Finite Group Databasebeta

← #13054765

Subgroups — #13054765

Order 1344.

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

normal subgroups only

This group has 292 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, complemented
2C3×#6510C211yesmaximal, characteristic, complemented
3#10620874C212yesmaximal, complemented
4C7×#836C211yesmaximal, characteristic, complemented
5#10620516C212yesmaximal, complemented
6#89525—31nomaximal, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8#66732—22no—
9Q8×D21—22no—
10#66728—22no—
11#66728C2212yes—
12C2×C3×D28C2211yescharacteristic
13C7×#208C2212yes—
14C3×(C2×D7)⋊C4—22no—
15C3×C4×Dic7C2211yescharacteristic
16#66565—22no—
17#66561—22no—
18#6551—31no—
19#6510D311yescharacteristic, complemented
20#6471—32no—
21C7×#101—31no—
22#6399—32no—
23#5062—71nomaximal, Hall
24C22×C3×D7—21no—
25C3×C7×Q8—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28#4268—22no—
29D84—22no—
30D84D412yes—
31C4×D21—24no—
32C4×D21—22no—
33Dic42—22no—
34Dic42D412yes—
35C2×C3×Dic7—21no—
36C3×D28—22no—
37C7×C3⋊C8—22no—
38C2×C7×Q8C2×D311yescharacteristic
39#1161—62no—
40Q8×D7—62no—
41#1157—32no—
42#1157—62no—
43C2×D28C2×D311yescharacteristic
44C7×M16—32no—
45(C2×D7)⋊C4—22no—
46C4×Dic7C2×D311yescharacteristic
47D56—62no—
48#1132—62no—
49#1007—71no—
50C3×#124—71no—
51#932—72no—
52#836D711yescharacteristic, complemented
53#805—72no—
54C22×C3×C7C2×D411yescharacteristic
55C2×D21—22no—
56C2×C3×D7—42no—
57C2×C3×D7—41no—
58C3×C4×C7—22no—
59C3×C4×C7C2×D412yes—
60Dic21—22no—
61C3×Dic7—41no—
62#454—211noSylow
63C22×D7—21no—
64C7×Q8—22no—
65C2×C4×C7#6712yes—
66C2×C4×C7C22×D311yescharacteristic
67C22⋊D7—62no—
68C2×Dic7—21no—
69D28—22no—
70D28—32no—
71D28—62no—
72C4×D7—64no—
73C4×D7—62no—
74Dic14—32no—
75Dic14—62no—
76C7×C8—62no—
77C2×C3×Q8C2×D711yescharacteristic
78C2×C3×D4—71no—
79#239—142no—
80D3×Q8—142no—
81#235—72no—
82#235—142no—
83C3×C22⋊C4—142no—
84C3×C42—71no—
85#214—142no—
86#213—142no—
87#208C2×D712yes—
88C2×C3×C7—21no—
89C2×C3×C7#12011yescharacteristic
90D21—42no—
91C3×D7—81no—
922−1+4—211no—
93AGL(1,ℤ/8)—212no—
94#124—71no—
95#104—212no—
96#101—31no—
97C22×C7C2×#6711yescharacteristic
98C2×D7—42no—
99C2×D7—41no—
100C2×D7—62no—
101C4×C7—22no—
102C4×C7D3×D412yes—
103Dic7—41no—
104Dic7—62no—
105C23×C3—141no—
106C3×Q8—22no—
107C3×D4—142no—
108C2×C3×C4—141no—
109C2×C3×C4D2812yes—
110C2×C3×C4C22×D711yescharacteristic
111#67—142no—
112D12—72no—
113D12—142no—
114C4×D3—144no—
115C4×D3—142no—
116Dic6—72no—
117Dic6—142no—
118C3⋊C8—22no—
119C3×C7#45411yescharacteristic, complemented, Hall
120#41—212no—
121#41—422no—
122#41—422no—
123C2×Q8—422no—
124C2×Q8D3×D711yescharacteristic
125C2×D4—71no—
126SD16—422no—
127D8—422no—
128M16—32no—
129C22⋊C4—142no—
130C42—71no—
131C2×C7—21no—
132C2×C7#93711yescharacteristic
133D7—81no—
134D7—122no—
135C22×C3—282no—
136C22×C3—281no—
137C22×C3C2×D2811yescharacteristic
138C2×D3—142no—
139C3×C4—22no—
140C3×C4—281no—
141C3×C4D4×D712yes—
142Dic3—142no—
143C23—141no—
144Q8—22no—
145Q8—212no—
146Q8—422no—
147D4—142no—
148D4—212no—
149D4—422no—
150D4—422no—
151C2×C4—141no—
152C2×C4—424no—
153C2×C4—422no—
154C2×C4#424612yes—
155C2×C4C2×D3×D711yescharacteristic
156C8—62no—
157C7#506211yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3—561no—
160C2×C3#644511yescharacteristic
161D3—282no—
162C22—282no—
163C22—281no—
164C22—422no—
165C22C2×#424611yescharacteristic
166C4—22no—
167C4—281no—
168C4—422no—
169C4#6669412yes—
170C3#8952511yescharacteristic, complemented, Sylow
171C2—21no—
172C2—561no—
173C2—842no—
174C2#1062113411yescharacteristic
subgroupG/H
centerC2#10621134
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. #10621660C2semidirect
2. C3×#6510C2semidirect
3. #10620874C2semidirect
4. C7×#836C2semidirect
5. #10620516C2semidirect
19. #6510D3semidirect
52. #836D7semidirect
119. C3×C7#454semidirect
157. C7#5062semidirect
170. C3#89525semidirect

Maximal subgroups

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

subgroupindex
#106216602
C3×#65102
#106208742
C7×#8362
#106205162
#895253
#50627