Finite Group Databasebeta

← #13054577

Subgroups — #13054577

Order 1344.

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

normal subgroups only

This group has 208 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#10621429C211yesmaximal, characteristic
2#10620999C211yesmaximal, characteristic
3C7×#835C211yesmaximal, characteristic, complemented
4C3×#6409C211yesmaximal, characteristic, complemented
5#10620525C211yesmaximal, characteristic, complemented
6#10620523C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic
8#89581—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C4×D21C2211yescharacteristic
11C2×D21⋊C4—21no—
12#66631—21no—
13C4⋊Dic21C2211yescharacteristic
14#66629—21no—
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19#66577—21no—
20#6509—31no—
21D7×C4⋊C4—31no—
22C7×#103—31no—
23#6409D311yescharacteristic, complemented
24(C2×D7)⋊C8—31no—
25#6392—31no—
26#6381—31no—
27#4811—71nomaximal, Hall
28C22×D21D411yescharacteristic
29C3×C7×Q8—21no—
30C3×C7×Q8—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21D411yescharacteristic
35C4×D21—21no—
36C4×D21—21no—
37C2×C7×Dic3—21no—
38C2×C3×Dic7—21no—
39D21⋊C4—42no—
40C3×C7⋊C8—21no—
41C7×C3⋊C8—21no—
42C2×C7×Q8C2×D311yescharacteristic
43C2×C4×D7—31no—
44C2×C4×D7—61no—
45C2×C7×C8—31no—
46C7×C4⋊C4—31no—
47(C2×D7)⋊C4—61no—
48C4⋊Dic7—31no—
49C4⋊Dic7C2×D311yescharacteristic
50Dic7⋊C4—61no—
51Dic7⋊C4—61no—
52C2×C7⋊C8C2×D311yescharacteristic
53#946—71no—
54D3×C4⋊C4—71no—
55C3×#103—71no—
56#835D711yescharacteristic, complemented
57(C2×D3)⋊C8—71no—
58#818—71no—
59#807—71no—
60C22×C3×C7C2×D411yescharacteristic
61C2×D21—21no—
62C2×D21—21no—
63C2×D21Q1612yes—
64C3×C4×C7—41no—
65C3×C4×C7#4111yescharacteristic
66C3×C4×C7#4111yescharacteristic
67Dic21—21no—
68Dic21—41no—
69C3×Dic7—41no—
70C7×Dic3—41no—
71#484—211noSylow
72C22×D7—31no—
73C7×Q8—21no—
74C7×Q8—21no—
75C2×C4×C7—21no—
76C2×C4×C7—61no—
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—21no—
79C2×Dic7—31no—
80C2×Dic7—61no—
81C4×D7—61no—
82C4×D7—61no—
83C4×D7—122no—
84C7×C8—61no—
85C7⋊C8—21no—
86C2×C3×Q8C2×D711yescharacteristic
87C2×C4×D3—71no—
88C2×C4×D3—141no—
89C2×C3×C8—71no—
90C3×C4⋊C4—71no—
91(C2×D3)⋊C4—141no—
92C4⋊Dic3—71no—
93C4⋊Dic3C2×D711yescharacteristic
94Dic3⋊C4—141no—
95Dic3⋊C4—141no—
96C2×C3⋊C8C2×D711yescharacteristic
97C2×C3×C7#12311yescharacteristic
98C2×C3×C7C2×Q1611yescharacteristic
99C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
100D21—22no—
101C22⋊Q8—211no—
102C2×C4⋊C4—211no—
103#107—211no—
104#107—211no—
105#103—31no—
106#103—71no—
107C22⋊C8—211no—
108C22×C7D3×D411yescharacteristic
109C2×D7—32no—
110C2×D7—61no—
111C2×D7—61no—
112C4×C7—41no—
113C4×C7—121no—
114C4×C7#23511yescharacteristic
115C4×C7#23711yescharacteristic
116Dic7—41no—
117Dic7—61no—
118Dic7—121no—
119C22×D3—71no—
120C3×Q8—21no—
121C3×Q8—21no—
122C2×C3×C4—21no—
123C2×C3×C4—141no—
124C2×C3×C4C22×D711yescharacteristic
125C2×Dic3—21no—
126C2×Dic3—71no—
127C2×Dic3—141no—
128C4×D3—141no—
129C4×D3—141no—
130C4×D3—282no—
131C3×C8—141no—
132C3⋊C8—21no—
133C3×C7#48411yescharacteristic, complemented, Hall
134C2×Q8D3×D711yescharacteristic
135C22×C4—211no—
136C22×C4—421no—
137C2×C8—31no—
138C2×C8—71no—
139C4⋊C4—31no—
140C4⋊C4—71no—
141C4⋊C4—211no—
142C4⋊C4—421no—
143C4⋊C4—421no—
144C22⋊C4—421no—
145C2×C7#88311yescharacteristic
146C2×C7#91411yescharacteristic
147C2×C7D3×Q1611yescharacteristic
148D7—62no—
149C22×C3D4×D711yescharacteristic
150C2×D3—72no—
151C2×D3—141no—
152C2×D3—141no—
153C3×C4—41no—
154C3×C4—281no—
155C3×C4#115711yescharacteristic
156C3×C4#115911yescharacteristic
157Dic3—41no—
158Dic3—141no—
159Dic3—281no—
160C23—211no—
161Q8—21no—
162Q8—21no—
163C2×C4—21no—
164C2×C4—61no—
165C2×C4—141no—
166C2×C4—211no—
167C2×C4—421no—
168C2×C4—421no—
169C2×C4—421no—
170C2×C4—842no—
171C2×C4C2×D3×D711yescharacteristic
172C8—61no—
173C8—141no—
174C7#481111yescharacteristic, complemented, Sylow
175C2×C3#644611yescharacteristic
176C2×C3#647711yescharacteristic
177C2×C3D7×Q1611yescharacteristic
178D3—142no—
179C22—212no—
180C22—421no—
181C22—421no—
182C22#6669411yescharacteristic
183C4—41no—
184C4—121no—
185C4—281no—
186C4—421no—
187C4—841no—
188C4#6668311yescharacteristic
189C4#6668611yescharacteristic
190C3#8958111yescharacteristic, complemented, Sylow
191C2—422no—
192C2#1062106811yescharacteristic
193C2#1062107311yescharacteristic
194C2#1062112411yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#484
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C7×#835C2semidirect
4. C3×#6409C2semidirect
5. #10620525C2semidirect
6. #10620523C2semidirect
23. #6409D3semidirect
56. #835D7semidirect
133. C3×C7#484semidirect
174. C7#4811semidirect
190. C3#89581semidirect

Maximal subgroups

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

subgroupindex
#106214292
#106209992
C7×#8352
C3×#64092
#106205252
#106205232
#106204922
#895813
#48117