Finite Group Databasebeta

← #13052850

Subgroups — #13052850

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#10620906C211yesmaximal, characteristic
3C2×#66571C211yesmaximal, characteristic
4C7×#810C211yesmaximal, characteristic
5C3×#6382C211yesmaximal, characteristic
6#10620509C211yesmaximal, characteristic
7#10620489C211yesmaximal, characteristic
8#89561—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C3×C7×C4⋊C4C2211yescharacteristic
12#66629—21no—
13C4×Dic21—21no—
14C4×Dic21C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C4⋊Dic7C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18Dic3⋊Dic7—21no—
19Dic3×Dic7—21no—
20#66571C414yescomplemented
21Q8×Dic7—31no—
22C2×#1144—31no—
23#6450—31no—
24C7×#103—31no—
25#6390—31no—
26#6389—31no—
27#6382D311yescharacteristic
28#4853—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—
37C7×Dic6—21no—
38C7×Dic6C2×C412yes—
39C2×C3×Dic7—21no—
40C3×C7⋊C8C2×C412yes—
41C7×C3⋊C8—21no—
42C2×C7×Q8—31no—
43C2×Dic14—31no—
44C2×C7×C8—31no—
45C7×C4⋊C4C2×D311yescharacteristic
46#1144—34no—
47C4⋊Dic7—61no—
48C4⋊Dic7C2×D311yescharacteristic
49Dic7⋊C4—61no—
50C4×Dic7—31no—
51C4×Dic7—61no—
52C4×Dic7—61no—
53C2×C7⋊C8C2×D311yescharacteristic
54C2×Dic12—71no—
55#887—71no—
56#887—71no—
57C3×#106—71no—
58#815—71no—
59#810—71no—
60#810D711yescharacteristic
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—41no—
63C3×C4×C7C22×C411yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—22no—
66Dic21—21no—
67Dic21—21no—
68C3×Dic7—41no—
69C7×Dic3—22no—
70#441—211noSylow
71C7×Q8—32no—
72C7×Q8—61no—
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—32no—
81Dic14—61no—
82C7×C8—61no—
83C7⋊C8C4×D312yes—
84C2×Dic6—71no—
85C2×Dic6C2×D711yescharacteristic
86C2×C3×C8—71no—
87C3×C4⋊C4—71no—
88C3×C4⋊C4C2×D711yescharacteristic
89Dic3⋊C4—141no—
90Dic3⋊C4—141no—
91C4×Dic3—71no—
92C4×Dic3—141no—
93C4×Dic3—141no—
94C2×C3⋊C8C2×D711yescharacteristic
95Dic12—74no—
96C2×C3×C7C4×D411yescharacteristic
97C2×C3×C7#13711yescharacteristic
98C2×C3×C7#13711yescharacteristic
99C2×Q16—211no—
100C4×Q8—211no—
101C4×Q8—211no—
102#106—71no—
103#103—31no—
104#103—211no—
105#97—211no—
106C22×C7D3×D411yescharacteristic
107C4×C7—41no—
108C4×C7—62no—
109C4×C7C2×C4×D311yescharacteristic
110C4×C7#23911yescharacteristic
111Dic7—41no—
112Dic7—62no—
113Dic7—61no—
114Dic7—61no—
115C2×C3×C4—21no—
116C2×C3×C4—141no—
117C2×C3×C4C22×D711yescharacteristic
118C2×Dic3—21no—
119C2×Dic3—71no—
120C2×Dic3—71no—
121C2×Dic3—141no—
122Dic6—21no—
123Dic6—72no—
124Dic6—141no—
125Dic6C4×D712yes—
126C3×C8—72no—
127C3⋊C8—21no—
128C3×C7#44111yescharacteristic, complemented, Hall
129C2×Q8—31no—
130C2×Q8—211no—
131Q16—214no—
132C2×C8—31no—
133C2×C8—71no—
134C4⋊C4—71no—
135C4⋊C4—421no—
136C4⋊C4—421no—
137C4⋊C4D3×D711yescharacteristic
138C42—211no—
139C42—421no—
140C42—421no—
141C2×C7#89311yescharacteristic
142C2×C7#91511yescharacteristic
143C2×C7#91511yescharacteristic
144C22×C3D4×D711yescharacteristic
145C3×C4—41no—
146C3×C4—281no—
147C3×C4C2×C4×D711yescharacteristic
148C3×C4#115911yescharacteristic
149Dic3—22no—
150Dic3—142no—
151Dic3—141no—
152Dic3—141no—
153Q8—32no—
154Q8—61no—
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—61no—
165C8—72no—
166C7#485311yescharacteristic, complemented, Sylow
167C2×C3(C22⋊D7)⋊C411yescharacteristic
168C2×C3#647811yescharacteristic
169C2×C3#648111yescharacteristic
170C22#6669411yescharacteristic
171C4—41no—
172C4—62no—
173C4—281no—
174C4—422no—
175C4—421no—
176C4—421no—
177C4#6667211yescharacteristic
178C4C4×D3×D711yescharacteristic
179C3#8956111yescharacteristic, complemented, Sylow
180C2#1062096811yescharacteristic
181C2#1062105011yescharacteristic
182C2#1062108411yescharacteristic
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. #66571C4semidirect
128. C3×C7#441semidirect
166. C7#4853semidirect
179. C3#89561semidirect

Maximal subgroups

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

subgroupindex
#106213702
#106209062
C2×#665712
C7×#8102
C3×#63822
#106205092
#106204892
#895613
#48537