Finite Group Databasebeta

← #13054203

Subgroups — #13054203

Order 1344.

376 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C22×C3×D28C211yesmaximal, characteristic
2#10621407C211yesmaximal, characteristic, complemented
3C2×#66565C212yesmaximal, complemented
4C7×#830C211yesmaximal, characteristic, complemented
5#10620500C212yesmaximal, complemented
6#89503—31nomaximal, Hall
7C23×C3×D7—21no—
8C22×C3×C4×C7C2211yescharacteristic
9C2×Dic42C2211yescharacteristic, complemented
10C2×C3×D28—21no—
11C2×C3×D28—24no—
12C2×C3×D28C2212yes—
13#66636—21no—
14C4⋊Dic21C2211yescharacteristic, complemented
15#66629—21no—
16C2×C7×C3⋊C8C2212yescomplemented
17#66565—24no—
18C22×D28D311yescharacteristic
19#6487—31no—
20C2×#1132—32no—
21C7×C22⋊C8—31no—
22#6395—32no—
23#5218—71nomaximal, Hall
24C23×C3×C7D411yescharacteristic
25C22×C3×D7—24no—
26C22×C3×D7—22no—
27C22×C3×D7—42no—
28C22×C3×D7—42no—
29C2×C3×C4×C7—22no—
30C2×C3×C4×C7D411yescharacteristic
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21—21no—
34Dic42—22no—
35C3×D28—24no—
36C3×D28—22no—
37C3×D28D414yes—
38C7×C3⋊C8—22no—
39C23×D7—21no—
40C22×C4×C7C2×D311yescharacteristic
41C2×D28—21no—
42C2×D28—24no—
43C2×D28C2×D312yes—
44C2×Dic14—31no—
45C2×C7×C8—32no—
46C22⋊Dic7—61no—
47C4⋊Dic7—31no—
48Dic7⋊C4—61no—
49#1132—64no—
50C22×C3×D4—71no—
51C2×#214—72no—
52#924—71no—
53#832—72no—
54#830D711yescharacteristic, complemented
55C22×C3×C7—21no—
56C22×C3×C7—21no—
57C22×C3×C7SD1612yescomplemented
58C22×C3×C7C2×D411yescharacteristic
59C2×C3×D7—24no—
60C2×C3×D7—48no—
61C2×C3×D7—42no—
62C2×C3×D7—42no—
63C3×C4×C7—21no—
64C3×C4×C7C2×D412yes—
65Dic21—41no—
66Dic21—41no—
67#450—211noSylow
68C23×C7#6711yescharacteristic
69C22×D7—24no—
70C22×D7—22no—
71C22×D7—42no—
72C22×D7—42no—
73C2×C4×C7—22no—
74C2×C4×C7#6711yescharacteristic
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—61no—
77C2×Dic7—61no—
78D28—24no—
79D28—22no—
80D28#6714yes—
81Dic14—62no—
82C7×C8—62no—
83C24×C3—141no—
84C2×C3×D4—72no—
85C2×C3×D4—144no—
86C2×C3×D4—141no—
87C22×C3×C4C2×D711yescharacteristic
88C2×Dic6—71no—
89#217—141no—
90#214—144no—
91C4⋊Dic3—71no—
92Dic3⋊C4—141no—
93C2×C3⋊C8C2×D712yescomplemented
94C2×C3×C7—22no—
95C2×C3×C7#12011yescharacteristic
96C2×C3×C7C2×SD1611yescharacteristic
97C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
98C3×D7—44no—
99C22×D4—71no—
100C2×SD16—212no—
101C22⋊Q8—211no—
102#102—212no—
103C22⋊C8—31no—
104C22×C7—21no—
105C22×C7—21no—
106C22×C7#21412yescomplemented
107C22×C7C2×#6711yescharacteristic
108C2×D7—24no—
109C2×D7—48no—
110C2×D7—42no—
111C2×D7—42no—
112C4×C7—21no—
113C4×C7C2×#6712yes—
114Dic7—121no—
115Dic7—121no—
116C23×C3—144no—
117C23×C3—142no—
118C23×C3—282no—
119C23×C3—282no—
120C23×C3D2811yescharacteristic
121C3×D4—74no—
122C3×D4—144no—
123C3×D4—142no—
124C2×C3×C4—22no—
125C2×C3×C4D2811yescharacteristic
126C2×C3×C4C22×D711yescharacteristic
127C2×Dic3—141no—
128C2×Dic3—141no—
129Dic6—142no—
130C3⋊C8—22no—
131C3×C7#45011yescharacteristic, complemented, Hall
132C24—141no—
133C2×Q8—211no—
134C2×D4—72no—
135C2×D4—144no—
136C2×D4—141no—
137C22×C4D3×D711yescharacteristic
138SD16—424no—
139C2×C8—32no—
140C4⋊C4—211no—
141C4⋊C4—421no—
142C22⋊C4—421no—
143C2×C7—22no—
144C2×C7#93211yescharacteristic
145C2×C7C2×#21411yescharacteristic
146C2×C7#95311yescharacteristic
147D7—44no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3—144no—
151C22×C3—288no—
152C22×C3—282no—
153C22×C3—282no—
154C22×C3#113212yescomplemented
155C22×C3C2×D2811yescharacteristic
156C3×C4—21no—
157C3×C4D4×D712yes—
158Dic3—281no—
159Dic3—281no—
160C23—144no—
161C23—142no—
162C23—282no—
163C23—282no—
164C23#424611yescharacteristic
165Q8—422no—
166D4—74no—
167D4—144no—
168D4—142no—
169C2×C4—22no—
170C2×C4—421no—
171C2×C4—421no—
172C2×C4#424611yescharacteristic
173C2×C4C2×D3×D711yescharacteristic
174C8—62no—
175C7#521811yescharacteristic, complemented, Sylow
176C2×C3—22no—
177C2×C3—284no—
178C2×C3#644511yescharacteristic
179C2×C3C2×#113211yescharacteristic
180C2×C3#647111yescharacteristic
181C22—21no—
182C22—21no—
183C22—144no—
184C22—288no—
185C22—282no—
186C22—282no—
187C22#6656512yescomplemented
188C22C2×#424611yescharacteristic
189C4—21no—
190C4—841no—
191C4—841no—
192C4D7×#6712yes—
193C3#8950311yescharacteristic, complemented, Sylow
194C2—22no—
195C2—284no—
196C2C2×#6656511yescharacteristic
197C2#1062087411yescharacteristic
198C2#1062112611yescharacteristic
subgroupG/H
centerC22C2×#4246
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
prime radicalC22×C3×D28C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#450
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #10621407C2semidirect
3. C2×#66565C2semidirect
4. C7×#830C2semidirect
5. #10620500C2semidirect
9. C2×Dic42C22semidirect
14. C4⋊Dic21C22semidirect
16. C2×C7×C3⋊C8C22semidirect
54. #830D7semidirect
57. C22×C3×C7SD16semidirect
93. C2×C3⋊C8C2×D7semidirect
106. C22×C7#214semidirect
131. C3×C7#450semidirect
154. C22×C3#1132semidirect
175. C7#5218semidirect
187. C22#66565semidirect
193. C3#89503semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D282
#106214072
C2×#665652
C7×#8302
#106205002
#895033
#52187