic
129C22×C3#115712yes—
130C22×C3D4×D711yescharacteristic
131C22×C3#115911yescharacteristic
132C22×C3Q8×D712yes—
133C22×C3C2×C22⋊D711yescharacteristic
134C3×C4—22no—
135C3×C4—42no—
136Dic3—144no—
137Dic3—142no—
138Dic3—281no—
139C23C22×D2111yescharacteristic
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—212no—
145C2×C4—214no—
146C2×C4—422no—
147C2×C4—422no—
148C2×C4—421no—
149C2×C4—421no—
150C2×C4—422no—
151C2×C4—421no—
152C2×C4#426812yes—
153C7#498311yescharacteristic, complemented, Sylow
154C2×C3#645212yes—
155C2×C3#645912yes—
156C2×C3#649211yescharacteristic
157C2×C3#650111yescharacteristic
158C2×C3#650711yescharacteristic
159C22#6672812yes—
160C22D4×D2111yescharacteristic
161C22#6673011yescharacteristic
162C22Q8×D2112yes—
163C22C2×#426811yescharacteristic
164C4—22no—
165C4—42no—
166C4—424no—
167C4—422no—
168C4—841no—
169C3#8975211yescharacteristic, complemented, Sylow
170C2#1062137212yes—
171C2#1062137912yes—
172C2#1062141211yescharacteristic
173C2#1062142111yescharacteristic
174C2#1062142711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, prime radical, cosocleC23×C3×C7C23
FittingC2×C3×C7×C4⋊C4C2
hypercenterC2×C4⋊C4D21
hypocenterC3×C7#398
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
115. C3×C7#398semidirect
153. C7#4983semidirect
169. C3#89752semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C7×C4⋊C42
C2×#666292
#106206702
#897523
#49837