Finite Group Databasebeta

← #13054608

Subgroups — #13054608

Order 1344.

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

normal subgroups only

This group has 220 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
1C2×#66635C211yesmaximal, characteristic
2C3×Q8×Dic7C211yesmaximal, characteristic
3#10620895C211yesmaximal, characteristic
4C7×#835C211yesmaximal, characteristic
5#10620524C211yesmaximal, characteristic
6#10620509C211yesmaximal, characteristic
7C3⋊C8×Dic7C211yesmaximal, characteristic
8#89562—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11#66635C414yescomplemented
12C2×C3⋊(C7⋊C8)C2211yescharacteristic
13C7×C4⋊Dic3C2211yescharacteristic
14C2×C7×C3⋊C8C2211yescharacteristic
15C3×C4⋊Dic7—21no—
16C3×C4⋊Dic7C2211yescharacteristic
17C3×C4×Dic7—21no—
18C3×C4×Dic7C2211yescharacteristic
19#66577—21no—
20Dic3×Dic7—21no—
21Q8×Dic7D311yescharacteristic
22C2×#1144—31no—
23#6450—31no—
24C7×#103—31no—
25#6390—31no—
26C8×Dic7—31no—
27#6381—31no—
28C4×#216—71nomaximal, Hall
29C3×C7×Q8—21no—
30C3×C7×Q8C2×C412yes—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34Dic42—21no—
35Dic42C2×C412yes—
36C2×C7×Dic3—21no—
37C2×C3×Dic7—21no—
38C2×C3×Dic7D411yescharacteristic
39C2×C3×Dic7D411yescharacteristic
40C3⋊(C7⋊C8)C2×C412yes—
41C7×C3⋊C8—21no—
42C2×C7×Q8C2×D311yescharacteristic
43C2×Dic14—31no—
44C2×C7×C8—31no—
45C7×C4⋊C4—31no—
46#1144—34no—
47C4⋊Dic7—21no—
48C4⋊Dic7C2×D311yescharacteristic
49Dic7⋊C4—61no—
50C4×Dic7—21no—
51C4×Dic7—61no—
52C4×Dic7C2×D311yescharacteristic
53C2×C7⋊C8—31no—
54C3×C4×Q8—71no—
55C2×#216—71no—
56C4×Dic6—71no—
57#835D711yescharacteristic
58#810—71no—
59#807—71no—
60C4×C3⋊C8—71no—
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—22no—
63C3×C4×C7C22×C411yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—22no—
66C3×Dic7—21no—
67C3×Dic7—41no—
68C3×Dic7Q1612yescomplemented
69C7×Dic3—41no—
70C4×Q16—211noSylow
71C7×Q8—21no—
72C7×Q8C4×D312yes—
73C2×C4×C7—21no—
74C2×C4×C7—61no—
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—21no—
77C2×Dic7—61no—
78C2×Dic7#6711yescharacteristic
79C2×Dic7#6711yescharacteristic
80Dic14—32no—
81Dic14—61no—
82C7×C8—61no—
83C7⋊C8—32no—
84C2×C3×Q8C2×D711yescharacteristic
85C2×Dic6—71no—
86C3×C4⋊C4—71no—
87C3×C4⋊C4—141no—
88C3×C42—71no—
89C3×C42—141no—
90#216—74no—
91C4⋊Dic3C2×D711yescharacteristic
92Dic3⋊C4—141no—
93C4×Dic3—141no—
94C2×C3⋊C8—71no—
95C2×C3⋊C8C2×D711yescharacteristic
96C2×C3×C7C4×D411yescharacteristic
97C2×C3×C7C2×Q1611yescharacteristic
98C2×C3×C7#13511yescharacteristic
99C2×Q16—211no—
100C4×Q8—71no—
101C4×Q8—211no—
102#107—211no—
103#103—31no—
104#103—211no—
105C4×C8—211no—
106C22×C7C2×#6711yescharacteristic
107C4×C7—22no—
108C4×C7—121no—
109C4×C7C2×C4×D311yescharacteristic
110C4×C7#23511yescharacteristic
111Dic7—21no—
112Dic7—41no—
113Dic7—62no—
114Dic7#21612yescomplemented
115C3×Q8—21no—
116C3×Q8C4×D712yes—
117C2×C3×C4—21no—
118C2×C3×C4—71no—
119C2×C3×C4—71no—
120C2×C3×C4—141no—
121C2×C3×C4C22×D711yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—141no—
124Dic6—72no—
125Dic6—141no—
126C3⋊C8—21no—
127C3⋊C8—72no—
128C3×C7C4×Q1611yescharacteristic, complemented, Hall
129C2×Q8—211no—
130C2×Q8D3×D711yescharacteristic
131Q16—214no—
132C2×C8—31no—
133C2×C8—211no—
134C4⋊C4—31no—
135C4⋊C4—71no—
136C4⋊C4—141no—
137C4⋊C4—421no—
138C42—71no—
139C42—141no—
140C42—421no—
141C2×C7C4×#6711yescharacteristic
142C2×C7C2×#21611yescharacteristic
143C2×C7#95011yescharacteristic
144C22×C3D4×D711yescharacteristic
145C3×C4—22no—
146C3×C4—72no—
147C3×C4—141no—
148C3×C4—281no—
149C3×C4C2×C4×D711yescharacteristic
150C3×C4#115911yescharacteristic
151Dic3—41no—
152Dic3—142no—
153Q8—21no—
154Q8—212no—
155Q8—421no—
156Q8D21⋊C412yes—
157C2×C4—21no—
158C2×C4—61no—
159C2×C4—71no—
160C2×C4—71no—
161C2×C4—141no—
162C2×C4—421no—
163C2×C4C2×D3×D711yescharacteristic
164C8—61no—
165C8—212no—
166C7C4×#21611yescharacteristic, complemented, Sylow
167C2×C3(C22⋊D7)⋊C411yescharacteristic
168C2×C3#647911yescharacteristic
169C2×C3D7×Q1611yescharacteristic
170C22D7×#6711yescharacteristic
171C4—22no—
172C4—72no—
173C4—121no—
174C4—141no—
175C4—281no—
176C4—422no—
177C4#6668611yescharacteristic
178C4C2×D21⋊C411yescharacteristic
179C3#8956211yescharacteristic, complemented, Sylow
180C2D7×#21611yescharacteristic
181C2#1062107611yescharacteristic
182C2#1062111411yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socle, prime radicalC22×C3×C7C2×D4
cosocleC2×C3×C4×C7C23
hypercenterC2×Q8D3×D7
hypocenterC3×C7C4×Q16
LCSG ▷ 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
11. #66635C4semidirect
68. C3×Dic7Q16semidirect
114. Dic7#216semidirect
128. C3×C7C4×Q16semidirect
166. C7C4×#216semidirect
179. C3#89562semidirect

Maximal subgroups

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

subgroupindex
C2×#666352
C3×Q8×Dic72
#106208952
C7×#8352
#106205242
#106205092
C3⋊C8×Dic72
#895623
C4×#2167