Finite Group Databasebeta

← #13052845

Subgroups — #13052845

Order 1344.

216 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 72 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#10621370C211yesmaximal, characteristic
2#10620893C211yesmaximal, characteristic
3C2×#66570C211yesmaximal, characteristic
4C7×#808C211yesmaximal, characteristic
5C3×#6384C211yesmaximal, characteristic
6#10620510C211yesmaximal, characteristic
7#10620489C211yesmaximal, characteristic
8#89625—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C3×Dic14C2211yescharacteristic
11C3×C7×C4⋊C4C2211yescharacteristic
12#66629—21no—
13C4×Dic21—21no—
14C4×Dic21C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18Dic7⋊Dic3—21no—
19Dic3×Dic7—21no—
20#66570C414yescomplemented
21C2×Dic28—31no—
22#6450—31no—
23#6450—31no—
24C7×#106—31no—
25#6389—31no—
26#6384—31no—
27#6384D311yescharacteristic
28#4789—71nomaximal, Hall
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7C2311yescharacteristic
31C2×Dic21—21no—
32C2×Dic21D411yescharacteristic
33C2×Dic21D411yescharacteristic
34Dic42—21no—
35Dic42C2×C412yes—
36C2×C7×Dic3—21no—
37C2×C3×Dic7—21no—
38C3×Dic14—21no—
39C3×Dic14C2×C412yes—
40C3×C7⋊C8—21no—
41C7×C3⋊C8C2×C412yes—
42C2×Dic14—31no—
43C2×Dic14C2×D311yescharacteristic
44C2×C7×C8—31no—
45C7×C4⋊C4—31no—
46C7×C4⋊C4C2×D311yescharacteristic
47Dic7⋊C4—61no—
48Dic7⋊C4—61no—
49C4×Dic7—31no—
50C4×Dic7—61no—
51C4×Dic7—61no—
52C2×C7⋊C8C2×D311yescharacteristic
53Dic28—34no—
54Q8×Dic3—71no—
55C2×#216—71no—
56#887—71no—
57C3×#103—71no—
58#816—71no—
59#815—71no—
60#808D711yescharacteristic
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—41no—
63C3×C4×C7C22×C411yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—22no—
66Dic21—21no—
67Dic21—21no—
68C3×Dic7—22no—
69C7×Dic3—41no—
70#441—211noSylow
71C2×C4×C7—21no—
72C2×C4×C7—61no—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—21no—
75C2×Dic7—31no—
76C2×Dic7—31no—
77C2×Dic7—61no—
78Dic14—21no—
79Dic14—32no—
80Dic14—61no—
81Dic14C4×D312yes—
82C7×C8—32no—
83C7⋊C8—21no—
84C2×C3×Q8—71no—
85C2×Dic6—71no—
86C2×C3×C8—71no—
87C3×C4⋊C4C2×D711yescharacteristic
88#216—74no—
89C4⋊Dic3—141no—
90C4⋊Dic3C2×D711yescharacteristic
91Dic3⋊C4—141no—
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—211no—
100C4×Q8—211no—
101C4×Q8—211no—
102#106—31no—
103#103—71no—
104#103—211no—
105#97—211no—
106C22×C7D3×D411yescharacteristic
107C4×C7—41no—
108C4×C7—121no—
109C4×C7C2×C4×D311yescharacteristic
110C4×C7#23711yescharacteristic
111Dic7—22no—
112Dic7—62no—
113Dic7—61no—
114Dic7—61no—
115C3×Q8—72no—
116C3×Q8—141no—
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—72no—
125Dic6—141no—
126C3×C8—141no—
127C3⋊C8C4×D712yes—
128C3×C7#44111yescharacteristic, complemented, Hall
129C2×Q8—71no—
130C2×Q8—211no—
131Q16—214no—
132C2×C8—31no—
133C2×C8—71no—
134C4⋊C4—31no—
135C4⋊C4—421no—
136C4⋊C4—421no—
137C4⋊C4D3×D711yescharacteristic
138C42—211no—
139C42—421no—
140C42—421no—
141C2×C7#67⋊C411yescharacteristic
142C2×C7#91511yescharacteristic
143C2×C7#91811yescharacteristic
144C22×C3D4×D711yescharacteristic
145C3×C4—41no—
146C3×C4—142no—
147C3×C4C2×C4×D711yescharacteristic
148C3×C4#116111yescharacteristic
149Dic3—41no—
150Dic3—142no—
151Dic3—141no—
152Dic3—141no—
153Q8—72no—
154Q8—141no—
155Q8—212no—
156Q8—421no—
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#478911yescharacteristic, complemented, Sylow
167C2×C3#645611yescharacteristic
168C2×C3#647811yescharacteristic
169C2×C3#647811yescharacteristic
170C22#6669411yescharacteristic
171C4—41no—
172C4—121no—
173C4—142no—
174C4—422no—
175C4—421no—
176C4—421no—
177C4#6667011yescharacteristic
178C4C4×D3×D711yescharacteristic
179C3#8962511yescharacteristic, complemented, Sylow
180C2#1062095811yescharacteristic
181C2#1062105011yescharacteristic
182C2#1062108611yescharacteristic
subgroupG/H
centerC22#66694
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×C7#441
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. #66570C4semidirect
128. C3×C7#441semidirect
166. C7#4789semidirect
179. C3#89625semidirect

Maximal subgroups

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

subgroupindex
#106213702
#106208932
C2×#665702
C7×#8082
C3×#63842
#106205102
#106204892
#896253
#47897