Finite Group Databasebeta

← #13052819

Subgroups — #13052819

Order 1344.

240 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C3×#6456C211yesmaximal, characteristic
2#10620895C211yesmaximal, characteristic, complemented
3C2×#66565C211yesmaximal, characteristic
4#10620649C211yesmaximal, characteristic, complemented
5C7×#807C211yesmaximal, characteristic, complemented
6#10620499C211yesmaximal, characteristic
7#10620487C211yesmaximal, characteristic, complemented
8#89661—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C3×D28C2211yescharacteristic
11C2×C3×C4×D7—21no—
12C3×C7×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×C4⋊Dic3C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×(C2×D7)⋊C4—21no—
17C3×C4×Dic7C2211yescharacteristic
18#66577—21no—
19Dic3×Dic7—21no—
20#66565C414yescomplemented
21C2×#1132—31no—
22#6456D311yescharacteristic
23#6450—31no—
24C7×#107—31no—
25#6389—31no—
26#6384—31no—
27#6383—31no—
28#5018—71nomaximal, Hall
29C22×C3×D7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33Dic42—21no—
34Dic42C2×C412yescomplemented
35C2×C7×Dic3—21no—
36C2×C3×Dic7D411yescharacteristic
37C2×C3×Dic7D411yescharacteristic
38C3×D28—21no—
39C3×D28C2×C412yes—
40C3×C4×D7—42no—
41C3⋊(C7⋊C8)—21no—
42C7×C3⋊C8C2×C412yescomplemented
43C2×D28C2×D311yescharacteristic
44C2×C4×D7—21no—
45C2×Dic14—31no—
46C2×C7×C8—31no—
47C7×C4⋊C4—31no—
48C7×C4⋊C4C2×D311yescharacteristic
49(C2×D7)⋊C4—21no—
50Dic7⋊C4—61no—
51C4×Dic7—61no—
52C4×Dic7C2×D311yescharacteristic
53C2×C7⋊C8—31no—
54#1132—34no—
55C3×C4×D4—71no—
56C2×#214—71no—
57C4×Dic6—71no—
58#832—71no—
59#810—71no—
60#807D711yescharacteristic, complemented
61#803—71no—
62C22×C3×C7C2×D411yescharacteristic
63C2×C3×D7—22no—
64C2×C3×D7—41no—
65C2×C3×D7—41no—
66C3×C4×C7—41no—
67C3×C4×C7C22×C411yescharacteristic
68C3×C4×C7#4111yescharacteristic
69Dic21—22no—
70C3×Dic7—21no—
71C3×Dic7—21no—
72C7×Dic3—41no—
73#440—211noSylow
74C22×D7—21no—
75C2×C4×C7—21no—
76C2×C4×C7—61no—
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—61no—
79C2×Dic7#6711yescharacteristic
80C2×Dic7#6711yescharacteristic
81D28—21no—
82D28C4×D312yes—
83C4×D7—42no—
84Dic14—32no—
85Dic14—61no—
86C7×C8—32no—
87C7⋊C8—61no—
88C2×C3×D4—71no—
89C22×C3×C4—141no—
90C2×Dic6—71no—
91C3×C4⋊C4C2×D711yescharacteristic
92C3×C22⋊C4—141no—
93C3×C42—71no—
94#214—74no—
95C4⋊Dic3C2×D711yescharacteristic
96Dic3⋊C4—141no—
97C4×Dic3—141no—
98C2×C3⋊C8—71no—
99C2×C3⋊C8C2×D711yescharacteristic
100C2×C3×C7C4×D411yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C2×C3×C7#13711yescharacteristic
103C3×D7—42no—
104C2×SD16—211no—
105C4×Q8—211no—
106C4×D4—71no—
107#107—31no—
108#103—211no—
109#102—211no—
110#97—211no—
111C22×C7C2×#6711yescharacteristic
112C2×D7—22no—
113C2×D7—41no—
114C2×D7—41no—
115C4×C7—41no—
116C4×C7—121no—
117C4×C7C2×C4×D311yescharacteristic
118C4×C7#23511yescharacteristic
119Dic7—21no—
120Dic7—21no—
121Dic7—62no—
122C23×C3—141no—
123C3×D4—72no—
124C3×D4—141no—
125C2×C3×C4—21no—
126C2×C3×C4—71no—
127C2×C3×C4—71no—
128C2×C3×C4—282no—
129C2×C3×C4C22×D711yescharacteristic
130C2×Dic3—21no—
131C2×Dic3—141no—
132Dic6—72no—
133Dic6—141no—
134C3⋊C8—141no—
135C3⋊C8C4×D712yescomplemented
136C3×C7#44011yescharacteristic, complemented, Hall
137C2×Q8—211no—
138C2×D4—71no—
139C22×C4—141no—
140SD16—214no—
141C2×C8—31no—
142C2×C8—211no—
143C4⋊C4—31no—
144C4⋊C4—421no—
145C4⋊C4D3×D711yescharacteristic
146C22⋊C4—141no—
147C42—71no—
148C42—421no—
149C2×C7C4×#6711yescharacteristic
150C2×C7#93211yescharacteristic
151C2×C7#95111yescharacteristic
152D7—42no—
153C22×C3—142no—
154C22×C3—281no—
155C22×C3—281no—
156C22×C3D4×D711yescharacteristic
157C3×C4—41no—
158C3×C4—141no—
159C3×C4—141no—
160C3×C4C2×C4×D711yescharacteristic
161C3×C4#116111yescharacteristic
162Dic3—41no—
163Dic3—142no—
164C23—141no—
165Q8—212no—
166Q8—421no—
167D4—72no—
168D4—141no—
169C2×C4—21no—
170C2×C4—61no—
171C2×C4—71no—
172C2×C4—71no—
173C2×C4—282no—
174C2×C4—421no—
175C2×C4C2×D3×D711yescharacteristic
176C8—32no—
177C8—421no—
178C7#501811yescharacteristic, complemented, Sylow
179C2×C3—282no—
180C2×C3#645611yescharacteristic
181C2×C3#647411yescharacteristic
182C2×C3#648111yescharacteristic
183C22—142no—
184C22—281no—
185C22—281no—
186C22D7×#6711yescharacteristic
187C4—41no—
188C4—121no—
189C4—141no—
190C4—141no—
191C4—422no—
192C4#6667711yescharacteristic
193C4C4×D3×D711yescharacteristic
194C3#8966111yescharacteristic, complemented, Sylow
195C2—282no—
196C2#1062100611yescharacteristic
197C2#1062105111yescharacteristic
198C2#1062107011yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×D28C22
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#440
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #10620895C2semidirect
4. #10620649C2semidirect
5. C7×#807C2semidirect
7. #10620487C2semidirect
20. #66565C4semidirect
34. Dic42C2×C4semidirect
42. C7×C3⋊C8C2×C4semidirect
60. #807D7semidirect
135. C3⋊C8C4×D7semidirect
136. C3×C7#440semidirect
178. C7#5018semidirect
194. C3#89661semidirect

Maximal subgroups

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

subgroupindex
C3×#64562
#106208952
C2×#665652
#106206492
C7×#8072
#106204992
#106204872
#896613
#50187