Finite Group Databasebeta

← #13056512

Subgroups — #13056512

Order 1344.

192 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 65 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#10621371C211yesmaximal, characteristic
2#10621370C211yesmaximal, characteristic
3C3×C7×#107C211yesmaximal, characteristic
4#10620655C211yesmaximal, characteristic
5#10620653C211yesmaximal, characteristic
6#10620649C211yesmaximal, characteristic
7#10620646C211yesmaximal, characteristic
8#89647—31nomaximal, Hall
9C2×Dic42—21no—
10C2×Dic42C2211yescharacteristic
11C2×C3×C7×C8C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C4⋊Dic21C2211yescharacteristic
15#66629—21no—
16#66629—21no—
17C4×Dic21—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6451—31no—
21#6450—31no—
22C7×#107D311yescharacteristic
23#6390—31no—
24Dic7⋊C8—31no—
25#6384—31no—
26#6381—31no—
27#4875—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—
36Dic42—42no—
37Dic42Q812yescomplemented
38C3×C7×C8—21no—
39C3⋊(C7⋊C8)—21no—
40C2×Dic14—31no—
41C2×Dic14—61no—
42C2×C7×C8C2×D311yescharacteristic
43C7×C4⋊C4C2×D311yescharacteristic
44C7×C4⋊C4C2×D311yescharacteristic
45C4⋊Dic7—31no—
46Dic7⋊C4—61no—
47Dic7⋊C4—61no—
48C4×Dic7—31no—
49C4×Dic7—61no—
50C2×C7⋊C8—31no—
51#888—71no—
52#887—71no—
53C3×#107D711yescharacteristic
54#816—71no—
55Dic3⋊C8—71no—
56#810—71no—
57#807—71no—
58C22×C3×C7C2×D411yescharacteristic
59C3×C4×C7—41no—
60C3×C4×C7—41no—
61C3×C4×C7C2×Q811yescharacteristic
62C3×C4×C7#4111yescharacteristic
63Dic21—22no—
64Dic21—21no—
65Dic21—41no—
66Dic21Q1612yes—
67#477—211noSylow
68C2×C4×C7—21no—
69C2×C4×C7—21no—
70C2×C4×C7C22×D311yescharacteristic
71C2×Dic7—31no—
72C2×Dic7—31no—
73C2×Dic7—61no—
74C2×Dic7—61no—
75Dic14—32no—
76Dic14—61no—
77Dic14—122no—
78C7×C8—21no—
79C7⋊C8—61no—
80C2×Dic6—71no—
81C2×Dic6—141no—
82C2×C3×C8C2×D711yescharacteristic
83C3×C4⋊C4C2×D711yescharacteristic
84C3×C4⋊C4C2×D711yescharacteristic
85C4⋊Dic3—71no—
86Dic3⋊C4—141no—
87Dic3⋊C4—141no—
88C4×Dic3—71no—
89C4×Dic3—141no—
90C2×C3⋊C8—71no—
91C2×C3×C7C22⋊Q811yescharacteristic
92C2×C3×C7C2×Q1611yescharacteristic
93C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
94C4⋊Q8—211no—
95C4×Q8—211no—
96#107—211no—
97#107D2111yescharacteristic
98C4⋊C8—211no—
99#103—211no—
100#103—211no—
101C22×C7D3×D411yescharacteristic
102C4×C7—41no—
103C4×C7—41no—
104C4×C7#23511yescharacteristic
105C4×C7D3×Q811yescharacteristic
106Dic7—32no—
107Dic7—62no—
108Dic7—61no—
109Dic7—121no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4C22×D711yescharacteristic
113C2×Dic3—71no—
114C2×Dic3—71no—
115C2×Dic3—141no—
116C2×Dic3—141no—
117Dic6—72no—
118Dic6—141no—
119Dic6—282no—
120C3×C8—21no—
121C3⋊C8—141no—
122C3×C7#47711yescharacteristic, complemented, Hall
123C2×Q8—211no—
124C2×Q8—421no—
125C2×C8—211no—
126C2×C8C2×D2111yescharacteristic
127C4⋊C4—211no—
128C4⋊C4—421no—
129C4⋊C4—421no—
130C4⋊C4C2×D2111yescharacteristic
131C4⋊C4C2×D2111yescharacteristic
132C42—211no—
133C42—421no—
134C2×C7#89611yescharacteristic
135C2×C7#91111yescharacteristic
136C2×C7D3×Q1611yescharacteristic
137C22×C3D4×D711yescharacteristic
138C3×C4—41no—
139C3×C4—41no—
140C3×C4#115711yescharacteristic
141C3×C4Q8×D711yescharacteristic
142Dic3—72no—
143Dic3—142no—
144Dic3—141no—
145Dic3—281no—
146Q8—212no—
147Q8—421no—
148Q8—842no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—211no—
152C2×C4—211no—
153C2×C4—421no—
154C2×C4—421no—
155C2×C4C22×D2111yescharacteristic
156C8—21no—
157C8—421no—
158C7#487511yescharacteristic, complemented, Sylow
159C2×C3#645911yescharacteristic
160C2×C3#647411yescharacteristic
161C2×C3D7×Q1611yescharacteristic
162C22D4×D2111yescharacteristic
163C4—41no—
164C4—41no—
165C4—212no—
166C4—422no—
167C4—421no—
168C4—841no—
169C4#6672811yescharacteristic
170C4Q8×D2111yescharacteristic
171C3#8964711yescharacteristic, complemented, Sylow
172C2#1062137911yescharacteristic
173C2#1062139411yescharacteristic
174C2Q16×D2111yescharacteristic
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#477
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
37. Dic42Q8semidirect
122. C3×C7#477semidirect
158. C7#4875semidirect
171. C3#89647semidirect

Maximal subgroups

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

subgroupindex
#106213712
#106213702
C3×C7×#1072
#106206552
#106206532
#106206492
#106206462
#896473
#48757