Finite Group Databasebeta

← #13054774

Subgroups — #13054774

Order 1344.

284 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 60 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
1#10621660C211yesmaximal, characteristic
2C3×#6507C211yesmaximal, characteristic, complemented
3#10620877C212yesmaximal, complemented
4C7×#836C211yesmaximal, characteristic, complemented
5#10620516C212yesmaximal, complemented
6#89524—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8#66732—22no—
9Q8×D21—22no—
10#66728—22no—
11#66728C2212yes—
12C2×C3×Dic14C2211yescharacteristic
13C7×#208C2212yes—
14C3×Dic7⋊C4—22no—
15C3×C4×Dic7C2211yescharacteristic
16#66570—22no—
17#66566—22no—
18#6551—31no—
19#6507D311yescharacteristic, complemented
20#6472—32no—
21C7×#101—31no—
22#6399—32no—
23#5095—71nomaximal, Hall
24C3×C7×Q8—22no—
25C2×C3×C4×C7D412yes—
26C2×C3×C4×C7C2311yescharacteristic
27#4268—22no—
28D84—22no—
29D84D412yes—
30C4×D21—24no—
31C4×D21—22no—
32Dic42—22no—
33Dic42D412yes—
34C2×C3×Dic7—21no—
35C2×C3×Dic7—21no—
36C3×Dic14—22no—
37C7×C3⋊C8—22no—
38C2×C7×Q8C2×D311yescharacteristic
39#1161—62no—
40Q8×D7—62no—
41#1157—32no—
42#1157—62no—
43C2×Dic14C2×D311yescharacteristic
44C7×M16—32no—
45Dic7⋊C4—22no—
46C4×Dic7C2×D311yescharacteristic
47Dic28—62no—
48#1132—62no—
49#1007—71no—
50C3×C4⋊Q8—71no—
51#942—72no—
52#836D711yescharacteristic, complemented
53#805—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D21—22no—
56C3×C4×C7—22no—
57C3×C4×C7C2×D412yes—
58Dic21—22no—
59C3×Dic7—22no—
60C3×Dic7—41no—
61#456—211noSylow
62C7×Q8—22no—
63C2×C4×C7#6712yes—
64C2×C4×C7C22×D311yescharacteristic
65C22⋊D7—62no—
66C2×Dic7—21no—
67C2×Dic7—21no—
68D28—32no—
69D28—62no—
70C4×D7—64no—
71C4×D7—62no—
72Dic14—22no—
73Dic14—32no—
74Dic14—62no—
75C7×C8—62no—
76C2×C3×Q8—71no—
77C2×C3×Q8C2×D711yescharacteristic
78#239—142no—
79D3×Q8—142no—
80#235—72no—
81#235—142no—
82C3×C4⋊C4—142no—
83C3×C42—71no—
84#216—142no—
85#215—142no—
86#208C2×D712yes—
87C2×C3×C7—21no—
88C2×C3×C7#12011yescharacteristic
89D21—42no—
902−1+4—211no—
91#137—212no—
92C4⋊Q8—71no—
93#104—212no—
94#101—31no—
95C22×C7C2×#6711yescharacteristic
96C2×D7—62no—
97C4×C7—22no—
98C4×C7D3×D412yes—
99Dic7—22no—
100Dic7—41no—
101Dic7—62no—
102C3×Q8—22no—
103C3×Q8—142no—
104C2×C3×C4—141no—
105C2×C3×C4—141no—
106C2×C3×C4D2812yes—
107C2×C3×C4C22×D711yescharacteristic
108#67—142no—
109D12—72no—
110D12—142no—
111C4×D3—144no—
112C4×D3—142no—
113Dic6—72no—
114Dic6—142no—
115C3⋊C8—22no—
116C3×C7#45611yescharacteristic, complemented, Hall
117#41—212no—
118#41—422no—
119#41—422no—
120C2×Q8—71no—
121C2×Q8—422no—
122C2×Q8D3×D711yescharacteristic
123Q16—422no—
124SD16—422no—
125M16—32no—
126C4⋊C4—142no—
127C42—71no—
128C2×C7—21no—
129C2×C7#93711yescharacteristic
130D7—122no—
131C22×C3C2×D2811yescharacteristic
132C2×D3—142no—
133C3×C4—22no—
134C3×C4—142no—
135C3×C4—281no—
136C3×C4D4×D712yes—
137Dic3—142no—
138Q8—22no—
139Q8—142no—
140Q8—212no—
141Q8—422no—
142D4—212no—
143D4—422no—
144D4—422no—
145C2×C4—141no—
146C2×C4—141no—
147C2×C4—424no—
148C2×C4—422no—
149C2×C4#424612yes—
150C2×C4C2×D3×D711yescharacteristic
151C8—62no—
152C7#509511yescharacteristic, complemented, Sylow
153C2×C3—21no—
154C2×C3#644511yescharacteristic
155D3—282no—
156C22—422no—
157C22C2×#424611yescharacteristic
158C4—22no—
159C4—142no—
160C4—281no—
161C4—422no—
162C4#6669412yes—
163C3#8952411yescharacteristic, complemented, Sylow
164C2—21no—
165C2—842no—
166C2#1062113411yescharacteristic
subgroupG/H
centerC2#10621134
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
prime radical#10621660C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#456
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#6507C2semidirect
3. #10620877C2semidirect
4. C7×#836C2semidirect
5. #10620516C2semidirect
19. #6507D3semidirect
52. #836D7semidirect
116. C3×C7#456semidirect
152. C7#5095semidirect
163. C3#89524semidirect

Maximal subgroups

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

subgroupindex
#106216602
C3×#65072
#106208772
C7×#8362
#106205162
#895243
#50957