Finite Group Databasebeta

← #13059343

Subgroups — #13059343

Order 1344.

536 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 126 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
1#10621582C212yesmaximal, complemented
2#10621402C211yesmaximal, characteristic, complemented
3C7×#919C211yesmaximal, characteristic, complemented
4C3×#6482C211yesmaximal, characteristic, complemented
5#10621068C212yesmaximal, complemented
6#10621066C212yesmaximal, complemented
7#10621064C212yesmaximal, complemented
8#10620827C211yesmaximal, characteristic, complemented
9#10620818C211yesmaximal, characteristic, complemented
10D3×D56C211yesmaximal, characteristic, complemented
11D7×D24C211yesmaximal, characteristic, complemented
12#90460—31nomaximal, Hall
13#66732C2212yescomplemented
14C7×#239C2212yescomplemented
15C3×#1161C2212yescomplemented
16#66681—22no—
17#66681C2212yescomplemented
18D3×D28—22no—
19D3×D28C2212yescomplemented
20D7×D12—22no—
21D7×D12C2212yescomplemented
22#66675—22no—
23#66675C2211yescharacteristic
24C3×C7×Q16C2211yescharacteristic, complemented
25#66634C2212yescomplemented
26D168C2211yescharacteristic, complemented
27C8×D21C2211yescharacteristic, complemented
28C7×#215C2212yescomplemented
29C7×D24C2211yescharacteristic, complemented
30D3×C7×C8C2211yescharacteristic, complemented
31C3×#1143C2212yescomplemented
32C3×D56C2211yescharacteristic, complemented
33C3×C8×D7C2211yescharacteristic, complemented
34#66562C2212yescomplemented
35#66561C2212yescomplemented
36#66560C2212yescomplemented
37#66559C2211yescharacteristic, complemented
38#66558C2211yescharacteristic, complemented
39#66557C2211yescharacteristic, complemented
40#6553—32no—
41C7×#135—31no—
42#6512—32no—
43#6482—31no—
44#6482D311yescharacteristic, complemented
45#6477—32no—
46#6477—32no—
47D7×D8—31no—
48D7×D8—31no—
49C2×D56—31no—
50#6464—31no—
51#5769—71nomaximal, Hall
52C2×D3×D7—22no—
53C2×D3×D7—22no—
54C2×D3×D7—22no—
55C3×C7×Q8C2312yes—
56D84—22no—
57D84C2312yes—
58C4×D21—22no—
59C4×D21C2311yescharacteristic
60C7×D12—22no—
61C7×D12C2312yes—
62C4×D3×C7—22no—
63C4×D3×C7C2311yescharacteristic
64C3×D28—22no—
65C3×D28C2312yes—
66C3×C4×D7—22no—
67C3×C4×D7C2311yescharacteristic
68#4248D411yescharacteristic
69#4247—22no—
70#4247D411yescharacteristic
71#4246—22no—
72#4246D411yescharacteristic
73#4245—22no—
74#4245D411yescharacteristic
75C3×C7×C8C2311yescharacteristic
76C3⋊(C7⋊C8)C2311yescharacteristic
77C3×C7⋊C8C2311yescharacteristic
78C7×C3⋊C8C2311yescharacteristic
79C7×#41—32no—
80#1161—32no—
81#1161C2×D312yescomplemented
82D4×D7—32no—
83D4×D7—32no—
84D4×D7—62no—
85D4×D7—62no—
86#1157—31no—
87#1157—62no—
88C2×D28—32no—
89C2×D28—62no—
90C7×Q16C2×D311yescharacteristic, complemented
91C7×SD16—32no—
92C7×D8—31no—
93C2×C7×C8—31no—
94#1143—32no—
95#1143C2×D312yescomplemented
96#1141—32no—
97#1141—32no—
98#1136—31no—
99D56—32no—
100D56—31no—
101D56C2×D311yescharacteristic, complemented
102#1131—31no—
103#1131—31no—
104C8×D7—31no—
105C8×D7C2×D311yescharacteristic, complemented
106#1009—72no—
107C3×#135—71no—
108#949—72no—
109#919—71no—
110#919D711yescharacteristic, complemented
111#914—72no—
112#914—72no—
113D3×D8—71no—
114D3×D8—71no—
115C2×D24—71no—
116#901—71no—
117C2×D21—22no—
118C2×D21C2×D411yescharacteristic
119C2×D3×C7—22no—
120C2×D3×C7C2×D411yescharacteristic
121C2×C3×D7—22no—
122C2×C3×D7C2×D411yescharacteristic
123D3×D7—82no—
124D3×D7—82no—
125D3×D7—82no—
126C3×C4×C7—22no—
127C3×C4×C7C2411yescharacteristic
128Dic21C2×D411yescharacteristic
129C3×Dic7C2×D411yescharacteristic
130C7×Dic3C2×D411yescharacteristic
131#576—211noSylow
132C22×D7—62no—
133C22×D7—62no—
134C22×D7—62no—
135C7×Q8C22×D312yes—
136C7×D4—32no—
137C7×D4—62no—
138C2×C4×C7—31no—
139C2×C4×C7—62no—
140C22⋊D7—31no—
141C22⋊D7—31no—
142C22⋊D7—62no—
143C22⋊D7—62no—
144D28—22no—
145D28—31no—
146D28—32no—
147D28—62no—
148D28—62no—
149D28C22×D312yes—
150C4×D7—22no—
151C4×D7—31no—
152C4×D7—62no—
153C4×D7C22×D311yescharacteristic
154Dic14—31no—
155C7×C8—31no—
156C7×C8C22×D311yescharacteristic
157C7⋊C8—31no—
158C7⋊C8C22×D311yescharacteristic
159C3×#41—72no—
160#239—72no—
161#239C2×D712yescomplemented
162D3×D4—72no—
163D3×D4—72no—
164D3×D4—142no—
165D3×D4—142no—
166#235—71no—
167#235—142no—
168C2×D12—72no—
169C2×D12—142no—
170C3×Q16C2×D711yescharacteristic, complemented
171C3×SD16—72no—
172C3×D8—71no—
173C2×C3×C8—71no—
174#215—72no—
175#215C2×D712yescomplemented
176#213—72no—
177#213—72no—
178#208—71no—
179D24—72no—
180D24—71no—
181D24C2×D711yescharacteristic, complemented
182ΓL(1,𝔽25)—71no—
183ΓL(1,𝔽25)—71no—
184D3×C8—71no—
185D3×C8C2×D711yescharacteristic, complemented
186C2×C3×C7C22×D411yescharacteristic
187D21—21no—
188D21—42no—
189C3×D7—21no—
190C3×D7—42no—
191D3×C7—21no—
192D3×C7—42no—
193H(5,𝔽2)—212no—
194AGL(1,ℤ/8)—212no—
195AGL(1,ℤ/8)—212no—
196AGL(1,ℤ/8)—212no—
197#135—31no—
198#135—71no—
199#135—211no—
200C2×D8—211no—
201C2×D8—211no—
202C2×D8—211no—
203#131—211no—
204C22×C7—31no—
205C22×C7—62no—
206C2×D7—22no—
207C2×D7—31no—
208C2×D7—62no—
209C2×D7—242no—
210C2×D7—242no—
211C2×D7—242no—
212C2×D7D3×D411yescharacteristic
213C4×C7—22no—
214C4×C7—31no—
215C4×C7C23×D311yescharacteristic
216Dic7—31no—
217Dic7D3×D411yescharacteristic
218C22×D3—142no—
219C22×D3—142no—
220C22×D3—142no—
221C3×Q8C22×D712yes—
222C3×D4—72no—
223C3×D4—142no—
224C2×C3×C4—71no—
225C2×C3×C4—142no—
226#67—71no—
227#67—71no—
228#67—142no—
229#67—142no—
230D12—22no—
231D12—71no—
232D12—72no—
233D12—142no—
234D12—142no—
235D12C22×D712yes—
236C4×D3—22no—
237C4×D3—71no—
238C4×D3—142no—
239C4×D3C22×D711yescharacteristic
240Dic6—71no—
241C3×C8—71no—
242C3×C8C22×D711yescharacteristic
243C3⋊C8—71no—
244C3⋊C8C22×D711yescharacteristic
245C3×C7#57611yescharacteristic, complemented, Hall
246#41—32no—
247#41—72no—
248#41—211no—
249#41—212no—
250#41—422no—
251C2×D4—212no—
252C2×D4—212no—
253C2×D4—212no—
254C2×D4—422no—
255C2×D4—422no—
256C2×D4—422no—
257Q16D3×D711yescharacteristic, complemented
258SD16—32no—
259SD16—72no—
260SD16—212no—
261D8—31no—
262D8—71no—
263D8—212no—
264D8—212no—
265D8—212no—
266D8—211no—
267M16—211no—
268M16—211no—
269M16—211no—
270C2×C8—31no—
271C2×C8—71no—
272C2×C8—211no—
273C2×C7—61no—
274C2×C7—122no—
275C2×C7C2×D3×D411yescharacteristic
276D7—21no—
277D7—42no—
278D7—61no—
279D7—122no—
280C22×C3—71no—
281C22×C3—142no—
282C2×D3—22no—
283C2×D3—71no—
284C2×D3—142no—
285C2×D3—562no—
286C2×D3—562no—
287C2×D3—562no—
288C2×D3D4×D711yescharacteristic
289C3×C4—22no—
290C3×C4—71no—
291C3×C4C23×D711yescharacteristic
292Dic3—71no—
293Dic3D4×D711yescharacteristic
294C23—422no—
295C23—422no—
296C23—422no—
297Q8—211no—
298Q8C2×D3×D712yes—
299D4—32no—
300D4—62no—
301D4—72no—
302D4—142no—
303D4—211no—
304D4—211no—
305D4—211no—
306D4—212no—
307D4—422no—
308D4—422no—
309D4—422no—
310D4—422no—
311C2×C4—31no—
312C2×C4—62no—
313C2×C4—71no—
314C2×C4—142no—
315C2×C4—211no—
316C2×C4—422no—
317C8—31no—
318C8—71no—
319C8—211no—
320C8C2×D3×D711yescharacteristic
321C7#576911yescharacteristic, complemented, Sylow
322C2×C3—141no—
323C2×C3—282no—
324C2×C3C2×D4×D711yescharacteristic
325D3—21no—
326D3—42no—
327D3—141no—
328D3—282no—
329C22—31no—
330C22—62no—
331C22—71no—
332C22—142no—
333C22—211no—
334C22—422no—
335C22—1682no—
336C22—1682no—
337C22—1682no—
338C4—22no—
339C4—31no—
340C4—71no—
341C4—211no—
342C4C22×D3×D711yescharacteristic
343C3#9046011yescharacteristic, complemented, Sylow
344C2—61no—
345C2—122no—
346C2—141no—
347C2—282no—
348C2—421no—
349C2—842no—
350C2D3×D4×D711yescharacteristic
subgroupG/H
centerC2D3×D4×D7
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC3×C7×Q16C22
socleC2×C3×C7C22×D4
hypercenterQ16D3×D7
hypocenterC3×C7#576
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #10621582C2semidirect
2. #10621402C2semidirect
3. C7×#919C2semidirect
4. C3×#6482C2semidirect
5. #10621068C2semidirect
6. #10621066C2semidirect
7. #10621064C2semidirect
8. #10620827C2semidirect
9. #10620818C2semidirect
10. D3×D56C2semidirect
11. D7×D24C2semidirect
13. #66732C22semidirect
14. C7×#239C22semidirect
15. C3×#1161C22semidirect
17. #66681C22semidirect
19. D3×D28C22semidirect
21. D7×D12C22semidirect
24. C3×C7×Q16C22semidirect
25. #66634C22semidirect
26. D168C22semidirect
27. C8×D21C22semidirect
28. C7×#215C22semidirect
29. C7×D24C22semidirect
30. D3×C7×C8C22semidirect
31. C3×#1143C22semidirect
32. C3×D56C22semidirect
33. C3×C8×D7C22semidirect
34. #66562C22semidirect
35. #66561C22semidirect
36. #66560C22semidirect
37. #66559C22semidirect
38. #66558C22semidirect
39. #66557C22semidirect
44. #6482D3semidirect
81. #1161C2×D3semidirect
90. C7×Q16C2×D3semidirect
95. #1143C2×D3semidirect
101. D56C2×D3semidirect
105. C8×D7C2×D3semidirect
110. #919D7semidirect
161. #239C2×D7semidirect
170. C3×Q16C2×D7semidirect
175. #215C2×D7semidirect
181. D24C2×D7semidirect
185. D3×C8C2×D7semidirect
245. C3×C7#576semidirect
257. Q16D3×D7semidirect
321. C7#5769semidirect
343. C3#90460semidirect

Maximal subgroups

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

subgroupindex
#106215822
#106214022
C7×#9192
C3×#64822
#106210682
#106210662
#106210642
#106208272
#106208182
D3×D562
D7×D242
#904603
#57697