Finite Group Databasebeta

← #13054622

Subgroups — #13054622

Order 1344.

220 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 84 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
1Q8×Dic21C211yesmaximal, characteristic
2C2×C7×#216C211yesmaximal, characteristic
3#10620906C211yesmaximal, characteristic
4C3×#6409C211yesmaximal, characteristic
5#10620525C211yesmaximal, characteristic
6#10620508C211yesmaximal, characteristic
7#10620486C211yesmaximal, characteristic
8Q16×Dic7—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C4⋊Dic21—21no—
12C4⋊Dic21C2211yescharacteristic
13C4×Dic21—21no—
14C4×Dic21C2211yescharacteristic
15C7×#216C414yescomplemented
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19Dic3⋊Dic7—21no—
20Dic3×Dic7—21no—
21C2×C7×Q16—31no—
22Q8×Dic7—31no—
23Q8×Dic7—31no—
24#6409—31no—
25#6409D311yescharacteristic
26#6392—31no—
27C8×Dic7—31no—
28#4790—71nomaximal, Hall
29C3×C7×Q8—21no—
30C3×C7×Q8C2×C412yes—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21D411yescharacteristic
35C2×Dic21D411yescharacteristic
36C2×C7×Dic3—21no—
37C7×Dic6—21no—
38C7×Dic6C2×C412yes—
39C2×C3×Dic7—21no—
40C3×C7⋊C8—21no—
41C7×C3⋊C8C2×C412yes—
42C2×C7×Q8—31no—
43C2×C7×Q8C2×D311yescharacteristic
44C7×Q16—34no—
45C2×C7×C8—31no—
46C4⋊Dic7—31no—
47C4⋊Dic7—61no—
48C4⋊Dic7—61no—
49C4⋊Dic7C2×D311yescharacteristic
50C4×Dic7—31no—
51C4×Dic7—61no—
52C4×Dic7—61no—
53C2×C7⋊C8C2×D311yescharacteristic
54Q8×Dic3—71no—
55C2×#216D711yescharacteristic
56#887—71no—
57C3×#103—71no—
58#816—71no—
59C8×Dic3—71no—
60#807—71no—
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—22no—
63C3×C4×C7C22×C411yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—21no—
66Dic21—41no—
67Dic21Q1612yescomplemented
68C3×Dic7—41no—
69C7×Dic3—22no—
70C4×Q16—211noSylow
71C7×Q8—21no—
72C7×Q8—32no—
73C7×Q8—61no—
74C7×Q8C4×D312yes—
75C2×C4×C7—21no—
76C2×C4×C7—61no—
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—21no—
79C2×Dic7—31no—
80C2×Dic7—31no—
81C2×Dic7—61no—
82C7×C8—32no—
83C7⋊C8—21no—
84C2×C3×Q8C2×D711yescharacteristic
85C2×Dic6C2×D711yescharacteristic
86C2×C3×C8—71no—
87C3×C4⋊C4—71no—
88#216Dic714yescomplemented
89C4⋊Dic3—71no—
90C4⋊Dic3—141no—
91Dic3⋊C4—141no—
92C4×Dic3—71no—
93C4×Dic3—141no—
94C4×Dic3—141no—
95C2×C3⋊C8C2×D711yescharacteristic
96C2×C3×C7C4×D411yescharacteristic
97C2×C3×C7C2×Q1611yescharacteristic
98C2×C3×C7#13511yescharacteristic
99C2×Q16—31no—
100C4×Q8—211no—
101C4×Q8—211no—
102#107—211no—
103#103—71no—
104#103—211no—
105C4×C8—211no—
106C22×C7D3×D411yescharacteristic
107C4×C7—22no—
108C4×C7—62no—
109C4×C7C2×C4×D311yescharacteristic
110C4×C7#23711yescharacteristic
111Dic7—32no—
112Dic7—41no—
113Dic7—61no—
114Dic7—121no—
115C3×Q8—21no—
116C3×Q8C2×Dic712yes—
117C2×C3×C4—21no—
118C2×C3×C4—141no—
119C2×C3×C4C22×D711yescharacteristic
120C2×Dic3—21no—
121C2×Dic3—71no—
122C2×Dic3—71no—
123C2×Dic3—141no—
124Dic6—21no—
125Dic6C2×Dic712yes—
126C3×C8—141no—
127C3⋊C8C2×Dic712yes—
128C3×C7C4×Q1611yescharacteristic, complemented, Hall
129C2×Q8—31no—
130C2×Q8D3×D711yescharacteristic
131Q16—34no—
132C2×C8—31no—
133C2×C8—71no—
134C4⋊C4—71no—
135C4⋊C4—211no—
136C4⋊C4—421no—
137C4⋊C4—421no—
138C42—211no—
139C42—421no—
140C42—421no—
141C2×C7#67⋊C411yescharacteristic
142C2×C7#91611yescharacteristic
143C2×C7D3×Q1611yescharacteristic
144C22×C3D4×D711yescharacteristic
145C3×C4—22no—
146C3×C4—281no—
147C3×C4#115911yescharacteristic
148C3×C4C22×Dic711yescharacteristic
149Dic3—22no—
150Dic3—72no—
151Dic3—141no—
152Dic3—281no—
153Q8—21no—
154Q8—32no—
155Q8—61no—
156Q8D3×Dic712yes—
157C2×C4—21no—
158C2×C4—61no—
159C2×C4—141no—
160C2×C4—211no—
161C2×C4—211no—
162C2×C4—421no—
163C2×C4C2×D3×D711yescharacteristic
164C8—32no—
165C8—141no—
166C7#479011yescharacteristic, complemented, Sylow
167C2×C3D7×Q1611yescharacteristic
168C2×C3#648211yescharacteristic
169C2×C3D4×Dic711yescharacteristic
170C22#6669411yescharacteristic
171C4—22no—
172C4—62no—
173C4—212no—
174C4—281no—
175C4—421no—
176C4—841no—
177C4#6668411yescharacteristic
178C4C2×D3×Dic711yescharacteristic
179C3Q16×Dic711yescharacteristic, complemented, Sylow
180C2#1062107311yescharacteristic
181C2#1062108311yescharacteristic
182C2#67⋊Dic711yescharacteristic
subgroupG/H
centerC22#66694
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
15. C7×#216C4semidirect
67. Dic21Q16semidirect
88. #216Dic7semidirect
128. C3×C7C4×Q16semidirect
166. C7#4790semidirect
179. C3Q16×Dic7semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic212
C2×C7×#2162
#106209062
C3×#64092
#106205252
#106205082
#106204862
Q16×Dic73
#47907