Finite Group Databasebeta

← #13054240

Subgroups — #13054240

Order 1344.

296 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 76 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×Dic42C211yesmaximal, characteristic
2C3×#6487C211yesmaximal, characteristic
3C2×#66570C212yesmaximal, complemented
4C7×#830C211yesmaximal, characteristic
5#10620509C212yesmaximal, complemented
6#89511—31nomaximal, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×Dic21—21no—
9C2×Dic42—21no—
10C2×Dic42—24no—
11C2×Dic42C2212yes—
12C2×C3×Dic14C2211yescharacteristic
13C2×C7×C3⋊C8C2212yes—
14C3×C22⋊Dic7—21no—
15C3×C4⋊Dic7C2211yescharacteristic
16C3×Dic7⋊C4—21no—
17#66570—24no—
18C22×Dic14—31no—
19#6487D311yescharacteristic
20C2×Dic28—32no—
21C7×C22⋊C8—31no—
22#6390—32no—
23#5050—71nomaximal, Hall
24C23×C3×C7D411yescharacteristic
25C2×C3×C4×C7—22no—
26C2×C3×C4×C7D411yescharacteristic
27C2×C3×C4×C7C2311yescharacteristic
28C2×Dic21—24no—
29C2×Dic21—22no—
30Dic42—24no—
31Dic42—22no—
32Dic42D414yes—
33C2×C3×Dic7—21no—
34C2×C3×Dic7—21no—
35C3×Dic14—22no—
36C7×C3⋊C8—22no—
37C22×C4×C7C2×D311yescharacteristic
38C22×Dic7—61no—
39C2×Dic14—32no—
40C2×Dic14—64no—
41C2×Dic14—61no—
42C2×Dic14C2×D311yescharacteristic
43C2×C7×C8—32no—
44C22⋊Dic7—21no—
45C4⋊Dic7C2×D311yescharacteristic
46Dic7⋊C4—21no—
47Dic28—64no—
48C22×Dic6—71no—
49C3×C22⋊Q8—71no—
50C2×#216—72no—
51#830D711yescharacteristic
52#810—72no—
53C22×C3×C7—21no—
54C22×C3×C7—21no—
55C22×C3×C7Q1612yescomplemented
56C22×C3×C7C2×D411yescharacteristic
57C3×C4×C7—21no—
58C3×C4×C7C2×D412yes—
59Dic21—24no—
60C3×Dic7—41no—
61C3×Dic7—41no—
62#451—211noSylow
63C23×C7#6711yescharacteristic
64C2×C4×C7—22no—
65C2×C4×C7#6711yescharacteristic
66C2×C4×C7C22×D311yescharacteristic
67C2×Dic7—21no—
68C2×Dic7—21no—
69C2×Dic7—64no—
70C2×Dic7—62no—
71Dic14—22no—
72Dic14—34no—
73Dic14—64no—
74Dic14—62no—
75C7×C8—62no—
76C2×C3×Q8—71no—
77C22×C3×C4C2×D711yescharacteristic
78C22×Dic3—141no—
79C2×Dic6—72no—
80C2×Dic6—144no—
81C2×Dic6—141no—
82C3×C4⋊C4—71no—
83C3×C4⋊C4—141no—
84C3×C22⋊C4—141no—
85#216—144no—
86C2×C3⋊C8C2×D712yes—
87C2×C3×C7—22no—
88C2×C3×C7#12011yescharacteristic
89C2×C3×C7C2×Q1611yescharacteristic
90C2×C3×C7#13711yescharacteristic
91C22×Q8—211no—
92C2×Q16—212no—
93C22⋊Q8—71no—
94#103—212no—
95C22⋊C8—31no—
96C22×C7—21no—
97C22×C7—21no—
98C22×C7#21612yescomplemented
99C22×C7C2×#6711yescharacteristic
100C4×C7—21no—
101C4×C7D3×D412yes—
102Dic7—41no—
103Dic7—41no—
104Dic7—64no—
105C23×C3D2811yescharacteristic
106C3×Q8—142no—
107C2×C3×C4—22no—
108C2×C3×C4—141no—
109C2×C3×C4—141no—
110C2×C3×C4D2811yescharacteristic
111C2×C3×C4C22×D711yescharacteristic
112C2×Dic3—144no—
113C2×Dic3—142no—
114Dic6—74no—
115Dic6—144no—
116Dic6—142no—
117C3⋊C8—22no—
118C3×C7#45111yescharacteristic, complemented, Hall
119C2×Q8—71no—
120C2×Q8—212no—
121C2×Q8—424no—
122C2×Q8—421no—
123C22×C4—421no—
124C22×C4D3×D711yescharacteristic
125Q16—424no—
126C2×C8—32no—
127C4⋊C4—71no—
128C4⋊C4—141no—
129C22⋊C4—141no—
130C2×C7—22no—
131C2×C7#93711yescharacteristic
132C2×C7C2×#21611yescharacteristic
133C2×C7#95111yescharacteristic
134C22×C3—21no—
135C22×C3—21no—
136C22×C3Dic2812yescomplemented
137C22×C3C2×D2811yescharacteristic
138C3×C4—21no—
139C3×C4—281no—
140C3×C4—281no—
141C3×C4D4×D712yes—
142Dic3—144no—
143C23#424611yescharacteristic
144Q8—142no—
145Q8—214no—
146Q8—424no—
147Q8—422no—
148C2×C4—22no—
149C2×C4—141no—
150C2×C4—141no—
151C2×C4—424no—
152C2×C4—422no—
153C2×C4#424611yescharacteristic
154C2×C4C2×D3×D711yescharacteristic
155C8—62no—
156C7#505011yescharacteristic, complemented, Sylow
157C2×C3—22no—
158C2×C3#644511yescharacteristic
159C2×C3C2×Dic2811yescharacteristic
160C2×C3#647211yescharacteristic
161C22—21no—
162C22—21no—
163C22#6657012yescomplemented
164C22C2×#424611yescharacteristic
165C4—21no—
166C4—281no—
167C4—281no—
168C4—424no—
169C4#6669412yes—
170C3#8951111yescharacteristic, complemented, Sylow
171C2—22no—
172C2#1062087511yescharacteristic
173C2C2×#6657011yescharacteristic
174C2#1062113411yescharacteristic
subgroupG/H
centerC22C2×#4246
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC22×C4D3×D7
hypocenterC3×C7#451
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#66570C2semidirect
5. #10620509C2semidirect
55. C22×C3×C7Q16semidirect
98. C22×C7#216semidirect
118. C3×C7#451semidirect
136. C22×C3Dic28semidirect
156. C7#5050semidirect
163. C22#66570semidirect
170. C3#89511semidirect

Maximal subgroups

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

subgroupindex
C22×Dic422
C3×#64872
C2×#665702
C7×#8302
#106205092
#895113
#50507