Finite Group Databasebeta

← #13060070

Subgroups — #13060070

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×D3×C22⋊D7C212yesmaximal, complemented
2C2×D7×#67C212yesmaximal, complemented
3#10621423C211yesmaximal, characteristic, complemented
4C7×#937C211yesmaximal, characteristic, complemented
5C3×#6500C211yesmaximal, characteristic, complemented
6#10621120C212yesmaximal, complemented
7#10621112C212yesmaximal, complemented
8#10621038C212yesmaximal, complemented
9#10621004C212yesmaximal, complemented
10#90281—31nomaximal, Hall
11C23×D3×C7C2211yescharacteristic, complemented
12C23×C3×D7C2211yescharacteristic, complemented
13C22×D3×D7C2212yescomplemented
14C2×C3×C7×D4C2211yescharacteristic, complemented
15C2×#4268C2212yescomplemented
16C2×#4268C2212yescomplemented
17C2×D84C2211yescharacteristic, complemented
18C2×C7×#67C2212yescomplemented
19C2×C3×C22⋊D7C2212yescomplemented
20D3×C22⋊D7—28no—
21D7×#67—28no—
22C2×#4247C2212yescomplemented
23C2×#4246C2212yescomplemented
24C2×#4245C2212yes—
25C2×D3×Dic7C2212yescomplemented
26C2×Dic3×D7C2212yescomplemented
27C4×Dic21C2211yescharacteristic, complemented
28C7×#217C2211yescharacteristic, complemented
29C7×(C2×D3)⋊C4C2212yescomplemented
30C3×C22⋊Dic7C2211yescharacteristic, complemented
31C3×(C2×D7)⋊C4C2212yescomplemented
32#66577C2212yescomplemented
33#66575C2212yescomplemented
34C22×C22⋊D7—32no—
35C2×D4×D7—32no—
36C7×#120—31no—
37#6503—31no—
38#6502—32no—
39#6500D311yescharacteristic, complemented
40#6447—32no—
41#6445—32no—
42(C22⋊D7)⋊C4—32no—
43#5590—71nomaximal, Hall
44C23×C3×C7C2311yescharacteristic
45C23×C3×C7C2311yescharacteristic
46C22×D21C2312yes—
47C22×D3×C7—24no—
48C22×D3×C7—41no—
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—
59#4268—24no—
60C2×Dic21C2312yes—
61D84—41no—
62C7×#67—24no—
63C2×C7×Dic3C2312yes—
64C3×C22⋊D7—24no—
65C2×C3×Dic7C2312yes—
66#4247—24no—
67#4246—24no—
68#4245D418yescomplemented
69D3×Dic7—24no—
70Dic3×D7—24no—
71C24×C7—31no—
72C23×D7—32no—
73C23×D7C2×D311yescharacteristic, complemented
74C2×C7×D4—32no—
75C2×C7×D4C2×D311yescharacteristic, complemented
76C2×C22⋊D7—32no—
77C2×C22⋊D7—32no—
78C2×C22⋊D7—32no—
79C2×C22⋊D7—32no—
80C2×C22⋊D7—68no—
81C2×C22⋊D7C2×D312yescomplemented
82C22×Dic7—32no—
83D4×D7—68no—
84C2×D28—32no—
85C2×D28—31no—
86C2×C4×D7—32no—
87C7×C22⋊C4—31no—
88C7×C22⋊C4—32no—
89C22⋊Dic7C2×D311yescharacteristic, complemented
90(C2×D7)⋊C4—32no—
91(C2×D7)⋊C4C2×D312yescomplemented
92Dic7⋊C4—32no—
93C4×Dic7—31no—
94C22×#67—72no—
95C2×D3×D4—72no—
96C3×#120—71no—
97#940—71no—
98#939—72no—
99#937D711yescharacteristic, complemented
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—22no—
111C2×D3×C7—22no—
112C2×D3×C7—22no—
113C2×D3×C7—44no—
114C2×D3×C7C2×D414yescomplemented
115C2×C3×D7—22no—
116C2×C3×D7—22no—
117C2×C3×D7—44no—
118C2×C3×D7C2×D414yescomplemented
119D3×D7—48no—
120C3×C4×C7—21no—
121Dic21C2×D414yescomplemented
122C3×Dic7—22no—
123C7×Dic3—22no—
124D42—211noSylow
125C23×C7—32no—
126C23×C7—64no—
127C23×C7—121no—
128C23×C7C22×D311yescharacteristic
129C23×C7C22×D311yescharacteristic
130C22×D7—24no—
131C22×D7—32no—
132C22×D7—41no—
133C22×D7—64no—
134C22×D7—64no—
135C22×D7—122no—
136C22×D7C22×D312yes—
137C7×D4—41no—
138C7×D4—64no—
139C2×C4×C7—32no—
140C2×C4×C7C22×D311yescharacteristic
141C22⋊D7—24no—
142C22⋊D7—38no—
143C22⋊D7—64no—
144C22⋊D7—64no—
145C22⋊D7—64no—
146C2×Dic7—32no—
147C2×Dic7—64no—
148C2×Dic7C22×D312yes—
149D28—64no—
150D28—121no—
151C4×D7—64no—
152C24×C3—71no—
153C23×D3—72no—
154C23×D3C2×D711yescharacteristic, complemented
155C2×C3×D4—72no—
156C2×C3×D4C2×D711yescharacteristic, complemented
157C2×#67—72no—
158C2×#67—72no—
159C2×#67—72no—
160C2×#67—72no—
161C2×#67—148no—
162C2×#67C2×D712yescomplemented
163C22×Dic3—72no—
164D3×D4—148no—
165C2×D12—72no—
166C2×D12—71no—
167C2×C4×D3—72no—
168C3×C22⋊C4—71no—
169C3×C22⋊C4—72no—
170#217C2×D711yescharacteristic, complemented
171(C2×D3)⋊C4—72no—
172(C2×D3)⋊C4C2×D712yescomplemented
173Dic3⋊C4—72no—
174C4×Dic3—71no—
175C2×C3×C7—41no—
176C2×C3×C7—41no—
177C2×C3×C7C22×D412yes—
178C2×C3×C7H(5,𝔽2)11yescharacteristic
179D21—42no—
180C3×D7—24no—
181D3×C7—24no—
182C22×D4—212no—
183C22×D4—212no—
184#127—211no—
185#121—212no—
186#121—212no—
187#120—31no—
188#120—71no—
189#120—212no—
190C4×D4—212no—
191C22×C7—22no—
192C22×C7—22no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—34no—
196C22×C7—62no—
197C22×C7—62no—
198C22×C7—124no—
199C22×C7C23×D311yescharacteristic
200C2×D7—22no—
201C2×D7—22no—
202C2×D7—44no—
203C2×D7—62no—
204C2×D7—62no—
205C2×D7—62no—
206C2×D7—128no—
207C2×D7D3×D414yescomplemented
208C4×C7—21no—
209C4×C7—62no—
210Dic7—22no—
211Dic7—34no—
212C23×C3—72no—
213C23×C3—144no—
214C23×C3—281no—
215C23×C3C22×D711yescharacteristic
216C23×C3C22×D711yescharacteristic
217C22×D3—24no—
218C22×D3—41no—
219C22×D3—72no—
220C22×D3—144no—
221C22×D3—144no—
222C22×D3—282no—
223C22×D3C22×D712yes—
224C3×D4—41no—
225C3×D4—144no—
226C2×C3×C4—72no—
227C2×C3×C4C22×D711yescharacteristic
228#67—24no—
229#67—78no—
230#67—144no—
231#67—144no—
232#67—144no—
233C2×Dic3—72no—
234C2×Dic3—144no—
235C2×Dic3C22×D712yes—
236D12—144no—
237D12—281no—
238C4×D3—144no—
239C3×C7D4211yescharacteristic, complemented, Hall
240C24—31no—
241C24—71no—
242C24—212no—
243C2×D4—32no—
244C2×D4—72no—
245C2×D4—212no—
246C2×D4—212no—
247C2×D4—212no—
248C2×D4—212no—
249C2×D4—212no—
250C2×D4—211no—
251C2×D4—428no—
252C2×D4—428no—
253C2×D4D3×D711yescharacteristic, complemented
254C22×C4—212no—
255C22×C4—212no—
256C4⋊C4—212no—
257C22⋊C4—31no—
258C22⋊C4—32no—
259C22⋊C4—71no—
260C22⋊C4—72no—
261C22⋊C4—212no—
262C42—211no—
263C2×C7—41no—
264C2×C7—41no—
265C2×C7—64no—
266C2×C7C2×D3×D412yes—
267C2×C7#100411yescharacteristic
268D7—24no—
269D7—122no—
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—44no—
282C2×D3—142no—
283C2×D3—142no—
284C2×D3—142no—
285C2×D3—288no—
286C2×D3D4×D714yescomplemented
287C3×C4—21no—
288C3×C4—142no—
289Dic3—22no—
290Dic3—74no—
291C23—32no—
292C23—64no—
293C23—72no—
294C23—121no—
295C23—144no—
296C23—212no—
297C23—281no—
298C23—424no—
299C23—424no—
300C23—842no—
301C23C2×D3×D711yescharacteristic
302C23C2×D3×D711yescharacteristic
303D4—41no—
304D4—64no—
305D4—144no—
306D4—218no—
307D4—424no—
308D4—424no—
309D4—424no—
310D4—424no—
311D4—841no—
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—24no—
325D3—282no—
326C22—22no—
327C22—22no—
328C22—21no—
329C22—21no—
330C22—34no—
331C22—62no—
332C22—62no—
333C22—74no—
334C22—124no—
335C22—142no—
336C22—142no—
337C22—284no—
338C22—422no—
339C22—422no—
340C22—422no—
341C22—848no—
342C22C22×D3×D711yescharacteristic
343C4—21no—
344C4—62no—
345C4—142no—
346C4—214no—
347C3#9028111yescharacteristic, complemented, Sylow
348C2—41no—
349C2—41no—
350C2—64no—
351C2—144no—
352C2—842no—
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×D3×C22⋊D7C2semidirect
2. C2×D7×#67C2semidirect
3. #10621423C2semidirect
4. C7×#937C2semidirect
5. C3×#6500C2semidirect
6. #10621120C2semidirect
7. #10621112C2semidirect
8. #10621038C2semidirect
9. #10621004C2semidirect
11. C23×D3×C7C22semidirect
12. C23×C3×D7C22semidirect
13. C22×D3×D7C22semidirect
14. C2×C3×C7×D4C22semidirect
15. C2×#4268C22semidirect
16. C2×#4268C22semidirect
17. C2×D84C22semidirect
18. C2×C7×#67C22semidirect
19. C2×C3×C22⋊D7C22semidirect
22. C2×#4247C22semidirect
23. C2×#4246C22semidirect
25. C2×D3×Dic7C22semidirect
26. C2×Dic3×D7C22semidirect
27. C4×Dic21C22semidirect
28. C7×#217C22semidirect
29. C7×(C2×D3)⋊C4C22semidirect
30. C3×C22⋊Dic7C22semidirect
31. C3×(C2×D7)⋊C4C22semidirect
32. #66577C22semidirect
33. #66575C22semidirect
39. #6500D3semidirect
68. #4245D4semidirect
73. C23×D7C2×D3semidirect
75. C2×C7×D4C2×D3semidirect
81. C2×C22⋊D7C2×D3semidirect
89. C22⋊Dic7C2×D3semidirect
91. (C2×D7)⋊C4C2×D3semidirect
99. #937D7semidirect
114. C2×D3×C7C2×D4semidirect
118. C2×C3×D7C2×D4semidirect
121. Dic21C2×D4semidirect
154. C23×D3C2×D7semidirect
156. C2×C3×D4C2×D7semidirect
162. C2×#67C2×D7semidirect
170. #217C2×D7semidirect
172. (C2×D3)⋊C4C2×D7semidirect
207. C2×D7D3×D4semidirect
239. C3×C7D42semidirect
253. C2×D4D3×D7semidirect
286. C2×D3D4×D7semidirect
318. C7#5590semidirect
347. C3#90281semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C22⋊D72
C2×D7×#672
#106214232
C7×#9372
C3×#65002
#106211202
#106211122
#106210382
#106210042
#902813
#55907