Finite Group Databasebeta

← #13056521

Subgroups — #13056521

Order 1344.

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

normal subgroups only

This group has 208 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#10621380C211yesmaximal, characteristic
2C4⋊C4×D21C211yesmaximal, characteristic
3C3×C7×#107C211yesmaximal, characteristic, complemented
4#10620659C211yesmaximal, characteristic
5#10620655C211yesmaximal, characteristic, complemented
6#10620649C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89656—31nomaximal, Hall
9C2×C4×D21—21no—
10C2×C4×D21C2211yescharacteristic
11C2×Dic42C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15#66631—21no—
16C4⋊Dic21—21no—
17C4⋊Dic21C2211yescharacteristic
18#66629—21no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6460—31no—
21D7×C4⋊C4—31no—
22C7×#107D311yescharacteristic, complemented
23(C2×D7)⋊C8—31no—
24#6390—31no—
25#6384—31no—
26#6381—31no—
27#4884—71nomaximal, Hall
28C22×D21D411yescharacteristic
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21—21no—
34C2×Dic21D411yescharacteristic
35C4×D21—21no—
36C4×D21—21no—
37C4×D21—42no—
38Dic42—21no—
39Dic42—21no—
40C3×C7×C8—21no—
41C3⋊(C7⋊C8)—21no—
42C2×C4×D7—31no—
43C2×C4×D7—61no—
44C2×Dic14—31no—
45C2×C7×C8C2×D311yescharacteristic
46C7×C4⋊C4C2×D311yescharacteristic
47C7×C4⋊C4C2×D311yescharacteristic
48(C2×D7)⋊C4—61no—
49C4⋊Dic7—31no—
50C4⋊Dic7—61no—
51Dic7⋊C4—61no—
52C2×C7⋊C8—31no—
53#897—71no—
54D3×C4⋊C4—71no—
55C3×#107D711yescharacteristic, complemented
56(C2×D3)⋊C8—71no—
57#816—71no—
58#810—71no—
59#807—71no—
60C22×C3×C7C2×D411yescharacteristic
61C2×D21—21no—
62C2×D21—21no—
63C2×D21Q1612yes—
64C3×C4×C7—41no—
65C3×C4×C7—41no—
66C3×C4×C7#4111yescharacteristic
67C3×C4×C7#4111yescharacteristic
68Dic21—21no—
69Dic21—41no—
70Dic21—41no—
71#484—211noSylow
72C22×D7—31no—
73C2×C4×C7—21no—
74C2×C4×C7—21no—
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—31no—
77C2×Dic7—61no—
78C2×Dic7—61no—
79C4×D7—61no—
80C4×D7—61no—
81C4×D7—122no—
82Dic14—61no—
83Dic14—61no—
84C7×C8—21no—
85C7⋊C8—61no—
86C2×C4×D3—71no—
87C2×C4×D3—141no—
88C2×Dic6—71no—
89C2×C3×C8C2×D711yescharacteristic
90C3×C4⋊C4C2×D711yescharacteristic
91C3×C4⋊C4C2×D711yescharacteristic
92(C2×D3)⋊C4—141no—
93C4⋊Dic3—71no—
94C4⋊Dic3—141no—
95Dic3⋊C4—141no—
96C2×C3⋊C8—71no—
97C2×C3×C7#12311yescharacteristic
98C2×C3×C7C2×Q1611yescharacteristic
99C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
100D21—22no—
101C22⋊Q8—211no—
102C2×C4⋊C4—211no—
103#107—211no—
104#107D2111yescharacteristic, complemented
105#103—211no—
106#103—211no—
107C22⋊C8—211no—
108C22×C7D3×D411yescharacteristic
109C2×D7—32no—
110C2×D7—61no—
111C2×D7—61no—
112C4×C7—41no—
113C4×C7—41no—
114C4×C7#23511yescharacteristic
115C4×C7#23911yescharacteristic
116Dic7—61no—
117Dic7—121no—
118Dic7—121no—
119C22×D3—71no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4C22×D711yescharacteristic
123C2×Dic3—71no—
124C2×Dic3—141no—
125C2×Dic3—141no—
126C4×D3—141no—
127C4×D3—141no—
128C4×D3—282no—
129Dic6—141no—
130Dic6—141no—
131C3×C8—21no—
132C3⋊C8—141no—
133C3×C7#48411yescharacteristic, complemented, Hall
134C2×Q8—211no—
135C22×C4—211no—
136C22×C4—421no—
137C2×C8—211no—
138C2×C8C2×D2111yescharacteristic
139C4⋊C4—211no—
140C4⋊C4—421no—
141C4⋊C4—421no—
142C4⋊C4C2×D2111yescharacteristic
143C4⋊C4C2×D2111yescharacteristic
144C22⋊C4—421no—
145C2×C7#89411yescharacteristic
146C2×C7#91111yescharacteristic
147C2×C7D3×Q1611yescharacteristic
148D7—62no—
149C22×C3D4×D711yescharacteristic
150C2×D3—72no—
151C2×D3—141no—
152C2×D3—141no—
153C3×C4—41no—
154C3×C4—41no—
155C3×C4#115711yescharacteristic
156C3×C4#116111yescharacteristic
157Dic3—141no—
158Dic3—281no—
159Dic3—281no—
160C23—211no—
161Q8—421no—
162Q8—421no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—211no—
166C2×C4—421no—
167C2×C4—421no—
168C2×C4—421no—
169C2×C4—421no—
170C2×C4—842no—
171C2×C4C22×D2111yescharacteristic
172C8—21no—
173C8—421no—
174C7#488411yescharacteristic, complemented, Sylow
175C2×C3#645711yescharacteristic
176C2×C3#647411yescharacteristic
177C2×C3D7×Q1611yescharacteristic
178D3—142no—
179C22—212no—
180C22—421no—
181C22—421no—
182C22D4×D2111yescharacteristic
183C4—41no—
184C4—41no—
185C4—421no—
186C4—841no—
187C4—841no—
188C4#6672811yescharacteristic
189C4#6673211yescharacteristic
190C3#8965611yescharacteristic, complemented, Sylow
191C2—422no—
192C2#1062137711yescharacteristic
193C2#1062139411yescharacteristic
194C2Q16×D2111yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#107C2
socleC22×C3×C7C2×D4
prime radicalC22×D21D4
hypercenter#107D21
hypocenterC3×C7#484
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
3. C3×C7×#107C2semidirect
5. #10620655C2semidirect
6. #10620649C2semidirect
7. #10620646C2semidirect
22. C7×#107D3semidirect
55. C3×#107D7semidirect
104. #107D21semidirect
133. C3×C7#484semidirect
174. C7#4884semidirect
190. C3#89656semidirect

Maximal subgroups

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

subgroupindex
#106213802
C4⋊C4×D212
C3×C7×#1072
#106206592
#106206552
#106206492
#106206462
#896563
#48847