Finite Group Databasebeta

← #13052899

Subgroups — #13052899

Order 1344.

304 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×#66672C211yesmaximal, characteristic
2#10621380C211yesmaximal, characteristic, complemented
3C2×#66570C211yesmaximal, characteristic, complemented
4C2×#66567C211yesmaximal, characteristic, complemented
5C7×#809C211yesmaximal, characteristic, complemented
6C3×#6384C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89555—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13C2×#4247—21no—
14C2×D3×Dic7—21no—
15#66672—22no—
16#66672—22no—
17C3×C7×C4⋊C4C2211yescharacteristic
18#66631—21no—
19C4⋊Dic21—21no—
20C2×C7×C3⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22#66570—22no—
23#66567—22no—
24C2×#1159—31no—
25C2×#1142—31no—
26C2×Dic28—31no—
27#6460—31no—
28C7×#102—31no—
29(C2×D7)⋊C8—31no—
30#6384D311yescharacteristic, complemented
31#4808—71nomaximal, Hall
32C22×D21D411yescharacteristic
33C22×D3×C7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×Dic21D411yescharacteristic
38C4×D21—21no—
39C4×D21—21no—
40Dic42—21no—
41Dic42—21no—
42C7×D12—21no—
43C7×D12D412yes—
44C2×C3×Dic7—21no—
45C3×Dic14—21no—
46C3×Dic14D412yes—
47#4247—24no—
48D3×Dic7—24no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C2×C7×D4—31no—
52C2×C22⋊D7—61no—
53C22×Dic7—61no—
54#1159—62no—
55#1159—62no—
56C2×C4×D7—31no—
57C2×Dic14—31no—
58C2×Dic14C2×D311yescharacteristic
59C2×C7×C8—31no—
60C7×C4⋊C4C2×D311yescharacteristic
61#1142—62no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—61no—
64C2×C7⋊C8C2×D311yescharacteristic
65Dic28—62no—
66C2×#239—71no—
67C2×#216—71no—
68C2×#204—71no—
69#897—71no—
70C3×#103—71no—
71(C2×D3)⋊C8—71no—
72#809D711yescharacteristic, complemented
73C22×C3×C7C2×D411yescharacteristic
74C2×D21—21no—
75C2×D21—21no—
76C2×D21—21no—
77C2×D3×C7—22no—
78C2×D3×C7—41no—
79C2×D3×C7—41no—
80C3×C4×C7—41no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—21no—
84Dic21—41no—
85C3×Dic7—22no—
86#452—211noSylow
87C23×C7—61no—
88C22×D7—31no—
89C7×D4—32no—
90C7×D4—61no—
91C2×C4×C7—21no—
92C2×C4×C7C22×D311yescharacteristic
93C22⋊D7—64no—
94C2×Dic7—21no—
95C2×Dic7—31no—
96C2×Dic7—64no—
97C2×Dic7—61no—
98C4×D7—61no—
99C4×D7—61no—
100Dic14—21no—
101Dic14—61no—
102Dic14—61no—
103Dic14D1212yes—
104C7×C8—61no—
105C7⋊C8—21no—
106C2×C3×Q8—71no—
107#239—142no—
108#239—142no—
109C2×D12—141no—
110C2×D12C2×D711yescharacteristic
111C2×C4×D3—71no—
112C2×C4×D3—141no—
113C2×Dic6—71no—
114C2×C3×C8—71no—
115C3×C4⋊C4C2×D711yescharacteristic
116#216—142no—
117(C2×D3)⋊C4—141no—
118C4⋊Dic3—141no—
119C2×C3⋊C8C2×D711yescharacteristic
120#204—142no—
121C2×C3×C7#12011yescharacteristic
122C2×C3×C7#13511yescharacteristic
123C2×C3×C7#13711yescharacteristic
124D21—41no—
125D3×C7—42no—
126C2×#41—211no—
127C2×Q16—211no—
128C2×SD16—211no—
129C22⋊Q8—211no—
130#103—71no—
131#102—31no—
132C22⋊C8—211no—
133C22×C7—62no—
134C22×C7—121no—
135C22×C7—121no—
136C22×C7D3×D411yescharacteristic
137C2×D7—61no—
138C2×D7—61no—
139C2×D7—61no—
140C4×C7—41no—
141C4×C7C2×D1211yescharacteristic
142C4×C7D3×D411yescharacteristic
143Dic7—22no—
144Dic7—61no—
145Dic7—121no—
146C22×D3—21no—
147C22×D3—71no—
148C3×Q8—72no—
149C3×Q8—141no—
150C2×C3×C4—21no—
151C2×C3×C4—141no—
152C2×C3×C4C22×D711yescharacteristic
153C2×Dic3—71no—
154C2×Dic3—141no—
155D12—21no—
156D12—144no—
157D12D2812yes—
158C4×D3—144no—
159C4×D3—141no—
160C4×D3—141no—
161Dic6—141no—
162Dic6—141no—
163C3×C8—141no—
164C3⋊C8—21no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—422no—
167#41—422no—
168C2×Q8—71no—
169C2×Q8—211no—
170C2×D4—31no—
171C2×D4—421no—
172C22×C4—211no—
173C22×C4—421no—
174Q16—422no—
175SD16—422no—
176C2×C8—31no—
177C2×C8—71no—
178C4⋊C4—421no—
179C4⋊C4D3×D711yescharacteristic
180C22⋊C4—421no—
181C2×C7—122no—
182C2×C7#88211yescharacteristic
183C2×C7#91611yescharacteristic
184C2×C7#91811yescharacteristic
185D7—121no—
186C22×C3D4×D711yescharacteristic
187C2×D3—22no—
188C2×D3—41no—
189C2×D3—41no—
190C2×D3—141no—
191C2×D3—141no—
192C2×D3—141no—
193C3×C4—41no—
194C3×C4—142no—
195C3×C4C2×D2811yescharacteristic
196C3×C4D4×D711yescharacteristic
197Dic3—141no—
198Dic3—281no—
199C23—61no—
200C23—211no—
201Q8—72no—
202Q8—141no—
203Q8—421no—
204Q8—421no—
205D4—32no—
206D4—61no—
207D4—424no—
208C2×C4—21no—
209C2×C4—141no—
210C2×C4—211no—
211C2×C4—424no—
212C2×C4—421no—
213C2×C4—421no—
214C2×C4—421no—
215C2×C4C2×D3×D711yescharacteristic
216C8—61no—
217C8—141no—
218C7#480811yescharacteristic, complemented, Sylow
219C2×C3#644511yescharacteristic
220C2×C3#647511yescharacteristic
221C2×C3#647811yescharacteristic
222D3—42no—
223D3—281no—
224C22—62no—
225C22—121no—
226C22—121no—
227C22—421no—
228C22—421no—
229C22—421no—
230C22#6669411yescharacteristic
231C4—41no—
232C4—142no—
233C4—421no—
234C4—841no—
235C4D7×D1211yescharacteristic
236C4D3×D2811yescharacteristic
237C3#8955511yescharacteristic, complemented, Sylow
238C2—122no—
239C2—841no—
240C2#1062103811yescharacteristic
241C2#1062106111yescharacteristic
242C2#1062108611yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66672C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#452
LCSG ▷ C2×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. #10621380C2semidirect
3. C2×#66570C2semidirect
4. C2×#66567C2semidirect
5. C7×#809C2semidirect
6. C3×#6384C2semidirect
7. #10620492C2semidirect
30. #6384D3semidirect
72. #809D7semidirect
165. C3×C7#452semidirect
218. C7#4808semidirect
237. C3#89555semidirect

Maximal subgroups

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

subgroupindex
C2×#666722
#106213802
C2×#665702
C2×#665672
C7×#8092
C3×#63842
#106204922
#895553
#48087