Finite Group Databasebeta

← #13062278

Subgroups — #13062278

Order 1344.

1620 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×D21C211yesmaximal, characteristic, complemented
2C7×#1009C211yesmaximal, characteristic, complemented
3C3×#6554C211yesmaximal, characteristic, complemented
4D7×#239C212yesmaximal, complemented
5#10621577C212yesmaximal, complemented
6#10621574C216yesmaximal, complemented
7D3×#1159C216yesmaximal, complemented
8#10621565C213yesmaximal, complemented
9#10621563C213yesmaximal, complemented
10#10621551C213yesmaximal, complemented
11C2×#66672C213yesmaximal, complemented
12#90618—31nomaximal, Hall
13C3×C7×#41C2211yescharacteristic, complemented
14#66732C2211yescharacteristic, complemented
15Q8×D21C2211yescharacteristic, complemented
16#66730C2213yescomplemented
17D4×D21C2213yescomplemented
18#66728C2213yescomplemented
19C2×C4×D21C2213yescomplemented
20C7×#239C2212yescomplemented
21D3×C7×D4C2216yescomplemented
22C7×#235C2213yescomplemented
23C2×C7×D12C2213yescomplemented
24C3×Q8×D7C2212yescomplemented
25C3×#1159C2216yescomplemented
26C3×#1157C2213yescomplemented
27C2×C3×Dic14C2213yescomplemented
28#66694C2216yescomplemented
29D3×C22⋊D7C22112yescomplemented
30C2×#4247C2216yescomplemented
31#66686C22112yescomplemented
32C2×D3×Dic7C2216yescomplemented
33#66684C2216yescomplemented
34#66681C2213yescomplemented
35D7×D12C2216yescomplemented
36C4×D3×D7C2216yescomplemented
37#66676C2216yescomplemented
38#66675C2216yescomplemented
39D21⋊Q8C2213yescomplemented
40#66673C2216yescomplemented
41#66672C2213yescomplemented
42#66672C2216yescomplemented
43#66672C2216yescomplemented
44#66672C2216yescomplemented
45D3×Dic14C2216yescomplemented
46#66670C2211yescharacteristic, complemented
47C7×H(5,𝔽2)—31no—
48#6554—32no—
49#6554—33no—
50#6554D311yescharacteristic, complemented
51D7×#41—31no—
52D7×#41—32no—
53D7×#41—33no—
54#6548—36no—
55#6548—33no—
56C2×#1159—36no—
57C2×#1159—33no—
58#5968—71nomaximal, Hall
59C22×D21C2313yes—
60C22×D3×C7C2316yes—
61C2×D3×D7C2316yes—
62C3×C7×Q8C2311yescharacteristic
63C3×C7×D4C2313yes—
64C2×C3×C4×C7C2313yes—
65#4268C2316yes—
66C2×Dic21C2313yes—
67D84C2313yes—
68C4×D21C2311yescharacteristic
69C4×D21C2313yes—
70C4×D21C2313yes—
71C4×D21C2313yes—
72Dic42C2313yes—
73C7×#67C2316yes—
74C7×D12C2313yes—
75C7×D12C2316yes—
76C4×D3×C7C2316yes—
77C7×Dic6C2311yescharacteristic
78C3×C22⋊D7C2316yes—
79C2×C3×Dic7C2316yes—
80C3×D28C2311yescharacteristic
81C3×C4×D7C2316yes—
82C3×Dic14C2316yes—
83C3×Dic14C2313yes—
84#4248C2316yes—
85#4247C2316yes—
86#4247C23112yes—
87#4246C2312yes—
88#4245C2316yes—
89D21⋊C4C2316yes—
90D3×Dic7C23112yes—
91D3×Dic7C2316yes—
92Dic3×D7C2312yes—
93C7×#41—32no—
94C7×#41—33no—
95C7×#41C2×D311yescharacteristic, complemented
96C2×C7×D4—36no—
97C2×C7×D4—33no—
98C2×C22⋊D7—312no—
99C2×C22⋊D7—36no—
100C22×Dic7—36no—
101#1161—31no—
102#1161—31no—
103Q8×D7—31no—
104Q8×D7—33no—
105Q8×D7C2×D312yescomplemented
106#1159—33no—
107#1159—36no—
108#1159—312no—
109#1159—36no—
110#1159—36no—
111#1159—33no—
112#1159—36no—
113#1159—36no—
114#1159C2×D316yescomplemented
115D4×D7—33no—
116D4×D7—36no—
117D4×D7—33no—
118D4×D7—36no—
119#1157—33no—
120#1157—36no—
121#1157—36no—
122#1157C2×D313yescomplemented
123C2×C4×D7—33no—
124C2×C4×D7—36no—
125C2×Dic14—36no—
126C2×Dic14C2×D313yescomplemented
127C3×2−1+4—71no—
128#1009—76no—
129#1009—73no—
130#1009D711yescharacteristic, complemented
131D3×#41—71no—
132D3×#41—76no—
133D3×#41—73no—
134#1007—72no—
135#1007—73no—
136C2×#239—72no—
137C2×#239—73no—
138C22×C3×C7C2413yes—
139C2×D21—43no—
140C2×D21C2413yes—
141C2×D21C2411yescharacteristic
142C2×D3×C7—46no—
143C2×D3×C7C2416yes—
144C2×C3×D7C2412yes—
145D3×D7—46no—
146C3×C4×C7C2411yescharacteristic
147C3×C4×C7C2413yes—
148Dic21C2411yescharacteristic
149Dic21C2413yes—
150C3×Dic7C2416yes—
151C7×Dic3C2412yes—
152#585—211noSylow
153C23×C7—36no—
154C22×D7—33no—
155C22×D7—36no—
156C7×Q8—31no—
157C7×Q8C22×D311yescharacteristic
158C7×D4—36no—
159C7×D4—36no—
160C7×D4—33no—
161C7×D4C22×D313yes—
162C2×C4×C7—36no—
163C2×C4×C7C22×D313yes—
164C22⋊D7—36no—
165C22⋊D7—36no—
166C22⋊D7—36no—
167C22⋊D7—312no—
168C22⋊D7C22×D316yes—
169C2×Dic7—33no—
170C2×Dic7—312no—
171C2×Dic7—36no—
172C2×Dic7C22×D316yes—
173D28—32no—
174D28—33no—
175D28C22×D311yescharacteristic
176C4×D7—32no—
177C4×D7—33no—
178C4×D7—31no—
179C4×D7—33no—
180C4×D7—33no—
181C4×D7—36no—
182C4×D7C22×D316yes—
183Dic14—33no—
184Dic14—36no—
185Dic14C22×D316yes—
186Dic14C22×D313yes—
187C3×#41—76no—
188C3×#41—73no—
189C3×#41C2×D711yescharacteristic, complemented
190C2×C3×Q8—72no—
191C2×C3×Q8—73no—
192#239—71no—
193#239—76no—
194#239—73no—
195#239—76no—
196#239—76no—
197#239—76no—
198#239C2×D712yescomplemented
199D3×Q8—71no—
200D3×Q8—76no—
201D3×Q8—73no—
202#237—73no—
203#237—76no—
204#237—71no—
205D3×D4—712no—
206D3×D4—73no—
207D3×D4—76no—
208D3×D4—73no—
209D3×D4C2×D716yescomplemented
210#235—73no—
211#235—712no—
212#235—76no—
213#235—76no—
214#235C2×D713yescomplemented
215C2×D12—76no—
216C2×D12—76no—
217C2×D12C2×D713yescomplemented
218C2×C4×D3—76no—
219C2×C4×D3—73no—
220C2×C4×D3—76no—
221C2×C3×C7—23no—
222C2×C3×C7C2511yescharacteristic
223D21—23no—
224D21—21no—
225C3×D7—22no—
226D3×C7—26no—
2272−1+4—71no—
2282−1+4—212no—
2292−1+4—213no—
230H(5,𝔽2)—31no—
231H(5,𝔽2)—216no—
232H(5,𝔽2)—213no—
233C2×#41—213no—
234C2×#41—216no—
235C2×#41—212no—
236C2×#41—213no—
237C2×#41—211no—
238C22×C7—36no—
239C22×C7—126no—
240C22×C7C23×D313yes—
241C2×D7—31no—
242C2×D7—33no—
243C2×D7—126no—
244C2×D7—123no—
245C2×D7C23×D312yes—
246C4×C7—32no—
247C4×C7C23×D313yes—
248C4×C7C23×D311yescharacteristic
249Dic7—33no—
250Dic7—31no—
251Dic7C23×D316yes—
252C22×D3—73no—
253C22×D3—76no—
254C22×D3C22×D716yes—
255C3×Q8—76no—
256C3×Q8—73no—
257C3×Q8C22×D711yescharacteristic
258C3×D4—76no—
259C3×D4—71no—
260C3×D4C22×D713yes—
261C2×C3×C4—76no—
262C2×C3×C4—76no—
263C2×C3×C4C22×D713yes—
264#67—76no—
265#67—76no—
266#67—72no—
267#67C22×D716yes—
268C2×Dic3—73no—
269C2×Dic3—72no—
270D12—76no—
271D12—712no—
272D12—73no—
273D12C22×D716yes—
274D12C22×D713yes—
275C4×D3—76no—
276C4×D3—712no—
277C4×D3—73no—
278C4×D3—71no—
279C4×D3—73no—
280C4×D3—73no—
281C4×D3—76no—
282C4×D3C22×D716yes—
283Dic6—73no—
284Dic6—76no—
285Dic6C22×D711yescharacteristic
286C3×C7#58511yescharacteristic, complemented, Hall
287#41—32no—
288#41—33no—
289#41—76no—
290#41—73no—
291#41—216no—
292#41—216no—
293#41—2112no—
294#41—216no—
295#41—216no—
296#41—213no—
297#41—216no—
298#41—216no—
299#41—216no—
300#41—211no—
301#41—211no—
302#41—213no—
303#41—213no—
304#41D3×D711yescharacteristic, complemented
305C2×Q8—72no—
306C2×Q8—73no—
307C2×Q8—216no—
308C2×Q8—213no—
309C2×Q8—211no—
310C2×D4—36no—
311C2×D4—33no—
312C2×D4—216no—
313C2×D4—216no—
314C2×D4—216no—
315C2×D4—213no—
316C2×D4—2112no—
317C2×D4—213no—
318C22×C4—216no—
319C22×C4—216no—
320C22×C4—213no—
321C2×C7—23no—
322C2×C7—66no—
323C2×C7C24×D311yescharacteristic
324D7—22no—
325D7—63no—
326D7—61no—
327C22×C3—72no—
328C22×C3C23×D713yes—
329C2×D3—46no—
330C2×D3—71no—
331C2×D3—73no—
332C2×D3—286no—
333C2×D3—283no—
334C2×D3C23×D716yes—
335C3×C4—76no—
336C3×C4C23×D713yes—
337C3×C4C23×D711yescharacteristic
338Dic3—73no—
339Dic3—71no—
340Dic3C23×D712yes—
341C23—36no—
342C23—216no—
343C23—213no—
344Q8—31no—
345Q8—76no—
346Q8—73no—
347Q8—216no—
348Q8—213no—
349Q8C2×D3×D711yescharacteristic
350D4—36no—
351D4—36no—
352D4—33no—
353D4—76no—
354D4—71no—
355D4—212no—
356D4—2112no—
357D4—216no—
358D4—216no—
359D4—216no—
360D4—213no—
361D4C2×D3×D713yes—
362C2×C4—36no—
363C2×C4—76no—
364C2×C4—76no—
365C2×C4—216no—
366C2×C4—212no—
367C2×C4—216no—
368C2×C4—2112no—
369C2×C4—213no—
370C2×C4—213no—
371C2×C4—211no—
372C2×C4—213no—
373C2×C4—213no—
374C2×C4C2×D3×D713yes—
375C7#596811yescharacteristic, complemented, Sylow
376C2×C3—23no—
377C2×C3—142no—
378C2×C3C24×D711yescharacteristic
379D3—26no—
380D3—143no—
381D3—141no—
382C22—36no—
383C22—72no—
384C22—126no—
385C22—213no—
386C22—211no—
387C22—843no—
388C22—846no—
389C22C22×D3×D713yes—
390C4—32no—
391C4—76no—
392C4—211no—
393C4—213no—
394C4C22×D3×D713yes—
395C4C22×D3×D711yescharacteristic
396C3#9061811yescharacteristic, complemented, Sylow
397C2—23no—
398C2—66no—
399C2—142no—
400C2—421no—
401C2—423no—
402C2C23×D3×D711yescharacteristic
subgroupG/H
center, FrattiniC2C23×D3×D7
commutator, socle, cosocleC2×C3×C7C25
FittingC3×C7×#41C22
hypercenter#41D3×D7
hypocenterC3×C7#585
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #41×D21C2semidirect
2. C7×#1009C2semidirect
3. C3×#6554C2semidirect
4. D7×#239C2semidirect
5. #10621577C2semidirect
6. #10621574C2semidirect
7. D3×#1159C2semidirect
8. #10621565C2semidirect
9. #10621563C2semidirect
10. #10621551C2semidirect
11. C2×#66672C2semidirect
13. C3×C7×#41C22semidirect
14. #66732C22semidirect
15. Q8×D21C22semidirect
16. #66730C22semidirect
17. D4×D21C22semidirect
18. #66728C22semidirect
19. C2×C4×D21C22semidirect
20. C7×#239C22semidirect
21. D3×C7×D4C22semidirect
22. C7×#235C22semidirect
23. C2×C7×D12C22semidirect
24. C3×Q8×D7C22semidirect
25. C3×#1159C22semidirect
26. C3×#1157C22semidirect
27. C2×C3×Dic14C22semidirect
28. #66694C22semidirect
29. D3×C22⋊D7C22semidirect
30. C2×#4247C22semidirect
31. #66686C22semidirect
32. C2×D3×Dic7C22semidirect
33. #66684C22semidirect
34. #66681C22semidirect
35. D7×D12C22semidirect
36. C4×D3×D7C22semidirect
37. #66676C22semidirect
38. #66675C22semidirect
39. D21⋊Q8C22semidirect
40. #66673C22semidirect
41. #66672C22semidirect
42. #66672C22semidirect
43. #66672C22semidirect
44. #66672C22semidirect
45. D3×Dic14C22semidirect
46. #66670C22semidirect
50. #6554D3semidirect
95. C7×#41C2×D3semidirect
105. Q8×D7C2×D3semidirect
114. #1159C2×D3semidirect
122. #1157C2×D3semidirect
126. C2×Dic14C2×D3semidirect
130. #1009D7semidirect
189. C3×#41C2×D7semidirect
198. #239C2×D7semidirect
209. D3×D4C2×D7semidirect
214. #235C2×D7semidirect
217. C2×D12C2×D7semidirect
286. C3×C7#585semidirect
304. #41D3×D7semidirect
375. C7#5968semidirect
396. C3#90618semidirect

Maximal subgroups

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

subgroupindex
#41×D212
C7×#10092
C3×#65542
D7×#2392
#106215772
#106215742
D3×#11592
#106215652
#106215632
#106215512
C2×#666722
#906183
#59687