Finite Group Databasebeta

← #13056513

Subgroups — #13056513

Order 1344.

180 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 71 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#10621371C211yesmaximal, characteristic
3C3×C7×#107C211yesmaximal, characteristic
4#10620656C211yesmaximal, characteristic
5#10620654C211yesmaximal, characteristic
6#10620647C211yesmaximal, characteristic
7#10620646C211yesmaximal, characteristic
8#89648—31nomaximal, Hall
9C2×Dic42—21no—
10C2×C3×C7×C8C2211yescharacteristic
11C3×C7×C4⋊C4C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C4⋊Dic21—21no—
14C4⋊Dic21C2211yescharacteristic
15C4⋊Dic21C2211yescharacteristic
16#66629—21no—
17#66629—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6453—31no—
21#6451—31no—
22C7×#107D311yescharacteristic
23#6391—31no—
24#6389—31no—
25#6382—31no—
26#6381—31no—
27#4876—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—42no—
36C3×C7×C8Q812yescomplemented
37C3⋊(C7⋊C8)Q812yescomplemented
38C2×Dic14—61no—
39C2×C7×C8C2×D311yescharacteristic
40C7×C4⋊C4C2×D311yescharacteristic
41C7×C4⋊C4C2×D311yescharacteristic
42C4⋊Dic7—31no—
43C4⋊Dic7—31no—
44C4⋊Dic7—61no—
45Dic7⋊C4—61no—
46Dic7⋊C4—61no—
47C4×Dic7—31no—
48C2×C7⋊C8—31no—
49#890—71no—
50#888—71no—
51C3×#107D711yescharacteristic
52#817—71no—
53#815—71no—
54#808—71no—
55#807—71no—
56C22×C3×C7C2×D411yescharacteristic
57C3×C4×C7—41no—
58C3×C4×C7—41no—
59C3×C4×C7C2×Q811yescharacteristic
60C3×C4×C7C2×Q811yescharacteristic
61Dic21—21no—
62Dic21—21no—
63Dic21—41no—
64Dic21—41no—
65#501—211noSylow
66C2×C4×C7—21no—
67C2×C4×C7—21no—
68C2×C4×C7C22×D311yescharacteristic
69C2×Dic7—31no—
70C2×Dic7—31no—
71C2×Dic7—61no—
72C2×Dic7—61no—
73Dic14—122no—
74C7×C8Dic612yescomplemented
75C7⋊C8—32no—
76C2×Dic6—141no—
77C2×C3×C8C2×D711yescharacteristic
78C3×C4⋊C4C2×D711yescharacteristic
79C3×C4⋊C4C2×D711yescharacteristic
80C4⋊Dic3—71no—
81C4⋊Dic3—71no—
82C4⋊Dic3—141no—
83Dic3⋊C4—141no—
84Dic3⋊C4—141no—
85C4×Dic3—71no—
86C2×C3⋊C8—71no—
87C2×C3×C7C4⋊Q811yescharacteristic
88C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
89C2×C3×C7#13711yescharacteristic
90C4⋊Q8—211no—
91#125—211no—
92#107—211no—
93#107D2111yescharacteristic
94#106—211no—
95#106—211no—
96#97—211no—
97C22×C7D3×D411yescharacteristic
98C4×C7—41no—
99C4×C7—41no—
100C4×C7C2×Dic611yescharacteristic
101C4×C7D3×Q811yescharacteristic
102Dic7—61no—
103Dic7—61no—
104Dic7—121no—
105Dic7—121no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4C22×D711yescharacteristic
109C2×Dic3—71no—
110C2×Dic3—71no—
111C2×Dic3—141no—
112C2×Dic3—141no—
113Dic6—282no—
114C3×C8Dic1412yescomplemented
115C3⋊C8—72no—
116C3×C7#50111yescharacteristic, complemented, Hall
117C2×Q8—421no—
118C2×C8—211no—
119C2×C8C2×D2111yescharacteristic
120C4⋊C4—211no—
121C4⋊C4—211no—
122C4⋊C4—421no—
123C4⋊C4—421no—
124C4⋊C4—421no—
125C4⋊C4C2×D2111yescharacteristic
126C4⋊C4C2×D2111yescharacteristic
127C42—211no—
128C2×C7#88811yescharacteristic
129C2×C7#91111yescharacteristic
130C2×C7#91811yescharacteristic
131C22×C3D4×D711yescharacteristic
132C3×C4—41no—
133C3×C4—41no—
134C3×C4C2×Dic1411yescharacteristic
135C3×C4Q8×D711yescharacteristic
136Dic3—141no—
137Dic3—141no—
138Dic3—281no—
139Dic3—281no—
140Q8—842no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—211no—
144C2×C4—211no—
145C2×C4—421no—
146C2×C4—421no—
147C2×C4C22×D2111yescharacteristic
148C8—212no—
149C8Dic4212yescomplemented
150C7#487611yescharacteristic, complemented, Sylow
151C2×C3#645111yescharacteristic
152C2×C3#647411yescharacteristic
153C2×C3#648111yescharacteristic
154C22D4×D2111yescharacteristic
155C4—41no—
156C4—41no—
157C4—421no—
158C4—421no—
159C4—841no—
160C4—841no—
161C4C2×Dic4211yescharacteristic
162C4Q8×D2111yescharacteristic
163C3#8964811yescharacteristic, complemented, Sylow
164C2#1062137111yescharacteristic
165C2#1062139411yescharacteristic
166C2#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#501
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
36. C3×C7×C8Q8semidirect
37. C3⋊(C7⋊C8)Q8semidirect
74. C7×C8Dic6semidirect
114. C3×C8Dic14semidirect
116. C3×C7#501semidirect
149. C8Dic42semidirect
150. C7#4876semidirect
163. C3#89648semidirect

Maximal subgroups

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

subgroupindex
#106213732
#106213712
C3×C7×#1072
#106206562
#106206542
#106206472
#106206462
#896483
#48767