Finite Group Databasebeta

← #13052840

Subgroups — #13052840

Order 1344.

220 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 78 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×#6450C211yesmaximal, characteristic
2Dic6×Dic7C211yesmaximal, characteristic
3C2×#66569C211yesmaximal, characteristic
4#10620646C211yesmaximal, characteristic
5C7×#810C211yesmaximal, characteristic
6#10620506C211yesmaximal, characteristic
7C3⋊C8×Dic7C211yesmaximal, characteristic
8#89562—31nomaximal, Hall
9C2×C7×Dic6C2211yescharacteristic
10C2×C3×Dic14C2211yescharacteristic
11C3×C7×C4⋊C4C2211yescharacteristic
12C4⋊Dic21C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C2×C7×C3⋊C8C2211yescharacteristic
15C3×Dic7⋊C4—21no—
16C3×C4×Dic7—21no—
17C3×C4×Dic7C2211yescharacteristic
18Dic3⋊Dic7—21no—
19Dic3×Dic7—21no—
20#66569C414yescomplemented
21Q8×Dic7—31no—
22C2×#1144—31no—
23#6450D311yescharacteristic
24C7×#103—31no—
25#6390—31no—
26C8×Dic7—31no—
27#6381—31no—
28C4×#216—71nomaximal, Hall
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7C2311yescharacteristic
31C2×Dic21—21no—
32C2×C7×Dic3—21no—
33C7×Dic6—21no—
34C7×Dic6C2×C412yes—
35C2×C3×Dic7—21no—
36C2×C3×Dic7D411yescharacteristic
37C2×C3×Dic7D411yescharacteristic
38C3×Dic14—21no—
39C3×Dic14C2×C412yes—
40C3⋊(C7⋊C8)C2×C412yes—
41C7×C3⋊C8—21no—
42C2×C7×Q8—31no—
43C2×Dic14C2×D311yescharacteristic
44C2×C7×C8—31no—
45C7×C4⋊C4C2×D311yescharacteristic
46#1144—34no—
47C4⋊Dic7—31no—
48C4⋊Dic7—61no—
49Dic7⋊C4—21no—
50C4×Dic7—21no—
51C4×Dic7—61no—
52C4×Dic7C2×D311yescharacteristic
53C2×C7⋊C8—31no—
54C3×C4×Q8—71no—
55C2×#216—71no—
56C4×Dic6—71no—
57#835—71no—
58#810D711yescharacteristic
59#807—71no—
60C4×C3⋊C8—71no—
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—41no—
63C3×C4×C7C22×C411yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—41no—
66C3×Dic7—22no—
67C3×Dic7—21no—
68C3×Dic7Q1612yescomplemented
69C7×Dic3—22no—
70C4×Q16—211noSylow
71C7×Q8—32no—
72C7×Q8—61no—
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—21no—
81Dic14C4×D312yes—
82C7×C8—61no—
83C7⋊C8—32no—
84C2×C3×Q8—71no—
85C2×Dic6C2×D711yescharacteristic
86C3×C4⋊C4—141no—
87C3×C4⋊C4C2×D711yescharacteristic
88C3×C42—71no—
89C3×C42—141no—
90#216—74no—
91C4⋊Dic3—71no—
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—41no—
108C4×C7—62no—
109C4×C7C2×C4×D311yescharacteristic
110C4×C7#23511yescharacteristic
111Dic7—22no—
112Dic7—21no—
113Dic7—121no—
114Dic7#21612yescomplemented
115C3×Q8—72no—
116C3×Q8—141no—
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—21no—
125Dic6C4×D712yes—
126C3⋊C8—21no—
127C3⋊C8—72no—
128C3×C7C4×Q1611yescharacteristic, complemented, Hall
129C2×Q8—31no—
130C2×Q8—71no—
131Q16—214no—
132C2×C8—31no—
133C2×C8—211no—
134C4⋊C4—141no—
135C4⋊C4—211no—
136C4⋊C4—421no—
137C4⋊C4D3×D711yescharacteristic
138C42—71no—
139C42—141no—
140C42—421no—
141C2×C7C4×#6711yescharacteristic
142C2×C7C2×#21611yescharacteristic
143C2×C7#95011yescharacteristic
144C22×C3D4×D711yescharacteristic
145C3×C4—41no—
146C3×C4—72no—
147C3×C4—142no—
148C3×C4—141no—
149C3×C4C2×C4×D711yescharacteristic
150C3×C4#115911yescharacteristic
151Dic3—22no—
152Dic3—281no—
153Q8—32no—
154Q8—61no—
155Q8—72no—
156Q8—141no—
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—41no—
172C4—62no—
173C4—72no—
174C4—142no—
175C4—141no—
176C4—841no—
177C4#6667611yescharacteristic
178C4C4×D3×D711yescharacteristic
179C3#8956211yescharacteristic, complemented, Sylow
180C2#1062100211yescharacteristic
181C2#1062105411yescharacteristic
182C2D7×#21611yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socle, prime radicalC22×C3×C7C2×D4
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7C4×Q16
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
20. #66569C4semidirect
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
C3×#64502
Dic6×Dic72
C2×#665692
#106206462
C7×#8102
#106205062
C3⋊C8×Dic72
#895623
C4×#2167