Finite Group Databasebeta

← #13053124

Subgroups — #13053124

Order 1344.

204 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 62 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#10620990C211yesmaximal, characteristic
2#10620943C211yesmaximal, characteristic, complemented
3#10620660C211yesmaximal, characteristic
4C7×Dic3⋊C8C211yesmaximal, characteristic, complemented
5C3×#6391C211yesmaximal, characteristic, complemented
6#10620525C211yesmaximal, characteristic, complemented
7#10620503C211yesmaximal, characteristic
8#89619—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×D21⋊C4—21no—
11C2×C3×C7×C8C2211yescharacteristic
12C4⋊Dic21C2211yescharacteristic
13C4×C7×Dic3C2211yescharacteristic
14C2×C7×C3⋊C8C2211yescharacteristic
15C3×C4⋊Dic7C2211yescharacteristic
16C3×C4⋊Dic7C2211yescharacteristic
17#66577—21no—
18Dic7⋊Dic3—21no—
19#66575—21no—
20C4×D28—31no—
21#6433—31no—
22C7×C4⋊C8—31no—
23#6395—31no—
24#6395—31no—
25#6392—31no—
26#6391D311yescharacteristic, complemented
27#4871—71nomaximal, Hall
28C22×D21—21no—
29C2×C3×C4×C7C2311yescharacteristic
30C2×Dic21—21no—
31D84—21no—
32D84Q812yes—
33C2×C7×Dic3D411yescharacteristic
34C2×C7×Dic3D411yescharacteristic
35C2×C3×Dic7—21no—
36C2×C3×Dic7—21no—
37D21⋊C4—42no—
38C3×C7×C8—21no—
39C7×C3⋊C8—21no—
40C2×D28—31no—
41C2×C4×D7—61no—
42C2×C7×C8—31no—
43C2×C7×C8C2×D311yescharacteristic
44C42×C7—31no—
45(C2×D7)⋊C4—61no—
46C4⋊Dic7—31no—
47C4⋊Dic7C2×D311yescharacteristic
48C4⋊Dic7C2×D311yescharacteristic
49Dic7⋊C4—61no—
50Dic7⋊C4—61no—
51#893—71no—
52#890—71no—
53C3×#106—71no—
54#821—71no—
55Dic3⋊C8D711yescharacteristic, complemented
56#809—71no—
57#807—71no—
58C22×C3×C7C2×D411yescharacteristic
59C2×D21—22no—
60C2×D21—41no—
61C2×D21—41no—
62C3×C4×C7C2×Q811yescharacteristic
63C3×C4×C7#4111yescharacteristic
64Dic21—41no—
65C3×Dic7—41no—
66C3×Dic7—41no—
67C7×Dic3—21no—
68C7×Dic3—21no—
69#478—211noSylow
70C22×D7—61no—
71C2×C4×C7—31no—
72C2×C4×C7—31no—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—21no—
75C2×Dic7—21no—
76C2×Dic7—61no—
77D28—32no—
78D28—61no—
79C4×D7—122no—
80C7×C8—21no—
81C7×C8—61no—
82C2×D12—71no—
83C2×C4×D3—141no—
84C2×C3×C8C2×D711yescharacteristic
85C3×C4⋊C4—71no—
86C3×C4⋊C4—71no—
87(C2×D3)⋊C4—141no—
88C4⋊Dic3—71no—
89C4⋊Dic3—141no—
90Dic3⋊C4—141no—
91C4×Dic3C2×D711yescharacteristic
92C2×C3⋊C8C2×D711yescharacteristic
93C2×C3×C7C22⋊Q811yescharacteristic
94C2×C3×C7#13511yescharacteristic
95C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
96D21—42no—
97#125—211no—
98C4×D4—211no—
99#107—211no—
100#106—71no—
101C4⋊C8—31no—
102#102—211no—
103#102—211no—
104C22×C7D3×D411yescharacteristic
105C2×D7—62no—
106C2×D7—121no—
107C2×D7—121no—
108C4×C7—61no—
109C4×C7—61no—
110C4×C7#23511yescharacteristic
111C4×C7D3×Q811yescharacteristic
112Dic7—41no—
113Dic7—41no—
114Dic7—121no—
115C22×D3—141no—
116C2×C3×C4—141no—
117C2×C3×C4—141no—
118C2×C3×C4C22×D711yescharacteristic
119C2×Dic3—141no—
120C2×Dic3D2811yescharacteristic
121C2×Dic3D2811yescharacteristic
122D12—72no—
123D12—141no—
124C4×D3—282no—
125C3×C8—21no—
126C3⋊C8—21no—
127C3×C7#47811yescharacteristic, complemented, Hall
128C2×D4—211no—
129C22×C4—421no—
130C2×C8—31no—
131C2×C8D3×D711yescharacteristic
132C4⋊C4—71no—
133C4⋊C4—71no—
134C4⋊C4—211no—
135C4⋊C4—421no—
136C4⋊C4—421no—
137C22⋊C4—421no—
138C42—31no—
139C2×C7#89611yescharacteristic
140C2×C7#91411yescharacteristic
141C2×C7#91611yescharacteristic
142D7—122no—
143C22×C3C2×D2811yescharacteristic
144C2×D3—142no—
145C2×D3—281no—
146C2×D3—281no—
147C3×C4—281no—
148C3×C4—281no—
149C3×C4#115911yescharacteristic
150C3×C4Q8×D711yescharacteristic
151Dic3—21no—
152Dic3—21no—
153Dic3—281no—
154C23—421no—
155D4—212no—
156D4—421no—
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—141no—
160C2×C4—141no—
161C2×C4—421no—
162C2×C4—842no—
163C2×C4C2×D3×D711yescharacteristic
164C8—21no—
165C8—61no—
166C7#487111yescharacteristic, complemented, Sylow
167C2×C3#646011yescharacteristic
168C2×C3#646711yescharacteristic
169C2×C3#647111yescharacteristic
170D3—282no—
171C22—422no—
172C22—841no—
173C22—841no—
174C22D3×D2811yescharacteristic
175C4—61no—
176C4—61no—
177C4—281no—
178C4—281no—
179C4—841no—
180C4D21⋊Q811yescharacteristic
181C4#6668611yescharacteristic
182C3#8961911yescharacteristic, complemented, Sylow
183C2—842no—
184C2#1062081511yescharacteristic
185C2#1062083411yescharacteristic
186C2#1062099111yescharacteristic
subgroupG/H
centerC22D3×D28
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×C8C22
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenterC2×C8D3×D7
hypocenterC3×C7#478
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #10620943C2semidirect
4. C7×Dic3⋊C8C2semidirect
5. C3×#6391C2semidirect
6. #10620525C2semidirect
26. #6391D3semidirect
55. Dic3⋊C8D7semidirect
127. C3×C7#478semidirect
166. C7#4871semidirect
182. C3#89619semidirect

Maximal subgroups

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

subgroupindex
#106209902
#106209432
#106206602
C7×Dic3⋊C82
C3×#63912
#106205252
#106205032
#896193
#48717