Finite Group Databasebeta

← C2×D7×#214 #13060216

Subgroups — C2×D7×#214 #13060216

Order 1344.

596 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 160 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×D7×Dic6C211yesmaximal, characteristic, complemented
3C2×#66633C211yesmaximal, characteristic, complemented
4C2×C7×#214C211yesmaximal, characteristic, complemented
5D7×#214C218yesmaximal, direct product factor
6C2×#66565C211yesmaximal, characteristic, complemented
7C2×#66563C211yesmaximal, characteristic, complemented
8C2×D7×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×D7×SD16—31nomaximal, Hall
10C23×C3×D7—21no—
11C2×C3×C7×D4C2211yescharacteristic
12C2×Dic42C2211yescharacteristic, complemented
13C2×C7×Dic6C2211yescharacteristic, complemented
14C2×C3×C22⋊D7—21no—
15C3×D4×D7—22no—
16C3×D4×D7C2214yes—
17C2×C3×D28C2211yescharacteristic
18C2×C3×C4×D7C2211yescharacteristic
19C2×#4248—21no—
20C2×Dic3×D7—21no—
21D7×Dic6—22no—
22D7×Dic6C2214yescomplemented
23#66633C2214yescomplemented
24C2×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
25C7×#214C2214yescomplemented
26C2×C7×C3⋊C8C2211yescharacteristic, complemented
27#66565C2214yescomplemented
28#66563C2214yescomplemented
29D7×C3⋊C8C2214yescomplemented
30C2×Q8×D7—31no—
31C2×D4×D7D311yescharacteristic
32C2×C7×SD16—31no—
33C2×#1143—31no—
34C2×#1142—31no—
35D7×SD16—38no—
36C2×#1132—31no—
37C2×C8×D7—31no—
38C22×#214—71nomaximal, Hall
39C23×C3×C7—21no—
40C22×C3×D7—21no—
41C22×C3×D7—24no—
42C22×C3×D7—44no—
43C22×C3×D7D411yescharacteristic
44C3×C7×D4—21no—
45C3×C7×D4C2312yes—
46C2×C3×C4×C7C2311yescharacteristic
47C2×Dic21—21no—
48Dic42—21no—
49Dic42C2312yescomplemented
50C2×C7×Dic3—21no—
51C7×Dic6—21no—
52C7×Dic6C2312yescomplemented
53C3×C22⋊D7—24no—
54C2×C3×Dic7D411yescharacteristic
55C3×D28—21no—
56C3×D28C2312yes—
57C3×C4×D7D412yes—
58C3×C4×D7C2312yes—
59#4248—24no—
60Dic3×D7—24no—
61C3⋊(C7⋊C8)C2312yescomplemented
62C7×C3⋊C8C2312yescomplemented
63C23×D7—21no—
64C2×C7×Q8—31no—
65C2×C7×D4C2×D311yescharacteristic
66C2×C22⋊D7—21no—
67Q8×D7—34no—
68Q8×D7—62no—
69D4×D7—22no—
70D4×D7C2×D314yes—
71C2×D28C2×D311yescharacteristic
72C2×C4×D7—61no—
73C2×C4×D7C2×D311yescharacteristic
74C2×Dic14—31no—
75C2×Dic14—61no—
76C7×SD16—34no—
77C2×C7×C8—31no—
78#1143—34no—
79#1142—34no—
80C2×C7⋊C8—31no—
81#1132—34no—
82C8×D7—34no—
83C22×C3×D4—71no—
84C22×Dic6—71no—
85C2×#214—71no—
86C2×#214—71no—
87C2×#214—78no—
88C2×#214—71no—
89C2×#214D711yescharacteristic, direct product factor
90C22×C3⋊C8—71no—
91C22×C3×C7—22no—
92C22×C3×C7—42no—
93C22×C3×C7C2×D411yescharacteristic
94C2×C3×D7—22no—
95C2×C3×D7—42no—
96C2×C3×D7—48no—
97C2×C3×D7SD1614yescomplemented
98C2×C3×D7C2×D412yes—
99C3×C4×C7C2×D411yescharacteristic
100C3×C4×C7C2411yescharacteristic
101Dic21—22no—
102C3×Dic7C2×D412yes—
103C7×Dic3—22no—
104C22×SD16—211noSylow
105C23×C7—21no—
106C22×D7—21no—
107C22×D7—24no—
108C22×D7—44no—
109C22×D7#6711yescharacteristic
110C7×Q8—32no—
111C7×Q8—61no—
112C7×D4—21no—
113C7×D4C22×D312yes—
114C2×C4×C7—61no—
115C2×C4×C7C22×D311yescharacteristic
116C22⋊D7—24no—
117C2×Dic7—61no—
118C2×Dic7#6711yescharacteristic
119D28—21no—
120D28C22×D312yes—
121C4×D7—64no—
122C4×D7#6712yes—
123C4×D7C22×D312yes—
124Dic14—32no—
125Dic14—61no—
126Dic14—64no—
127C7×C8—32no—
128C7⋊C8—32no—
129C24×C3—141no—
130C2×C3×D4—74no—
131C2×C3×D4—71no—
132C2×C3×D4—142no—
133C2×C3×D4—141no—
134C2×C3×D4C2×D711yescharacteristic
135C22×C3×C4—71no—
136C22×Dic3—141no—
137C2×Dic6—71no—
138C2×Dic6—74no—
139C2×Dic6—142no—
140C2×Dic6—141no—
141C2×Dic6C2×D711yescharacteristic, complemented
142#214—74no—
143#214—74no—
144#214—74no—
145#214C2×D714yesdirect product factor
146C2×C3⋊C8—71no—
147C2×C3⋊C8—74no—
148C2×C3⋊C8C2×D711yescharacteristic, complemented
149C2×C3×C7—42no—
150C2×C3×C7C2×SD1612yescomplemented
151C2×C3×C7C22×D411yescharacteristic
152C3×D7—42no—
153C3×D7C2×SD1614yescomplemented
154C22×Q8—211no—
155C22×D4—71no—
156C2×SD16—31no—
157C2×SD16—211no—
158C2×SD16—218no—
159C2×SD16—211no—
160C2×SD16—211no—
161C22×C8—211no—
162C22×C7—22no—
163C22×C7—42no—
164C22×C7C2×#6711yescharacteristic
165C2×D7—22no—
166C2×D7—42no—
167C2×D7—48no—
168C2×D7#21414yesdirect product factor
169C2×D7C2×#6712yes—
170C4×C7—62no—
171C4×C7C2×#6711yescharacteristic
172C4×C7C23×D311yescharacteristic
173Dic7—62no—
174Dic7C2×#6712yes—
175C23×C3—21no—
176C23×C3—71no—
177C23×C3—144no—
178C23×C3—141no—
179C23×C3—284no—
180C3×D4—21no—
181C3×D4—72no—
182C3×D4—144no—
183C3×D4—141no—
184C3×D4C22×D712yes—
185C2×C3×C4—71no—
186C2×C3×C4—72no—
187C2×C3×C4—72no—
188C2×C3×C4C22×D711yescharacteristic
189C2×Dic3—21no—
190C2×Dic3—144no—
191C2×Dic3—141no—
192Dic6—21no—
193Dic6—72no—
194Dic6—144no—
195Dic6—141no—
196Dic6C22×D712yescomplemented
197C3⋊C8—72no—
198C3⋊C8C22×D712yescomplemented
199C3×C7C22×SD1611yescharacteristic, complemented, Hall
200C24—141no—
201C2×Q8—31no—
202C2×Q8—214no—
203C2×Q8—211no—
204C2×Q8—421no—
205C2×Q8—422no—
206C2×D4—74no—
207C2×D4—71no—
208C2×D4—142no—
209C2×D4—141no—
210C2×D4D3×D711yescharacteristic
211C22×C4—71no—
212C22×C4—421no—
213SD16—34no—
214SD16—214no—
215SD16—214no—
216SD16—214no—
217C2×C8—31no—
218C2×C8—214no—
219C2×C8—211no—
220C2×C7—42no—
221C2×C7C2×#21412yescomplemented
222C2×C7C22×#6711yescharacteristic
223D7—42no—
224D7C2×#21414yesdirect product factor
225C22×C3—22no—
226C22×C3—42no—
227C22×C3—74no—
228C22×C3—72no—
229C22×C3—142no—
230C22×C3—288no—
231C22×C3—282no—
232C22×C3D4×D711yescharacteristic
233C3×C4—72no—
234C3×C4D4×D711yescharacteristic
235C3×C4C23×D711yescharacteristic
236Dic3—22no—
237Dic3—142no—
238C23—21no—
239C23—71no—
240C23—144no—
241C23—141no—
242C23—284no—
243Q8—32no—
244Q8—61no—
245Q8—212no—
246Q8—424no—
247Q8—421no—
248D4—21no—
249D4—72no—
250D4—144no—
251D4—141no—
252D4C2×D3×D712yes—
253C2×C4—61no—
254C2×C4—71no—
255C2×C4—72no—
256C2×C4—72no—
257C2×C4—424no—
258C2×C4—421no—
259C2×C4C2×D3×D711yescharacteristic
260C8—32no—
261C8—212no—
262C7C22×#21411yescharacteristic, complemented, Sylow
263C2×C3—42no—
264C2×C3—74no—
265C2×C3—282no—
266C2×C3D7×SD1612yescomplemented
267C2×C3C2×D4×D711yescharacteristic
268C22—22no—
269C22—42no—
270C22—74no—
271C22—72no—
272C22—142no—
273C22—288no—
274C22—282no—
275C22D7×#6711yescharacteristic
276C4—62no—
277C4—72no—
278C4—422no—
279C4D7×#6711yescharacteristic
280C4C22×D3×D711yescharacteristic
281C3C2×D7×SD1611yescharacteristic, complemented, Sylow
282C2—42no—
283C2—74no—
284C2—282no—
285C2D7×#21412yesdirect product factor
286C2C2×D7×#6711yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×D4×D7C2
hypercenterC2×D4D3×D7
hypocenterC3×C7C22×SD16
LCSG ▷ 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×D7×Dic6C2semidirect
3. C2×#66633C2semidirect
4. C2×C7×#214C2semidirect
5. D7×#214C2direct (N × H)
6. C2×#66565C2semidirect
7. C2×#66563C2semidirect
8. C2×D7×C3⋊C8C2semidirect
12. C2×Dic42C22semidirect
13. C2×C7×Dic6C22semidirect
22. D7×Dic6C22semidirect
23. #66633C22semidirect
24. C2×C3⋊(C7⋊C8)C22semidirect
25. C7×#214C22semidirect
26. C2×C7×C3⋊C8C22semidirect
27. #66565C22semidirect
28. #66563C22semidirect
29. D7×C3⋊C8C22semidirect
49. Dic42C23semidirect
52. C7×Dic6C23semidirect
61. C3⋊(C7⋊C8)C23semidirect
62. C7×C3⋊C8C23semidirect
89. C2×#214D7direct (N × H)
97. C2×C3×D7SD16semidirect
141. C2×Dic6C2×D7semidirect
145. #214C2×D7direct (N × H)
148. C2×C3⋊C8C2×D7semidirect
150. C2×C3×C7C2×SD16semidirect
153. C3×D7C2×SD16semidirect
168. C2×D7#214direct (N × H)
196. Dic6C22×D7semidirect
198. C3⋊C8C22×D7semidirect
199. C3×C7C22×SD16semidirect
221. C2×C7C2×#214semidirect
224. D7C2×#214direct (N × H)
262. C7C22×#214semidirect
266. C2×C3D7×SD16semidirect
281. C3C2×D7×SD16semidirect
285. C2D7×#214direct (N × H)

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×D7×Dic62
C2×#666332
C2×C7×#2142
D7×#2142
C2×#665652
C2×#665632
C2×D7×C3⋊C82
C2×D7×SD163
C22×#2147