Finite Group Databasebeta

← #13056620

Subgroups — #13056620

Order 1344.

300 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 111 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×C4⋊Dic21C211yesmaximal, characteristic
3C2×#66629C212yesmaximal
4#10620670C212yesmaximal
5#10620670C211yesmaximal, characteristic
6#89755—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21C2212yes—
11C22×Dic21C2212yes—
12C3×C7×C4⋊C4—22no—
13C4⋊Dic21—22no—
14#66629—24no—
15C2×C7×C4⋊C4D311yescharacteristic
16C2×C4⋊Dic7—31no—
17C2×Dic7⋊C4—32no—
18C7⋊#95—31no—
19C7⋊#95—32no—
20#4981—71nomaximal, Hall
21C23×C3×C7C2311yescharacteristic
22C2×C3×C4×C7—21no—
23C2×C3×C4×C7—21no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—21no—
26C2×C3×C4×C7—22no—
27C2×C3×C4×C7D412yes—
28C2×C3×C4×C7D412yes—
29C2×Dic21—22no—
30C2×Dic21—22no—
31C2×Dic21—22no—
32C2×Dic21—22no—
33C2×Dic21—22no—
34C2×Dic21Q814yes—
35C22×C4×C7C2×D311yescharacteristic
36C22×C4×C7C2×D311yescharacteristic
37C22×C4×C7C2×D311yescharacteristic
38C22×Dic7—32no—
39C22×Dic7—32no—
40C7×C4⋊C4—22no—
41C4⋊Dic7—62no—
42Dic7⋊C4—64no—
43C2×C3×C4⋊C4D711yescharacteristic
44C2×C4⋊Dic3—71no—
45C2×Dic3⋊C4—72no—
46C3⋊#95—71no—
47C3⋊#95—72no—
48C22×C3×C7C2×D411yescharacteristic
49C22×C3×C7C2×D411yescharacteristic
50C22×C3×C7C2×Q812yes—
51C22×C3×C7#4111yescharacteristic
52C22×C3×C7#4112yes—
53C3×C4×C7—22no—
54C3×C4×C7—22no—
55C3×C4×C7—41no—
56Dic21—24no—
57Dic21—42no—
58#398—211noSylow
59C23×C7C22×D311yescharacteristic
60C2×C4×C7—21no—
61C2×C4×C7—22no—
62C2×C4×C7—21no—
63C2×C4×C7—22no—
64C2×C4×C7—21no—
65C2×C4×C7D1212yes—
66C2×C4×C7#6712yes—
67C2×Dic7—34no—
68C2×Dic7—62no—
69C2×Dic7—62no—
70C2×Dic7—62no—
71C2×Dic7—62no—
72C2×Dic7—62no—
73C22×C3×C4C2×D711yescharacteristic
74C22×C3×C4C2×D711yescharacteristic
75C22×C3×C4C2×D711yescharacteristic
76C22×Dic3—72no—
77C22×Dic3—72no—
78C3×C4⋊C4—22no—
79C4⋊Dic3—142no—
80Dic3⋊C4—144no—
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×C7C2×D1211yescharacteristic
92C22×C7#23511yescharacteristic
93C22×C7#23712yes—
94C22×C7D3×Q812yes—
95C22×C7C2×#6711yescharacteristic
96C4×C7—22no—
97C4×C7—22no—
98C4×C7—41no—
99Dic7—64no—
100Dic7—122no—
101C23×C3C22×D711yescharacteristic
102C2×C3×C4—21no—
103C2×C3×C4—22no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—21no—
107C2×C3×C4D2812yes—
108C2×C3×C4C22⋊D712yes—
109C2×Dic3—74no—
110C2×Dic3—142no—
111C2×Dic3—142no—
112C2×Dic3—142no—
113C2×Dic3—142no—
114C2×Dic3—142no—
115C3×C7#39811yescharacteristic, complemented, Hall
116C22×C4—212no—
117C22×C4—212no—
118C22×C4C2×D2111yescharacteristic
119C22×C4C2×D2111yescharacteristic
120C22×C4C2×D2111yescharacteristic
121C4⋊C4—22no—
122C4⋊C4—424no—
123C4⋊C4—422no—
124C2×C7#88912yes—
125C2×C7#89712yes—
126C2×C7#93011yescharacteristic
127C2×C7#93511yescharacteristic
128C2×C7#94411yescharacteristic
129C22×C3C2×D2811yescharacteristic
130C22×C3#115711yescharacteristic
131C22×C3#115912yes—
132C22×C3Q8×D712yes—
133C22×C3C2×C22⋊D711yescharacteristic
134C3×C4—22no—
135C3×C4—22no—
136C3×C4—41no—
137Dic3—144no—
138Dic3—282no—
139C23C22×D2111yescharacteristic
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—214no—
146C2×C4—422no—
147C2×C4—422no—
148C2×C4—422no—
149C2×C4—422no—
150C2×C4—422no—
151C2×C4D8412yes—
152C2×C4#426812yes—
153C7#498111yescharacteristic, complemented, Sylow
154C2×C3#645212yes—
155C2×C3#646012yes—
156C2×C3#649311yescharacteristic
157C2×C3#649811yescharacteristic
158C2×C3#650711yescharacteristic
159C22C2×D8411yescharacteristic
160C22#6672811yescharacteristic
161C22#6673012yes—
162C22Q8×D2112yes—
163C22C2×#426811yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—424no—
168C4—842no—
169C3#8975511yescharacteristic, complemented, Sylow
170C2#1062137212yes—
171C2#1062138012yes—
172C2#1062141311yescharacteristic
173C2#1062141811yescharacteristic
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#4981semidirect
169. C3#89755semidirect

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×C4⋊Dic212
C2×#666292
#106206702
#897553
#49817