Finite Group Databasebeta

← #13054621

Subgroups — #13054621

Order 1344.

216 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 82 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×C3×#1144C211yesmaximal, characteristic
3#10620893C211yesmaximal, characteristic
4C7×#835C211yesmaximal, characteristic
5#10620520C211yesmaximal, characteristic
6#10620506C211yesmaximal, characteristic
7#10620489C211yesmaximal, characteristic
8#89561—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C3×Dic14C2211yescharacteristic
11C4⋊Dic21—21no—
12C4⋊Dic21C2211yescharacteristic
13C4×Dic21—21no—
14C4×Dic21C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×#1144C414yescomplemented
18C2×C3×C7⋊C8C2211yescharacteristic
19Dic7⋊Dic3—21no—
20Dic3×Dic7—21no—
21Q8×Dic7—31no—
22C2×#1144D311yescharacteristic
23#6450—31no—
24C7×#103—31no—
25#6390—31no—
26#6389—31no—
27#6382—31no—
28#5184—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—
37C2×C3×Dic7—21no—
38C3×Dic14—21no—
39C3×Dic14C2×C412yes—
40C3×C7⋊C8C2×C412yes—
41C7×C3⋊C8—21no—
42C2×C7×Q8C2×D311yescharacteristic
43C2×Dic14C2×D311yescharacteristic
44C2×C7×C8—31no—
45C7×C4⋊C4—31no—
46#1144Dic314yescomplemented
47C4⋊Dic7—31no—
48C4⋊Dic7—61no—
49Dic7⋊C4—61no—
50C4×Dic7—31no—
51C4×Dic7—61no—
52C4×Dic7—61no—
53C2×C7⋊C8C2×D311yescharacteristic
54C2×C3×Q16—71no—
55Q8×Dic3—71no—
56Q8×Dic3—71no—
57#835—71no—
58#835D711yescharacteristic
59#817—71no—
60#815—71no—
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—22no—
63C3×C4×C7C22×C411yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—21no—
66Dic21—21no—
67Dic21—41no—
68C3×Dic7—22no—
69C7×Dic3—41no—
70#441—211noSylow
71C7×Q8—21no—
72C7×Q8C2×Dic312yes—
73C2×C4×C7—21no—
74C2×C4×C7—61no—
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—21no—
77C2×Dic7—31no—
78C2×Dic7—31no—
79C2×Dic7—61no—
80Dic14—21no—
81Dic14C2×Dic312yes—
82C7×C8—61no—
83C7⋊C8C2×Dic312yes—
84C2×C3×Q8—71no—
85C2×C3×Q8C2×D711yescharacteristic
86C3×Q16—74no—
87C2×C3×C8—71no—
88C4⋊Dic3—71no—
89C4⋊Dic3—141no—
90C4⋊Dic3—141no—
91C4⋊Dic3C2×D711yescharacteristic
92C4×Dic3—71no—
93C4×Dic3—141no—
94C4×Dic3—141no—
95C2×C3⋊C8C2×D711yescharacteristic
96C2×C3×C7C4×D411yescharacteristic
97C2×C3×C7#13711yescharacteristic
98C2×C3×C7#13711yescharacteristic
99C2×Q16—71no—
100C4×Q8—211no—
101C4×Q8—211no—
102#106—211no—
103#103—31no—
104#103—211no—
105#97—211no—
106C22×C7D3×D411yescharacteristic
107C4×C7—22no—
108C4×C7—121no—
109C4×C7#23711yescharacteristic
110C4×C7C22×Dic311yescharacteristic
111Dic7—22no—
112Dic7—61no—
113Dic7—61no—
114Dic7—121no—
115C3×Q8—21no—
116C3×Q8—72no—
117C3×Q8—141no—
118C3×Q8C4×D712yes—
119C2×C3×C4—21no—
120C2×C3×C4—141no—
121C2×C3×C4C22×D711yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—71no—
124C2×Dic3—71no—
125C2×Dic3—141no—
126C3×C8—72no—
127C3⋊C8—21no—
128C3×C7#44111yescharacteristic, complemented, Hall
129C2×Q8—71no—
130C2×Q8D3×D711yescharacteristic
131Q16—74no—
132C2×C8—31no—
133C2×C8—71no—
134C4⋊C4—31no—
135C4⋊C4—211no—
136C4⋊C4—421no—
137C4⋊C4—421no—
138C42—211no—
139C42—421no—
140C42—421no—
141C2×C7#91811yescharacteristic
142C2×C7#91811yescharacteristic
143C2×C7D4×Dic311yescharacteristic
144C22×C3D4×D711yescharacteristic
145C3×C4—22no—
146C3×C4—142no—
147C3×C4C2×C4×D711yescharacteristic
148C3×C4#115911yescharacteristic
149Dic3—41no—
150Dic3—141no—
151Dic3—141no—
152Dic3—281no—
153Q8—21no—
154Q8—72no—
155Q8—141no—
156Q8Dic3×D712yes—
157C2×C4—21no—
158C2×C4—61no—
159C2×C4—141no—
160C2×C4—211no—
161C2×C4—211no—
162C2×C4—421no—
163C2×C4C2×D3×D711yescharacteristic
164C8—61no—
165C8—72no—
166C7#518411yescharacteristic, complemented, Sylow
167C2×C3(C22⋊D7)⋊C411yescharacteristic
168C2×C3#647811yescharacteristic
169C2×C3#648111yescharacteristic
170C22#6669411yescharacteristic
171C4—22no—
172C4—121no—
173C4—142no—
174C4—421no—
175C4—421no—
176C4—841no—
177C4C2×Dic3×D711yescharacteristic
178C4#6668411yescharacteristic
179C3#8956111yescharacteristic, complemented, Sylow
180C2#1062107411yescharacteristic
181C2#1062108611yescharacteristic
182C2(C22⋊D7)⋊Dic311yescharacteristic
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×C7#441
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
17. C3×#1144C4semidirect
46. #1144Dic3semidirect
128. C3×C7#441semidirect
166. C7#5184semidirect
179. C3#89561semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic212
C2×C3×#11442
#106208932
C7×#8352
#106205202
#106205062
#106204892
#895613
#51847