Finite Group Databasebeta

← #13052863

Subgroups — #13052863

Order 1344.

376 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 68 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
2#10621380C211yesmaximal, characteristic, complemented
3C2×#66567C211yesmaximal, characteristic, complemented
4C2×#66565C211yesmaximal, characteristic, complemented
5C7×#809C211yesmaximal, characteristic, complemented
6C3×#6383C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89548—31nomaximal, Hall
9C22×D3×D7—21no—
10C2×C4×D21C2211yescharacteristic
11C2×Dic42C2211yescharacteristic, complemented
12C2×C7×D12C2211yescharacteristic
13C2×C3×D28C2211yescharacteristic
14C2×#4245—21no—
15#66681—22no—
16#66681—22no—
17C3×C7×C4⋊C4C2211yescharacteristic, complemented
18#66631—21no—
19C4⋊Dic21—21no—
20C2×C7×C3⋊C8C2211yescharacteristic, complemented
21C2×C3×C7⋊C8C2211yescharacteristic, complemented
22#66567—22no—
23#66565—22no—
24C2×D4×D7—31no—
25C2×#1142—31no—
26C2×#1132—31no—
27#6460—31no—
28C7×#102—31no—
29(C2×D7)⋊C8—31no—
30#6383D311yescharacteristic, complemented
31#4776—71nomaximal, Hall
32C22×D21D411yescharacteristic
33C22×D3×C7—21no—
34C22×C3×D7—21no—
35C2×D3×D7—24no—
36C2×D3×D7—42no—
37C2×D3×D7—42no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7C2311yescharacteristic
40C2×Dic21—21no—
41C2×Dic21D411yescharacteristic
42C4×D21—21no—
43C4×D21—21no—
44Dic42—21no—
45Dic42—21no—
46C7×D12—21no—
47C7×D12D412yes—
48C3×D28—21no—
49C3×D28D412yes—
50#4245—24no—
51C3×C7⋊C8—21no—
52C7×C3⋊C8—21no—
53C23×D7—61no—
54C2×C7×D4—31no—
55C2×C22⋊D7—61no—
56D4×D7—62no—
57D4×D7—62no—
58C2×D28C2×D311yescharacteristic
59C2×C4×D7—31no—
60C2×Dic14—31no—
61C2×C7×C8—31no—
62C7×C4⋊C4C2×D311yescharacteristic, complemented
63#1142—62no—
64(C2×D7)⋊C4—61no—
65C4⋊Dic7—61no—
66C2×C7⋊C8C2×D311yescharacteristic, complemented
67#1132—62no—
68C2×D3×D4—71no—
69C2×#214—71no—
70C2×#204—71no—
71#897—71no—
72C3×#102—71no—
73(C2×D3)⋊C8—71no—
74#809D711yescharacteristic, complemented
75C22×C3×C7C2×D411yescharacteristic
76C2×D21—21no—
77C2×D21—21no—
78C2×D21SD1612yescomplemented
79C2×D3×C7—22no—
80C2×D3×C7—41no—
81C2×D3×C7—41no—
82C2×C3×D7—22no—
83C2×C3×D7—41no—
84C2×C3×D7—41no—
85D3×D7—48no—
86C3×C4×C7—41no—
87C3×C4×C7C2×D411yescharacteristic
88C3×C4×C7C2×D411yescharacteristic
89Dic21—21no—
90Dic21—41no—
91#450—211noSylow
92C23×C7—61no—
93C22×D7—21no—
94C22×D7—31no—
95C22×D7—64no—
96C22×D7—122no—
97C22×D7—122no—
98C7×D4—32no—
99C7×D4—61no—
100C2×C4×C7—21no—
101C2×C4×C7C22×D311yescharacteristic
102C22⋊D7—64no—
103C2×Dic7—31no—
104C2×Dic7—61no—
105D28—21no—
106D28D1212yes—
107C4×D7—61no—
108C4×D7—61no—
109Dic14—61no—
110Dic14—61no—
111C7×C8—61no—
112C7⋊C8—21no—
113C23×D3—141no—
114C2×C3×D4—71no—
115C2×#67—141no—
116D3×D4—142no—
117D3×D4—142no—
118C2×D12C2×D711yescharacteristic
119C2×C4×D3—71no—
120C2×Dic6—71no—
121C2×C3×C8—71no—
122C3×C4⋊C4C2×D711yescharacteristic, complemented
123#214—142no—
124(C2×D3)⋊C4—141no—
125C4⋊Dic3—141no—
126C2×C3⋊C8C2×D711yescharacteristic, complemented
127#204—142no—
128C2×C3×C7#12011yescharacteristic
129C2×C3×C7C2×SD1611yescharacteristic
130C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
131D21—22no—
132C3×D7—42no—
133D3×C7—42no—
134C22×D4—211no—
135C2×SD16—211no—
136C2×SD16—211no—
137C22⋊Q8—211no—
138#102—31no—
139#102—71no—
140C22⋊C8—211no—
141C22×C7—62no—
142C22×C7—121no—
143C22×C7—121no—
144C22×C7D3×D411yescharacteristic
145C2×D7—22no—
146C2×D7—32no—
147C2×D7—41no—
148C2×D7—41no—
149C2×D7—61no—
150C2×D7—61no—
151C2×D7—128no—
152C4×C7—41no—
153C4×C7C2×D1211yescharacteristic
154C4×C7D3×D411yescharacteristic
155Dic7—61no—
156Dic7—121no—
157C23×C3—141no—
158C22×D3—21no—
159C22×D3—71no—
160C22×D3—144no—
161C22×D3—282no—
162C22×D3—282no—
163C3×D4—72no—
164C3×D4—141no—
165C2×C3×C4—21no—
166C2×C3×C4C22×D711yescharacteristic
167#67—144no—
168C2×Dic3—71no—
169C2×Dic3—141no—
170D12—21no—
171D12D2812yes—
172C4×D3—141no—
173C4×D3—141no—
174Dic6—141no—
175Dic6—141no—
176C3×C8—141no—
177C3⋊C8—21no—
178C3×C7#45011yescharacteristic, complemented, Hall
179C24—421no—
180C2×Q8—211no—
181C2×D4—31no—
182C2×D4—71no—
183C2×D4—421no—
184C2×D4—422no—
185C2×D4—422no—
186C22×C4—211no—
187SD16—422no—
188SD16—422no—
189C2×C8—31no—
190C2×C8—71no—
191C4⋊C4—421no—
192C4⋊C4D3×D711yescharacteristic, complemented
193C22⋊C4—421no—
194C2×C7—122no—
195C2×C7#88211yescharacteristic
196C2×C7#91111yescharacteristic
197C2×C7D3×SD1611yescharacteristic
198D7—42no—
199D7—62no—
200C22×C3—142no—
201C22×C3—281no—
202C22×C3—281no—
203C22×C3D4×D711yescharacteristic
204C2×D3—22no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—72no—
208C2×D3—141no—
209C2×D3—141no—
210C2×D3—288no—
211C3×C4—41no—
212C3×C4C2×D2811yescharacteristic
213C3×C4D4×D711yescharacteristic
214Dic3—141no—
215Dic3—281no—
216C23—61no—
217C23—141no—
218C23—211no—
219C23—424no—
220C23—842no—
221C23—842no—
222Q8—421no—
223Q8—421no—
224D4—32no—
225D4—61no—
226D4—72no—
227D4—141no—
228D4—424no—
229C2×C4—21no—
230C2×C4—211no—
231C2×C4—421no—
232C2×C4—421no—
233C2×C4—421no—
234C2×C4C2×D3×D711yescharacteristic
235C8—61no—
236C8—141no—
237C7#477611yescharacteristic, complemented, Sylow
238C2×C3—282no—
239C2×C3#644511yescharacteristic
240C2×C3#647411yescharacteristic
241C2×C3D7×SD1611yescharacteristic
242D3—42no—
243D3—142no—
244C22—62no—
245C22—121no—
246C22—121no—
247C22—142no—
248C22—212no—
249C22—281no—
250C22—281no—
251C22—421no—
252C22—421no—
253C22—848no—
254C22#6669411yescharacteristic
255C4—41no—
256C4—421no—
257C4—841no—
258C4D7×D1211yescharacteristic
259C4D3×D2811yescharacteristic
260C3#8954811yescharacteristic, complemented, Sylow
261C2—122no—
262C2—282no—
263C2—422no—
264C2#1062103811yescharacteristic
265C2#1062104411yescharacteristic
266C2#1062106711yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66681C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#450
LCSG ▷ C2×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. #10621380C2semidirect
3. C2×#66567C2semidirect
4. C2×#66565C2semidirect
5. C7×#809C2semidirect
6. C3×#6383C2semidirect
7. #10620492C2semidirect
11. C2×Dic42C22semidirect
17. C3×C7×C4⋊C4C22semidirect
20. C2×C7×C3⋊C8C22semidirect
21. C2×C3×C7⋊C8C22semidirect
30. #6383D3semidirect
62. C7×C4⋊C4C2×D3semidirect
66. C2×C7⋊C8C2×D3semidirect
74. #809D7semidirect
78. C2×D21SD16semidirect
122. C3×C4⋊C4C2×D7semidirect
126. C2×C3⋊C8C2×D7semidirect
178. C3×C7#450semidirect
192. C4⋊C4D3×D7semidirect
237. C7#4776semidirect
260. C3#89548semidirect

Maximal subgroups

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

subgroupindex
C2×#666812
#106213802
C2×#665672
C2×#665652
C7×#8092
C3×#63832
#106204922
#895483
#47767