Finite Group Databasebeta

← #13056689

Subgroups — #13056689

Order 1344.

296 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 79 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×C7×C22⋊Q8C211yesmaximal, characteristic
3C2×#66635C212yesmaximal, complemented
4#10620669C211yesmaximal, characteristic
5#10620649C212yesmaximal, complemented
6#89822—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic
9C22×Dic21—21no—
10C2×Dic42—21no—
11C2×Dic42—24no—
12C2×Dic42C2212yes—
13C3×C7×C4⋊C4—21no—
14C3×C7×C4⋊C4C2211yescharacteristic
15C3×C7×C22⋊C4—21no—
16#66635—24no—
17C2×C3⋊(C7⋊C8)C2212yes—
18C22×Dic14—31no—
19C7×C22⋊Q8D311yescharacteristic
20C2×#1144—32no—
21C22⋊(C7⋊C8)—31no—
22#6384—32no—
23#5050—71nomaximal, Hall
24C23×C3×C7D411yescharacteristic
25C3×C7×Q8—22no—
26C2×C3×C4×C7—22no—
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7—21no—
29C2×C3×C4×C7D411yescharacteristic
30C2×C3×C4×C7C2311yescharacteristic
31C2×Dic21—24no—
32C2×Dic21—22no—
33Dic42—24no—
34Dic42—22no—
35Dic42D414yes—
36C3⋊(C7⋊C8)—22no—
37C2×C7×Q8C2×D311yescharacteristic
38C22×C4×C7C2×D311yescharacteristic
39C22×Dic7—61no—
40C2×Dic14—32no—
41C2×Dic14—64no—
42C2×Dic14—61no—
43C7×C4⋊C4—21no—
44C7×C4⋊C4C2×D311yescharacteristic
45C7×C22⋊C4—21no—
46#1144—64no—
47C2×C7⋊C8—32no—
48C22×Dic6—71no—
49C3×C22⋊Q8D711yescharacteristic
50C2×#216—72no—
51#830—71no—
52#810—72no—
53C22×C3×C7—21no—
54C22×C3×C7—21no—
55C22×C3×C7Q1612yescomplemented
56C22×C3×C7C2×D411yescharacteristic
57C3×C4×C7—21no—
58C3×C4×C7—41no—
59C3×C4×C7—41no—
60C3×C4×C7C2×D412yes—
61Dic21—24no—
62#451—211noSylow
63C23×C7#6711yescharacteristic
64C7×Q8—22no—
65C2×C4×C7—22no—
66C2×C4×C7—21no—
67C2×C4×C7—21no—
68C2×C4×C7#6711yescharacteristic
69C2×C4×C7C22×D311yescharacteristic
70C2×Dic7—64no—
71C2×Dic7—62no—
72Dic14—34no—
73Dic14—64no—
74Dic14—62no—
75C7⋊C8—62no—
76C2×C3×Q8C2×D711yescharacteristic
77C22×C3×C4C2×D711yescharacteristic
78C22×Dic3—141no—
79C2×Dic6—72no—
80C2×Dic6—144no—
81C2×Dic6—141no—
82C3×C4⋊C4—21no—
83C3×C4⋊C4C2×D711yescharacteristic
84C3×C22⋊C4—21no—
85#216—144no—
86C2×C3⋊C8—72no—
87C2×C3×C7—22no—
88C2×C3×C7#12011yescharacteristic
89C2×C3×C7C2×Q1611yescharacteristic
90C2×C3×C7#13711yescharacteristic
91C22×Q8—211no—
92C2×Q16—212no—
93C22⋊Q8D2111yescharacteristic
94#103—212no—
95C22⋊C8—211no—
96C22×C7—21no—
97C22×C7—21no—
98C22×C7#21612yescomplemented
99C22×C7C2×#6711yescharacteristic
100C4×C7—21no—
101C4×C7—41no—
102C4×C7—41no—
103C4×C7D3×D412yes—
104Dic7—64no—
105C23×C3C22⋊D711yescharacteristic
106C3×Q8—22no—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4C22⋊D711yescharacteristic
111C2×C3×C4C22×D711yescharacteristic
112C2×Dic3—144no—
113C2×Dic3—142no—
114Dic6—74no—
115Dic6—144no—
116Dic6—142no—
117C3⋊C8—142no—
118C3×C7#45111yescharacteristic, complemented, Hall
119C2×Q8—212no—
120C2×Q8—424no—
121C2×Q8—421no—
122C2×Q8C2×D2111yescharacteristic
123C22×C4—421no—
124C22×C4C2×D2111yescharacteristic
125Q16—424no—
126C2×C8—212no—
127C4⋊C4—21no—
128C4⋊C4C2×D2111yescharacteristic
129C22⋊C4—21no—
130C2×C7—22no—
131C2×C7#93711yescharacteristic
132C2×C7C2×#21611yescharacteristic
133C2×C7#95111yescharacteristic
134C22×C3—21no—
135C22×C3—21no—
136C22×C3#114412yescomplemented
137C22×C3C2×C22⋊D711yescharacteristic
138C3×C4—21no—
139C3×C4—41no—
140C3×C4—41no—
141C3×C4D4×D712yes—
142Dic3—144no—
143C23#426811yescharacteristic
144Q8—22no—
145Q8—214no—
146Q8—424no—
147Q8—422no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—424no—
152C2×C4—422no—
153C2×C4#426811yescharacteristic
154C2×C4C22×D2111yescharacteristic
155C8—422no—
156C7#505011yescharacteristic, complemented, Sylow
157C2×C3—22no—
158C2×C3#650011yescharacteristic
159C2×C3C2×#114411yescharacteristic
160C2×C3#651411yescharacteristic
161C22—21no—
162C22—21no—
163C22#6663512yescomplemented
164C22C2×#426811yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—424no—
169C4D4×D2112yes—
170C3#8982211yescharacteristic, complemented, Sylow
171C2—22no—
172C2#1062142011yescharacteristic
173C2C2×#6663511yescharacteristic
174C2#1062143411yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC22⋊Q8D21
hypocenterC3×C7#451
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#66635C2semidirect
5. #10620649C2semidirect
55. C22×C3×C7Q16semidirect
98. C22×C7#216semidirect
118. C3×C7#451semidirect
136. C22×C3#1144semidirect
156. C7#5050semidirect
163. C22#66635semidirect
170. C3#89822semidirect

Maximal subgroups

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

subgroupindex
C22×Dic422
C3×C7×C22⋊Q82
C2×#666352
#106206692
#106206492
#898223
#50507