Finite Group Databasebeta

← #13054331

Subgroups — #13054331

Order 1344.

404 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 88 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×D4×D21C211yesmaximal, characteristic
2#10620999C211yesmaximal, characteristic, complemented
3C2×D21⋊C8C211yesmaximal, characteristic, complemented
4C7×#832C211yesmaximal, characteristic, complemented
5C3×#6406C211yesmaximal, characteristic, complemented
6#10620504C211yesmaximal, characteristic, complemented
7#10620503C211yesmaximal, characteristic, complemented
8D7×#102—31nomaximal, Hall
9C23×D21—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C2×#4268—21no—
12D4×D21—22no—
13D4×D21C414yescomplemented
14C2×D84C2211yescharacteristic
15C2×C4×D21C2211yescharacteristic
16C2×D21⋊C4—21no—
17C7×C4⋊Dic3C2211yescharacteristic, complemented
18C2×C7×C3⋊C8C2211yescharacteristic, complemented
19C3×C4⋊Dic7C2211yescharacteristic, complemented
20C2×C3×C7⋊C8C2211yescharacteristic, complemented
21#66577—21no—
22D21⋊C8—22no—
23C2×D4×D7—31no—
24C2×C8×D7—31no—
25D7×C4⋊C4—31no—
26C7×#102—31no—
27#6406D311yescharacteristic, complemented
28#6395—31no—
29#6383—31no—
30D3×#102—71nomaximal, Hall
31C23×C3×C7—21no—
32C22×D21—21no—
33C22×D21—24no—
34C22×D21—42no—
35C22×D21—42no—
36C22×D21D411yescharacteristic
37C3×C7×D4—21no—
38C3×C7×D4C2×C412yescomplemented
39C2×C3×C4×C7C2311yescharacteristic
40#4268—24no—
41C2×Dic21D411yescharacteristic
42D84—21no—
43D84C2×C412yescomplemented
44C4×D21C2×C412yes—
45C4×D21D412yes—
46C2×C7×Dic3—21no—
47C2×C3×Dic7—21no—
48D21⋊C4—42no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C23×D7—61no—
52C2×C7×D4C2×D311yescharacteristic
53C2×C22⋊D7—61no—
54D4×D7—34no—
55D4×D7—62no—
56C2×D28—31no—
57C2×C4×D7—31no—
58C2×C4×D7—61no—
59C2×C7×C8—31no—
60C7×C4⋊C4—31no—
61C4⋊Dic7C2×D311yescharacteristic, complemented
62Dic7⋊C4—61no—
63C2×C7⋊C8C2×D311yescharacteristic, complemented
64C8×D7—62no—
65C2×D3×D4—71no—
66C2×D3×C8—71no—
67D3×C4⋊C4—71no—
68C3×#102—71no—
69#832D711yescharacteristic, complemented
70#821—71no—
71#809—71no—
72C22×C3×C7—22no—
73C22×C3×C7—41no—
74C22×C3×C7—41no—
75C22×C3×C7C2×D411yescharacteristic
76C2×D21—22no—
77C2×D21—41no—
78C2×D21—41no—
79C2×D21—48no—
80C2×D21C22⋊C412yes—
81C2×D21D812yes—
82C2×D21SD1612yes—
83C3×C4×C7C22×C411yescharacteristic
84C3×C4×C7C2×D411yescharacteristic
85Dic21C22⋊C412yes—
86C3×Dic7—41no—
87C7×Dic3—41no—
88C2×#102—211noSylow
89C23×C7—21no—
90C22×D7—31no—
91C22×D7—64no—
92C22×D7—61no—
93C22×D7—122no—
94C22×D7—122no—
95C7×D4—21no—
96C7×D4C4×D312yescomplemented
97C2×C4×C7—61no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—64no—
100C2×Dic7—21no—
101C2×Dic7—31no—
102D28—32no—
103D28—61no—
104C4×D7—32no—
105C4×D7—32no—
106C4×D7—122no—
107C7×C8—61no—
108C7⋊C8—21no—
109C23×D3—141no—
110C2×C3×D4C2×D711yescharacteristic
111C2×#67—141no—
112D3×D4—74no—
113D3×D4—142no—
114C2×D12—71no—
115C2×C4×D3—71no—
116C2×C4×D3—141no—
117C2×C3×C8—71no—
118C3×C4⋊C4—71no—
119C4⋊Dic3C2×D711yescharacteristic, complemented
120Dic3⋊C4—141no—
121C2×C3⋊C8C2×D711yescharacteristic, complemented
122D3×C8—142no—
123C2×C3×C7—42no—
124C2×C3×C7C2×C22⋊C411yescharacteristic
125C2×C3×C7C2×D811yescharacteristic
126C2×C3×C7C2×SD1611yescharacteristic
127D21—42no—
128D21#10214yescomplemented
129C22×D4—211no—
130C22×C8—211no—
131C2×C4⋊C4—211no—
132#102—31no—
133#102—71no—
134#102—211no—
135#102—211no—
136C22×C7—22no—
137C22×C7—41no—
138C22×C7—41no—
139C22×C7D3×D411yescharacteristic
140C2×D7—32no—
141C2×D7—32no—
142C2×D7—32no—
143C2×D7—62no—
144C2×D7—128no—
145C2×D7—121no—
146C2×D7—121no—
147C4×C7—121no—
148C4×C7C2×C4×D311yescharacteristic
149C4×C7D3×D411yescharacteristic
150Dic7—32no—
151Dic7—41no—
152C23×C3—21no—
153C22×D3—71no—
154C22×D3—144no—
155C22×D3—141no—
156C22×D3—282no—
157C22×D3—282no—
158C3×D4—21no—
159C3×D4C4×D712yescomplemented
160C2×C3×C4—141no—
161C2×C3×C4C22×D711yescharacteristic
162#67—144no—
163C2×Dic3—21no—
164C2×Dic3—71no—
165D12—72no—
166D12—141no—
167C4×D3—72no—
168C4×D3—72no—
169C4×D3—282no—
170C3×C8—141no—
171C3⋊C8—21no—
172C3×C7C2×#10211yescharacteristic, complemented, Hall
173C24—421no—
174C2×D4—214no—
175C2×D4—211no—
176C2×D4—422no—
177C2×D4—421no—
178C2×D4D3×D711yescharacteristic
179C22×C4—211no—
180C22×C4—421no—
181C2×C8—31no—
182C2×C8—71no—
183C2×C8—422no—
184C4⋊C4—31no—
185C4⋊C4—71no—
186C4⋊C4—421no—
187C2×C7—42no—
188C2×C7D3×C22⋊C411yescharacteristic
189C2×C7D3×D811yescharacteristic
190C2×C7D3×SD1611yescharacteristic
191D7—34no—
192D7—122no—
193C22×C3—22no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3D4×D711yescharacteristic
197C2×D3—72no—
198C2×D3—72no—
199C2×D3—72no—
200C2×D3—142no—
201C2×D3—288no—
202C2×D3—281no—
203C2×D3—281no—
204C3×C4—281no—
205C3×C4C2×C4×D711yescharacteristic
206C3×C4D4×D711yescharacteristic
207Dic3—41no—
208Dic3—72no—
209C23—21no—
210C23—211no—
211C23—424no—
212C23—421no—
213C23—842no—
214C23—842no—
215D4—21no—
216D4—212no—
217D4—424no—
218D4—421no—
219D4D21⋊C412yescomplemented
220C2×C4—61no—
221C2×C4—141no—
222C2×C4—211no—
223C2×C4—212no—
224C2×C4—212no—
225C2×C4—842no—
226C2×C4C2×D3×D711yescharacteristic
227C8—61no—
228C8—141no—
229C7D3×#10211yescharacteristic, complemented, Sylow
230C2×C3—42no—
231C2×C3D7×C22⋊C411yescharacteristic
232C2×C3D7×D811yescharacteristic
233C2×C3D7×SD1611yescharacteristic
234D3—74no—
235D3—282no—
236C22—22no—
237C22—41no—
238C22—41no—
239C22—212no—
240C22—212no—
241C22—212no—
242C22—422no—
243C22—848no—
244C22—841no—
245C22—841no—
246C22#6669411yescharacteristic
247C4—121no—
248C4—212no—
249C4—281no—
250C4C2×D21⋊C411yescharacteristic
251C4#6669411yescharacteristic
252C3D7×#10211yescharacteristic, complemented, Sylow
253C2—42no—
254C2—214no—
255C2—842no—
256C2#1062104311yescharacteristic
257C2#1062104911yescharacteristic
258C2#1062112311yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×D4×D21C2
cosocleC2×C3×C4×C7C23
hypercenterC2×D4D3×D7
hypocenterC3×C7C2×#102
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. #10620999C2semidirect
3. C2×D21⋊C8C2semidirect
4. C7×#832C2semidirect
5. C3×#6406C2semidirect
6. #10620504C2semidirect
7. #10620503C2semidirect
13. D4×D21C4semidirect
17. C7×C4⋊Dic3C22semidirect
18. C2×C7×C3⋊C8C22semidirect
19. C3×C4⋊Dic7C22semidirect
20. C2×C3×C7⋊C8C22semidirect
27. #6406D3semidirect
38. C3×C7×D4C2×C4semidirect
43. D84C2×C4semidirect
61. C4⋊Dic7C2×D3semidirect
63. C2×C7⋊C8C2×D3semidirect
69. #832D7semidirect
96. C7×D4C4×D3semidirect
119. C4⋊Dic3C2×D7semidirect
121. C2×C3⋊C8C2×D7semidirect
128. D21#102semidirect
159. C3×D4C4×D7semidirect
172. C3×C7C2×#102semidirect
219. D4D21⋊C4semidirect
229. C7D3×#102semidirect
252. C3D7×#102semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D212
#106209992
C2×D21⋊C82
C7×#8322
C3×#64062
#106205042
#106205032
D7×#1023
D3×#1027