Finite Group Databasebeta

← #13054539

Subgroups — #13054539

Order 1344.

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

normal subgroups only

This group has 304 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×#6548C211yesmaximal, characteristic
2C7×#936C211yesmaximal, characteristic, complemented
3#10620869C212yesmaximal, complemented
4#10620672C211yesmaximal, characteristic, complemented
5#10620512C212yesmaximal, complemented
6#89833—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×C7×Dic6C2211yescharacteristic, complemented
9C2×C3×C22⋊D7—22no—
10C3×#1159—22no—
11C3×D4×D7—22no—
12C3×#1157C2212yes—
13#66627C2212yescomplemented
14C7×#217—22no—
15C4×C7×Dic3C2211yescharacteristic, complemented
16#66569—22no—
17#66563—22no—
18#6548D311yescharacteristic
19C7×#124—31no—
20#6505—32no—
21#6407—31no—
22#6379—32no—
23#5242—71nomaximal, Hall
24C23×C3×C7D412yes—
25C22×C3×D7—22no—
26C3×C7×D4—22no—
27C2×C3×C4×C7C2311yescharacteristic
28C2×C7×Dic3—21no—
29C2×C7×Dic3—21no—
30C7×Dic6—22no—
31C3×C22⋊D7—24no—
32C3×C22⋊D7—22no—
33C2×C3×Dic7—22no—
34C3×D28D412yes—
35C3×C4×D7—22no—
36C3×Dic14D412yes—
37C3⋊(C7⋊C8)—22no—
38C2×C7×Q8—31no—
39C2×C7×D4C2×D311yescharacteristic
40C2×C22⋊D7—22no—
41#1159—22no—
42D4×D7—22no—
43#1157C2×D312yes—
44C7×C22⋊C4—62no—
45C42×C7—31no—
46#1144—62no—
47#1143—62no—
48#1136—32no—
49C3×H(5,𝔽2)—71no—
50#951—72no—
51#936D711yescharacteristic, complemented
52#837—72no—
53#833—71no—
54C22×C3×C7—22no—
55C22×C3×C7—42no—
56C22×C3×C7C2×D411yescharacteristic
57C2×C3×D7—22no—
58C2×C3×D7—82no—
59C3×C4×C7C2×D412yes—
60C3×Dic7—22no—
61C7×Dic3—41no—
62C7×Dic3—41no—
63#455—211noSylow
64C23×C7#6712yes—
65C22×D7—22no—
66C7×Q8—62no—
67C7×D4—22no—
68C2×C4×C7—61no—
69C2×C4×C7—61no—
70C2×C4×C7C22×D311yescharacteristic
71C22⋊D7—24no—
72C22⋊D7—22no—
73C2×Dic7—22no—
74D28#6712yes—
75C4×D7—22no—
76Dic14#6712yes—
77C7⋊C8—62no—
78C3×#41—72no—
79C3×#41—142no—
80C2×C3×D4—142no—
81C2×C3×D4—142no—
82C2×C3×D4C2×D711yescharacteristic
83C2×Dic6C2×D711yescharacteristic, complemented
84#217—22no—
85#216—142no—
86#214—142no—
87C4×Dic3C2×D711yescharacteristic, complemented
88#208—72no—
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7#12011yescharacteristic
92C3×D7—42no—
93H(5,𝔽2)—71no—
94#137—212no—
95#124—31no—
96#104—212no—
97#100—211no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7C2×#6711yescharacteristic
101C2×D7—22no—
102C2×D7—82no—
103C4×C7—121no—
104C4×C7—121no—
105C4×C7C2×#6712yes—
106Dic7—22no—
107C23×C3—142no—
108C23×C3C22⋊D712yes—
109C3×Q8—72no—
110C3×D4—22no—
111C3×D4—72no—
112C3×D4—144no—
113C3×D4—142no—
114C2×C3×C4—142no—
115C2×C3×C4—142no—
116C2×C3×C4C22×D711yescharacteristic
117C2×Dic3—21no—
118C2×Dic3—21no—
119Dic6—22no—
120C3⋊C8—142no—
121C3×C7#45511yescharacteristic, complemented, Hall
122#41—72no—
123#41—142no—
124C2×Q8—31no—
125C2×D4—142no—
126C2×D4—142no—
127C2×D4D3×D711yescharacteristic
128Q16—422no—
129SD16—422no—
130M16—212no—
131C22⋊C4—62no—
132C42—31no—
133C2×C7—21no—
134C2×C7—42no—
135C2×C7#95311yescharacteristic
136D7—42no—
137C22×C3—22no—
138C22×C3—42no—
139C22×C3—142no—
140C22×C3—562no—
141C22×C3C2×C22⋊D711yescharacteristic
142C3×C4—142no—
143C3×C4D4×D712yes—
144Dic3—41no—
145Dic3—41no—
146C23—142no—
147C23#424512yes—
148Q8—62no—
149Q8—72no—
150D4—22no—
151D4—72no—
152D4—144no—
153D4—142no—
154C2×C4—61no—
155C2×C4—61no—
156C2×C4—142no—
157C2×C4—142no—
158C2×C4C2×D3×D711yescharacteristic
159C8—422no—
160C7#524211yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3—282no—
164C2×C3#650011yescharacteristic
165C22—22no—
166C22—42no—
167C22—142no—
168C22—562no—
169C22C2×#424511yescharacteristic
170C4—121no—
171C4—121no—
172C4—142no—
173C4D7×#6712yes—
174C3#8983311yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—282no—
178C2#1062112511yescharacteristic
subgroupG/H
centerC2#10621125
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#120
prime radicalC3×#6548C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#455
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C7×#936C2semidirect
3. #10620869C2semidirect
4. #10620672C2semidirect
5. #10620512C2semidirect
8. C2×C7×Dic6C22semidirect
13. #66627C22semidirect
15. C4×C7×Dic3C22semidirect
51. #936D7semidirect
83. C2×Dic6C2×D7semidirect
87. C4×Dic3C2×D7semidirect
121. C3×C7#455semidirect
160. C7#5242semidirect
174. C3#89833semidirect

Maximal subgroups

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

subgroupindex
C3×#65482
C7×#9362
#106208692
#106206722
#106205122
#898333
#52427