Finite Group Databasebeta

← #13060127

Subgroups — #13060127

Order 1344.

536 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 128 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#10621657C211yesmaximal, characteristic, complemented
2#10621564C211yesmaximal, characteristic, complemented
3C7×#932C211yesmaximal, characteristic, complemented
4C2×C3×#1141C211yesmaximal, characteristic, complemented
5#10621060C212yesmaximal, complemented
6#10621059C212yesmaximal, complemented
7#10621044C212yesmaximal, complemented
8#10621043C212yesmaximal, complemented
9#10620874C211yesmaximal, characteristic, complemented
10#10620865C211yesmaximal, characteristic, complemented
11#10620852C211yesmaximal, characteristic, complemented
12#90465—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic, complemented
14C2×#4268—22no—
15#66730—21no—
16#66730C2212yescomplemented
17D4×D21—21no—
18D4×D21C2212yescomplemented
19#66728C2211yescharacteristic
20#66728C2211yescharacteristic, complemented
21C7×#235C2211yescharacteristic, complemented
22C2×C3×D28C2211yescharacteristic, complemented
23D7×#67—22no—
24#66681C2212yescomplemented
25D3×D28—21no—
26#66670C2212yescomplemented
27#66669—21no—
28C7×#214C2212yescomplemented
29C7×#213C2212yescomplemented
30C7×#208C2211yescharacteristic, complemented
31C3×#1141C2212yescomplemented
32C3×#1141C2212yescomplemented
33C2×C3×C7⋊C8C2211yescharacteristic, complemented
34#66571C2211yescharacteristic, complemented
35#66568C2211yescharacteristic, complemented
36#66567C2211yescharacteristic, complemented
37#66565C2212yescomplemented
38#66562C2211yescharacteristic, complemented
39#66561C2212yescomplemented
40#66559C2212yescomplemented
41D21⋊C8C2212yescomplemented
42#6553—31no—
43#6548—31no—
44C7×AGL(1,ℤ/8)—31no—
45#6513—31no—
46C2×#1141D311yescharacteristic, complemented
47#6479—32no—
48#6477—32no—
49#6474—32no—
50D7×D8—32no—
51#6471—31no—
52#6470—31no—
53#5757—71nomaximal, Hall
54C23×C3×C7—21no—
55C22×D21—22no—
56C22×C3×D7—21no—
57C2×D3×D7—22no—
58C3×C7×D4—21no—
59C3×C7×D4C2312yes—
60C2×C3×C4×C7C2311yescharacteristic
61#4268—21no—
62#4268—22no—
63#4268—22no—
64#4268D412yes—
65C2×Dic21—22no—
66D84D411yescharacteristic
67D84C2311yescharacteristic
68C4×D21—21no—
69C4×D21C2312yes—
70Dic42D411yescharacteristic
71Dic42C2311yescharacteristic
72C7×#67—21no—
73C7×D12C2311yescharacteristic
74C4×D3×C7—21no—
75C7×Dic6C2311yescharacteristic
76C3×D28—21no—
77C3×D28C2312yes—
78#4246—22no—
79#4245—22no—
80Dic3×D7—22no—
81C3×C7⋊C8C2311yescharacteristic
82C3×C7⋊C8C2311yescharacteristic
83C7×C3⋊C8C2312yes—
84C7×#41—31no—
85C2×C7×D4C2×D311yescharacteristic, complemented
86C2×C22⋊D7—62no—
87#1161—32no—
88#1159—32no—
89#1159—61no—
90D4×D7—32no—
91D4×D7—32no—
92D4×D7—61no—
93D4×D7—62no—
94#1157—31no—
95#1157—31no—
96#1157—61no—
97C2×D28—61no—
98C2×D28C2×D311yescharacteristic, complemented
99C7×SD16—32no—
100C7×D8—32no—
101C7×M16—31no—
102#1144—31no—
103#1143—31no—
104#1142—31no—
105#1141—31no—
106#1141C2×D312yescomplemented
107#1141C2×D312yescomplemented
108C2×C7⋊C8C2×D311yescharacteristic, complemented
109D56—32no—
110#1132—32no—
111#1131—32no—
112C8×D7—32no—
113#1004—71no—
114#1004—71no—
115C2×C3×D8—71no—
116#932—71no—
117#932D711yescharacteristic, complemented
118#912—72no—
119#911—72no—
120#911—72no—
121D3×D8—72no—
122#904—71no—
123#901—71no—
124C22×C3×C7—22no—
125C22×C3×C7—81no—
126C22×C3×C7C2×D411yescharacteristic
127C2×D21—21no—
128C2×D21—82no—
129C2×D21C2×D412yes—
130C2×D3×C7—21no—
131C2×C3×D7—22no—
132C2×C3×D7—81no—
133D3×D7—82no—
134C3×C4×C7C2×D411yescharacteristic
135C3×C4×C7C2411yescharacteristic
136Dic21—21no—
137Dic21C2×D412yes—
138C7×Dic3—21no—
139#576—211noSylow
140C23×C7—21no—
141C22×D7—21no—
142C22×D7—62no—
143C22×D7—62no—
144C7×Q8—31no—
145C7×D4—21no—
146C7×D4—31no—
147C7×D4—61no—
148C7×D4C22×D312yes—
149C2×C4×C7—61no—
150C2×C4×C7C22×D311yescharacteristic
151C22⋊D7—32no—
152C22⋊D7—62no—
153C22⋊D7—62no—
154C22⋊D7—62no—
155C22⋊D7—61no—
156C2×Dic7—62no—
157D28—21no—
158D28—31no—
159D28—31no—
160D28—62no—
161D28C22×D312yes—
162C4×D7—32no—
163C4×D7—62no—
164C4×D7—61no—
165Dic14—31no—
166Dic14—31no—
167C7×C8—32no—
168C7⋊C8C22×D311yescharacteristic
169C7⋊C8C22×D311yescharacteristic
170C2×C3×D4—71no—
171C2×C3×D4C2×D711yescharacteristic, complemented
172C2×#67—142no—
173C2×#67—142no—
174#237—72no—
175#237—72no—
176#237—141no—
177#237—141no—
178D3×D4—72no—
179D3×D4—72no—
180D3×D4—141no—
181D3×D4—141no—
182#235—71no—
183#235—71no—
184#235C2×D711yescharacteristic, complemented
185C3×D8—72no—
186C3×D8—72no—
187C2×C3×C8—71no—
188#214—72no—
189#214C2×D712yescomplemented
190#213—72no—
191#213C2×D712yescomplemented
192#208C2×D711yescharacteristic, complemented
193Dic12—71no—
194D24—71no—
195#204—71no—
196#204—71no—
197ΓL(1,𝔽25)—72no—
198D3×C8—72no—
199C2×C3×C7—21no—
200C2×C3×C7—42no—
201C2×C3×C7C22×D411yescharacteristic
202D21—22no—
203D21—41no—
204C3×D7—42no—
205D3×C7—41no—
206H(5,𝔽2)—211no—
207H(5,𝔽2)—211no—
208AGL(1,ℤ/8)—31no—
209AGL(1,ℤ/8)—211no—
210AGL(1,ℤ/8)—212no—
211AGL(1,ℤ/8)—212no—
212#135—211no—
213#135—212no—
214C2×D8—71no—
215C2×D8—212no—
216#131—211no—
217C22×C7—22no—
218C22×C7—61no—
219C22×C7—81no—
220C22×C7D3×D411yescharacteristic
221C2×D7—22no—
222C2×D7—32no—
223C2×D7—61no—
224C2×D7—81no—
225C2×D7—242no—
226C2×D7—242no—
227C4×C7—61no—
228C4×C7D3×D411yescharacteristic
229C4×C7C23×D311yescharacteristic
230Dic7—32no—
231Dic7—61no—
232C23×C3—21no—
233C23×C3—141no—
234C22×D3—142no—
235C22×D3—142no—
236C3×D4—21no—
237C3×D4—72no—
238C3×D4—141no—
239C3×D4C22×D712yes—
240C2×C3×C4C22×D711yescharacteristic
241#67—21no—
242#67—72no—
243#67—142no—
244#67—142no—
245#67—142no—
246#67—142no—
247#67—141no—
248C2×Dic3—142no—
249C2×Dic3—142no—
250D12—71no—
251D12—71no—
252D12C22×D711yescharacteristic
253C4×D3—21no—
254C4×D3—72no—
255C4×D3—141no—
256Dic6—71no—
257Dic6—71no—
258Dic6C22×D711yescharacteristic
259C3×C8—71no—
260C3×C8—71no—
261C3⋊C8C22×D712yes—
262C3×C7#57611yescharacteristic, complemented, Hall
263#41—31no—
264#41—211no—
265#41—212no—
266#41—212no—
267#41—211no—
268#41—421no—
269#41—421no—
270C2×D4—71no—
271C2×D4—212no—
272C2×D4—212no—
273C2×D4—421no—
274C2×D4—421no—
275C2×D4—422no—
276C2×D4—422no—
277C2×D4D3×D711yescharacteristic, complemented
278Q16—211no—
279SD16—32no—
280SD16—212no—
281SD16—211no—
282SD16—211no—
283D8—32no—
284D8—72no—
285D8—72no—
286D8—212no—
287D8—211no—
288M16—31no—
289M16—212no—
290C2×C8—71no—
291C2×C8—212no—
292C2×C7—21no—
293C2×C7—42no—
294C2×C7—121no—
295C2×C7C2×D3×D411yescharacteristic
296D7—42no—
297D7—62no—
298D7—121no—
299C22×C3—22no—
300C22×C3—81no—
301C22×C3—142no—
302C22×C3—561no—
303C22×C3D4×D711yescharacteristic
304C2×D3—21no—
305C2×D3—72no—
306C2×D3—141no—
307C2×D3—562no—
308C2×D3—562no—
309C3×C4D4×D711yescharacteristic
310C3×C4C23×D711yescharacteristic
311Dic3—21no—
312Dic3—72no—
313Dic3—141no—
314C23—21no—
315C23—141no—
316C23—422no—
317C23—422no—
318Q8—31no—
319Q8—211no—
320Q8—211no—
321D4—21no—
322D4—31no—
323D4—61no—
324D4—72no—
325D4—141no—
326D4—212no—
327D4—211no—
328D4—211no—
329D4—422no—
330D4—422no—
331D4—422no—
332D4—422no—
333D4—421no—
334D4C2×D3×D712yes—
335C2×C4—61no—
336C2×C4—212no—
337C2×C4—422no—
338C2×C4—422no—
339C2×C4—421no—
340C2×C4C2×D3×D711yescharacteristic
341C8—32no—
342C8—71no—
343C8—71no—
344C7#575711yescharacteristic, complemented, Sylow
345C2×C3—21no—
346C2×C3—42no—
347C2×C3—282no—
348C2×C3C2×D4×D711yescharacteristic
349D3—41no—
350D3—142no—
351D3—281no—
352C22—22no—
353C22—61no—
354C22—81no—
355C22—142no—
356C22—212no—
357C22—421no—
358C22—561no—
359C22—1682no—
360C22—1682no—
361C22#6669411yescharacteristic
362C4—61no—
363C4—212no—
364C4—421no—
365C4#6669411yescharacteristic
366C4C22×D3×D711yescharacteristic
367C3#9046511yescharacteristic, complemented, Sylow
368C2—21no—
369C2—42no—
370C2—121no—
371C2—282no—
372C2—422no—
373C2—841no—
374C2C2×#6669411yescharacteristic
subgroupG/H
centerC2C2×#66694
commutator, cosocleC3×C4×C7C24
FrattiniC4C22×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7C22×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#576
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621657C2semidirect
2. #10621564C2semidirect
3. C7×#932C2semidirect
4. C2×C3×#1141C2semidirect
5. #10621060C2semidirect
6. #10621059C2semidirect
7. #10621044C2semidirect
8. #10621043C2semidirect
9. #10620874C2semidirect
10. #10620865C2semidirect
11. #10620852C2semidirect
13. C2×C3×C7×D4C22semidirect
16. #66730C22semidirect
18. D4×D21C22semidirect
20. #66728C22semidirect
21. C7×#235C22semidirect
22. C2×C3×D28C22semidirect
24. #66681C22semidirect
26. #66670C22semidirect
28. C7×#214C22semidirect
29. C7×#213C22semidirect
30. C7×#208C22semidirect
31. C3×#1141C22semidirect
32. C3×#1141C22semidirect
33. C2×C3×C7⋊C8C22semidirect
34. #66571C22semidirect
35. #66568C22semidirect
36. #66567C22semidirect
37. #66565C22semidirect
38. #66562C22semidirect
39. #66561C22semidirect
40. #66559C22semidirect
41. D21⋊C8C22semidirect
46. C2×#1141D3semidirect
85. C2×C7×D4C2×D3semidirect
98. C2×D28C2×D3semidirect
106. #1141C2×D3semidirect
107. #1141C2×D3semidirect
108. C2×C7⋊C8C2×D3semidirect
117. #932D7semidirect
171. C2×C3×D4C2×D7semidirect
184. #235C2×D7semidirect
189. #214C2×D7semidirect
191. #213C2×D7semidirect
192. #208C2×D7semidirect
262. C3×C7#576semidirect
277. C2×D4D3×D7semidirect
344. C7#5757semidirect
367. C3#90465semidirect

Maximal subgroups

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

subgroupindex
#106216572
#106215642
C7×#9322
C2×C3×#11412
#106210602
#106210592
#106210442
#106210432
#106208742
#106208652
#106208522
#904653
#57577