Finite Group Databasebeta

← #13056699

Subgroups — #13056699

Order 1344.

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

normal subgroups only

This group has 292 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, complemented
2C3×C7×#124C211yesmaximal, characteristic, complemented
3#10621415C212yesmaximal, complemented
4#10620675C211yesmaximal, characteristic, complemented
5#10620644C212yesmaximal, complemented
6#89834—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C2×C3×C7×D4C2211yescharacteristic
9#66732—22no—
10Q8×D21—22no—
11#66728—22no—
12#66728C2212yes—
13C3×C7×C22⋊C4—22no—
14C3×C42×C7C2211yescharacteristic
15#66633—22no—
16#66632—22no—
17#66627C2212yes—
18#6551—31no—
19C7×#124D311yescharacteristic, complemented
20#6495—32no—
21#6410—31no—
22#6379—32no—
23#5062—71nomaximal, Hall
24C23×C3×C7—21no—
25C3×C7×Q8—22no—
26C3×C7×D4—22no—
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7D412yes—
29C2×C3×C4×C7C2311yescharacteristic
30#4268—22no—
31D84—22no—
32D84D412yes—
33C4×D21—24no—
34C4×D21—22no—
35Dic42—22no—
36Dic42D412yes—
37C3⋊(C7⋊C8)—22no—
38C2×C7×Q8C2×D311yescharacteristic
39C2×C7×D4C2×D311yescharacteristic
40#1161—62no—
41Q8×D7—62no—
42#1157—32no—
43#1157—62no—
44C7×C22⋊C4—22no—
45C42×C7C2×D311yescharacteristic
46#1142—62no—
47#1141—62no—
48#1136—32no—
49#1007—71no—
50C3×#124D711yescharacteristic, complemented
51#932—72no—
52#836—71no—
53#805—72no—
54C22×C3×C7—42no—
55C22×C3×C7—41no—
56C22×C3×C7C2×D411yescharacteristic
57C2×D21—22no—
58C3×C4×C7—22no—
59C3×C4×C7—41no—
60C3×C4×C7C2×D412yes—
61Dic21—22no—
62#454—211noSylow
63C23×C7—21no—
64C7×Q8—22no—
65C7×D4—22no—
66C2×C4×C7—21no—
67C2×C4×C7#6712yes—
68C2×C4×C7C22×D311yescharacteristic
69C22⋊D7—62no—
70D28—32no—
71D28—62no—
72C4×D7—64no—
73C4×D7—62no—
74Dic14—32no—
75Dic14—62no—
76C7⋊C8—62no—
77C2×C3×Q8C2×D711yescharacteristic
78C2×C3×D4C2×D711yescharacteristic
79#239—142no—
80D3×Q8—142no—
81#235—72no—
82#235—142no—
83C3×C22⋊C4—22no—
84C3×C42C2×D711yescharacteristic
85#214—142no—
86#213—142no—
87#208—72no—
88C2×C3×C7—21no—
89C2×C3×C7—81no—
90C2×C3×C7#12011yescharacteristic
91D21—42no—
922−1+4—211no—
93AGL(1,ℤ/8)—212no—
94#124D2111yescharacteristic, complemented
95#104—212no—
96#101—211no—
97C22×C7—42no—
98C22×C7—41no—
99C22×C7C2×#6711yescharacteristic
100C2×D7—62no—
101C4×C7—22no—
102C4×C7—41no—
103C4×C7D3×D412yes—
104Dic7—62no—
105C23×C3—21no—
106C3×Q8—22no—
107C3×D4—22no—
108C2×C3×C4—21no—
109C2×C3×C4C22⋊D712yes—
110C2×C3×C4C22×D711yescharacteristic
111#67—142no—
112D12—72no—
113D12—142no—
114C4×D3—144no—
115C4×D3—142no—
116Dic6—72no—
117Dic6—142no—
118C3⋊C8—142no—
119C3×C7#45411yescharacteristic, complemented, Hall
120#41—212no—
121#41—422no—
122#41—422no—
123C2×Q8—422no—
124C2×Q8C2×D2111yescharacteristic
125C2×D4C2×D2111yescharacteristic
126SD16—422no—
127D8—422no—
128M16—212no—
129C22⋊C4—22no—
130C42C2×D2111yescharacteristic
131C2×C7—21no—
132C2×C7—81no—
133C2×C7#93711yescharacteristic
134D7—122no—
135C22×C3—42no—
136C22×C3—41no—
137C22×C3C2×C22⋊D711yescharacteristic
138C2×D3—142no—
139C3×C4—22no—
140C3×C4—41no—
141C3×C4D4×D712yes—
142Dic3—142no—
143C23—21no—
144Q8—22no—
145Q8—212no—
146Q8—422no—
147D4—22no—
148D4—212no—
149D4—422no—
150D4—422no—
151C2×C4—21no—
152C2×C4—424no—
153C2×C4—422no—
154C2×C4#426812yes—
155C2×C4C22×D2111yescharacteristic
156C8—422no—
157C7#506211yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3—81no—
160C2×C3#650011yescharacteristic
161D3—282no—
162C22—42no—
163C22—41no—
164C22—422no—
165C22C2×#426811yescharacteristic
166C4—22no—
167C4—41no—
168C4—422no—
169C4D4×D2112yes—
170C3#8983411yescharacteristic, complemented, Sylow
171C2—21no—
172C2—81no—
173C2—842no—
174C2#1062142011yescharacteristic
subgroupG/H
centerC2#10621420
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#124C2
socleC2×C3×C7#120
hypercenter#124D21
hypocenterC3×C7#454
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #10621660C2semidirect
2. C3×C7×#124C2semidirect
3. #10621415C2semidirect
4. #10620675C2semidirect
5. #10620644C2semidirect
19. C7×#124D3semidirect
50. C3×#124D7semidirect
94. #124D21semidirect
119. C3×C7#454semidirect
157. C7#5062semidirect
170. C3#89834semidirect

Maximal subgroups

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

subgroupindex
#106216602
C3×C7×#1242
#106214152
#106206752
#106206442
#898343
#50627