Finite Group Databasebeta

← #13054541

Subgroups — #13054541

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
2#10621419C211yesmaximal, characteristic, complemented
3#10620875C212yesmaximal, complemented
4C7×#833C211yesmaximal, characteristic, complemented
5#10620511C212yesmaximal, complemented
6#89518—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C2×Dic42C2211yescharacteristic, complemented
9C2×C3×C22⋊D7—22no—
10C3×#1159—22no—
11C3×D4×D7—22no—
12C3×#1157C2212yes—
13#66636—22no—
14C4×Dic21C2211yescharacteristic, complemented
15C7×#208C2212yescomplemented
16#66570—22no—
17#66565—22no—
18#6548D311yescharacteristic
19#6499—31no—
20#6472—32no—
21C7×#100—31no—
22#6399—32no—
23#5242—71nomaximal, Hall
24C23×C3×C7D412yes—
25C22×C3×D7—22no—
26C3×C7×D4—22no—
27C2×C3×C4×C7C2311yescharacteristic
28C2×Dic21—21no—
29C2×Dic21—21no—
30Dic42—22no—
31C3×C22⋊D7—24no—
32C3×C22⋊D7—22no—
33C2×C3×Dic7—22no—
34C3×D28D412yes—
35C3×C4×D7—22no—
36C3×Dic14D412yes—
37C7×C3⋊C8—22no—
38C2×C7×D4C2×D311yescharacteristic
39C2×C22⋊D7—22no—
40#1159—22no—
41D4×D7—22no—
42#1157C2×D312yes—
43C2×Dic14—31no—
44C7×M16—32no—
45C22⋊Dic7—62no—
46C4×Dic7—31no—
47Dic28—62no—
48#1132—62no—
49C3×H(5,𝔽2)—71no—
50#951—72no—
51#936—71no—
52#837—72no—
53#833D711yescharacteristic, complemented
54C22×C3×C7—22no—
55C22×C3×C7—42no—
56C22×C3×C7C2×D411yescharacteristic
57C2×C3×D7—22no—
58C2×C3×D7—82no—
59C3×C4×C7C2×D412yes—
60Dic21—41no—
61Dic21—41no—
62C3×Dic7—22no—
63#455—211noSylow
64C23×C7#6712yes—
65C22×D7—22no—
66C7×D4—22no—
67C2×C4×C7C22×D311yescharacteristic
68C22⋊D7—24no—
69C22⋊D7—22no—
70C2×Dic7—22no—
71C2×Dic7—61no—
72C2×Dic7—61no—
73D28#6712yes—
74C4×D7—22no—
75Dic14—62no—
76Dic14#6712yes—
77C7×C8—62no—
78C3×#41—72no—
79C3×#41—142no—
80C2×C3×D4—142no—
81C2×C3×D4—142no—
82C2×C3×D4C2×D711yescharacteristic
83C2×Dic6—71no—
84#217—142no—
85#216—142no—
86#214—142no—
87C4×Dic3—71no—
88#208C2×D712yescomplemented
89C2×C3×C7—21no—
90C2×C3×C7—42no—
91C2×C3×C7#12011yescharacteristic
92C3×D7—42no—
93H(5,𝔽2)—71no—
94#137—212no—
95#124—211no—
96#104—212no—
97#100—31no—
98C22×C7—22no—
99C22×C7—42no—
100C22×C7C2×#6711yescharacteristic
101C2×D7—22no—
102C2×D7—82no—
103C4×C7C2×#6712yes—
104Dic7—22no—
105Dic7—121no—
106Dic7—121no—
107C23×C3—142no—
108C23×C3D2812yes—
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—141no—
118C2×Dic3—141no—
119Dic6—142no—
120C3⋊C8—22no—
121C3×C7#45511yescharacteristic, complemented, Hall
122#41—72no—
123#41—142no—
124C2×Q8—211no—
125C2×D4—142no—
126C2×D4—142no—
127C2×D4D3×D711yescharacteristic
128Q16—422no—
129SD16—422no—
130M16—32no—
131C22⋊C4—422no—
132C42—211no—
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×D2811yescharacteristic
142C3×C4—142no—
143C3×C4D4×D712yes—
144Dic3—281no—
145Dic3—281no—
146C23—142no—
147C23#424612yes—
148Q8—72no—
149Q8—422no—
150D4—22no—
151D4—72no—
152D4—144no—
153D4—142no—
154C2×C4—142no—
155C2×C4—142no—
156C2×C4—421no—
157C2×C4—421no—
158C2×C4C2×D3×D711yescharacteristic
159C8—62no—
160C7#524211yescharacteristic, complemented, Sylow
161C2×C3—21no—
162C2×C3—42no—
163C2×C3—282no—
164C2×C3#644511yescharacteristic
165C22—22no—
166C22—42no—
167C22—142no—
168C22—562no—
169C22C2×#424611yescharacteristic
170C4—142no—
171C4—841no—
172C4—841no—
173C4D7×#6712yes—
174C3#8951811yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—282no—
178C2#1062112611yescharacteristic
subgroupG/H
centerC2#10621126
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. #10621419C2semidirect
3. #10620875C2semidirect
4. C7×#833C2semidirect
5. #10620511C2semidirect
8. C2×Dic42C22semidirect
14. C4×Dic21C22semidirect
15. C7×#208C22semidirect
53. #833D7semidirect
88. #208C2×D7semidirect
121. C3×C7#455semidirect
160. C7#5242semidirect
174. C3#89518semidirect

Maximal subgroups

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

subgroupindex
C3×#65482
#106214192
#106208752
C7×#8332
#106205112
#895183
#52427