Finite Group Databasebeta

← C2×#10621044 #13060124

Subgroups — C2×#10621044 #13060124

Order 1344.

596 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 136 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×#66730C211yesmaximal, characteristic, complemented
2C2×#66681C211yesmaximal, characteristic, complemented
3C2×C7×#213C211yesmaximal, characteristic, complemented
4C2×C3×#1141C211yesmaximal, characteristic, complemented
5#10621044C218yesmaximal, direct product factor
6C2×#66567C211yesmaximal, characteristic, complemented
7C2×#66565C211yesmaximal, characteristic, complemented
8C2×#66559C211yesmaximal, characteristic, complemented
9C2×#6474—31nomaximal, Hall
10C22×D3×D7—21no—
11C2×C3×C7×D4C2211yescharacteristic, complemented
12C2×#4268—21no—
13C22×Dic21—21no—
14#66730—22no—
15#66730C2214yescomplemented
16C2×C4×D21C2211yescharacteristic
17C2×Dic42C2211yescharacteristic, complemented
18C2×C7×D12C2211yescharacteristic
19C2×C3×D28C2211yescharacteristic
20C2×#4245—21no—
21#66681—22no—
22#66681C2214yescomplemented
23C7×#213C2214yescomplemented
24C2×C7×C3⋊C8C2211yescharacteristic, complemented
25C3×#1141C2214yescomplemented
26C2×C3×C7⋊C8C2211yescharacteristic, complemented
27#66567C2214yescomplemented
28#66565C2214yescomplemented
29#66559C2214yescomplemented
30C2×#1159—31no—
31C2×D4×D7—31no—
32C2×C7×D8—31no—
33C2×#1142—31no—
34C2×#1141D311yescharacteristic, complemented
35#6474—38no—
36C2×#1132—31no—
37C2×#1131—31no—
38C2×#911—71nomaximal, Hall
39C23×C3×C7—21no—
40C22×D21D411yescharacteristic
41C22×D3×C7—21no—
42C22×C3×D7—21no—
43C2×D3×D7—24no—
44C2×D3×D7—44no—
45C3×C7×D4—21no—
46C3×C7×D4C2312yescomplemented
47C2×C3×C4×C7C2311yescharacteristic
48#4268—24no—
49C2×Dic21—21no—
50C2×Dic21—24no—
51C2×Dic21D411yescharacteristic
52C4×D21D412yes—
53C4×D21C2312yes—
54Dic42—21no—
55Dic42C2312yescomplemented
56C7×D12—21no—
57C7×D12C2312yes—
58C3×D28—21no—
59C3×D28C2312yes—
60#4245—24no—
61C3×C7⋊C8C2312yescomplemented
62C7×C3⋊C8C2312yescomplemented
63C23×D7—61no—
64C2×C7×D4—31no—
65C2×C7×D4C2×D311yescharacteristic, complemented
66C2×C22⋊D7—61no—
67C2×C22⋊D7—61no—
68C22×Dic7—61no—
69#1159—34no—
70#1159—62no—
71D4×D7—34no—
72D4×D7—62no—
73C2×D28C2×D311yescharacteristic
74C2×C4×D7—31no—
75C2×Dic14—31no—
76C7×D8—34no—
77C2×C7×C8—31no—
78#1142—34no—
79#1141C2×D314yescomplemented
80C2×C7⋊C8C2×D311yescharacteristic, complemented
81#1132—34no—
82#1131—34no—
83C2×#237—71no—
84C2×D3×D4—71no—
85C2×C3×D8—71no—
86C2×#214—71no—
87C2×#213D711yescharacteristic, complemented
88#911—78no—
89C2×#204—71no—
90C2×ΓL(1,𝔽25)—71no—
91C22×C3×C7—22no—
92C22×C3×C7—42no—
93C22×C3×C7C2×D411yescharacteristic
94C2×D21—22no—
95C2×D21C2×D412yes—
96C2×D3×C7—22no—
97C2×D3×C7—42no—
98C2×C3×D7—22no—
99C2×C3×D7—42no—
100D3×D7—48no—
101C3×C4×C7C2×D411yescharacteristic
102C3×C4×C7C2411yescharacteristic
103Dic21—22no—
104Dic21C2×D412yes—
105C2×AGL(1,ℤ/8)—211noSylow
106C23×C7—21no—
107C23×C7—61no—
108C22×D7—21no—
109C22×D7—31no—
110C22×D7—64no—
111C22×D7—124no—
112C7×D4—21no—
113C7×D4—32no—
114C7×D4—61no—
115C7×D4C22×D312yescomplemented
116C2×C4×C7C22×D311yescharacteristic
117C22⋊D7—64no—
118C22⋊D7—64no—
119C2×Dic7—31no—
120C2×Dic7—64no—
121C2×Dic7—61no—
122D28—21no—
123D28C22×D312yes—
124C4×D7—32no—
125C4×D7—32no—
126Dic14—32no—
127Dic14—61no—
128C7×C8—32no—
129C7⋊C8C22×D312yescomplemented
130C23×D3—141no—
131C2×C3×D4—71no—
132C2×C3×D4C2×D711yescharacteristic, complemented
133C2×#67—141no—
134C2×#67—141no—
135C22×Dic3—141no—
136#237—74no—
137#237—142no—
138D3×D4—74no—
139D3×D4—142no—
140C2×D12C2×D711yescharacteristic
141C2×C4×D3—71no—
142C2×Dic6—71no—
143C3×D8—74no—
144C2×C3×C8—71no—
145#214—74no—
146#213C2×D714yescomplemented
147C2×C3⋊C8C2×D711yescharacteristic, complemented
148#204—74no—
149ΓL(1,𝔽25)—74no—
150C2×C3×C7—42no—
151C2×C3×C7AGL(1,ℤ/8)12yescomplemented
152C2×C3×C7C22×D411yescharacteristic
153D21—22no—
154C3×D7—42no—
155D3×C7—42no—
156C2×#41—211no—
157C22×D4—211no—
158AGL(1,ℤ/8)—218no—
159C2×SD16—211no—
160C2×SD16—211no—
161C2×D8—31no—
162C2×D8—71no—
163C2×M16—211no—
164C22×C7—22no—
165C22×C7—42no—
166C22×C7—62no—
167C22×C7—122no—
168C22×C7D3×D411yescharacteristic
169C2×D7—22no—
170C2×D7—32no—
171C2×D7—42no—
172C2×D7—62no—
173C2×D7—128no—
174C4×C7D3×D411yescharacteristic
175C4×C7C23×D311yescharacteristic
176Dic7—32no—
177Dic7—62no—
178C23×C3—21no—
179C23×C3—141no—
180C22×D3—21no—
181C22×D3—71no—
182C22×D3—144no—
183C22×D3—284no—
184C3×D4—21no—
185C3×D4—72no—
186C3×D4—141no—
187C3×D4C22×D712yescomplemented
188C2×C3×C4C22×D711yescharacteristic
189#67—144no—
190#67—144no—
191C2×Dic3—71no—
192C2×Dic3—144no—
193C2×Dic3—141no—
194D12—21no—
195D12C22×D712yes—
196C4×D3—72no—
197C4×D3—72no—
198Dic6—72no—
199Dic6—141no—
200C3×C8—72no—
201C3⋊C8C22×D712yescomplemented
202C3×C7C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
203C24—421no—
204#41—214no—
205#41—422no—
206C2×Q8—211no—
207C2×D4—31no—
208C2×D4—71no—
209C2×D4—214no—
210C2×D4—422no—
211C2×D4—421no—
212C2×D4—421no—
213C2×D4D3×D711yescharacteristic, complemented
214C22×C4—211no—
215C22×C4—421no—
216SD16—214no—
217SD16—214no—
218D8—34no—
219D8—74no—
220M16—214no—
221C2×C8—31no—
222C2×C8—71no—
223C2×C7—42no—
224C2×C7—122no—
225C2×C7#91112yescomplemented
226C2×C7C2×D3×D411yescharacteristic
227D7—42no—
228D7—62no—
229C22×C3—22no—
230C22×C3—42no—
231C22×C3—142no—
232C22×C3—282no—
233C22×C3D4×D711yescharacteristic
234C2×D3—22no—
235C2×D3—42no—
236C2×D3—72no—
237C2×D3—142no—
238C2×D3—288no—
239C3×C4D4×D711yescharacteristic
240C3×C4C23×D711yescharacteristic
241Dic3—72no—
242Dic3—142no—
243C23—21no—
244C23—61no—
245C23—141no—
246C23—211no—
247C23—424no—
248C23—844no—
249Q8—212no—
250Q8—421no—
251D4—21no—
252D4—32no—
253D4—61no—
254D4—72no—
255D4—141no—
256D4—424no—
257D4—424no—
258D4C2×D3×D712yescomplemented
259C2×C4—211no—
260C2×C4—212no—
261C2×C4—212no—
262C2×C4—424no—
263C2×C4—421no—
264C2×C4C2×D3×D711yescharacteristic
265C8—32no—
266C8—72no—
267C7C2×#91111yescharacteristic, complemented, Sylow
268C2×C3—42no—
269C2×C3—282no—
270C2×C3#647412yescomplemented
271C2×C3C2×D4×D711yescharacteristic
272D3—42no—
273D3—142no—
274C22—22no—
275C22—42no—
276C22—62no—
277C22—122no—
278C22—142no—
279C22—212no—
280C22—282no—
281C22—422no—
282C22—848no—
283C22#6669411yescharacteristic
284C4—212no—
285C4—422no—
286C4#6669411yescharacteristic
287C4C22×D3×D711yescharacteristic
288C3C2×#647411yescharacteristic, complemented, Sylow
289C2—42no—
290C2—122no—
291C2—282no—
292C2—422no—
293C2#1062104412yesdirect product factor
294C2C2×#6669411yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7C2×AGL(1,ℤ/8)
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
1. C2×#66730C2semidirect
2. C2×#66681C2semidirect
3. C2×C7×#213C2semidirect
4. C2×C3×#1141C2semidirect
5. #10621044C2direct (N × H)
6. C2×#66567C2semidirect
7. C2×#66565C2semidirect
8. C2×#66559C2semidirect
11. C2×C3×C7×D4C22semidirect
15. #66730C22semidirect
17. C2×Dic42C22semidirect
22. #66681C22semidirect
23. C7×#213C22semidirect
24. C2×C7×C3⋊C8C22semidirect
25. C3×#1141C22semidirect
26. C2×C3×C7⋊C8C22semidirect
27. #66567C22semidirect
28. #66565C22semidirect
29. #66559C22semidirect
34. C2×#1141D3semidirect
46. C3×C7×D4C23semidirect
55. Dic42C23semidirect
61. C3×C7⋊C8C23semidirect
62. C7×C3⋊C8C23semidirect
65. C2×C7×D4C2×D3semidirect
79. #1141C2×D3semidirect
80. C2×C7⋊C8C2×D3semidirect
87. C2×#213D7semidirect
115. C7×D4C22×D3semidirect
129. C7⋊C8C22×D3semidirect
132. C2×C3×D4C2×D7semidirect
146. #213C2×D7semidirect
147. C2×C3⋊C8C2×D7semidirect
151. C2×C3×C7AGL(1,ℤ/8)semidirect
187. C3×D4C22×D7semidirect
201. C3⋊C8C22×D7semidirect
202. C3×C7C2×AGL(1,ℤ/8)semidirect
213. C2×D4D3×D7semidirect
225. C2×C7#911semidirect
258. D4C2×D3×D7semidirect
267. C7C2×#911semidirect
270. C2×C3#6474semidirect
288. C3C2×#6474semidirect
293. C2#10621044direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#667302
C2×#666812
C2×C7×#2132
C2×C3×#11412
#106210442
C2×#665672
C2×#665652
C2×#665592
C2×#64743
C2×#9117