Finite Group Databasebeta

← #13054422

Subgroups — #13054422

Order 1344.

376 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 70 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
2C2×C7×#214C211yesmaximal, characteristic, complemented
3#10620945C211yesmaximal, characteristic, complemented
4C2×#66568C211yesmaximal, characteristic, complemented
5C3×#6406C211yesmaximal, characteristic, complemented
6#10620503C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89941—31nomaximal, Hall
9C23×D21—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C2×#4268—21no—
12D4×D21—22no—
13D4×D21—22no—
14C2×D84C2211yescharacteristic
15C2×C4×D21C2211yescharacteristic
16C2×C7×Dic6C2211yescharacteristic, complemented
17C7×#214—22no—
18C2×C7×C3⋊C8C2211yescharacteristic, complemented
19C3×C4⋊Dic7C2211yescharacteristic, complemented
20C2×C3×C7⋊C8C2211yescharacteristic, complemented
21Dic7⋊Dic3—21no—
22#66575—21no—
23#66568—22no—
24C2×D4×D7—31no—
25C2×C7×SD16—31no—
26#6509—31no—
27C2×#1143—31no—
28#6406D311yescharacteristic, complemented
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#4776—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21—24no—
34C22×D21—21no—
35C22×D21—42no—
36C22×D21—42no—
37C22×D21D411yescharacteristic
38C3×C7×D4—21no—
39C3×C7×D4D412yes—
40C2×C3×C4×C7C2311yescharacteristic
41#4268—24no—
42C2×Dic21D411yescharacteristic
43D84—21no—
44D84D412yes—
45C4×D21—21no—
46C4×D21—21no—
47C2×C7×Dic3—21no—
48C7×Dic6—21no—
49C7×Dic6—21no—
50C2×C3×Dic7—21no—
51C3×C7⋊C8—21no—
52C7×C3⋊C8—21no—
53C23×D7—61no—
54C2×C7×Q8—31no—
55C2×C7×D4C2×D311yescharacteristic
56C2×C22⋊D7—61no—
57D4×D7—62no—
58D4×D7—62no—
59C2×D28—31no—
60C2×C4×D7—31no—
61C7×SD16—62no—
62C2×C7×C8—31no—
63#1143—62no—
64(C2×D7)⋊C4—61no—
65C4⋊Dic7C2×D311yescharacteristic, complemented
66Dic7⋊C4—61no—
67C2×C7⋊C8C2×D311yescharacteristic, complemented
68C2×D3×D4—71no—
69C2×#214D711yescharacteristic, complemented
70C2×#204—71no—
71#897—71no—
72C3×#102—71no—
73(C2×D3)⋊C8—71no—
74#809—71no—
75C22×C3×C7—22no—
76C22×C3×C7—41no—
77C22×C3×C7—41no—
78C22×C3×C7C2×D411yescharacteristic
79C2×D21—21no—
80C2×D21—21no—
81C2×D21—22no—
82C2×D21—48no—
83C2×D21—41no—
84C2×D21—41no—
85C2×D21SD1612yescomplemented
86C3×C4×C7C2×D411yescharacteristic
87C3×C4×C7C2×D411yescharacteristic
88Dic21—21no—
89C3×Dic7—41no—
90C7×Dic3—41no—
91#450—211noSylow
92C23×C7—21no—
93C22×D7—31no—
94C22×D7—64no—
95C22×D7—61no—
96C22×D7—122no—
97C22×D7—122no—
98C7×Q8—61no—
99C7×Q8—61no—
100C7×D4—21no—
101C7×D4D1212yes—
102C2×C4×C7—61no—
103C2×C4×C7C22×D311yescharacteristic
104C22⋊D7—64no—
105C2×Dic7—21no—
106C2×Dic7—31no—
107D28—32no—
108D28—61no—
109C4×D7—61no—
110C4×D7—61no—
111C7×C8—61no—
112C7⋊C8—21no—
113C23×D3—141no—
114C2×C3×D4C2×D711yescharacteristic
115C2×#67—141no—
116D3×D4—142no—
117D3×D4—142no—
118C2×D12—71no—
119C2×C4×D3—71no—
120C2×Dic6C2×D711yescharacteristic, complemented
121C2×C3×C8—71no—
122C3×C4⋊C4—71no—
123#214—22no—
124(C2×D3)⋊C4—141no—
125C4⋊Dic3—141no—
126C2×C3⋊C8C2×D711yescharacteristic, complemented
127#204—142no—
128C2×C3×C7—42no—
129C2×C3×C7#12011yescharacteristic
130C2×C3×C7C2×SD1611yescharacteristic
131C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
132D21—22no—
133D21—42no—
134C22×D4—211no—
135C2×SD16—31no—
136C2×SD16—211no—
137C22⋊Q8—211no—
138#102—71no—
139#102—211no—
140C22⋊C8—211no—
141C22×C7—22no—
142C22×C7—41no—
143C22×C7—41no—
144C22×C7D3×D411yescharacteristic
145C2×D7—32no—
146C2×D7—61no—
147C2×D7—61no—
148C2×D7—62no—
149C2×D7—128no—
150C2×D7—121no—
151C2×D7—121no—
152C4×C7—121no—
153C4×C7C2×D1211yescharacteristic
154C4×C7D3×D411yescharacteristic
155Dic7—41no—
156Dic7—61no—
157C23×C3—21no—
158C22×D3—71no—
159C22×D3—144no—
160C22×D3—141no—
161C22×D3—282no—
162C22×D3—282no—
163C3×D4—21no—
164C3×D4C22⋊D712yes—
165C2×C3×C4—141no—
166C2×C3×C4C22×D711yescharacteristic
167#67—144no—
168C2×Dic3—21no—
169C2×Dic3—71no—
170D12—72no—
171D12—141no—
172C4×D3—141no—
173C4×D3—141no—
174Dic6—21no—
175Dic6—21no—
176C3×C8—141no—
177C3⋊C8—21no—
178C3×C7#45011yescharacteristic, complemented, Hall
179C24—421no—
180C2×Q8—31no—
181C2×D4—211no—
182C2×D4—421no—
183C2×D4—422no—
184C2×D4—422no—
185C2×D4D3×D711yescharacteristic
186C22×C4—211no—
187SD16—62no—
188SD16—422no—
189C2×C8—31no—
190C2×C8—71no—
191C4⋊C4—71no—
192C4⋊C4—421no—
193C22⋊C4—421no—
194C2×C7—42no—
195C2×C7#88211yescharacteristic
196C2×C7#91111yescharacteristic
197C2×C7D3×SD1611yescharacteristic
198D7—62no—
199D7—122no—
200C22×C3—22no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3D4×D711yescharacteristic
204C2×D3—72no—
205C2×D3—141no—
206C2×D3—141no—
207C2×D3—142no—
208C2×D3—288no—
209C2×D3—281no—
210C2×D3—281no—
211C3×C4—281no—
212C3×C4D4×D711yescharacteristic
213C3×C4C2×C22⋊D711yescharacteristic
214Dic3—41no—
215Dic3—141no—
216C23—21no—
217C23—211no—
218C23—424no—
219C23—421no—
220C23—842no—
221C23—842no—
222Q8—61no—
223Q8—61no—
224D4—21no—
225D4—212no—
226D4—424no—
227D4—421no—
228D4#424712yes—
229C2×C4—61no—
230C2×C4—141no—
231C2×C4—211no—
232C2×C4—421no—
233C2×C4—421no—
234C2×C4C2×D3×D711yescharacteristic
235C8—61no—
236C8—141no—
237C7#477611yescharacteristic, complemented, Sylow
238C2×C3—42no—
239C2×C3D7×SD1611yescharacteristic
240C2×C3#647711yescharacteristic
241C2×C3#650011yescharacteristic
242D3—142no—
243D3—282no—
244C22—22no—
245C22—41no—
246C22—41no—
247C22—212no—
248C22—421no—
249C22—421no—
250C22—422no—
251C22—848no—
252C22—841no—
253C22—841no—
254C22#6669411yescharacteristic
255C4—121no—
256C4—281no—
257C4—421no—
258C4C2×#424711yescharacteristic
259C4#6669411yescharacteristic
260C3#8994111yescharacteristic, complemented, Sylow
261C2—42no—
262C2—422no—
263C2—842no—
264C2#1062104911yescharacteristic
265C2#1062106011yescharacteristic
266C2#1062113511yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×D4×D21C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#450
LCSG ▷ C2×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. C2×C7×#214C2semidirect
3. #10620945C2semidirect
4. C2×#66568C2semidirect
5. C3×#6406C2semidirect
6. #10620503C2semidirect
7. #10620492C2semidirect
16. C2×C7×Dic6C22semidirect
18. C2×C7×C3⋊C8C22semidirect
19. C3×C4⋊Dic7C22semidirect
20. C2×C3×C7⋊C8C22semidirect
28. #6406D3semidirect
65. C4⋊Dic7C2×D3semidirect
67. C2×C7⋊C8C2×D3semidirect
69. C2×#214D7semidirect
85. C2×D21SD16semidirect
120. C2×Dic6C2×D7semidirect
126. C2×C3⋊C8C2×D7semidirect
178. C3×C7#450semidirect
237. C7#4776semidirect
260. C3#89941semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D212
C2×C7×#2142
#106209452
C2×#665682
C3×#64062
#106205032
#106204922
#899413
#47767