Finite Group Databasebeta

← C2×#10621042 #13060111

Subgroups — C2×#10621042 #13060111

Order 1344.

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

normal subgroups only

This group has 596 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×C7×#237C211yesmaximal, characteristic, complemented
2C2×D3×D28C211yesmaximal, characteristic, complemented
3C2×#66632C211yesmaximal, characteristic, complemented
4C2×C3×#1141C211yesmaximal, characteristic, complemented
5#10621042C218yesmaximal, direct product factor
6C2×#66568C211yesmaximal, characteristic, complemented
7C2×#66563C211yesmaximal, characteristic, complemented
8C2×#66558C211yesmaximal, characteristic, complemented
9C2×#6512—31nomaximal, Hall
10C22×D3×D7—21no—
11C2×C3×C7×D4C2211yescharacteristic, complemented
12C2×D84C2211yescharacteristic
13C2×C7×#67—21no—
14C22×C7×Dic3—21no—
15C7×#237—22no—
16C7×#237C2214yescomplemented
17C2×C4×D3×C7C2211yescharacteristic
18C2×C7×Dic6C2211yescharacteristic, complemented
19C2×C3×D28C2211yescharacteristic
20C2×#4246—21no—
21D3×D28—22no—
22D3×D28C2214yescomplemented
23#66632C2214yescomplemented
24C2×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
25C3×#1141C2214yescomplemented
26C2×C3×C7⋊C8C2211yescharacteristic, complemented
27#66568C2214yescomplemented
28#66563C2214yescomplemented
29#66558C2214yescomplemented
30C2×C7×#41—31no—
31C22×D28—31no—
32#6512—38no—
33C2×#1143—31no—
34C2×#1143—31no—
35C2×#1141—31no—
36C2×#1141D311yescharacteristic, complemented
37C2×#1136—31no—
38C2×#911—71nomaximal, Hall
39C23×C3×C7—21no—
40C22×D21—21no—
41C22×D3×C7D411yescharacteristic
42C22×C3×D7—21no—
43C2×D3×D7—24no—
44C2×D3×D7—44no—
45C3×C7×D4—21no—
46C3×C7×D4C2312yescomplemented
47C2×C3×C4×C7C2311yescharacteristic
48D84—21no—
49D84C2312yes—
50C7×#67—24no—
51C2×C7×Dic3—21no—
52C2×C7×Dic3—24no—
53C2×C7×Dic3D411yescharacteristic
54C4×D3×C7D412yes—
55C4×D3×C7C2312yes—
56C7×Dic6—21no—
57C7×Dic6C2312yescomplemented
58C3×D28—21no—
59C3×D28C2312yes—
60#4246—24no—
61C3⋊(C7⋊C8)C2312yescomplemented
62C3×C7⋊C8C2312yescomplemented
63C23×D7—61no—
64C7×#41—34no—
65C7×#41—62no—
66C2×C7×Q8—31no—
67C2×C7×D4—61no—
68C2×C7×D4C2×D311yescharacteristic, complemented
69C22×C4×C7—31no—
70C22×C4×C7—61no—
71C2×D28—31no—
72C2×D28—34no—
73C2×D28—62no—
74C2×D28—61no—
75C2×D28C2×D311yescharacteristic
76#1143—34no—
77#1143—34no—
78#1141—34no—
79#1141C2×D314yescomplemented
80#1136—34no—
81C2×C7⋊C8—31no—
82C2×C7⋊C8C2×D311yescharacteristic, complemented
83C2×#237D711yescharacteristic, complemented
84C2×D3×D4—71no—
85C2×C3×D8—71no—
86C2×#214—71no—
87C2×#213—71no—
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×D21—42no—
96C2×D3×C7—22no—
97C2×D3×C7C2×D412yes—
98C2×C3×D7—22no—
99C2×C3×D7—42no—
100D3×D7—48no—
101C3×C4×C7C2×D411yescharacteristic
102C3×C4×C7C2411yescharacteristic
103C7×Dic3—22no—
104C7×Dic3C2×D412yes—
105C2×AGL(1,ℤ/8)—211noSylow
106C23×C7—21no—
107C23×C7—31no—
108C22×D7—21no—
109C22×D7—64no—
110C22×D7—61no—
111C22×D7—124no—
112C7×Q8—32no—
113C7×Q8—61no—
114C7×D4—21no—
115C7×D4—64no—
116C7×D4C22×D312yescomplemented
117C2×C4×C7—31no—
118C2×C4×C7—32no—
119C2×C4×C7—32no—
120C2×C4×C7—64no—
121C2×C4×C7—61no—
122C2×C4×C7C22×D311yescharacteristic
123D28—21no—
124D28—32no—
125D28—64no—
126D28—61no—
127D28C22×D312yes—
128C7⋊C8—32no—
129C7⋊C8C22×D312yescomplemented
130C23×D3—141no—
131C2×C3×D4—71no—
132C2×C3×D4C2×D711yescharacteristic, complemented
133C2×#67—21no—
134C2×#67—141no—
135C22×Dic3—21no—
136#237—22no—
137#237C2×D714yescomplemented
138D3×D4—74no—
139D3×D4—142no—
140C2×D12—71no—
141C2×C4×D3C2×D711yescharacteristic
142C2×Dic6C2×D711yescharacteristic, complemented
143C3×D8—74no—
144C2×C3×C8—71no—
145#214—74no—
146#213—74no—
147C2×C3⋊C8—71no—
148#204—74no—
149ΓL(1,𝔽25)—74no—
150C2×C3×C7—42no—
151C2×C3×C7AGL(1,ℤ/8)12yescomplemented
152C2×C3×C7C22×D411yescharacteristic
153D21—42no—
154C3×D7—42no—
155D3×C7—22no—
156C2×#41—31no—
157C22×D4—211no—
158AGL(1,ℤ/8)—218no—
159C2×SD16—211no—
160C2×SD16—211no—
161C2×D8—71no—
162C2×D8—211no—
163C2×M16—211no—
164C22×C7—22no—
165C22×C7—32no—
166C22×C7—42no—
167C22×C7—62no—
168C22×C7D3×D411yescharacteristic
169C2×D7—22no—
170C2×D7—42no—
171C2×D7—62no—
172C2×D7—128no—
173C2×D7—122no—
174C4×C7—32no—
175C4×C7—62no—
176C4×C7D3×D411yescharacteristic
177C4×C7C23×D311yescharacteristic
178C23×C3—21no—
179C23×C3—141no—
180C22×D3—141no—
181C22×D3—144no—
182C22×D3—284no—
183C22×D3C22⋊D711yescharacteristic
184C3×D4—21no—
185C3×D4—72no—
186C3×D4—141no—
187C3×D4C22×D712yescomplemented
188C2×C3×C4C22×D711yescharacteristic
189#67—24no—
190#67—144no—
191C2×Dic3—21no—
192C2×Dic3—24no—
193C2×Dic3C22⋊D711yescharacteristic
194D12—72no—
195D12—141no—
196C4×D3C22⋊D712yes—
197C4×D3C22×D712yes—
198Dic6—21no—
199Dic6C22×D712yescomplemented
200C3×C8—72no—
201C3⋊C8—72no—
202C3×C7C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
203C24—421no—
204#41—34no—
205#41—62no—
206C2×Q8—31no—
207C2×D4—61no—
208C2×D4—71no—
209C2×D4—214no—
210C2×D4—211no—
211C2×D4—422no—
212C2×D4—421no—
213C2×D4D3×D711yescharacteristic, complemented
214C22×C4—31no—
215C22×C4—61no—
216SD16—214no—
217SD16—214no—
218D8—74no—
219D8—214no—
220M16—214no—
221C2×C8—71no—
222C2×C8—211no—
223C2×C7—42no—
224C2×C7—62no—
225C2×C7#91112yescomplemented
226C2×C7C2×D3×D411yescharacteristic
227D7—42no—
228D7—122no—
229C22×C3—22no—
230C22×C3—42no—
231C22×C3—142no—
232C22×C3—282no—
233C22×C3C2×C22⋊D711yescharacteristic
234C2×D3—22no—
235C2×D3—142no—
236C2×D3—288no—
237C2×D3—282no—
238C2×D3C2×C22⋊D712yes—
239C3×C4C2×C22⋊D711yescharacteristic
240C3×C4C23×D711yescharacteristic
241Dic3—22no—
242Dic3C2×C22⋊D712yes—
243C23—21no—
244C23—31no—
245C23—141no—
246C23—424no—
247C23—421no—
248C23—844no—
249Q8—32no—
250Q8—61no—
251D4—21no—
252D4—64no—
253D4—72no—
254D4—141no—
255D4—212no—
256D4—424no—
257D4—421no—
258D4C2×D3×D712yescomplemented
259C2×C4—31no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—64no—
263C2×C4—61no—
264C2×C4C2×D3×D711yescharacteristic
265C8—72no—
266C8—212no—
267C7C2×#91111yescharacteristic, complemented, Sylow
268C2×C3—42no—
269C2×C3—282no—
270C2×C3#651212yescomplemented
271C2×C3C22×C22⋊D711yescharacteristic
272D3—22no—
273D3—282no—
274C22—22no—
275C22—32no—
276C22—42no—
277C22—62no—
278C22—142no—
279C22—282no—
280C22—422no—
281C22—848no—
282C22—842no—
283C22D3×C22⋊D711yescharacteristic
284C4—32no—
285C4—62no—
286C4D3×C22⋊D711yescharacteristic
287C4C22×D3×D711yescharacteristic
288C3C2×#651211yescharacteristic, complemented, Sylow
289C2—42no—
290C2—62no—
291C2—282no—
292C2—842no—
293C2#1062104212yesdirect product factor
294C2C2×D3×C22⋊D711yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
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×C7×#237C2semidirect
2. C2×D3×D28C2semidirect
3. C2×#66632C2semidirect
4. C2×C3×#1141C2semidirect
5. #10621042C2direct (N × H)
6. C2×#66568C2semidirect
7. C2×#66563C2semidirect
8. C2×#66558C2semidirect
11. C2×C3×C7×D4C22semidirect
16. C7×#237C22semidirect
18. C2×C7×Dic6C22semidirect
22. D3×D28C22semidirect
23. #66632C22semidirect
24. C2×C3⋊(C7⋊C8)C22semidirect
25. C3×#1141C22semidirect
26. C2×C3×C7⋊C8C22semidirect
27. #66568C22semidirect
28. #66563C22semidirect
29. #66558C22semidirect
36. C2×#1141D3semidirect
46. C3×C7×D4C23semidirect
57. C7×Dic6C23semidirect
61. C3⋊(C7⋊C8)C23semidirect
62. C3×C7⋊C8C23semidirect
68. C2×C7×D4C2×D3semidirect
79. #1141C2×D3semidirect
82. C2×C7⋊C8C2×D3semidirect
83. C2×#237D7semidirect
116. C7×D4C22×D3semidirect
129. C7⋊C8C22×D3semidirect
132. C2×C3×D4C2×D7semidirect
137. #237C2×D7semidirect
142. C2×Dic6C2×D7semidirect
151. C2×C3×C7AGL(1,ℤ/8)semidirect
187. C3×D4C22×D7semidirect
199. Dic6C22×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#6512semidirect
288. C3C2×#6512semidirect
293. C2#10621042direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#2372
C2×D3×D282
C2×#666322
C2×C3×#11412
#106210422
C2×#665682
C2×#665632
C2×#665582
C2×#65123
C2×#9117