Finite Group Databasebeta

← #13054423

Subgroups — #13054423

Order 1344.

304 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 68 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×#66730C211yesmaximal, characteristic
2C2×C3×#1142C211yesmaximal, characteristic, complemented
3#10620952C211yesmaximal, characteristic, complemented
4C2×#66570C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620510C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89555—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C22×Dic21—21no—
12#66730—22no—
13#66730—22no—
14C2×C4×D21C2211yescharacteristic
15C2×Dic42C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C7×C4⋊Dic3C2211yescharacteristic
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×#1142—22no—
20C2×C3×C7⋊C8C2211yescharacteristic
21Dic3⋊Dic7—21no—
22#66575—21no—
23#66570—22no—
24C2×#1159—31no—
25C2×#1142D311yescharacteristic, complemented
26C2×Dic28—31no—
27#6460—31no—
28C7×#102—31no—
29(C2×D7)⋊C8—31no—
30#6384—31no—
31#5173—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21D411yescharacteristic
34C3×C7×D4—21no—
35C3×C7×D4D412yes—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—24no—
38C2×Dic21—24no—
39C2×Dic21—21no—
40C2×Dic21D411yescharacteristic
41C4×D21—21no—
42C4×D21—21no—
43Dic42—21no—
44Dic42D412yes—
45C2×C7×Dic3—21no—
46C2×C3×Dic7—21no—
47C3×Dic14—21no—
48C3×Dic14—21no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C2×C7×D4C2×D311yescharacteristic
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⋊C4—31no—
61#1142—22no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—61no—
64C2×C7⋊C8C2×D311yescharacteristic
65Dic28—62no—
66C2×#237—71no—
67C2×C3×SD16—71no—
68#946—71no—
69C2×#216—71no—
70#832D711yescharacteristic, complemented
71(C2×D3)⋊C8—71no—
72#816—71no—
73C22×C3×C7—22no—
74C22×C3×C7—41no—
75C22×C3×C7—41no—
76C22×C3×C7C2×D411yescharacteristic
77C2×D21—21no—
78C2×D21—21no—
79C2×D21—21no—
80C3×C4×C7C2×D411yescharacteristic
81C3×C4×C7C2×D411yescharacteristic
82Dic21—22no—
83Dic21—21no—
84C3×Dic7—41no—
85C7×Dic3—41no—
86#452—211noSylow
87C23×C7—21no—
88C22×D7—31no—
89C7×D4—21no—
90C7×D4#6712yes—
91C2×C4×C7—61no—
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—21no—
102Dic14—32no—
103Dic14—61no—
104C7×C8—61no—
105C7⋊C8—21no—
106C2×C3×Q8—71no—
107C2×C3×D4C2×D711yescharacteristic
108C2×#67—141no—
109C22×Dic3—141no—
110#237—142no—
111#237—142no—
112C2×C4×D3—71no—
113C2×Dic6—71no—
114C3×SD16—142no—
115C2×C3×C8—71no—
116#216—142no—
117(C2×D3)⋊C4—141no—
118C4⋊Dic3C2×D711yescharacteristic
119Dic3⋊C4—141no—
120C2×C3⋊C8C2×D711yescharacteristic
121C2×C3×C7—42no—
122C2×C3×C7#12011yescharacteristic
123C2×C3×C7#13511yescharacteristic
124C2×C3×C7#13711yescharacteristic
125D21—41no—
126C2×#41—211no—
127C2×Q16—211no—
128C2×SD16—71no—
129C22⋊Q8—211no—
130#103—211no—
131#102—31no—
132C22⋊C8—211no—
133C22×C7—22no—
134C22×C7—41no—
135C22×C7—41no—
136C22×C7D3×D411yescharacteristic
137C2×D7—61no—
138C2×D7—61no—
139C2×D7—61no—
140C4×C7—121no—
141C4×C7D3×D411yescharacteristic
142C4×C7C2×#6711yescharacteristic
143Dic7—41no—
144Dic7—62no—
145Dic7—61no—
146C23×C3—21no—
147C22×D3—71no—
148C3×Q8—141no—
149C3×Q8—141no—
150C3×D4—21no—
151C3×D4D2812yes—
152C2×C3×C4—141no—
153C2×C3×C4C22×D711yescharacteristic
154#67—144no—
155C2×Dic3—21no—
156C2×Dic3—71no—
157C2×Dic3—144no—
158C2×Dic3—141no—
159C4×D3—141no—
160C4×D3—141no—
161Dic6—72no—
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—421no—
171C2×D4D3×D711yescharacteristic
172C22×C4—211no—
173C22×C4—421no—
174Q16—422no—
175SD16—142no—
176C2×C8—31no—
177C2×C8—71no—
178C4⋊C4—31no—
179C4⋊C4—421no—
180C22⋊C4—421no—
181C2×C7—42no—
182C2×C7#91511yescharacteristic
183C2×C7#91611yescharacteristic
184C2×C7#93711yescharacteristic
185D7—121no—
186C22×C3—22no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3D4×D711yescharacteristic
190C2×D3—141no—
191C2×D3—141no—
192C2×D3—141no—
193C3×C4—281no—
194C3×C4C2×D2811yescharacteristic
195C3×C4D4×D711yescharacteristic
196Dic3—41no—
197Dic3—142no—
198Dic3—141no—
199C23—21no—
200C23—211no—
201Q8—141no—
202Q8—141no—
203Q8—212no—
204Q8—421no—
205D4—21no—
206D4—424no—
207D4#424612yes—
208C2×C4—61no—
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#517311yescharacteristic, complemented, Sylow
219C2×C3—42no—
220C2×C3#644511yescharacteristic
221C2×C3#647511yescharacteristic
222C2×C3#647811yescharacteristic
223D3—281no—
224C22—22no—
225C22—41no—
226C22—41no—
227C22—421no—
228C22—421no—
229C22—421no—
230C22#6669411yescharacteristic
231C4—121no—
232C4—281no—
233C4—421no—
234C4—422no—
235C4C2×#424611yescharacteristic
236C4#6669411yescharacteristic
237C3#8955511yescharacteristic, complemented, Sylow
238C2—42no—
239C2—841no—
240C2#1062105011yescharacteristic
241C2#1062106111yescharacteristic
242C2#1062113411yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66730C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#452
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#1142C2semidirect
3. #10620952C2semidirect
4. C2×#66570C2semidirect
5. C7×#832C2semidirect
6. #10620510C2semidirect
7. #10620492C2semidirect
25. C2×#1142D3semidirect
70. #832D7semidirect
165. C3×C7#452semidirect
218. C7#5173semidirect
237. C3#89555semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C2×C3×#11422
#106209522
C2×#665702
C7×#8322
#106205102
#106204922
#895553
#51737