Finite Group Databasebeta

← #13054758

Subgroups — #13054758

Order 1344.

292 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#6551C211yesmaximal, characteristic, complemented
2C7×#947C211yesmaximal, characteristic, complemented
3#10620861C212yesmaximal, complemented
4#10620675C211yesmaximal, characteristic, complemented
5#10620512C212yesmaximal, complemented
6#89834—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C2×C7×D12C2211yescharacteristic
9C3×#1161—22no—
10C3×Q8×D7—22no—
11C3×#1157—22no—
12C3×#1157C2212yes—
13#66627C2212yes—
14C7×(C2×D3)⋊C4—22no—
15C4×C7×Dic3C2211yescharacteristic
16#66564—22no—
17#66560—22no—
18#6551D311yescharacteristic, complemented
19C7×#124—31no—
20#6495—32no—
21#6410—31no—
22#6379—32no—
23#5245—71nomaximal, Hall
24C22×D3×C7—21no—
25C3×C7×Q8—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28C2×C7×Dic3—21no—
29C7×D12—22no—
30C3×C22⋊D7—22no—
31C3×D28—22no—
32C3×D28D412yes—
33C3×C4×D7—24no—
34C3×C4×D7—22no—
35C3×Dic14—22no—
36C3×Dic14D412yes—
37C3⋊(C7⋊C8)—22no—
38C2×C7×Q8C2×D311yescharacteristic
39C2×C7×D4—31no—
40#1161—22no—
41Q8×D7—22no—
42#1157—22no—
43#1157C2×D312yes—
44C7×C22⋊C4—62no—
45C42×C7—31no—
46#1142—62no—
47#1141—62no—
48#1136—32no—
49C3×2−1+4—71no—
50#949—72no—
51#947D711yescharacteristic, complemented
52#837—72no—
53#836—71no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D3×C7—42no—
56C2×D3×C7—41no—
57C2×C3×D7—22no—
58C3×C4×C7—22no—
59C3×C4×C7C2×D412yes—
60C3×Dic7—22no—
61C7×Dic3—41no—
62#454—211noSylow
63C23×C7—61no—
64C7×Q8—22no—
65C7×D4—62no—
66C2×C4×C7—61no—
67C2×C4×C7#6712yes—
68C2×C4×C7C22×D311yescharacteristic
69C22⋊D7—22no—
70D28—22no—
71D28#6712yes—
72C4×D7—24no—
73C4×D7—22no—
74Dic14—22no—
75Dic14#6712yes—
76C7⋊C8—62no—
77C3×#41—72no—
78C3×#41—142no—
79C3×#41—142no—
80C2×C3×Q8—142no—
81C2×C3×Q8C2×D711yescharacteristic
82C2×D12C2×D711yescharacteristic
83#215—142no—
84#213—142no—
85(C2×D3)⋊C4—22no—
86C4×Dic3C2×D711yescharacteristic
87#208—72no—
88C2×C3×C7—21no—
89C2×C3×C7#12011yescharacteristic
90C3×D7—42no—
91D3×C7—81no—
922−1+4—71no—
93AGL(1,ℤ/8)—212no—
94#124—31no—
95#104—212no—
96#101—211no—
97C22×C7—122no—
98C22×C7—121no—
99C22×C7C2×#6711yescharacteristic
100C2×D7—22no—
101C4×C7—22no—
102C4×C7—121no—
103C4×C7C2×#6712yes—
104Dic7—22no—
105C22×D3—21no—
106C3×Q8—22no—
107C3×Q8—72no—
108C3×Q8—142no—
109C3×D4—72no—
110C3×D4—142no—
111C3×D4—142no—
112C2×C3×C4—144no—
113C2×C3×C4—142no—
114C2×C3×C4C22⋊D712yes—
115C2×C3×C4C22×D711yescharacteristic
116C2×Dic3—21no—
117D12—22no—
118C3⋊C8—142no—
119C3×C7#45411yescharacteristic, complemented, Hall
120#41—72no—
121#41—142no—
122#41—142no—
123C2×Q8—142no—
124C2×Q8D3×D711yescharacteristic
125C2×D4—31no—
126SD16—422no—
127D8—422no—
128M16—212no—
129C22⋊C4—62no—
130C42—31no—
131C2×C7—21no—
132C2×C7—241no—
133C2×C7#95311yescharacteristic
134D7—42no—
135C22×C3—142no—
136C22×C3C2×C22⋊D711yescharacteristic
137C2×D3—42no—
138C2×D3—41no—
139C3×C4—22no—
140C3×C4—142no—
141C3×C4D4×D712yes—
142Dic3—41no—
143C23—61no—
144Q8—22no—
145Q8—72no—
146Q8—142no—
147D4—62no—
148D4—72no—
149D4—142no—
150D4—142no—
151C2×C4—61no—
152C2×C4—144no—
153C2×C4—142no—
154C2×C4#424512yes—
155C2×C4C2×D3×D711yescharacteristic
156C8—422no—
157C7#524511yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3—282no—
160C2×C3#650011yescharacteristic
161D3—81no—
162C22—122no—
163C22—121no—
164C22—142no—
165C22C2×#424511yescharacteristic
166C4—22no—
167C4—121no—
168C4—142no—
169C4D7×#6712yes—
170C3#8983411yescharacteristic, complemented, Sylow
171C2—21no—
172C2—241no—
173C2—282no—
174C2#1062112511yescharacteristic
subgroupG/H
centerC2#10621125
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
hypercenterC2×Q8D3×D7
hypocenterC3×C7#454
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
1. C3×#6551C2semidirect
2. C7×#947C2semidirect
3. #10620861C2semidirect
4. #10620675C2semidirect
5. #10620512C2semidirect
18. #6551D3semidirect
51. #947D7semidirect
119. C3×C7#454semidirect
157. C7#5245semidirect
170. C3#89834semidirect

Maximal subgroups

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

subgroupindex
C3×#65512
C7×#9472
#106208612
#106206752
#106205122
#898343
#52457