Finite Group Databasebeta

← #13060081

Subgroups — #13060081

Order 1344.

856 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C2×#66694C212yesmaximal, complemented
2C2×D7×#67C212yesmaximal, complemented
3#10621420C211yesmaximal, characteristic, complemented
4C7×#940C211yesmaximal, characteristic, complemented
5C3×#6500C211yesmaximal, characteristic, complemented
6#10621119C212yesmaximal, complemented
7#10621111C212yesmaximal, complemented
8#10621037C212yesmaximal, complemented
9#10620957C212yesmaximal, complemented
10#90408—31nomaximal, Hall
11C23×D21C2211yescharacteristic, complemented
12C23×C3×D7C2211yescharacteristic, complemented
13C22×D3×D7C2212yescomplemented
14C2×C3×C7×D4C2211yescharacteristic, complemented
15C2×#4268C2212yescomplemented
16C2×C7×#67C2212yescomplemented
17C2×C7×#67C2212yescomplemented
18C2×C7×D12C2211yescharacteristic, complemented
19C2×C3×C22⋊D7C2212yescomplemented
20#66694—28no—
21D7×#67—28no—
22C2×#4247C2212yescomplemented
23C2×#4246C2212yes—
24C2×#4245C2212yescomplemented
25C2×D21⋊C4C2212yescomplemented
26C2×Dic3×D7C2212yescomplemented
27#66636C2211yescharacteristic, complemented
28#66631C2212yescomplemented
29C4×C7×Dic3C2211yescharacteristic, complemented
30C3×C22⋊Dic7C2211yescharacteristic, complemented
31C3×(C2×D7)⋊C4C2212yescomplemented
32Dic3⋊Dic7C2212yescomplemented
33(C2×D3)⋊Dic7C2212yescomplemented
34C2×D4×D7—32no—
35C2×D4×D7—32no—
36C7×#127—31no—
37#6501—32no—
38#6501—32no—
39#6500—31no—
40#6500—32no—
41#6500D311yescharacteristic, complemented
42C4×D28—32no—
43#5590—71nomaximal, Hall
44C23×C3×C7C2311yescharacteristic
45C23×C3×C7C2311yescharacteristic
46C22×D21—24no—
47C22×D21—41no—
48C22×D21C2312yes—
49C22×D3×C7C2312yes—
50C22×C3×D7—24no—
51C22×C3×D7—41no—
52C22×C3×D7C2312yes—
53C2×D3×D7—24no—
54C2×D3×D7—24no—
55C2×D3×D7—42no—
56C3×C7×D4—41no—
57C2×C3×C4×C7C2311yescharacteristic
58#4268—24no—
59C2×Dic21C2312yes—
60C7×#67—24no—
61C7×#67—24no—
62C2×C7×Dic3C2312yes—
63C7×D12—41no—
64C3×C22⋊D7—24no—
65C2×C3×Dic7C2312yes—
66#4247—24no—
67#4246D418yescomplemented
68#4245—24no—
69D21⋊C4—24no—
70Dic3×D7—24no—
71C23×D7—31no—
72C23×D7—32no—
73C23×D7C2×D311yescharacteristic, complemented
74C2×C7×D4—32no—
75C2×C7×D4—32no—
76C2×C7×D4—31no—
77C2×C7×D4C2×D311yescharacteristic, complemented
78C2×C22⋊D7—32no—
79C2×C22⋊D7—32no—
80C2×C22⋊D7—32no—
81C2×C22⋊D7C2×D312yescomplemented
82D4×D7—68no—
83D4×D7—68no—
84C2×D28—32no—
85C2×C4×D7—32no—
86C2×C4×D7—32no—
87C42×C7—31no—
88C22⋊Dic7—31no—
89C22⋊Dic7—32no—
90C22⋊Dic7C2×D311yescharacteristic, complemented
91(C2×D7)⋊C4—32no—
92(C2×D7)⋊C4C2×D312yescomplemented
93C4⋊Dic7—32no—
94C22×#67—72no—
95C2×D3×D4—72no—
96C3×#120—71no—
97#940D711yescharacteristic, complemented
98#939—72no—
99#937—71no—
100#884—72no—
101#882—72no—
102#67⋊C4—72no—
103C22×C3×C7—22no—
104C22×C3×C7—22no—
105C22×C3×C7—21no—
106C22×C3×C7—21no—
107C22×C3×C7C2411yescharacteristic
108C2×D21—22no—
109C2×D21—22no—
110C2×D21—44no—
111C2×D21C2×D414yescomplemented
112C2×D3×C7—22no—
113C2×D3×C7—22no—
114C2×D3×C7—22no—
115C2×C3×D7—22no—
116C2×C3×D7—22no—
117C2×C3×D7—44no—
118C2×C3×D7C2×D414yescomplemented
119D3×D7—48no—
120C3×C4×C7—21no—
121Dic21—22no—
122C3×Dic7—22no—
123C7×Dic3C2×D414yescomplemented
124D42—211noSylow
125C23×C7—32no—
126C23×C7C22×D311yescharacteristic
127C23×C7C22×D311yescharacteristic
128C22×D7—24no—
129C22×D7—32no—
130C22×D7—41no—
131C22×D7—64no—
132C22×D7—64no—
133C22×D7—64no—
134C22×D7—122no—
135C22×D7—121no—
136C22×D7C22×D312yes—
137C7×D4—41no—
138C7×D4—64no—
139C7×D4—64no—
140C7×D4—121no—
141C2×C4×C7—32no—
142C2×C4×C7C22×D311yescharacteristic
143C22⋊D7—24no—
144C22⋊D7—64no—
145C22⋊D7—64no—
146C22⋊D7—64no—
147C2×Dic7—32no—
148C2×Dic7C22×D312yes—
149D28—38no—
150C4×D7—64no—
151C4×D7—64no—
152C24×C3—71no—
153C23×D3—71no—
154C23×D3—72no—
155C2×C3×D4—72no—
156C2×C3×D4C2×D711yescharacteristic, complemented
157C2×#67—72no—
158C2×#67—72no—
159C2×#67—72no—
160C2×#67—148no—
161C2×#67C2×D712yescomplemented
162C2×#67C2×D712yescomplemented
163C22×Dic3—72no—
164D3×D4—148no—
165C2×D12—72no—
166C2×D12C2×D711yescharacteristic, complemented
167C2×C4×D3—72no—
168C3×C22⋊C4—71no—
169C3×C22⋊C4—72no—
170#217—71no—
171(C2×D3)⋊C4—72no—
172(C2×D3)⋊C4—72no—
173Dic3⋊C4—72no—
174C4×Dic3C2×D711yescharacteristic, complemented
175C2×C3×C7—41no—
176C2×C3×C7—41no—
177C2×C3×C7C22×D412yes—
178C2×C3×C7H(5,𝔽2)11yescharacteristic
179D21—24no—
180C3×D7—24no—
181D3×C7—42no—
182C22×D4—212no—
183C22×D4—212no—
184#127—31no—
185#121—212no—
186#121—212no—
187#120—71no—
188#120—212no—
189#120—211no—
190C4×D4—212no—
191C22×C7—22no—
192C22×C7—22no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—62no—
196C22×C7—62no—
197C22×C7—62no—
198C22×C7C23×D311yescharacteristic
199C2×D7—22no—
200C2×D7—22no—
201C2×D7—34no—
202C2×D7—44no—
203C2×D7—62no—
204C2×D7—62no—
205C2×D7—128no—
206C2×D7—124no—
207C2×D7D3×D414yescomplemented
208C4×C7—21no—
209C4×C7—34no—
210Dic7—22no—
211Dic7—62no—
212C23×C3—72no—
213C23×C3—144no—
214C23×C3—281no—
215C23×C3C22×D711yescharacteristic
216C23×C3C22×D711yescharacteristic
217C22×D3—72no—
218C22×D3—144no—
219C22×D3—144no—
220C22×D3—144no—
221C22×D3—282no—
222C22×D3—281no—
223C22×D3C22×D712yes—
224C3×D4—41no—
225C3×D4—144no—
226C2×C3×C4—72no—
227C2×C3×C4C22×D711yescharacteristic
228#67—24no—
229#67—24no—
230#67—78no—
231#67—144no—
232#67—144no—
233C2×Dic3—72no—
234C2×Dic3—144no—
235C2×Dic3C22×D712yes—
236D12—41no—
237D12—144no—
238C4×D3—144no—
239C3×C7D4211yescharacteristic, complemented, Hall
240C24—71no—
241C24—212no—
242C24—211no—
243C2×D4—32no—
244C2×D4—32no—
245C2×D4—31no—
246C2×D4—72no—
247C2×D4—212no—
248C2×D4—212no—
249C2×D4—212no—
250C2×D4—212no—
251C2×D4—428no—
252C2×D4—428no—
253C2×D4D3×D711yescharacteristic, complemented
254C22×C4—212no—
255C22×C4—212no—
256C4⋊C4—212no—
257C22⋊C4—71no—
258C22⋊C4—72no—
259C22⋊C4—212no—
260C22⋊C4—211no—
261C22⋊C4—212no—
262C42—31no—
263C2×C7—41no—
264C2×C7—41no—
265C2×C7—122no—
266C2×C7C2×D3×D412yes—
267C2×C7#100411yescharacteristic
268D7—24no—
269D7—64no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—74no—
275C22×C3—142no—
276C22×C3—142no—
277C22×C3—284no—
278C22×C3C23×D711yescharacteristic
279C2×D3—22no—
280C2×D3—22no—
281C2×D3—22no—
282C2×D3—74no—
283C2×D3—142no—
284C2×D3—142no—
285C2×D3—288no—
286C2×D3—284no—
287C3×C4—21no—
288C3×C4—142no—
289Dic3—142no—
290Dic3D4×D714yescomplemented
291C23—32no—
292C23—72no—
293C23—144no—
294C23—212no—
295C23—281no—
296C23—424no—
297C23—424no—
298C23—424no—
299C23—841no—
300C23—842no—
301C23C2×D3×D711yescharacteristic
302C23C2×D3×D711yescharacteristic
303D4—41no—
304D4—64no—
305D4—64no—
306D4—121no—
307D4—144no—
308D4—218no—
309D4—424no—
310D4—424no—
311D4—424no—
312C2×C4—32no—
313C2×C4—72no—
314C2×C4—212no—
315C2×C4—424no—
316C2×C4—424no—
317C2×C4C2×D3×D711yescharacteristic
318C7#559011yescharacteristic, complemented, Sylow
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—144no—
322C2×C3C2×D4×D712yes—
323C2×C3#654811yescharacteristic
324D3—42no—
325D3—144no—
326C22—22no—
327C22—22no—
328C22—21no—
329C22—21no—
330C22—62no—
331C22—62no—
332C22—62no—
333C22—74no—
334C22—142no—
335C22—142no—
336C22—214no—
337C22—284no—
338C22—422no—
339C22—422no—
340C22—844no—
341C22—848no—
342C22C22×D3×D711yescharacteristic
343C4—21no—
344C4—34no—
345C4—142no—
346C4—422no—
347C3#9040811yescharacteristic, complemented, Sylow
348C2—41no—
349C2—41no—
350C2—122no—
351C2—144no—
352C2—424no—
353C2D3×D4×D712yes—
354C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7D42
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66694C2semidirect
2. C2×D7×#67C2semidirect
3. #10621420C2semidirect
4. C7×#940C2semidirect
5. C3×#6500C2semidirect
6. #10621119C2semidirect
7. #10621111C2semidirect
8. #10621037C2semidirect
9. #10620957C2semidirect
11. C23×D21C22semidirect
12. C23×C3×D7C22semidirect
13. C22×D3×D7C22semidirect
14. C2×C3×C7×D4C22semidirect
15. C2×#4268C22semidirect
16. C2×C7×#67C22semidirect
17. C2×C7×#67C22semidirect
18. C2×C7×D12C22semidirect
19. C2×C3×C22⋊D7C22semidirect
22. C2×#4247C22semidirect
24. C2×#4245C22semidirect
25. C2×D21⋊C4C22semidirect
26. C2×Dic3×D7C22semidirect
27. #66636C22semidirect
28. #66631C22semidirect
29. C4×C7×Dic3C22semidirect
30. C3×C22⋊Dic7C22semidirect
31. C3×(C2×D7)⋊C4C22semidirect
32. Dic3⋊Dic7C22semidirect
33. (C2×D3)⋊Dic7C22semidirect
41. #6500D3semidirect
67. #4246D4semidirect
73. C23×D7C2×D3semidirect
77. C2×C7×D4C2×D3semidirect
81. C2×C22⋊D7C2×D3semidirect
90. C22⋊Dic7C2×D3semidirect
92. (C2×D7)⋊C4C2×D3semidirect
97. #940D7semidirect
111. C2×D21C2×D4semidirect
118. C2×C3×D7C2×D4semidirect
123. C7×Dic3C2×D4semidirect
156. C2×C3×D4C2×D7semidirect
161. C2×#67C2×D7semidirect
162. C2×#67C2×D7semidirect
166. C2×D12C2×D7semidirect
174. C4×Dic3C2×D7semidirect
207. C2×D7D3×D4semidirect
239. C3×C7D42semidirect
253. C2×D4D3×D7semidirect
290. Dic3D4×D7semidirect
318. C7#5590semidirect
347. C3#90408semidirect

Maximal subgroups

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

subgroupindex
C2×#666942
C2×D7×#672
#106214202
C7×#9402
C3×#65002
#106211192
#106211112
#106210372
#106209572
#904083
#55907