Finite Group Databasebeta

← #13054783

Subgroups — #13054783

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×#66670C211yesmaximal, characteristic
2#10621429C211yesmaximal, characteristic, complemented
3C2×C7×#216C211yesmaximal, characteristic, complemented
4C2×C3×#1143C211yesmaximal, characteristic, complemented
5#10620508C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89959—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C4×D21C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C2×C3×D28C2211yescharacteristic
13C2×#4246—21no—
14C2×Dic3×D7—21no—
15#66670—22no—
16#66670—22no—
17#66631—21no—
18C4⋊Dic21C2211yescharacteristic
19#66629—21no—
20C7×#216—22no—
21C2×C7×C3⋊C8C2211yescharacteristic
22C3×#1143—22no—
23C2×C3×C7⋊C8C2211yescharacteristic
24C2×#1161—31no—
25C2×C7×Q16—31no—
26#6509—31no—
27C2×#1143D311yescharacteristic, complemented
28#6409—31no—
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#5173—71nomaximal, Hall
32C22×D21D411yescharacteristic
33C22×C3×D7—21no—
34C3×C7×Q8—21no—
35C3×C7×Q8—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21—21no—
39C2×Dic21D411yescharacteristic
40C4×D21—21no—
41C4×D21—21no—
42C2×C7×Dic3—21no—
43C7×Dic6—21no—
44C7×Dic6D412yes—
45C3×D28—21no—
46C3×D28D412yes—
47#4246—24no—
48Dic3×D7—24no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C2×C7×Q8—31no—
52C2×C7×Q8C2×D311yescharacteristic
53#1161—62no—
54#1161—62no—
55C2×D28—61no—
56C2×D28C2×D311yescharacteristic
57C2×C4×D7—31no—
58C2×C4×D7—61no—
59C7×Q16—62no—
60C2×C7×C8—31no—
61#1143—22no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—31no—
64Dic7⋊C4—61no—
65C2×C7⋊C8C2×D311yescharacteristic
66C2×#237—71no—
67C2×C3×SD16—71no—
68#946—71no—
69C2×#216D711yescharacteristic, complemented
70#832—71no—
71(C2×D3)⋊C8—71no—
72#816—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×D21—21no—
75C2×D21—21no—
76C2×D21—21no—
77C2×C3×D7—22no—
78C2×C3×D7—41no—
79C2×C3×D7—41no—
80C3×C4×C7—41no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—21no—
84Dic21—41no—
85C7×Dic3—22no—
86#452—211noSylow
87C22×D7—21no—
88C22×D7—31no—
89C7×Q8—21no—
90C7×Q8—21no—
91C7×Q8—32no—
92C7×Q8—61no—
93C2×C4×C7—21no—
94C2×C4×C7—61no—
95C2×C4×C7C22×D311yescharacteristic
96C2×Dic7—31no—
97C2×Dic7—61no—
98D28—21no—
99D28—64no—
100D28#6712yes—
101C4×D7—64no—
102C4×D7—61no—
103C4×D7—61no—
104C7×C8—61no—
105C7⋊C8—21no—
106C2×C3×Q8C2×D711yescharacteristic
107C2×C3×D4—71no—
108C2×#67—141no—
109C22×Dic3—141no—
110#237—142no—
111#237—142no—
112C2×C4×D3—71no—
113C2×Dic6C2×D711yescharacteristic
114C3×SD16—142no—
115C2×C3×C8—71no—
116#216—22no—
117(C2×D3)⋊C4—141no—
118C4⋊Dic3—71no—
119Dic3⋊C4—141no—
120C2×C3⋊C8C2×D711yescharacteristic
121C2×C3×C7#12011yescharacteristic
122C2×C3×C7#13511yescharacteristic
123C2×C3×C7#13711yescharacteristic
124D21—41no—
125C3×D7—42no—
126C2×#41—211no—
127C2×Q16—31no—
128C2×SD16—71no—
129C22⋊Q8—211no—
130#103—211no—
131#102—211no—
132C22⋊C8—211no—
133C22×C7D3×D411yescharacteristic
134C2×D7—22no—
135C2×D7—41no—
136C2×D7—41no—
137C2×D7—61no—
138C2×D7—61no—
139C2×D7—61no—
140C4×C7—41no—
141C4×C7—62no—
142C4×C7D3×D411yescharacteristic
143C4×C7C2×#6711yescharacteristic
144Dic7—61no—
145Dic7—121no—
146C23×C3—141no—
147C22×D3—71no—
148C3×Q8—21no—
149C3×Q8—21no—
150C3×D4—72no—
151C3×D4—141no—
152C2×C3×C4—21no—
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—21no—
162Dic6C22⋊D712yes—
163C3×C8—141no—
164C3⋊C8—21no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—422no—
167#41—422no—
168C2×Q8—31no—
169C2×Q8D3×D711yescharacteristic
170C2×D4—71no—
171C2×D4—421no—
172C22×C4—211no—
173C22×C4—421no—
174Q16—62no—
175SD16—142no—
176C2×C8—31no—
177C2×C8—71no—
178C4⋊C4—211no—
179C4⋊C4—421no—
180C22⋊C4—421no—
181C2×C7#91511yescharacteristic
182C2×C7#91611yescharacteristic
183C2×C7#93711yescharacteristic
184D7—42no—
185D7—121no—
186C22×C3—142no—
187C22×C3—281no—
188C22×C3—281no—
189C22×C3D4×D711yescharacteristic
190C2×D3—141no—
191C2×D3—141no—
192C2×D3—141no—
193C3×C4—41no—
194C3×C4D4×D711yescharacteristic
195C3×C4C2×C22⋊D711yescharacteristic
196Dic3—22no—
197Dic3—141no—
198Dic3—281no—
199C23—141no—
200C23—211no—
201Q8—21no—
202Q8—21no—
203Q8—32no—
204Q8—61no—
205D4—72no—
206D4—141no—
207D4—424no—
208C2×C4—21no—
209C2×C4—61no—
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—282no—
220C2×C3#648111yescharacteristic
221C2×C3#648211yescharacteristic
222C2×C3#650011yescharacteristic
223D3—281no—
224C22—142no—
225C22—281no—
226C22—281no—
227C22—421no—
228C22—421no—
229C22—421no—
230C22#6669411yescharacteristic
231C4—41no—
232C4—62no—
233C4—421no—
234C4—841no—
235C4D7×#6711yescharacteristic
236C4D3×C22⋊D711yescharacteristic
237C3#8995911yescharacteristic, complemented, Sylow
238C2—282no—
239C2—841no—
240C2#1062108311yescharacteristic
241C2#1062108411yescharacteristic
242C2#1062113911yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×#66670C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#452
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #10621429C2semidirect
3. C2×C7×#216C2semidirect
4. C2×C3×#1143C2semidirect
5. #10620508C2semidirect
6. #10620500C2semidirect
7. #10620492C2semidirect
27. C2×#1143D3semidirect
69. C2×#216D7semidirect
165. C3×C7#452semidirect
218. C7#5173semidirect
237. C3#89959semidirect

Maximal subgroups

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

subgroupindex
C2×#666702
#106214292
C2×C7×#2162
C2×C3×#11432
#106205082
#106205002
#106204922
#899593
#51737