Finite Group Databasebeta

← C2×#10621040 #13060096

Subgroups — C2×#10621040 #13060096

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×C3×#1159C211yesmaximal, characteristic, complemented
2C2×D7×D12C211yesmaximal, characteristic, complemented
3C2×#66632C211yesmaximal, characteristic, complemented
4C2×C7×#213C211yesmaximal, characteristic, complemented
5#10621040C218yesmaximal, direct product factor
6C2×#66566C211yesmaximal, characteristic, complemented
7C2×#66564C211yesmaximal, characteristic, complemented
8C2×#66557C211yesmaximal, characteristic, complemented
9C2×#6474—31nomaximal, Hall
10C22×D3×D7—21no—
11C2×C3×C7×D4C2211yescharacteristic, complemented
12C2×D84C2211yescharacteristic
13C2×C7×D12C2211yescharacteristic
14C2×C3×C22⋊D7—21no—
15C22×C3×Dic7—21no—
16C3×#1159—22no—
17C3×#1159C2214yescomplemented
18C2×C3×C4×D7C2211yescharacteristic
19C2×C3×Dic14C2211yescharacteristic, complemented
20C2×#4247—21no—
21D7×D12—22no—
22D7×D12C2214yescomplemented
23#66632C2214yescomplemented
24C2×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
25C7×#213C2214yescomplemented
26C2×C7×C3⋊C8C2211yescharacteristic, complemented
27#66566C2214yescomplemented
28#66564C2214yescomplemented
29#66557C2214yescomplemented
30C2×#1159D311yescharacteristic, complemented
31C2×D4×D7—31no—
32C2×C7×D8—31no—
33C2×#1142—31no—
34C2×#1141—31no—
35#6474—38no—
36C2×#1132—31no—
37C2×#1131—31no—
38C2×#949—71nomaximal, Hall
39C23×C3×C7—21no—
40C22×D21—21no—
41C22×D3×C7—21no—
42C22×C3×D7D411yescharacteristic
43C2×D3×D7—24no—
44C2×D3×D7—44no—
45C3×C7×D4—21no—
46C3×C7×D4C2312yescomplemented
47C2×C3×C4×C7C2311yescharacteristic
48D84—21no—
49D84C2312yes—
50C7×D12—21no—
51C7×D12C2312yes—
52C3×C22⋊D7—24no—
53C2×C3×Dic7—21no—
54C2×C3×Dic7—24no—
55C2×C3×Dic7D411yescharacteristic
56C3×C4×D7D412yes—
57C3×C4×D7C2312yes—
58C3×Dic14—21no—
59C3×Dic14C2312yescomplemented
60#4247—24no—
61C3⋊(C7⋊C8)C2312yescomplemented
62C7×C3⋊C8C2312yescomplemented
63C23×D7—61no—
64C2×C7×D4—31no—
65C2×C7×D4C2×D311yescharacteristic, complemented
66C2×C22⋊D7—21no—
67C2×C22⋊D7—61no—
68C22×Dic7—21no—
69#1159—22no—
70#1159C2×D314yescomplemented
71D4×D7—34no—
72D4×D7—62no—
73C2×D28—31no—
74C2×C4×D7C2×D311yescharacteristic
75C2×Dic14C2×D311yescharacteristic, complemented
76C7×D8—34no—
77C2×C7×C8—31no—
78#1142—34no—
79#1141—34no—
80C2×C7⋊C8—31no—
81#1132—34no—
82#1131—34no—
83C2×C3×#41—71no—
84C22×D12—71no—
85#949—78no—
86C2×#215—71no—
87C2×#215—71no—
88C2×#213—71no—
89C2×#213D711yescharacteristic, complemented
90C2×#208—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×C7—42no—
98C2×C3×D7—22no—
99C2×C3×D7C2×D412yes—
100D3×D7—48no—
101C3×C4×C7C2×D411yescharacteristic
102C3×C4×C7C2411yescharacteristic
103C3×Dic7—22no—
104C3×Dic7C2×D412yes—
105C2×AGL(1,ℤ/8)—211noSylow
106C23×C7—21no—
107C23×C7—61no—
108C22×D7—61no—
109C22×D7—64no—
110C22×D7—124no—
111C22×D7#6711yescharacteristic
112C7×D4—21no—
113C7×D4—32no—
114C7×D4—61no—
115C7×D4C22×D312yescomplemented
116C2×C4×C7C22×D311yescharacteristic
117C22⋊D7—24no—
118C22⋊D7—64no—
119C2×Dic7—21no—
120C2×Dic7—24no—
121C2×Dic7#6711yescharacteristic
122D28—32no—
123D28—61no—
124C4×D7#6712yes—
125C4×D7C22×D312yes—
126Dic14—21no—
127Dic14C22×D312yescomplemented
128C7×C8—32no—
129C7⋊C8—32no—
130C23×D3—141no—
131C3×#41—74no—
132C3×#41—142no—
133C2×C3×Q8—71no—
134C2×C3×D4—141no—
135C2×C3×D4C2×D711yescharacteristic, complemented
136C22×C3×C4—71no—
137C22×C3×C4—141no—
138C2×D12—71no—
139C2×D12—74no—
140C2×D12—142no—
141C2×D12—141no—
142C2×D12C2×D711yescharacteristic
143#215—74no—
144#215—74no—
145#213—74no—
146#213C2×D714yescomplemented
147#208—74no—
148C2×C3⋊C8—71no—
149C2×C3⋊C8C2×D711yescharacteristic, complemented
150C2×C3×C7—42no—
151C2×C3×C7AGL(1,ℤ/8)12yescomplemented
152C2×C3×C7C22×D411yescharacteristic
153D21—42no—
154C3×D7—22no—
155D3×C7—42no—
156C2×#41—71no—
157C22×D4—211no—
158AGL(1,ℤ/8)—218no—
159C2×SD16—211no—
160C2×SD16—211no—
161C2×D8—31no—
162C2×D8—211no—
163C2×M16—211no—
164C22×C7—22no—
165C22×C7—42no—
166C22×C7—62no—
167C22×C7—122no—
168C22×C7C2×#6711yescharacteristic
169C2×D7—22no—
170C2×D7—62no—
171C2×D7—128no—
172C2×D7—122no—
173C2×D7C2×#6712yes—
174C4×C7C2×#6711yescharacteristic
175C4×C7C23×D311yescharacteristic
176Dic7—22no—
177Dic7C2×#6712yes—
178C23×C3—21no—
179C23×C3—71no—
180C22×D3—21no—
181C22×D3—144no—
182C22×D3—141no—
183C22×D3—284no—
184C3×Q8—72no—
185C3×Q8—141no—
186C3×D4—21no—
187C3×D4—144no—
188C3×D4C22×D712yescomplemented
189C2×C3×C4—71no—
190C2×C3×C4—72no—
191C2×C3×C4—72no—
192C2×C3×C4—144no—
193C2×C3×C4—141no—
194C2×C3×C4C22×D711yescharacteristic
195D12—21no—
196D12—72no—
197D12—144no—
198D12—141no—
199D12C22×D712yes—
200C3⋊C8—72no—
201C3⋊C8C22×D712yescomplemented
202C3×C7C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
203C24—421no—
204#41—74no—
205#41—142no—
206C2×Q8—71no—
207C2×D4—31no—
208C2×D4—141no—
209C2×D4—214no—
210C2×D4—211no—
211C2×D4—422no—
212C2×D4—421no—
213C2×D4D3×D711yescharacteristic, complemented
214C22×C4—71no—
215C22×C4—141no—
216SD16—214no—
217SD16—214no—
218D8—34no—
219D8—214no—
220M16—214no—
221C2×C8—31no—
222C2×C8—211no—
223C2×C7—42no—
224C2×C7—122no—
225C2×C7#94912yescomplemented
226C2×C7C22×#6711yescharacteristic
227D7—22no—
228D7—122no—
229C22×C3—22no—
230C22×C3—42no—
231C22×C3—72no—
232C22×C3—142no—
233C22×C3D4×D711yescharacteristic
234C2×D3—22no—
235C2×D3—42no—
236C2×D3—142no—
237C2×D3—288no—
238C2×D3—282no—
239C3×C4—72no—
240C3×C4—142no—
241C3×C4D4×D711yescharacteristic
242C3×C4C23×D711yescharacteristic
243C23—21no—
244C23—61no—
245C23—71no—
246C23—424no—
247C23—421no—
248C23—844no—
249Q8—72no—
250Q8—141no—
251D4—21no—
252D4—32no—
253D4—61no—
254D4—144no—
255D4—212no—
256D4—424no—
257D4—421no—
258D4C2×D3×D712yescomplemented
259C2×C4—71no—
260C2×C4—72no—
261C2×C4—72no—
262C2×C4—144no—
263C2×C4—141no—
264C2×C4C2×D3×D711yescharacteristic
265C8—32no—
266C8—212no—
267C7C2×#94911yescharacteristic, complemented, Sylow
268C2×C3—42no—
269C2×C3—142no—
270C2×C3#647412yescomplemented
271C2×C3C2×D4×D711yescharacteristic
272D3—42no—
273D3—282no—
274C22—22no—
275C22—42no—
276C22—62no—
277C22—72no—
278C22—122no—
279C22—142no—
280C22—422no—
281C22—848no—
282C22—842no—
283C22D7×#6711yescharacteristic
284C4—72no—
285C4—142no—
286C4D7×#6711yescharacteristic
287C4C22×D3×D711yescharacteristic
288C3C2×#647411yescharacteristic, complemented, Sylow
289C2—42no—
290C2—122no—
291C2—142no—
292C2—842no—
293C2#1062104012yesdirect product factor
294C2C2×D7×#6711yescharacteristic
subgroupG/H
centerC22D7×#67
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×C3×#1159C2semidirect
2. C2×D7×D12C2semidirect
3. C2×#66632C2semidirect
4. C2×C7×#213C2semidirect
5. #10621040C2direct (N × H)
6. C2×#66566C2semidirect
7. C2×#66564C2semidirect
8. C2×#66557C2semidirect
11. C2×C3×C7×D4C22semidirect
17. C3×#1159C22semidirect
19. C2×C3×Dic14C22semidirect
22. D7×D12C22semidirect
23. #66632C22semidirect
24. C2×C3⋊(C7⋊C8)C22semidirect
25. C7×#213C22semidirect
26. C2×C7×C3⋊C8C22semidirect
27. #66566C22semidirect
28. #66564C22semidirect
29. #66557C22semidirect
30. C2×#1159D3semidirect
46. C3×C7×D4C23semidirect
59. C3×Dic14C23semidirect
61. C3⋊(C7⋊C8)C23semidirect
62. C7×C3⋊C8C23semidirect
65. C2×C7×D4C2×D3semidirect
70. #1159C2×D3semidirect
75. C2×Dic14C2×D3semidirect
89. C2×#213D7semidirect
115. C7×D4C22×D3semidirect
127. Dic14C22×D3semidirect
135. C2×C3×D4C2×D7semidirect
146. #213C2×D7semidirect
149. C2×C3⋊C8C2×D7semidirect
151. C2×C3×C7AGL(1,ℤ/8)semidirect
188. C3×D4C22×D7semidirect
201. C3⋊C8C22×D7semidirect
202. C3×C7C2×AGL(1,ℤ/8)semidirect
213. C2×D4D3×D7semidirect
225. C2×C7#949semidirect
258. D4C2×D3×D7semidirect
267. C7C2×#949semidirect
270. C2×C3#6474semidirect
288. C3C2×#6474semidirect
293. C2#10621040direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#11592
C2×D7×D122
C2×#666322
C2×C7×#2132
#106210402
C2×#665662
C2×#665642
C2×#665572
C2×#64743
C2×#9497