Finite Group Databasebeta

← C2×#10621043 #13060123

Subgroups — C2×#10621043 #13060123

Order 1344.

676 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 144 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, complemented
2C2×#66681C211yesmaximal, characteristic, complemented
3C2×C7×#213C211yesmaximal, characteristic, complemented
4C2×C3×#1141C211yesmaximal, characteristic, complemented
5#10621043C218yesmaximal, direct product factor
6C2×#66562C211yesmaximal, characteristic, complemented
7C2×#66561C211yesmaximal, characteristic, complemented
8C2×D21⋊C8C211yesmaximal, characteristic, complemented
9C2×D7×D8—31nomaximal, Hall
10C23×D21—21no—
11C22×D3×D7—21no—
12C2×C3×C7×D4C2211yescharacteristic, complemented
13C2×#4268—21no—
14D4×D21—22no—
15D4×D21C2214yescomplemented
16C2×D84C2211yescharacteristic, complemented
17C2×C4×D21C2211yescharacteristic
18C2×C7×D12C2211yescharacteristic, complemented
19C2×C3×D28C2211yescharacteristic, complemented
20C2×#4245—21no—
21#66681—22no—
22#66681C2214yescomplemented
23C7×#213C2214yescomplemented
24C2×C7×C3⋊C8C2211yescharacteristic, complemented
25C3×#1141C2214yescomplemented
26C2×C3×C7⋊C8C2211yescharacteristic, complemented
27#66562C2214yescomplemented
28#66561C2214yescomplemented
29D21⋊C8C2214yescomplemented
30C2×D4×D7—31no—
31C2×D4×D7—31no—
32C2×C7×D8—31no—
33C2×#1141—31no—
34C2×#1141D311yescharacteristic, complemented
35D7×D8—38no—
36C2×D56—31no—
37C2×C8×D7—31no—
38C2×D3×D8—71nomaximal, Hall
39C23×C3×C7—21no—
40C22×D21—21no—
41C22×D21—24no—
42C22×D21—44no—
43C22×D21D411yescharacteristic
44C22×D3×C7—21no—
45C22×C3×D7—21no—
46C2×D3×D7—24no—
47C2×D3×D7—44no—
48C3×C7×D4—21no—
49C3×C7×D4C2312yescomplemented
50C2×C3×C4×C7C2311yescharacteristic
51#4268—24no—
52C2×Dic21D411yescharacteristic
53D84—21no—
54D84C2312yescomplemented
55C4×D21D412yes—
56C4×D21C2312yes—
57C7×D12—21no—
58C7×D12C2312yescomplemented
59C3×D28—21no—
60C3×D28C2312yescomplemented
61#4245—24no—
62C3×C7⋊C8C2312yescomplemented
63C7×C3⋊C8C2312yescomplemented
64C23×D7—61no—
65C23×D7—61no—
66C2×C7×D4—31no—
67C2×C7×D4C2×D311yescharacteristic, complemented
68C2×C22⋊D7—61no—
69C2×C22⋊D7—61no—
70D4×D7—34no—
71D4×D7—34no—
72D4×D7—62no—
73D4×D7—62no—
74C2×D28—31no—
75C2×D28C2×D311yescharacteristic, complemented
76C2×C4×D7—31no—
77C7×D8—34no—
78C2×C7×C8—31no—
79#1141—34no—
80#1141C2×D314yescomplemented
81C2×C7⋊C8C2×D311yescharacteristic, complemented
82D56—34no—
83C8×D7—34no—
84C2×D3×D4—71no—
85C2×D3×D4—71no—
86C2×C3×D8—71no—
87C2×#213—71no—
88C2×#213D711yescharacteristic, complemented
89D3×D8—78no—
90C2×D24—71no—
91C2×D3×C8—71no—
92C22×C3×C7—22no—
93C22×C3×C7—42no—
94C22×C3×C7C2×D411yescharacteristic
95C2×D21—22no—
96C2×D21—42no—
97C2×D21—48no—
98C2×D21D814yescomplemented
99C2×D21C2×D412yes—
100C2×D3×C7—22no—
101C2×D3×C7—42no—
102C2×C3×D7—22no—
103C2×C3×D7—42no—
104D3×D7—48no—
105C3×C4×C7C2×D411yescharacteristic
106C3×C4×C7C2411yescharacteristic
107Dic21C2×D412yes—
108C22×D8—211noSylow
109C23×C7—21no—
110C23×C7—61no—
111C22×D7—21no—
112C22×D7—31no—
113C22×D7—64no—
114C22×D7—64no—
115C22×D7—61no—
116C22×D7—124no—
117C22×D7—124no—
118C7×D4—21no—
119C7×D4—32no—
120C7×D4—61no—
121C7×D4C22×D312yescomplemented
122C2×C4×C7C22×D311yescharacteristic
123C22⋊D7—64no—
124C22⋊D7—64no—
125C2×Dic7—31no—
126D28—21no—
127D28—32no—
128D28—61no—
129D28C22×D312yescomplemented
130C4×D7—32no—
131C4×D7—32no—
132C7×C8—32no—
133C7⋊C8C22×D312yescomplemented
134C23×D3—141no—
135C23×D3—141no—
136C2×C3×D4—71no—
137C2×C3×D4C2×D711yescharacteristic, complemented
138C2×#67—141no—
139C2×#67—141no—
140D3×D4—74no—
141D3×D4—74no—
142D3×D4—142no—
143D3×D4—142no—
144C2×D12—71no—
145C2×D12C2×D711yescharacteristic, complemented
146C2×C4×D3—71no—
147C3×D8—74no—
148C2×C3×C8—71no—
149#213—74no—
150#213C2×D714yescomplemented
151C2×C3⋊C8C2×D711yescharacteristic, complemented
152D24—74no—
153D3×C8—74no—
154C2×C3×C7—42no—
155C2×C3×C7C2×D812yescomplemented
156C2×C3×C7C22×D411yescharacteristic
157D21—42no—
158D21C2×D814yescomplemented
159C3×D7—42no—
160D3×C7—42no—
161C22×D4—211no—
162C22×D4—211no—
163C2×D8—31no—
164C2×D8—71no—
165C2×D8—211no—
166C2×D8—211no—
167C2×D8—218no—
168C22×C8—211no—
169C22×C7—22no—
170C22×C7—42no—
171C22×C7—62no—
172C22×C7—122no—
173C22×C7D3×D411yescharacteristic
174C2×D7—22no—
175C2×D7—34no—
176C2×D7—32no—
177C2×D7—42no—
178C2×D7—62no—
179C2×D7—128no—
180C2×D7—128no—
181C2×D7—122no—
182C4×C7D3×D411yescharacteristic
183C4×C7C23×D311yescharacteristic
184Dic7—32no—
185C23×C3—21no—
186C23×C3—141no—
187C22×D3—21no—
188C22×D3—71no—
189C22×D3—144no—
190C22×D3—144no—
191C22×D3—141no—
192C22×D3—284no—
193C22×D3—284no—
194C3×D4—21no—
195C3×D4—72no—
196C3×D4—141no—
197C3×D4C22×D712yescomplemented
198C2×C3×C4C22×D711yescharacteristic
199#67—144no—
200#67—144no—
201C2×Dic3—71no—
202D12—21no—
203D12—72no—
204D12—141no—
205D12C22×D712yescomplemented
206C4×D3—72no—
207C4×D3—72no—
208C3×C8—72no—
209C3⋊C8C22×D712yescomplemented
210C3×C7C22×D811yescharacteristic, complemented, Hall
211C24—421no—
212C24—421no—
213C2×D4—31no—
214C2×D4—71no—
215C2×D4—214no—
216C2×D4—214no—
217C2×D4—211no—
218C2×D4—422no—
219C2×D4—422no—
220C2×D4—421no—
221C2×D4—421no—
222C2×D4D3×D711yescharacteristic, complemented
223C22×C4—211no—
224D8—34no—
225D8—74no—
226D8—214no—
227D8—214no—
228C2×C8—31no—
229C2×C8—71no—
230C2×C8—214no—
231C2×C7—42no—
232C2×C7—122no—
233C2×C7D3×D812yescomplemented
234C2×C7C2×D3×D411yescharacteristic
235D7—34no—
236D7—42no—
237D7—122no—
238C22×C3—22no—
239C22×C3—42no—
240C22×C3—142no—
241C22×C3—282no—
242C22×C3D4×D711yescharacteristic
243C2×D3—22no—
244C2×D3—42no—
245C2×D3—74no—
246C2×D3—72no—
247C2×D3—142no—
248C2×D3—288no—
249C2×D3—288no—
250C2×D3—282no—
251C3×C4D4×D711yescharacteristic
252C3×C4C23×D711yescharacteristic
253Dic3—72no—
254C23—21no—
255C23—61no—
256C23—141no—
257C23—211no—
258C23—424no—
259C23—424no—
260C23—421no—
261C23—844no—
262C23—844no—
263D4—21no—
264D4—32no—
265D4—61no—
266D4—72no—
267D4—141no—
268D4—212no—
269D4—424no—
270D4—424no—
271D4—421no—
272D4C2×D3×D712yescomplemented
273C2×C4—211no—
274C2×C4—212no—
275C2×C4—212no—
276C2×C4C2×D3×D711yescharacteristic
277C8—32no—
278C8—72no—
279C7C2×D3×D811yescharacteristic, complemented, Sylow
280C2×C3—42no—
281C2×C3—282no—
282C2×C3D7×D812yescomplemented
283C2×C3C2×D4×D711yescharacteristic
284D3—42no—
285D3—74no—
286D3—282no—
287C22—22no—
288C22—42no—
289C22—62no—
290C22—122no—
291C22—142no—
292C22—214no—
293C22—212no—
294C22—282no—
295C22—422no—
296C22—848no—
297C22—848no—
298C22—842no—
299C22#6669411yescharacteristic
300C4—212no—
301C4#6669411yescharacteristic
302C4C22×D3×D711yescharacteristic
303C3C2×D7×D811yescharacteristic, complemented, Sylow
304C2—42no—
305C2—122no—
306C2—214no—
307C2—282no—
308C2—842no—
309C2#1062104312yesdirect product factor
310C2C2×#6669411yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7C22×D8
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×D4×D21C2semidirect
2. C2×#66681C2semidirect
3. C2×C7×#213C2semidirect
4. C2×C3×#1141C2semidirect
5. #10621043C2direct (N × H)
6. C2×#66562C2semidirect
7. C2×#66561C2semidirect
8. C2×D21⋊C8C2semidirect
12. C2×C3×C7×D4C22semidirect
15. D4×D21C22semidirect
16. C2×D84C22semidirect
18. C2×C7×D12C22semidirect
19. C2×C3×D28C22semidirect
22. #66681C22semidirect
23. C7×#213C22semidirect
24. C2×C7×C3⋊C8C22semidirect
25. C3×#1141C22semidirect
26. C2×C3×C7⋊C8C22semidirect
27. #66562C22semidirect
28. #66561C22semidirect
29. D21⋊C8C22semidirect
34. C2×#1141D3semidirect
49. C3×C7×D4C23semidirect
54. D84C23semidirect
58. C7×D12C23semidirect
60. C3×D28C23semidirect
62. C3×C7⋊C8C23semidirect
63. C7×C3⋊C8C23semidirect
67. C2×C7×D4C2×D3semidirect
75. C2×D28C2×D3semidirect
80. #1141C2×D3semidirect
81. C2×C7⋊C8C2×D3semidirect
88. C2×#213D7semidirect
98. C2×D21D8semidirect
121. C7×D4C22×D3semidirect
129. D28C22×D3semidirect
133. C7⋊C8C22×D3semidirect
137. C2×C3×D4C2×D7semidirect
145. C2×D12C2×D7semidirect
150. #213C2×D7semidirect
151. C2×C3⋊C8C2×D7semidirect
155. C2×C3×C7C2×D8semidirect
158. D21C2×D8semidirect
197. C3×D4C22×D7semidirect
205. D12C22×D7semidirect
209. C3⋊C8C22×D7semidirect
210. C3×C7C22×D8semidirect
222. C2×D4D3×D7semidirect
233. C2×C7D3×D8semidirect
272. D4C2×D3×D7semidirect
279. C7C2×D3×D8semidirect
282. C2×C3D7×D8semidirect
303. C3C2×D7×D8semidirect
309. C2#10621043direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×D212
C2×#666812
C2×C7×#2132
C2×C3×#11412
#106210432
C2×#665622
C2×#665612
C2×D21⋊C82
C2×D7×D83
C2×D3×D87