Finite Group Databasebeta

← #13056389

Subgroups — #13056389

Order 1344.

284 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 63 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
2#10621427C211yesmaximal, characteristic, complemented
3#10621392C212yesmaximal, complemented
4C3×C7×#101C211yesmaximal, characteristic, complemented
5#10620664C212yesmaximal, complemented
6#89524—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8#66732—22no—
9Q8×D21—22no—
10#66728—22no—
11#66728C2212yes—
12C2×Dic42C2211yescharacteristic
13C3×C7×M16C2212yes—
14#66629—22no—
15C4×Dic21C2211yescharacteristic
16Dic84—22no—
17#66623—22no—
18#6551—31no—
19#6507—31no—
20#6472—32no—
21C7×#101D311yescharacteristic, complemented
22#6399—32no—
23#4752—71nomaximal, Hall
24C3×C7×Q8—22no—
25C2×C3×C4×C7D412yes—
26C2×C3×C4×C7C2311yescharacteristic
27#4268—22no—
28C2×Dic21—21no—
29C2×Dic21—21no—
30D84—22no—
31D84D412yes—
32C4×D21—24no—
33C4×D21—22no—
34Dic42—22no—
35Dic42—22no—
36Dic42D412yes—
37C3×C7×C8—22no—
38C2×C7×Q8C2×D311yescharacteristic
39#1161—62no—
40Q8×D7—62no—
41#1157—32no—
42#1157—62no—
43C2×Dic14—31no—
44C7×M16C2×D312yes—
45Dic7⋊C4—62no—
46C4×Dic7—31no—
47Dic28—62no—
48#1132—62no—
49#1007—71no—
50#944—71no—
51#909—72no—
52C3×#101D711yescharacteristic, complemented
53#825—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D21—22no—
56C3×C4×C7—22no—
57C3×C4×C7C2×D412yes—
58Dic21—22no—
59Dic21—22no—
60Dic21—41no—
61#456—211noSylow
62C7×Q8—22no—
63C2×C4×C7D1212yes—
64C2×C4×C7C22×D311yescharacteristic
65C22⋊D7—62no—
66C2×Dic7—61no—
67C2×Dic7—61no—
68D28—32no—
69D28—62no—
70C4×D7—64no—
71C4×D7—62no—
72Dic14—32no—
73Dic14—62no—
74Dic14—62no—
75C7×C8—22no—
76C2×C3×Q8C2×D711yescharacteristic
77#239—142no—
78D3×Q8—142no—
79#235—72no—
80#235—142no—
81C2×Dic6—71no—
82C3×M16C2×D712yes—
83Dic3⋊C4—142no—
84C4×Dic3—71no—
85Dic12—142no—
86#204—142no—
87C2×C3×C7—21no—
88C2×C3×C7#12011yescharacteristic
89D21—42no—
902−1+4—211no—
91#137—212no—
92C4⋊Q8—211no—
93#104—212no—
94#101D2111yescharacteristic, complemented
95C22×C7C2×D1211yescharacteristic
96C2×D7—62no—
97C4×C7—22no—
98C4×C7D3×D412yes—
99Dic7—62no—
100Dic7—62no—
101Dic7—121no—
102C3×Q8—22no—
103C2×C3×C4D2812yes—
104C2×C3×C4C22×D711yescharacteristic
105#67—142no—
106C2×Dic3—141no—
107C2×Dic3—141no—
108D12—72no—
109D12—142no—
110C4×D3—144no—
111C4×D3—142no—
112Dic6—72no—
113Dic6—142no—
114Dic6—142no—
115C3×C8—22no—
116C3×C7#45611yescharacteristic, complemented, Hall
117#41—212no—
118#41—422no—
119#41—422no—
120C2×Q8—211no—
121C2×Q8—422no—
122C2×Q8C2×D2111yescharacteristic
123Q16—422no—
124SD16—422no—
125M16C2×D2112yes—
126C4⋊C4—422no—
127C42—211no—
128C2×C7—21no—
129C2×C7#88211yescharacteristic
130D7—122no—
131C22×C3C2×D2811yescharacteristic
132C2×D3—142no—
133C3×C4—22no—
134C3×C4D4×D712yes—
135Dic3—142no—
136Dic3—142no—
137Dic3—281no—
138Q8—22no—
139Q8—212no—
140Q8—422no—
141Q8—422no—
142D4—212no—
143D4—422no—
144D4—422no—
145C2×C4—424no—
146C2×C4—421no—
147C2×C4—422no—
148C2×C4—421no—
149C2×C4D8412yes—
150C2×C4C22×D2111yescharacteristic
151C8—22no—
152C7#475211yescharacteristic, complemented, Sylow
153C2×C3—21no—
154C2×C3#644511yescharacteristic
155D3—282no—
156C22—422no—
157C22C2×D8411yescharacteristic
158C4—22no—
159C4—422no—
160C4—422no—
161C4—841no—
162C4D4×D2112yes—
163C3#8952411yescharacteristic, complemented, Sylow
164C2—21no—
165C2—842no—
166C2#1062136511yescharacteristic
subgroupG/H
centerC2#10621365
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#101C2
socleC2×C3×C7#120
prime radical#10621660C2
hypercenter#101D21
hypocenterC3×C7#456
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. #10621427C2semidirect
3. #10621392C2semidirect
4. C3×C7×#101C2semidirect
5. #10620664C2semidirect
21. C7×#101D3semidirect
52. C3×#101D7semidirect
94. #101D21semidirect
116. C3×C7#456semidirect
152. C7#4752semidirect
163. C3#89524semidirect

Maximal subgroups

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

subgroupindex
#106216602
#106214272
#106213922
C3×C7×#1012
#106206642
#895243
#47527