Finite Group Databasebeta

← #13056514

Subgroups — #13056514

Order 1344.

180 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621373C211yesmaximal, characteristic
2#10621370C211yesmaximal, characteristic
3C3×C7×#107C211yesmaximal, characteristic
4#10620655C211yesmaximal, characteristic
5#10620653C211yesmaximal, characteristic
6#10620649C211yesmaximal, characteristic
7#10620647C211yesmaximal, characteristic
8#89649—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C3×C7×C8C2211yescharacteristic
11C3×C7×C4⋊C4C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C4⋊Dic21—21no—
14C4⋊Dic21C2211yescharacteristic
15#66629—21no—
16#66629—21no—
17C4×Dic21—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6453—31no—
21#6450—31no—
22C7×#107D311yescharacteristic
23#6390—31no—
24Dic7⋊C8—31no—
25#6384—31no—
26#6382—31no—
27#4877—71nomaximal, Hall
28C2×C3×C4×C7—21no—
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7C2311yescharacteristic
31C2×Dic21—21no—
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C2×Dic21D411yescharacteristic
35Dic42—21no—
36Dic42Q812yes—
37C3×C7×C8—21no—
38C3⋊(C7⋊C8)—21no—
39C2×Dic14—31no—
40C2×C7×C8C2×D311yescharacteristic
41C7×C4⋊C4C2×D311yescharacteristic
42C7×C4⋊C4C2×D311yescharacteristic
43C4⋊Dic7—31no—
44C4⋊Dic7—61no—
45Dic7⋊C4—61no—
46Dic7⋊C4—61no—
47C4×Dic7—31no—
48C4×Dic7—61no—
49C2×C7⋊C8—31no—
50#890—71no—
51#887—71no—
52C3×#107D711yescharacteristic
53#816—71no—
54Dic3⋊C8—71no—
55#810—71no—
56#808—71no—
57C22×C3×C7C2×D411yescharacteristic
58C3×C4×C7—41no—
59C3×C4×C7—41no—
60C3×C4×C7C2×Q811yescharacteristic
61C3×C4×C7#4111yescharacteristic
62Dic21—22no—
63Dic21—21no—
64Dic21—21no—
65Dic21—41no—
66#479—211noSylow
67C2×C4×C7—21no—
68C2×C4×C7—21no—
69C2×C4×C7C22×D311yescharacteristic
70C2×Dic7—31no—
71C2×Dic7—31no—
72C2×Dic7—61no—
73C2×Dic7—61no—
74Dic14—32no—
75Dic14—61no—
76C7×C8—21no—
77C7⋊C8—61no—
78C2×Dic6—71no—
79C2×C3×C8C2×D711yescharacteristic
80C3×C4⋊C4C2×D711yescharacteristic
81C3×C4⋊C4C2×D711yescharacteristic
82C4⋊Dic3—71no—
83C4⋊Dic3—141no—
84Dic3⋊C4—141no—
85Dic3⋊C4—141no—
86C4×Dic3—71no—
87C4×Dic3—141no—
88C2×C3⋊C8—71no—
89C2×C3×C7C22⋊Q811yescharacteristic
90C2×C3×C7#13511yescharacteristic
91C2×C3×C7#13711yescharacteristic
92#125—211no—
93C4×Q8—211no—
94#107D2111yescharacteristic
95#106—211no—
96C4⋊C8—211no—
97#103—211no—
98#103—211no—
99C22×C7D3×D411yescharacteristic
100C4×C7—41no—
101C4×C7—41no—
102C4×C7#23511yescharacteristic
103C4×C7D3×Q811yescharacteristic
104Dic7—62no—
105Dic7—61no—
106Dic7—61no—
107Dic7—121no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4C22×D711yescharacteristic
111C2×Dic3—71no—
112C2×Dic3—71no—
113C2×Dic3—141no—
114C2×Dic3—141no—
115Dic6—72no—
116Dic6—141no—
117C3×C8—21no—
118C3⋊C8—141no—
119C3×C7#47911yescharacteristic, complemented, Hall
120C2×Q8—211no—
121C2×C8—211no—
122C2×C8C2×D2111yescharacteristic
123C4⋊C4—211no—
124C4⋊C4—421no—
125C4⋊C4—421no—
126C4⋊C4—421no—
127C4⋊C4C2×D2111yescharacteristic
128C4⋊C4C2×D2111yescharacteristic
129C42—211no—
130C42—421no—
131C2×C7#89611yescharacteristic
132C2×C7#91211yescharacteristic
133C2×C7#91811yescharacteristic
134C22×C3D4×D711yescharacteristic
135C3×C4—41no—
136C3×C4—41no—
137C3×C4#115711yescharacteristic
138C3×C4Q8×D711yescharacteristic
139Dic3—142no—
140Dic3—141no—
141Dic3—141no—
142Dic3—281no—
143Q8—212no—
144Q8—421no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—211no—
148C2×C4—211no—
149C2×C4—421no—
150C2×C4—421no—
151C2×C4C22×D2111yescharacteristic
152C8—21no—
153C8—421no—
154C7#487711yescharacteristic, complemented, Sylow
155C2×C3#645911yescharacteristic
156C2×C3#647511yescharacteristic
157C2×C3#648111yescharacteristic
158C22D4×D2111yescharacteristic
159C4—41no—
160C4—41no—
161C4—422no—
162C4—421no—
163C4—421no—
164C4—841no—
165C4#6672811yescharacteristic
166C4Q8×D2111yescharacteristic
167C3#8964911yescharacteristic, complemented, Sylow
168C2#1062137911yescharacteristic
169C2#1062139511yescharacteristic
170C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#107C2
socle, prime radicalC22×C3×C7C2×D4
hypercenter#107D21
hypocenterC3×C7#479
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
119. C3×C7#479semidirect
154. C7#4877semidirect
167. C3#89649semidirect

Maximal subgroups

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

subgroupindex
#106213732
#106213702
C3×C7×#1072
#106206552
#106206532
#106206492
#106206472
#896493
#48777