Finite Group Databasebeta

← #13054343

Subgroups — #13054343

Order 1344.

316 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 80 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
2#10620909C211yesmaximal, characteristic, complemented
3C2×D21⋊C8C211yesmaximal, characteristic, complemented
4C7×#832C211yesmaximal, characteristic, complemented
5C3×#6406C211yesmaximal, characteristic, complemented
6#10620510C211yesmaximal, characteristic, complemented
7#10620509C211yesmaximal, characteristic, complemented
8#89544—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C22×Dic21—21no—
12#66730—22no—
13#66730C414yescomplemented
14C2×C4×D21C2211yescharacteristic
15C2×Dic42C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C4⋊Dic7C2211yescharacteristic
19C2×C3×C7⋊C8C2211yescharacteristic
20#66575—21no—
21Dic3×Dic7—21no—
22D21⋊C8—22no—
23C2×#1159—31no—
24C2×C8×D7—31no—
25#6455—31no—
26C7×#102—31no—
27#6406D311yescharacteristic, complemented
28#6390—31no—
29#6384—31no—
30#4772—71nomaximal, Hall
31C23×C3×C7—21no—
32C22×D21D411yescharacteristic
33C3×C7×D4—21no—
34C3×C7×D4C2×C412yes—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—24no—
37C2×Dic21—21no—
38C2×Dic21—24no—
39C2×Dic21D411yescharacteristic
40C4×D21C2×C412yes—
41C4×D21D412yes—
42Dic42—21no—
43Dic42C2×C412yes—
44C2×C7×Dic3—21no—
45C2×C3×Dic7—21no—
46C3×C7⋊C8—21no—
47C7×C3⋊C8—21no—
48C2×C7×D4C2×D311yescharacteristic
49C2×C22⋊D7—61no—
50C22×Dic7—61no—
51#1159—34no—
52#1159—62no—
53C2×C4×D7—31no—
54C2×Dic14—31no—
55C2×C7×C8—31no—
56C7×C4⋊C4—31no—
57(C2×D7)⋊C4—61no—
58C4⋊Dic7C2×D311yescharacteristic
59C4×Dic7—61no—
60C2×C7⋊C8C2×D311yescharacteristic
61C8×D7—62no—
62C2×#237—71no—
63C2×D3×C8—71no—
64#892—71no—
65C3×#102—71no—
66#832D711yescharacteristic, complemented
67#816—71no—
68#810—71no—
69C22×C3×C7—22no—
70C22×C3×C7—41no—
71C22×C3×C7—41no—
72C22×C3×C7C2×D411yescharacteristic
73C2×D21—21no—
74C2×D21—21no—
75C2×D21C22⋊C412yes—
76C3×C4×C7C22×C411yescharacteristic
77C3×C4×C7C2×D411yescharacteristic
78Dic21—22no—
79Dic21C22⋊C412yes—
80C3×Dic7—41no—
81C7×Dic3—41no—
82#416—211noSylow
83C23×C7—21no—
84C22×D7—31no—
85C7×D4—21no—
86C7×D4C4×D312yes—
87C2×C4×C7—61no—
88C2×C4×C7C22×D311yescharacteristic
89C22⋊D7—64no—
90C2×Dic7—21no—
91C2×Dic7—31no—
92C2×Dic7—64no—
93C2×Dic7—61no—
94C4×D7—32no—
95C4×D7—32no—
96Dic14—32no—
97Dic14—61no—
98C7×C8—61no—
99C7⋊C8—21no—
100C2×C3×D4C2×D711yescharacteristic
101C2×#67—141no—
102C22×Dic3—141no—
103#237—74no—
104#237—142no—
105C2×C4×D3—71no—
106C2×Dic6—71no—
107C2×C3×C8—71no—
108C3×C4⋊C4—71no—
109(C2×D3)⋊C4—141no—
110C4⋊Dic3C2×D711yescharacteristic
111C4×Dic3—141no—
112C2×C3⋊C8C2×D711yescharacteristic
113D3×C8—142no—
114C2×C3×C7—42no—
115C2×C3×C7C2×C22⋊C411yescharacteristic
116C2×C3×C7#13511yescharacteristic
117C2×C3×C7#13511yescharacteristic
118D21—22no—
119C2×#41—211no—
120C22×C8—211no—
121#117—211no—
122#103—211no—
123#103—211no—
124#102—31no—
125#102—71no—
126C22×C7—22no—
127C22×C7—41no—
128C22×C7—41no—
129C22×C7D3×D411yescharacteristic
130C2×D7—32no—
131C2×D7—61no—
132C2×D7—61no—
133C4×C7—121no—
134C4×C7C2×C4×D311yescharacteristic
135C4×C7D3×D411yescharacteristic
136Dic7—32no—
137Dic7—41no—
138Dic7—62no—
139C23×C3—21no—
140C22×D3—71no—
141C3×D4—21no—
142C3×D4C4×D712yes—
143C2×C3×C4—141no—
144C2×C3×C4C22×D711yescharacteristic
145#67—144no—
146C2×Dic3—21no—
147C2×Dic3—71no—
148C2×Dic3—144no—
149C2×Dic3—141no—
150C4×D3—72no—
151C4×D3—72no—
152Dic6—72no—
153Dic6—141no—
154C3×C8—141no—
155C3⋊C8—21no—
156C3×C7#41611yescharacteristic, complemented, Hall
157#41—214no—
158#41—422no—
159C2×Q8—211no—
160C2×D4—421no—
161C2×D4D3×D711yescharacteristic
162C22×C4—211no—
163C22×C4—421no—
164C2×C8—31no—
165C2×C8—71no—
166C2×C8—422no—
167C4⋊C4—31no—
168C4⋊C4—71no—
169C22⋊C4—421no—
170C42—421no—
171C2×C7—42no—
172C2×C7D3×C22⋊C411yescharacteristic
173C2×C7#91211yescharacteristic
174C2×C7#91611yescharacteristic
175D7—62no—
176C22×C3—22no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3D4×D711yescharacteristic
180C2×D3—72no—
181C2×D3—141no—
182C2×D3—141no—
183C3×C4—281no—
184C3×C4C2×C4×D711yescharacteristic
185C3×C4D4×D711yescharacteristic
186Dic3—41no—
187Dic3—72no—
188Dic3—142no—
189C23—21no—
190C23—211no—
191Q8—212no—
192Q8—421no—
193D4—21no—
194D4—424no—
195D4D21⋊C412yes—
196C2×C4—61no—
197C2×C4—141no—
198C2×C4—211no—
199C2×C4—212no—
200C2×C4—212no—
201C2×C4—424no—
202C2×C4—421no—
203C2×C4C2×D3×D711yescharacteristic
204C8—61no—
205C8—141no—
206C7#477211yescharacteristic, complemented, Sylow
207C2×C3—42no—
208C2×C3D7×C22⋊C411yescharacteristic
209C2×C3#647511yescharacteristic
210C2×C3#647911yescharacteristic
211D3—142no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22—212no—
216C22—421no—
217C22—421no—
218C22#6669411yescharacteristic
219C4—121no—
220C4—212no—
221C4—281no—
222C4—422no—
223C4C2×D21⋊C411yescharacteristic
224C4#6669411yescharacteristic
225C3#8954411yescharacteristic, complemented, Sylow
226C2—42no—
227C2—422no—
228C2#1062105911yescharacteristic
229C2#1062106111yescharacteristic
230C2#1062112311yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66730C2
cosocleC2×C3×C4×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7#416
LCSG ▷ 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. #10620909C2semidirect
3. C2×D21⋊C8C2semidirect
4. C7×#832C2semidirect
5. C3×#6406C2semidirect
6. #10620510C2semidirect
7. #10620509C2semidirect
13. #66730C4semidirect
27. #6406D3semidirect
66. #832D7semidirect
156. C3×C7#416semidirect
206. C7#4772semidirect
225. C3#89544semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
#106209092
C2×D21⋊C82
C7×#8322
C3×#64062
#106205102
#106205092
#895443
#47727