Finite Group Databasebeta

← #13054408

Subgroups — #13054408

Order 1344.

376 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×D4×D7C211yesmaximal, characteristic
2C2×C7×#214C211yesmaximal, characteristic, complemented
3#10620921C211yesmaximal, characteristic, complemented
4C2×#66563C211yesmaximal, characteristic, complemented
5#10620671C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89941—31nomaximal, Hall
9C23×C3×D7—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic, complemented
12C2×C3×C22⋊D7—21no—
13C3×D4×D7—22no—
14C3×D4×D7—22no—
15C2×C3×D28C2211yescharacteristic
16C2×C3×C4×D7C2211yescharacteristic
17C4⋊Dic21C2211yescharacteristic, complemented
18C2×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
19C7×#214—22no—
20C2×C7×C3⋊C8C2211yescharacteristic, complemented
21#66577—21no—
22(C2×D7)⋊Dic3—21no—
23#66563—22no—
24C2×D4×D7D311yescharacteristic
25C2×C7×SD16—31no—
26#6509—31no—
27C2×#1143—31no—
28#6406—31no—
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#5218—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×C3×D7—24no—
34C22×C3×D7—21no—
35C22×C3×D7—42no—
36C22×C3×D7—42no—
37C22×C3×D7D411yescharacteristic
38C3×C7×D4—21no—
39C3×C7×D4D412yes—
40C2×C3×C4×C7C2311yescharacteristic
41C2×Dic21—21no—
42C2×C7×Dic3—21no—
43C7×Dic6—21no—
44C7×Dic6—21no—
45C3×C22⋊D7—24no—
46C2×C3×Dic7D411yescharacteristic
47C3×D28—21no—
48C3×D28D412yes—
49C3×C4×D7—21no—
50C3×C4×D7—21no—
51C3⋊(C7⋊C8)—21no—
52C7×C3⋊C8—21no—
53C23×D7—21no—
54C2×C7×Q8—31no—
55C2×C7×D4C2×D311yescharacteristic
56C2×C22⋊D7—21no—
57D4×D7—22no—
58D4×D7—22no—
59C2×D28C2×D311yescharacteristic
60C2×C4×D7C2×D311yescharacteristic
61C7×SD16—62no—
62C2×C7×C8—31no—
63#1143—62no—
64(C2×D7)⋊C4—61no—
65C4⋊Dic7—31no—
66Dic7⋊C4—61no—
67C2×C7⋊C8—31no—
68C22×C3×D4—71no—
69C2×#214—71no—
70C2×#214D711yescharacteristic, complemented
71#924—71no—
72#832—71no—
73#832—71no—
74#830—71no—
75C22×C3×C7—22no—
76C22×C3×C7—41no—
77C22×C3×C7—41no—
78C22×C3×C7C2×D411yescharacteristic
79C2×C3×D7—21no—
80C2×C3×D7—21no—
81C2×C3×D7—22no—
82C2×C3×D7—48no—
83C2×C3×D7—41no—
84C2×C3×D7—41no—
85C2×C3×D7SD1612yescomplemented
86C3×C4×C7C2×D411yescharacteristic
87C3×C4×C7C2×D411yescharacteristic
88Dic21—41no—
89C3×Dic7—21no—
90C7×Dic3—41no—
91#450—211noSylow
92C23×C7—21no—
93C22×D7—24no—
94C22×D7—21no—
95C22×D7—42no—
96C22×D7—42no—
97C22×D7#6711yescharacteristic
98C7×Q8—61no—
99C7×Q8—61no—
100C7×D4—21no—
101C7×D4#6712yes—
102C2×C4×C7—61no—
103C2×C4×C7C22×D311yescharacteristic
104C22⋊D7—24no—
105C2×Dic7—61no—
106C2×Dic7#6711yescharacteristic
107D28—21no—
108D28#6712yes—
109C4×D7—21no—
110C4×D7—21no—
111C7×C8—61no—
112C7⋊C8—61no—
113C24×C3—141no—
114C2×C3×D4—71no—
115C2×C3×D4—141no—
116C2×C3×D4—142no—
117C2×C3×D4—142no—
118C2×C3×D4C2×D711yescharacteristic
119C22×C3×C4—71no—
120C2×Dic6C2×D711yescharacteristic, complemented
121#217—141no—
122#214—22no—
123#214—142no—
124C4⋊Dic3—71no—
125Dic3⋊C4—141no—
126C2×C3⋊C8—71no—
127C2×C3⋊C8C2×D711yescharacteristic, complemented
128C2×C3×C7—42no—
129C2×C3×C7#12011yescharacteristic
130C2×C3×C7C2×SD1611yescharacteristic
131C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
132C3×D7—22no—
133C3×D7—42no—
134C22×D4—71no—
135C2×SD16—31no—
136C2×SD16—211no—
137C22⋊Q8—211no—
138#102—211no—
139#102—211no—
140C22⋊C8—211no—
141C22×C7—22no—
142C22×C7—41no—
143C22×C7—41no—
144C22×C7C2×#6711yescharacteristic
145C2×D7—21no—
146C2×D7—21no—
147C2×D7—22no—
148C2×D7—48no—
149C2×D7—41no—
150C2×D7—41no—
151C2×D7#21412yescomplemented
152C4×C7—121no—
153C4×C7C2×#6711yescharacteristic
154C4×C7C2×#6711yescharacteristic
155Dic7—21no—
156Dic7—121no—
157C23×C3—21no—
158C23×C3—71no—
159C23×C3—144no—
160C23×C3—141no—
161C23×C3—282no—
162C23×C3—282no—
163C3×D4—21no—
164C3×D4—72no—
165C3×D4—144no—
166C3×D4—141no—
167C3×D4C22⋊D712yes—
168C2×C3×C4—71no—
169C2×C3×C4—141no—
170C2×C3×C4—141no—
171C2×C3×C4C22×D711yescharacteristic
172C2×Dic3—21no—
173C2×Dic3—141no—
174Dic6—21no—
175Dic6—21no—
176C3⋊C8—21no—
177C3⋊C8—141no—
178C3×C7#45011yescharacteristic, complemented, Hall
179C24—141no—
180C2×Q8—31no—
181C2×D4—71no—
182C2×D4—141no—
183C2×D4—142no—
184C2×D4—142no—
185C2×D4D3×D711yescharacteristic
186C22×C4—71no—
187SD16—62no—
188SD16—422no—
189C2×C8—31no—
190C2×C8—211no—
191C4⋊C4—211no—
192C4⋊C4—421no—
193C22⋊C4—421no—
194C2×C7—42no—
195C2×C7#93211yescharacteristic
196C2×C7C2×#21411yescharacteristic
197C2×C7#95311yescharacteristic
198D7—22no—
199D7—42no—
200C22×C3—22no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—72no—
204C22×C3—141no—
205C22×C3—141no—
206C22×C3—142no—
207C22×C3—288no—
208C22×C3—281no—
209C22×C3—281no—
210C22×C3D4×D711yescharacteristic
211C3×C4—141no—
212C3×C4D4×D711yescharacteristic
213C3×C4C2×C22⋊D711yescharacteristic
214Dic3—41no—
215Dic3—281no—
216C23—21no—
217C23—71no—
218C23—144no—
219C23—141no—
220C23—282no—
221C23—282no—
222Q8—61no—
223Q8—61no—
224D4—21no—
225D4—72no—
226D4—144no—
227D4—141no—
228D4#424512yes—
229C2×C4—61no—
230C2×C4—71no—
231C2×C4—141no—
232C2×C4—141no—
233C2×C4—421no—
234C2×C4C2×D3×D711yescharacteristic
235C8—61no—
236C8—421no—
237C7#521811yescharacteristic, complemented, Sylow
238C2×C3—42no—
239C2×C3—142no—
240C2×C3—282no—
241C2×C3D7×SD1611yescharacteristic
242C2×C3#647711yescharacteristic
243C2×C3#650011yescharacteristic
244C22—22no—
245C22—41no—
246C22—41no—
247C22—72no—
248C22—141no—
249C22—141no—
250C22—142no—
251C22—288no—
252C22—281no—
253C22—281no—
254C22D7×#6711yescharacteristic
255C4—121no—
256C4—141no—
257C4—841no—
258C4C2×#424511yescharacteristic
259C4D7×#6711yescharacteristic
260C3#8994111yescharacteristic, complemented, Sylow
261C2—42no—
262C2—142no—
263C2—282no—
264C2D7×#21411yescharacteristic
265C2#1062105311yescharacteristic
266C2#1062112511yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×D4×D7C2
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. #10620921C2semidirect
4. C2×#66563C2semidirect
5. #10620671C2semidirect
6. #10620500C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
11. C2×C7×Dic6C22semidirect
17. C4⋊Dic21C22semidirect
18. C2×C3⋊(C7⋊C8)C22semidirect
20. C2×C7×C3⋊C8C22semidirect
70. C2×#214D7semidirect
85. C2×C3×D7SD16semidirect
120. C2×Dic6C2×D7semidirect
127. C2×C3⋊C8C2×D7semidirect
151. C2×D7#214semidirect
178. C3×C7#450semidirect
237. C7#5218semidirect
260. C3#89941semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D72
C2×C7×#2142
#106209212
C2×#665632
#106206712
#106205002
(C2×D7)⋊(C3⋊C8)2
#899413
#52187