Finite Group Databasebeta

← #13052857

Subgroups — #13052857

Order 1344.

404 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 86 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×#66681C211yesmaximal, characteristic
2C4⋊C4×D21C211yesmaximal, characteristic, complemented
3C2×D21⋊C8C211yesmaximal, characteristic, complemented
4C7×#809C211yesmaximal, characteristic, complemented
5C3×#6383C211yesmaximal, characteristic, complemented
6#10620502C211yesmaximal, characteristic, complemented
7#10620500C211yesmaximal, characteristic, complemented
8D7×#102—31nomaximal, Hall
9C22×D3×D7—21no—
10C2×C4×D21—21no—
11C2×C4×D21C2211yescharacteristic
12C2×C7×D12C2211yescharacteristic
13C2×C3×D28C2211yescharacteristic
14C2×#4245—21no—
15#66681—22no—
16#66681C414yescomplemented
17C3×C7×C4⋊C4C2211yescharacteristic, complemented
18C4⋊Dic21C2211yescharacteristic, complemented
19#66629—21no—
20C2×C7×C3⋊C8C2211yescharacteristic, complemented
21C2×C3×C7⋊C8C2211yescharacteristic, complemented
22D21⋊C8—22no—
23C2×D4×D7—31no—
24C2×C8×D7—31no—
25D7×C4⋊C4—31no—
26C7×#102—31no—
27#6406—31no—
28#6395—31no—
29#6383D311yescharacteristic, complemented
30D3×#102—71nomaximal, Hall
31C22×D21D411yescharacteristic
32C22×D3×C7—21no—
33C22×C3×D7—21no—
34C2×D3×D7—24no—
35C2×D3×D7—42no—
36C2×D3×D7—42no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39C2×Dic21—21no—
40C2×Dic21D411yescharacteristic
41C4×D21—42no—
42C4×D21C2×C412yes—
43C4×D21D412yes—
44C7×D12—21no—
45C7×D12C2×C412yescomplemented
46C3×D28—21no—
47C3×D28C2×C412yescomplemented
48#4245—24no—
49C3×C7⋊C8—21no—
50C7×C3⋊C8—21no—
51C23×D7—61no—
52C2×C7×D4—31no—
53C2×C22⋊D7—61no—
54D4×D7—34no—
55D4×D7—62no—
56C2×D28C2×D311yescharacteristic
57C2×C4×D7—31no—
58C2×C4×D7—61no—
59C2×C7×C8—31no—
60C7×C4⋊C4C2×D311yescharacteristic, complemented
61C4⋊Dic7—31no—
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#832—71no—
70#821—71no—
71#809D711yescharacteristic, complemented
72C22×C3×C7C2×D411yescharacteristic
73C2×D21C22⋊C412yes—
74C2×D21D812yes—
75C2×D21SD1612yes—
76C2×D3×C7—22no—
77C2×D3×C7—41no—
78C2×D3×C7—41no—
79C2×C3×D7—22no—
80C2×C3×D7—41no—
81C2×C3×D7—41no—
82D3×D7—48no—
83C3×C4×C7—41no—
84C3×C4×C7C22×C411yescharacteristic
85C3×C4×C7C2×D411yescharacteristic
86Dic21—41no—
87Dic21C22⋊C412yes—
88C2×#102—211noSylow
89C23×C7—61no—
90C22×D7—21no—
91C22×D7—31no—
92C22×D7—64no—
93C22×D7—122no—
94C22×D7—122no—
95C7×D4—32no—
96C7×D4—61no—
97C2×C4×C7—21no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—64no—
100C2×Dic7—31no—
101C2×Dic7—61no—
102D28—21no—
103D28C4×D312yescomplemented
104C4×D7—32no—
105C4×D7—32no—
106C4×D7—122no—
107C7×C8—61no—
108C7⋊C8—21no—
109C23×D3—141no—
110C2×C3×D4—71no—
111C2×#67—141no—
112D3×D4—74no—
113D3×D4—142no—
114C2×D12C2×D711yescharacteristic
115C2×C4×D3—71no—
116C2×C4×D3—141no—
117C2×C3×C8—71no—
118C3×C4⋊C4C2×D711yescharacteristic, complemented
119C4⋊Dic3—71no—
120Dic3⋊C4—141no—
121C2×C3⋊C8C2×D711yescharacteristic, complemented
122D3×C8—142no—
123C2×C3×C7C2×C22⋊C411yescharacteristic
124C2×C3×C7C2×D811yescharacteristic
125C2×C3×C7C2×SD1611yescharacteristic
126D21#10214yescomplemented
127C3×D7—42no—
128D3×C7—42no—
129C22×D4—211no—
130C22×C8—211no—
131C2×C4⋊C4—211no—
132#102—31no—
133#102—71no—
134#102—211no—
135#102—211no—
136C22×C7—62no—
137C22×C7—121no—
138C22×C7—121no—
139C22×C7D3×D411yescharacteristic
140C2×D7—22no—
141C2×D7—32no—
142C2×D7—32no—
143C2×D7—32no—
144C2×D7—41no—
145C2×D7—41no—
146C2×D7—128no—
147C4×C7—41no—
148C4×C7C2×C4×D311yescharacteristic
149C4×C7D3×D411yescharacteristic
150Dic7—32no—
151Dic7—121no—
152C23×C3—141no—
153C22×D3—21no—
154C22×D3—71no—
155C22×D3—144no—
156C22×D3—282no—
157C22×D3—282no—
158C3×D4—72no—
159C3×D4—141no—
160C2×C3×C4—21no—
161C2×C3×C4C22×D711yescharacteristic
162#67—144no—
163C2×Dic3—71no—
164C2×Dic3—141no—
165D12—21no—
166D12C4×D712yescomplemented
167C4×D3—72no—
168C4×D3—72no—
169C4×D3—282no—
170C3×C8—141no—
171C3⋊C8—21no—
172C3×C7C2×#10211yescharacteristic, complemented, Hall
173C24—421no—
174C2×D4—31no—
175C2×D4—71no—
176C2×D4—214no—
177C2×D4—422no—
178C2×D4—421no—
179C22×C4—211no—
180C22×C4—421no—
181C2×C8—31no—
182C2×C8—71no—
183C2×C8—422no—
184C4⋊C4—211no—
185C4⋊C4—421no—
186C4⋊C4D3×D711yescharacteristic, complemented
187C2×C7—122no—
188C2×C7D3×C22⋊C411yescharacteristic
189C2×C7D3×D811yescharacteristic
190C2×C7D3×SD1611yescharacteristic
191D7—34no—
192D7—42no—
193C22×C3—142no—
194C22×C3—281no—
195C22×C3—281no—
196C22×C3D4×D711yescharacteristic
197C2×D3—22no—
198C2×D3—41no—
199C2×D3—41no—
200C2×D3—72no—
201C2×D3—72no—
202C2×D3—72no—
203C2×D3—288no—
204C3×C4—41no—
205C3×C4C2×C4×D711yescharacteristic
206C3×C4D4×D711yescharacteristic
207Dic3—72no—
208Dic3—281no—
209C23—61no—
210C23—141no—
211C23—211no—
212C23—424no—
213C23—842no—
214C23—842no—
215D4—32no—
216D4—61no—
217D4—72no—
218D4—141no—
219D4—424no—
220C2×C4—21no—
221C2×C4—211no—
222C2×C4—212no—
223C2×C4—212no—
224C2×C4—421no—
225C2×C4—842no—
226C2×C4C2×D3×D711yescharacteristic
227C8—61no—
228C8—141no—
229C7D3×#10211yescharacteristic, complemented, Sylow
230C2×C3—282no—
231C2×C3D7×C22⋊C411yescharacteristic
232C2×C3D7×D811yescharacteristic
233C2×C3D7×SD1611yescharacteristic
234D3—42no—
235D3—74no—
236C22—62no—
237C22—121no—
238C22—121no—
239C22—142no—
240C22—212no—
241C22—212no—
242C22—212no—
243C22—281no—
244C22—281no—
245C22—848no—
246C22#6669411yescharacteristic
247C4—41no—
248C4—212no—
249C4—841no—
250C4C4×D3×D711yescharacteristic
251C4#6668111yescharacteristic
252C3D7×#10211yescharacteristic, complemented, Sylow
253C2—122no—
254C2—214no—
255C2—282no—
256C2#1062103511yescharacteristic
257C2#1062104311yescharacteristic
258C2#1062106711yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66681C2
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7C2×#102
LCSG ▷ 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. C4⋊C4×D21C2semidirect
3. C2×D21⋊C8C2semidirect
4. C7×#809C2semidirect
5. C3×#6383C2semidirect
6. #10620502C2semidirect
7. #10620500C2semidirect
16. #66681C4semidirect
17. C3×C7×C4⋊C4C22semidirect
18. C4⋊Dic21C22semidirect
20. C2×C7×C3⋊C8C22semidirect
21. C2×C3×C7⋊C8C22semidirect
29. #6383D3semidirect
45. C7×D12C2×C4semidirect
47. C3×D28C2×C4semidirect
60. C7×C4⋊C4C2×D3semidirect
63. C2×C7⋊C8C2×D3semidirect
71. #809D7semidirect
103. D28C4×D3semidirect
118. C3×C4⋊C4C2×D7semidirect
121. C2×C3⋊C8C2×D7semidirect
126. D21#102semidirect
166. D12C4×D7semidirect
172. C3×C7C2×#102semidirect
186. C4⋊C4D3×D7semidirect
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×#666812
C4⋊C4×D212
C2×D21⋊C82
C7×#8092
C3×#63832
#106205022
#106205002
D7×#1023
D3×#1027