Finite Group Databasebeta

← #13052888

Subgroups — #13052888

Order 1344.

324 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 78 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×#66672C211yesmaximal, characteristic
2C4⋊C4×D21C211yesmaximal, characteristic, complemented
3C2×#66559C211yesmaximal, characteristic, complemented
4C7×#809C211yesmaximal, characteristic, complemented
5C3×#6384C211yesmaximal, characteristic, complemented
6#10620506C211yesmaximal, characteristic, complemented
7#10620502C211yesmaximal, characteristic, complemented
8#89543—31nomaximal, Hall
9C2×C4×D21—21no—
10C2×C4×D21C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13C2×#4247—21no—
14C2×D3×Dic7—21no—
15#66672—22no—
16#66672C414yescomplemented
17C3×C7×C4⋊C4C2211yescharacteristic
18C4⋊Dic21C2211yescharacteristic
19#66629—21no—
20C2×C7×C3⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22#66559—22no—
23C2×#1159—31no—
24C2×#1131—31no—
25D7×C4⋊C4—31no—
26C7×#102—31no—
27#6406—31no—
28#6390—31no—
29#6384D311yescharacteristic, complemented
30#4803—71nomaximal, Hall
31C22×D21D411yescharacteristic
32C22×D3×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C4×D21—42no—
38C4×D21C2×C412yes—
39C4×D21D412yes—
40C7×D12—21no—
41C7×D12C2×C412yescomplemented
42C2×C3×Dic7—21no—
43C3×Dic14—21no—
44C3×Dic14C2×C412yescomplemented
45#4247—24no—
46D3×Dic7—24no—
47C3×C7⋊C8—21no—
48C7×C3⋊C8—21no—
49C2×C7×D4—31no—
50C2×C22⋊D7—61no—
51C22×Dic7—61no—
52#1159—34no—
53#1159—62no—
54C2×C4×D7—31no—
55C2×C4×D7—61no—
56C2×Dic14C2×D311yescharacteristic
57C2×C7×C8—31no—
58C7×C4⋊C4C2×D311yescharacteristic
59C4⋊Dic7—31no—
60Dic7⋊C4—61no—
61C2×C7⋊C8C2×D311yescharacteristic
62#1131—62no—
63C2×#239—71no—
64C2×ΓL(1,𝔽25)—71no—
65D3×C4⋊C4—71no—
66C3×#103—71no—
67#835—71no—
68#821—71no—
69#809D711yescharacteristic, complemented
70C22×C3×C7C2×D411yescharacteristic
71C2×D21—21no—
72C2×D21—21no—
73C2×D21C22⋊C412yes—
74C2×D3×C7—22no—
75C2×D3×C7—41no—
76C2×D3×C7—41no—
77C3×C4×C7—41no—
78C3×C4×C7C22×C411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80Dic21—41no—
81Dic21C22⋊C412yes—
82C3×Dic7—22no—
83#417—211noSylow
84C23×C7—61no—
85C22×D7—31no—
86C7×D4—32no—
87C7×D4—61no—
88C2×C4×C7—21no—
89C2×C4×C7C22×D311yescharacteristic
90C22⋊D7—64no—
91C2×Dic7—21no—
92C2×Dic7—31no—
93C2×Dic7—64no—
94C2×Dic7—61no—
95C4×D7—32no—
96C4×D7—32no—
97C4×D7—122no—
98Dic14—21no—
99Dic14C4×D312yescomplemented
100C7×C8—61no—
101C7⋊C8—21no—
102C2×C3×Q8—71no—
103#239—74no—
104#239—142no—
105C2×D12—141no—
106C2×D12C2×D711yescharacteristic
107C2×C4×D3—71no—
108C2×C4×D3—141no—
109C2×C4×D3—141no—
110C2×C3×C8—71no—
111C3×C4⋊C4C2×D711yescharacteristic
112C4⋊Dic3—71no—
113Dic3⋊C4—141no—
114C2×C3⋊C8C2×D711yescharacteristic
115ΓL(1,𝔽25)—142no—
116C2×C3×C7C2×C22⋊C411yescharacteristic
117C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
118C2×C3×C7#13711yescharacteristic
119D21—22no—
120D3×C7—42no—
121C2×#41—211no—
122C2×M16—211no—
123C2×C4⋊C4—211no—
124#103—71no—
125#103—211no—
126#102—31no—
127#102—211no—
128C22×C7—62no—
129C22×C7—121no—
130C22×C7—121no—
131C22×C7D3×D411yescharacteristic
132C2×D7—32no—
133C2×D7—61no—
134C2×D7—61no—
135C4×C7—41no—
136C4×C7C2×C4×D311yescharacteristic
137C4×C7D3×D411yescharacteristic
138Dic7—22no—
139Dic7—32no—
140Dic7—121no—
141C22×D3—21no—
142C22×D3—71no—
143C3×Q8—72no—
144C3×Q8—141no—
145C2×C3×C4—21no—
146C2×C3×C4—141no—
147C2×C3×C4C22×D711yescharacteristic
148C2×Dic3—71no—
149C2×Dic3—141no—
150D12—21no—
151D12—144no—
152D12C4×D712yescomplemented
153C4×D3—72no—
154C4×D3—72no—
155C4×D3—144no—
156C4×D3—282no—
157C3×C8—141no—
158C3⋊C8—21no—
159C3×C7#41711yescharacteristic, complemented, Hall
160#41—214no—
161#41—422no—
162C2×Q8—71no—
163C2×D4—31no—
164C2×D4—421no—
165C22×C4—211no—
166C22×C4—421no—
167C22×C4—421no—
168M16—422no—
169C2×C8—31no—
170C2×C8—71no—
171C4⋊C4—211no—
172C4⋊C4—421no—
173C4⋊C4D3×D711yescharacteristic
174C2×C7—122no—
175C2×C7D3×C22⋊C411yescharacteristic
176C2×C7#91411yescharacteristic
177C2×C7#91811yescharacteristic
178D7—62no—
179C22×C3D4×D711yescharacteristic
180C2×D3—22no—
181C2×D3—41no—
182C2×D3—41no—
183C2×D3—72no—
184C2×D3—141no—
185C2×D3—141no—
186C3×C4—41no—
187C3×C4—142no—
188C3×C4C2×C4×D711yescharacteristic
189C3×C4D4×D711yescharacteristic
190Dic3—72no—
191Dic3—281no—
192C23—61no—
193C23—211no—
194Q8—72no—
195Q8—141no—
196D4—32no—
197D4—61no—
198D4—424no—
199C2×C4—21no—
200C2×C4—141no—
201C2×C4—211no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—424no—
205C2×C4—421no—
206C2×C4—842no—
207C2×C4C2×D3×D711yescharacteristic
208C8—61no—
209C8—141no—
210C7#480311yescharacteristic, complemented, Sylow
211C2×C3D7×C22⋊C411yescharacteristic
212C2×C3#647411yescharacteristic
213C2×C3#647811yescharacteristic
214D3—42no—
215D3—142no—
216C22—62no—
217C22—121no—
218C22—121no—
219C22—212no—
220C22—421no—
221C22—421no—
222C22#6669411yescharacteristic
223C4—41no—
224C4—142no—
225C4—212no—
226C4—841no—
227C4C4×D3×D711yescharacteristic
228C4#6668111yescharacteristic
229C3#8954311yescharacteristic, complemented, Sylow
230C2—122no—
231C2—422no—
232C2#1062103511yescharacteristic
233C2#1062106211yescharacteristic
234C2#1062108611yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66672C2
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#417
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×#66559C2semidirect
4. C7×#809C2semidirect
5. C3×#6384C2semidirect
6. #10620506C2semidirect
7. #10620502C2semidirect
16. #66672C4semidirect
29. #6384D3semidirect
41. C7×D12C2×C4semidirect
44. C3×Dic14C2×C4semidirect
69. #809D7semidirect
99. Dic14C4×D3semidirect
152. D12C4×D7semidirect
159. C3×C7#417semidirect
210. C7#4803semidirect
229. C3#89543semidirect

Maximal subgroups

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

subgroupindex
C2×#666722
C4⋊C4×D212
C2×#665592
C7×#8092
C3×#63842
#106205062
#106205022
#895433
#48037