Finite Group Databasebeta

← #13059282

Subgroups — #13059282

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#10621572C212yesmaximal, complemented
2#10621395C211yesmaximal, characteristic, complemented
3D3×C7×D8C211yesmaximal, characteristic, complemented
4C3×D7×D8C211yesmaximal, characteristic, complemented
5#10621056C212yesmaximal, complemented
6#10621051C212yesmaximal, complemented
7#10621044C212yesmaximal, complemented
8#10620835C211yesmaximal, characteristic, complemented
9#10620832C211yesmaximal, characteristic, complemented
10#10620827C211yesmaximal, characteristic, complemented
11#10620818C211yesmaximal, characteristic, complemented
12#90448—31nomaximal, Hall
13#66730C2212yescomplemented
14D3×C7×D4C2212yescomplemented
15C3×D4×D7C2212yescomplemented
16D3×C22⋊D7—22no—
17D7×#67—22no—
18C2×#4245—22no—
19#66684—22no—
20#66681C2212yescomplemented
21#66676C2212yescomplemented
22#66675C2211yescharacteristic
23#66669C2212yescomplemented
24C3×C7×D8C2211yescharacteristic, complemented
25#66633C2212yescomplemented
26Dic84C2211yescharacteristic, complemented
27C8×D21C2211yescharacteristic, complemented
28C7×#213C2212yescomplemented
29C7×D24C2211yescharacteristic, complemented
30D3×C7×C8C2211yescharacteristic, complemented
31C3×#1141C2212yescomplemented
32C3×D56C2211yescharacteristic, complemented
33C3×C8×D7C2211yescharacteristic, complemented
34#66567C2212yescomplemented
35#66565C2212yescomplemented
36#66560C2212yescomplemented
37#66559C2211yescharacteristic, complemented
38#66558C2211yescharacteristic, complemented
39#66557C2211yescharacteristic, complemented
40#6548—32no—
41C2×C7×D8—31no—
42#6495—32no—
43#6475—31no—
44#6475—31no—
45#6474—32no—
46#6474—32no—
47D7×D8—31no—
48D7×D8D311yescharacteristic, complemented
49#6467—31no—
50#6464—31no—
51#5757—71nomaximal, Hall
52C22×D3×C7—22no—
53C22×C3×D7—22no—
54C2×D3×D7—22no—
55C3×C7×D4C2312yes—
56#4268—22no—
57C2×Dic21—22no—
58C4×D21C2311yescharacteristic
59Dic42C2312yes—
60C7×#67—22no—
61C7×D12C2312yes—
62C4×D3×C7C2311yescharacteristic
63C3×C22⋊D7—22no—
64C3×D28C2312yes—
65C3×C4×D7C2311yescharacteristic
66#4248D411yescharacteristic
67#4247D411yescharacteristic
68#4246D411yescharacteristic
69#4245—22no—
70#4245—22no—
71#4245—22no—
72#4245D411yescharacteristic
73D3×Dic7—22no—
74Dic3×D7—22no—
75C3×C7×C8C2311yescharacteristic
76C3⋊(C7⋊C8)C2311yescharacteristic
77C3×C7⋊C8C2311yescharacteristic
78C7×C3⋊C8C2311yescharacteristic
79C2×C7×D4—32no—
80C2×C22⋊D7—62no—
81C2×C22⋊D7—62no—
82#1159—32no—
83#1159—32no—
84#1159—62no—
85D4×D7—32no—
86D4×D7—62no—
87D4×D7C2×D312yescomplemented
88#1157—31no—
89#1157—32no—
90C7×D8—32no—
91C7×D8—31no—
92C7×D8C2×D311yescharacteristic, complemented
93C2×C7×C8—31no—
94#1142—32no—
95#1142—32no—
96#1141—32no—
97#1141C2×D312yescomplemented
98#1136—31no—
99Dic28—31no—
100D56C2×D311yescharacteristic, complemented
101#1132—32no—
102#1131—31no—
103#1131—31no—
104C8×D7—31no—
105C8×D7C2×D311yescharacteristic, complemented
106#1004—72no—
107C2×C3×D8—71no—
108#932—72no—
109#912—71no—
110#912—71no—
111#911—72no—
112#911—72no—
113D3×D8—71no—
114D3×D8D711yescharacteristic, complemented
115#904—71no—
116#901—71no—
117C22×C3×C7—22no—
118C2×D21C2×D411yescharacteristic
119C2×D3×C7—22no—
120C2×D3×C7—82no—
121C2×D3×C7C2×D411yescharacteristic
122C2×C3×D7—22no—
123C2×C3×D7—82no—
124C2×C3×D7C2×D411yescharacteristic
125D3×D7—82no—
126C3×C4×C7C2411yescharacteristic
127Dic21—22no—
128Dic21C2×D411yescharacteristic
129C3×Dic7C2×D411yescharacteristic
130C7×Dic3C2×D411yescharacteristic
131#576—211noSylow
132C23×C7—62no—
133C22×D7—22no—
134C22×D7—62no—
135C7×D4—32no—
136C7×D4—62no—
137C7×D4C22×D312yes—
138C2×C4×C7—31no—
139C22⋊D7—22no—
140C22⋊D7—31no—
141C22⋊D7—31no—
142C22⋊D7—62no—
143C22⋊D7—62no—
144C22⋊D7—62no—
145C22⋊D7—62no—
146C2×Dic7—62no—
147C2×Dic7—62no—
148D28—31no—
149D28C22×D312yes—
150C4×D7—31no—
151C4×D7—62no—
152C4×D7C22×D311yescharacteristic
153Dic14—31no—
154Dic14—32no—
155C7×C8—31no—
156C7×C8C22×D311yescharacteristic
157C7⋊C8—31no—
158C7⋊C8C22×D311yescharacteristic
159C2×C3×D4—72no—
160C2×#67—142no—
161C2×#67—142no—
162#237—72no—
163#237—72no—
164#237—142no—
165D3×D4—72no—
166D3×D4—142no—
167D3×D4C2×D712yescomplemented
168#235—71no—
169#235—72no—
170C3×D8—72no—
171C3×D8—71no—
172C3×D8C2×D711yescharacteristic, complemented
173C2×C3×C8—71no—
174#214—72no—
175#214—72no—
176#213—72no—
177#213C2×D712yescomplemented
178#208—71no—
179Dic12—71no—
180D24C2×D711yescharacteristic, complemented
181#204—72no—
182ΓL(1,𝔽25)—71no—
183ΓL(1,𝔽25)—71no—
184D3×C8—71no—
185D3×C8C2×D711yescharacteristic, complemented
186C2×C3×C7—42no—
187C2×C3×C7C22×D411yescharacteristic
188D21—21no—
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—211no—
198#135—211no—
199#135—211no—
200C2×D8—31no—
201C2×D8—71no—
202C2×D8—211no—
203#131—211no—
204C22×C7—22no—
205C22×C7—31no—
206C22×C7—62no—
207C22×C7—242no—
208C2×D7—22no—
209C2×D7—31no—
210C2×D7—82no—
211C2×D7—242no—
212C2×D7D3×D411yescharacteristic
213C4×C7—31no—
214C4×C7C23×D311yescharacteristic
215Dic7—31no—
216Dic7—62no—
217Dic7D3×D411yescharacteristic
218C23×C3—142no—
219C22×D3—22no—
220C22×D3—142no—
221C3×D4—72no—
222C3×D4—142no—
223C3×D4C22×D712yes—
224C2×C3×C4—71no—
225#67—22no—
226#67—71no—
227#67—71no—
228#67—142no—
229#67—142no—
230#67—142no—
231#67—142no—
232C2×Dic3—142no—
233C2×Dic3—142no—
234D12—71no—
235D12C22×D712yes—
236C4×D3—71no—
237C4×D3—142no—
238C4×D3C22×D711yescharacteristic
239Dic6—71no—
240Dic6—72no—
241C3×C8—71no—
242C3×C8C22×D711yescharacteristic
243C3⋊C8—71no—
244C3⋊C8C22×D711yescharacteristic
245C3×C7#57611yescharacteristic, complemented, Hall
246#41—211no—
247#41—212no—
248#41—212no—
249#41—212no—
250#41—422no—
251C2×D4—32no—
252C2×D4—72no—
253C2×D4—212no—
254C2×D4—422no—
255C2×D4—422no—
256C2×D4—422no—
257Q16—211no—
258SD16—212no—
259SD16—212no—
260SD16—212no—
261D8—32no—
262D8—31no—
263D8—72no—
264D8—71no—
265D8—212no—
266D8D3×D711yescharacteristic, complemented
267M16—211no—
268M16—211no—
269M16—211no—
270C2×C8—31no—
271C2×C8—71no—
272C2×C8—211no—
273C2×C7—42no—
274C2×C7—61no—
275C2×C7—122no—
276C2×C7C2×D3×D411yescharacteristic
277D7—21no—
278D7—42no—
279D7—61no—
280C22×C3—22no—
281C22×C3—71no—
282C22×C3—142no—
283C22×C3—562no—
284C2×D3—22no—
285C2×D3—71no—
286C2×D3—82no—
287C2×D3—562no—
288C2×D3D4×D711yescharacteristic
289C3×C4—71no—
290C3×C4C23×D711yescharacteristic
291Dic3—71no—
292Dic3—142no—
293Dic3D4×D711yescharacteristic
294C23—62no—
295C23—142no—
296C23—422no—
297Q8—211no—
298Q8—212no—
299D4—32no—
300D4—62no—
301D4—72no—
302D4—142no—
303D4—211no—
304D4—211no—
305D4—211no—
306D4—422no—
307D4—422no—
308D4—422no—
309D4—422no—
310D4C2×D3×D712yes—
311C2×C4—31no—
312C2×C4—71no—
313C2×C4—211no—
314C2×C4—422no—
315C2×C4—422no—
316C2×C4—422no—
317C8—31no—
318C8—71no—
319C8—211no—
320C8C2×D3×D711yescharacteristic
321C7#575711yescharacteristic, complemented, Sylow
322C2×C3—42no—
323C2×C3—141no—
324C2×C3—282no—
325C2×C3C2×D4×D711yescharacteristic
326D3—21no—
327D3—42no—
328D3—141no—
329C22—22no—
330C22—31no—
331C22—62no—
332C22—71no—
333C22—142no—
334C22—211no—
335C22—242no—
336C22—562no—
337C22—1682no—
338C4—31no—
339C4—71no—
340C4—211no—
341C4—422no—
342C4C22×D3×D711yescharacteristic
343C3#9044811yescharacteristic, complemented, Sylow
344C2—42no—
345C2—61no—
346C2—122no—
347C2—141no—
348C2—282no—
349C2—421no—
350C2D3×D4×D711yescharacteristic
subgroupG/H
centerC2D3×D4×D7
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC3×C7×D8C22
socleC2×C3×C7C22×D4
hypercenterD8D3×D7
hypocenterC3×C7#576
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #10621572C2semidirect
2. #10621395C2semidirect
3. D3×C7×D8C2semidirect
4. C3×D7×D8C2semidirect
5. #10621056C2semidirect
6. #10621051C2semidirect
7. #10621044C2semidirect
8. #10620835C2semidirect
9. #10620832C2semidirect
10. #10620827C2semidirect
11. #10620818C2semidirect
13. #66730C22semidirect
14. D3×C7×D4C22semidirect
15. C3×D4×D7C22semidirect
20. #66681C22semidirect
21. #66676C22semidirect
23. #66669C22semidirect
24. C3×C7×D8C22semidirect
25. #66633C22semidirect
26. Dic84C22semidirect
27. C8×D21C22semidirect
28. C7×#213C22semidirect
29. C7×D24C22semidirect
30. D3×C7×C8C22semidirect
31. C3×#1141C22semidirect
32. C3×D56C22semidirect
33. C3×C8×D7C22semidirect
34. #66567C22semidirect
35. #66565C22semidirect
36. #66560C22semidirect
37. #66559C22semidirect
38. #66558C22semidirect
39. #66557C22semidirect
48. D7×D8D3semidirect
87. D4×D7C2×D3semidirect
92. C7×D8C2×D3semidirect
97. #1141C2×D3semidirect
100. D56C2×D3semidirect
105. C8×D7C2×D3semidirect
114. D3×D8D7semidirect
167. D3×D4C2×D7semidirect
172. C3×D8C2×D7semidirect
177. #213C2×D7semidirect
180. D24C2×D7semidirect
185. D3×C8C2×D7semidirect
245. C3×C7#576semidirect
266. D8D3×D7semidirect
321. C7#5757semidirect
343. C3#90448semidirect

Maximal subgroups

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

subgroupindex
#106215722
#106213952
D3×C7×D82
C3×D7×D82
#106210562
#106210512
#106210442
#106208352
#106208322
#106208272
#106208182
#904483
#57577