Finite Group Databasebeta

← #13060128

Subgroups — #13060128

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#10621565C211yesmaximal, characteristic, complemented
3C2×C7×#213C211yesmaximal, characteristic, complemented
4C3×#6495C211yesmaximal, characteristic, complemented
5#10621062C212yesmaximal, complemented
6#10621061C212yesmaximal, complemented
7#10621044C212yesmaximal, complemented
8#10621043C212yesmaximal, complemented
9#10620873C211yesmaximal, characteristic, complemented
10#10620866C211yesmaximal, characteristic, complemented
11#10620853C211yesmaximal, characteristic, complemented
12#90448—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
21C2×C7×D12C2211yescharacteristic, complemented
22C3×#1157C2211yescharacteristic, complemented
23D3×C22⋊D7—22no—
24#66681C2212yescomplemented
25D7×D12—21no—
26#66676—21no—
27#66672C2212yescomplemented
28C7×#213C2212yescomplemented
29C7×#213C2212yescomplemented
30C2×C7×C3⋊C8C2211yescharacteristic, complemented
31C3×#1142C2212yescomplemented
32C3×#1141C2212yescomplemented
33C3×#1136C2211yescharacteristic, complemented
34#66570C2211yescharacteristic, complemented
35#66567C2212yescomplemented
36#66566C2211yescharacteristic, complemented
37#66565C2211yescharacteristic, complemented
38#66562C2212yescomplemented
39#66561C2211yescharacteristic, complemented
40#66559C2212yescomplemented
41D21⋊C8C2212yescomplemented
42#6548—31no—
43#6548—31no—
44C2×C7×D8—31no—
45#6495—31no—
46#6495D311yescharacteristic, complemented
47#6475—32no—
48#6474—32no—
49#6474—32no—
50D7×D8—32no—
51#6467—31no—
52#6464—31no—
53#5774—71nomaximal, Hall
54C23×C3×C7—21no—
55C22×D21—22no—
56C22×D3×C7—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×D12—21no—
73C7×D12C2312yes—
74C3×C22⋊D7—21no—
75C3×D28C2311yescharacteristic
76C3×C4×D7—21no—
77C3×Dic14C2311yescharacteristic
78#4247—22no—
79#4245—22no—
80D3×Dic7—22no—
81C3×C7⋊C8C2312yes—
82C7×C3⋊C8C2311yescharacteristic
83C7×C3⋊C8C2311yescharacteristic
84C2×C7×D4—31no—
85C2×C7×D4C2×D311yescharacteristic, complemented
86C2×C22⋊D7—62no—
87C2×C22⋊D7—62no—
88#1159—32no—
89#1159—32no—
90#1159—61no—
91#1159—61no—
92D4×D7—32no—
93D4×D7—32no—
94D4×D7—61no—
95D4×D7—61no—
96#1157—31no—
97#1157—31no—
98#1157C2×D311yescharacteristic, complemented
99C7×D8—32no—
100C7×D8—32no—
101C2×C7×C8—31no—
102#1142—32no—
103#1142C2×D312yescomplemented
104#1141—32no—
105#1141C2×D312yescomplemented
106#1136C2×D311yescharacteristic, complemented
107Dic28—31no—
108D56—31no—
109#1132—31no—
110#1132—31no—
111#1131—32no—
112C8×D7—32no—
113#1009—71no—
114#1004—71no—
115C3×AGL(1,ℤ/8)—71no—
116#950—71no—
117C2×#213D711yescharacteristic, complemented
118#916—72no—
119#914—72no—
120#911—72no—
121D3×D8—72no—
122#908—71no—
123#907—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—22no—
131C2×D3×C7—81no—
132C2×C3×D7—21no—
133D3×D7—82no—
134C3×C4×C7C2×D411yescharacteristic
135C3×C4×C7C2411yescharacteristic
136Dic21—21no—
137Dic21C2×D412yes—
138C3×Dic7—21no—
139#576—211noSylow
140C23×C7—21no—
141C23×C7—61no—
142C22×D7—62no—
143C22×D7—62no—
144C7×D4—21no—
145C7×D4—32no—
146C7×D4—61no—
147C7×D4C22×D312yes—
148C2×C4×C7C22×D311yescharacteristic
149C22⋊D7—21no—
150C22⋊D7—32no—
151C22⋊D7—62no—
152C22⋊D7—62no—
153C22⋊D7—62no—
154C22⋊D7—62no—
155C22⋊D7—61no—
156C2×Dic7—62no—
157C2×Dic7—62no—
158D28—31no—
159D28—31no—
160D28C22×D311yescharacteristic
161C4×D7—21no—
162C4×D7—32no—
163C4×D7—61no—
164Dic14—31no—
165Dic14—31no—
166Dic14C22×D311yescharacteristic
167C7×C8—31no—
168C7×C8—31no—
169C7⋊C8C22×D312yes—
170C3×#41—71no—
171C2×C3×D4C2×D711yescharacteristic, complemented
172C2×#67—142no—
173#239—72no—
174#237—72no—
175#237—141no—
176D3×D4—72no—
177D3×D4—72no—
178D3×D4—141no—
179D3×D4—142no—
180#235—71no—
181#235—71no—
182#235—141no—
183C2×D12—141no—
184C2×D12C2×D711yescharacteristic, complemented
185C3×SD16—72no—
186C3×D8—72no—
187C3×M16—71no—
188#216—71no—
189#215—71no—
190#214—71no—
191#213—71no—
192#213C2×D712yescomplemented
193#213C2×D712yescomplemented
194C2×C3⋊C8C2×D711yescharacteristic, complemented
195D24—72no—
196#204—72no—
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—41no—
205D3×C7—42no—
206H(5,𝔽2)—211no—
207H(5,𝔽2)—211no—
208AGL(1,ℤ/8)—71no—
209AGL(1,ℤ/8)—211no—
210AGL(1,ℤ/8)—212no—
211AGL(1,ℤ/8)—212no—
212#135—211no—
213#135—212no—
214C2×D8—31no—
215C2×D8—212no—
216#131—211no—
217C22×C7—22no—
218C22×C7—62no—
219C22×C7—81no—
220C22×C7—241no—
221C22×C7D3×D411yescharacteristic
222C2×D7—21no—
223C2×D7—32no—
224C2×D7—61no—
225C2×D7—242no—
226C2×D7—242no—
227C4×C7D3×D411yescharacteristic
228C4×C7C23×D311yescharacteristic
229Dic7—21no—
230Dic7—32no—
231Dic7—61no—
232C23×C3—21no—
233C22×D3—21no—
234C22×D3—142no—
235C22×D3—142no—
236C3×Q8—71no—
237C3×D4—21no—
238C3×D4—71no—
239C3×D4—141no—
240C3×D4C22×D712yes—
241C2×C3×C4—141no—
242C2×C3×C4C22×D711yescharacteristic
243#67—72no—
244#67—142no—
245#67—142no—
246#67—142no—
247#67—141no—
248C2×Dic3—142no—
249D12—21no—
250D12—71no—
251D12—71no—
252D12—142no—
253D12C22×D712yes—
254C4×D3—72no—
255C4×D3—142no—
256C4×D3—141no—
257Dic6—71no—
258Dic6—71no—
259C3×C8—72no—
260C3⋊C8C22×D711yescharacteristic
261C3⋊C8C22×D711yescharacteristic
262C3×C7#57611yescharacteristic, complemented, Hall
263#41—71no—
264#41—211no—
265#41—212no—
266#41—212no—
267#41—211no—
268#41—421no—
269#41—421no—
270C2×D4—31no—
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—72no—
280SD16—211no—
281SD16—211no—
282SD16—212no—
283D8—32no—
284D8—32no—
285D8—72no—
286D8—211no—
287D8—212no—
288M16—71no—
289M16—212no—
290C2×C8—31no—
291C2×C8—212no—
292C2×C7—21no—
293C2×C7—42no—
294C2×C7—122no—
295C2×C7C2×D3×D411yescharacteristic
296D7—41no—
297D7—62no—
298D7—121no—
299C22×C3—22no—
300C22×C3—81no—
301C22×C3—141no—
302C22×C3D4×D711yescharacteristic
303C2×D3—22no—
304C2×D3—72no—
305C2×D3—81no—
306C2×D3—141no—
307C2×D3—562no—
308C2×D3—562no—
309C3×C4—141no—
310C3×C4D4×D711yescharacteristic
311C3×C4C23×D711yescharacteristic
312Dic3—72no—
313Dic3—141no—
314C23—21no—
315C23—61no—
316C23—422no—
317C23—422no—
318Q8—71no—
319Q8—211no—
320Q8—211no—
321D4—21no—
322D4—32no—
323D4—61no—
324D4—71no—
325D4—141no—
326D4—212no—
327D4—211no—
328D4—211no—
329D4—422no—
330D4—422no—
331D4—422no—
332D4—422no—
333D4—421no—
334D4C2×D3×D712yes—
335C2×C4—141no—
336C2×C4—212no—
337C2×C4—422no—
338C2×C4—422no—
339C2×C4—421no—
340C2×C4C2×D3×D711yescharacteristic
341C8—31no—
342C8—31no—
343C8—72no—
344C7#577411yescharacteristic, complemented, Sylow
345C2×C3—21no—
346C2×C3—42no—
347C2×C3—281no—
348C2×C3C2×D4×D711yescharacteristic
349D3—42no—
350D3—142no—
351D3—281no—
352C22—22no—
353C22—62no—
354C22—81no—
355C22—141no—
356C22—212no—
357C22—241no—
358C22—421no—
359C22—1682no—
360C22—1682no—
361C22#6669411yescharacteristic
362C4—141no—
363C4—212no—
364C4—421no—
365C4#6669411yescharacteristic
366C4C22×D3×D711yescharacteristic
367C3#9044811yescharacteristic, complemented, Sylow
368C2—21no—
369C2—42no—
370C2—122no—
371C2—281no—
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. #10621565C2semidirect
3. C2×C7×#213C2semidirect
4. C3×#6495C2semidirect
5. #10621062C2semidirect
6. #10621061C2semidirect
7. #10621044C2semidirect
8. #10621043C2semidirect
9. #10620873C2semidirect
10. #10620866C2semidirect
11. #10620853C2semidirect
13. C2×C3×C7×D4C22semidirect
16. #66730C22semidirect
18. D4×D21C22semidirect
20. #66728C22semidirect
21. C2×C7×D12C22semidirect
22. C3×#1157C22semidirect
24. #66681C22semidirect
27. #66672C22semidirect
28. C7×#213C22semidirect
29. C7×#213C22semidirect
30. C2×C7×C3⋊C8C22semidirect
31. C3×#1142C22semidirect
32. C3×#1141C22semidirect
33. C3×#1136C22semidirect
34. #66570C22semidirect
35. #66567C22semidirect
36. #66566C22semidirect
37. #66565C22semidirect
38. #66562C22semidirect
39. #66561C22semidirect
40. #66559C22semidirect
41. D21⋊C8C22semidirect
46. #6495D3semidirect
85. C2×C7×D4C2×D3semidirect
98. #1157C2×D3semidirect
103. #1142C2×D3semidirect
105. #1141C2×D3semidirect
106. #1136C2×D3semidirect
117. C2×#213D7semidirect
171. C2×C3×D4C2×D7semidirect
184. C2×D12C2×D7semidirect
192. #213C2×D7semidirect
193. #213C2×D7semidirect
194. C2×C3⋊C8C2×D7semidirect
262. C3×C7#576semidirect
277. C2×D4D3×D7semidirect
344. C7#5774semidirect
367. C3#90448semidirect

Maximal subgroups

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

subgroupindex
#106216572
#106215652
C2×C7×#2132
C3×#64952
#106210622
#106210612
#106210442
#106210432
#106208732
#106208662
#106208532
#904483
#57747