Finite Group Databasebeta

← D4×D21⋊D7 #72246554

Subgroups — D4×D21⋊D7 #72246554

Order 4704.

1106 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 159 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#72246553C212yesmaximal, complemented
2#72246551C214yesmaximal, complemented
3C7×D4×D21C212yesmaximal, complemented
4C4×D21⋊D7C211yesmaximal, characteristic, complemented
5#72246460C211yesmaximal, characteristic, complemented
6C22×D21⋊D7C212yesmaximal, complemented
7C3×D4×C7⋊D7C211yesmaximal, characteristic, complemented
8#72222120C212yesmaximal, complemented
9D4×D72—31nomaximal, Hall
10C22×C7×D21C2214yescomplemented
11C22×C3×C7⋊D7C2212yescomplemented
12C2×D21⋊D7—28no—
13C2×D21⋊D7C2211yescharacteristic
14C2×D21⋊D7C2212yescomplemented
15C2×D21⋊D7C2214yescomplemented
16C3×C72×D4C2211yescharacteristic, complemented
17C7×#4268C2214yescomplemented
18C7×D84C2212yescomplemented
19C4×C7×D21C2212yescomplemented
20C3×C22⋊(C7⋊D7)C2212yescomplemented
21C3×C4⋊(C7⋊D7)C2211yescharacteristic, complemented
22C3×C4×C7⋊D7C2211yescharacteristic, complemented
23#11924254C2212yescomplemented
24#11924253C2214yescomplemented
25#11924252C2211yescharacteristic, complemented
26D21⋊Dic7C2212yescomplemented
27C22×D72—32no—
28D4×C7⋊D7D311yescharacteristic, complemented
29C7×D4×D7—32no—
30#11716115—32no—
31D7×C22⋊D7—34no—
32#11716106—31no—
33D7×D28—32no—
34C4×D72—31no—
35D3×D4×D7—72nomaximal
36C22×C3×C72C2312yescomplemented
37C2×C7×D21—28no—
38C2×C7×D21C2312yes—
39C2×C7×D21C2314yescomplemented
40C2×C3×C7⋊D7—24no—
41C2×C3×C7⋊D7C2311yescharacteristic
42C2×C3×C7⋊D7C2312yescomplemented
43D21⋊D7—28no—
44D21⋊D7—24no—
45D21⋊D7D414yesdirect product factor
46C3×C4×C72C2311yescharacteristic, complemented
47C7×Dic21C2312yescomplemented
48C3×C7⋊Dic7C2311yescharacteristic, complemented
49C22×C7⋊D7C2×D312yescomplemented
50C22×C7×D7—34no—
51C2×D72—31no—
52C2×D72—32no—
53C2×D72—34no—
54C2×D72—68no—
55C72×D4C2×D311yescharacteristic, complemented
56C22⋊(C7⋊D7)C2×D312yescomplemented
57C4⋊(C7⋊D7)C2×D311yescharacteristic, complemented
58C4×C7⋊D7C2×D311yescharacteristic, complemented
59C7×C22⋊D7—34no—
60C7×D28—32no—
61C4×C7×D7—32no—
62#87641—34no—
63#87640—32no—
64#87639—31no—
65D7×Dic7—32no—
66C22×D3×D7—74no—
67D4×D21D712yescomplemented
68D3×C7×D4—72no—
69C3×D4×D7—72no—
70C3×D4×D7—143no—
71#66694—74no—
72D3×C22⋊D7—74no—
73D7×#67—74no—
74#66681—72no—
75D3×D28—72no—
76D7×D12—72no—
77C4×D3×D7—72no—
78C2×C3×C72—22no—
79C2×C3×C72C2411yescharacteristic
80C7×D21—24no—
81C7×D21C2×D414yescomplemented
82C3×C7⋊D7—22no—
83C3×C7⋊D7C2×D412yescomplemented
84C2×D4×D7—212no—
85C22×C72C22×D312yescomplemented
86C2×C7⋊D7—24no—
87C2×C7⋊D7C22×D311yescharacteristic
88C2×C7⋊D7C22×D312yescomplemented
89C2×C7×D7—32no—
90C2×C7×D7—34no—
91C2×C7×D7—68no—
92D72—34no—
93D72—68no—
94D72—64no—
95C4×C72C22×D311yescharacteristic, complemented
96C7⋊Dic7C22×D311yescharacteristic, complemented
97C7×Dic7—32no—
98C22×D21C2×D714yescomplemented
99C22×D3×C7—74no—
100C22×C3×D7—74no—
101C22×C3×D7—146no—
102C2×D3×D7—72no—
103C2×D3×D7—74no—
104C2×D3×D7—74no—
105C2×D3×D7—74no—
106C2×D3×D7—1416no—
107C3×C7×D4—23no—
108C3×C7×D4C2×D712yescomplemented
109#4268C2×D714yescomplemented
110D84C2×D712yescomplemented
111C4×D21C2×D712yescomplemented
112C7×#67—74no—
113C7×D12—72no—
114C4×D3×C7—72no—
115C3×C22⋊D7—74no—
116C3×C22⋊D7—146no—
117C3×D28—72no—
118C3×D28—143no—
119C3×C4×D7—72no—
120C3×C4×D7—143no—
121#4247—74no—
122#4246—74no—
123#4245—74no—
124D21⋊C4—72no—
125D3×Dic7—72no—
126Dic3×D7—72no—
127C3×C72C22×D411yescharacteristic, complemented, Hall
128C23×D7—214no—
129C2×C7×D4—212no—
130C2×C22⋊D7—214no—
131D4×D7—32no—
132D4×D7—72no—
133D4×D7—143no—
134D4×D7—212no—
135D4×D7—212no—
136D4×D7—214no—
137D4×D7—214no—
138C2×D28—212no—
139C2×C4×D7—212no—
140C2×C72—22no—
141C2×C72C23×D311yescharacteristic
142C7⋊D7—22no—
143C7⋊D7D3×D412yescomplemented
144C7×D7—34no—
145C7×D7—64no—
146C2×D3×D4—491noHall
147C22×C3×C7—26no—
148C22×C3×C7C22×D714yescomplemented
149C2×D21—28no—
150C2×D21C22×D712yes—
151C2×D21C22×D714yescomplemented
152C2×D3×C7—72no—
153C2×D3×C7—74no—
154C2×D3×C7—148no—
155C2×C3×D7—72no—
156C2×C3×D7—74no—
157C2×C3×D7—143no—
158C2×C3×D7—146no—
159C2×C3×D7—148no—
160C2×C3×D7—2812no—
161D3×D7—78no—
162D3×D7—148no—
163D3×D7—148no—
164D3×D7—148no—
165C3×C4×C7—23no—
166C3×C4×C7C22×D712yescomplemented
167Dic21C22×D712yescomplemented
168C3×Dic7—72no—
169C3×Dic7—143no—
170C7×Dic3—72no—
171C23×C7—214no—
172C22×D7—34no—
173C22×D7—74no—
174C22×D7—146no—
175C22×D7—212no—
176C22×D7—214no—
177C22×D7—214no—
178C22×D7—214no—
179C22×D7—4216no—
180C7×D4—23no—
181C7×D4—214no—
182C7×D4—212no—
183C7×D4D3×D712yescomplemented
184C2×C4×C7—212no—
185C22⋊D7—34no—
186C22⋊D7—74no—
187C22⋊D7—146no—
188C22⋊D7—214no—
189C22⋊D7—214no—
190C2×Dic7—212no—
191D28—32no—
192D28—72no—
193D28—143no—
194D28—214no—
195C4×D7—32no—
196C4×D7—72no—
197C4×D7—143no—
198C4×D7—212no—
199C4×D7—212no—
200C72C2×D3×D411yescharacteristic, complemented, Sylow
201C23×D3—492no—
202C2×C3×D4—491no—
203C2×#67—492no—
204D3×D4—72no—
205D3×D4—492no—
206D3×D4—494no—
207C2×D12—491no—
208C2×C4×D3—491no—
209C2×C3×C7—24no—
210C2×C3×C7—23no—
211C2×C3×C7—46no—
212C2×C3×C7C23×D712yes—
213D21—24no—
214D21D4×D714yescomplemented
215C3×D7—74no—
216C3×D7—146no—
217C3×D7—144no—
218C3×D7—286no—
219D3×C7—74no—
220D3×C7—144no—
221C22×D4—1471noSylow
222C22×C7—26no—
223C22×C7—212no—
224C22×C7—214no—
225C22×C7—428no—
226C22×C7C2×D3×D714yescomplemented
227C2×D7—32no—
228C2×D7—34no—
229C2×D7—68no—
230C2×D7—72no—
231C2×D7—74no—
232C2×D7—143no—
233C2×D7—146no—
234C2×D7—148no—
235C2×D7—218no—
236C2×D7—2812no—
237C2×D7—428no—
238C2×D7—428no—
239C2×D7—428no—
240C4×C7—23no—
241C4×C7—212no—
242C4×C7C2×D3×D712yescomplemented
243Dic7—32no—
244Dic7—72no—
245Dic7—143no—
246C23×C3—492no—
247C22×D3—74no—
248C22×D3—491no—
249C22×D3—492no—
250C22×D3—494no—
251C22×D3—988no—
252C3×D4—492no—
253C3×D4—491no—
254C3×D4D7211yescharacteristic, complemented
255C2×C3×C4—491no—
256#67—74no—
257#67—494no—
258C2×Dic3—491no—
259D12—72no—
260D12—492no—
261C4×D3—72no—
262C4×D3—492no—
263C3×C7—23no—
264C3×C7C2×D4×D712yescomplemented
265C24—1472no—
266C2×D4—212no—
267C2×D4—491no—
268C2×D4—1472no—
269C2×D4—1471no—
270C2×D4—1472no—
271C2×D4—1474no—
272C22×C4—1471no—
273C2×C7—24no—
274C2×C7—23no—
275C2×C7—46no—
276C2×C7—214no—
277C2×C7—424no—
278C2×C7C22×D3×D712yes—
279D7—34no—
280D7—64no—
281D7—74no—
282D7—146no—
283D7—144no—
284D7—286no—
285C22×C3—491no—
286C22×C3—492no—
287C22×C3—984no—
288C22×C3C2×D7212yescomplemented
289C2×D3—72no—
290C2×D3—74no—
291C2×D3—148no—
292C2×D3—494no—
293C2×D3—988no—
294C2×D3—984no—
295C3×C4—491no—
296C3×C4C2×D7211yescharacteristic, complemented
297Dic3—72no—
298C23—214no—
299C23—2948no—
300C23—492no—
301C23—1474no—
302C23—1472no—
303C23—1471no—
304D4—214no—
305D4—212no—
306D4—492no—
307D4—491no—
308D4—1474no—
309D4—1472no—
310D4D21⋊D711yescharacteristic, direct product factor
311C2×C4—212no—
312C2×C4—491no—
313C2×C4—1471no—
314C2×C4—1472no—
315C7—23no—
316C7D3×D4×D712yescomplemented
317C2×C3—22no—
318C2×C3—492no—
319C2×C3—982no—
320C2×C3C22×D7211yescharacteristic
321D3—74no—
322D3—144no—
323C22—212no—
324C22—214no—
325C22—2948no—
326C22—2944no—
327C22—428no—
328C22—491no—
329C22—492no—
330C22—984no—
331C22—1474no—
332C22C2×D21⋊D712yescomplemented
333C4—212no—
334C4—491no—
335C4C2×D21⋊D711yescharacteristic, complemented
336C3D4×D7211yescharacteristic, complemented, Sylow
337C2—22no—
338C2—214no—
339C2—424no—
340C2—492no—
341C2—982no—
342C2C22×D21⋊D711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D21⋊D7
commutator, socle, cosocleC2×C3×C72C24
FittingC3×C72×D4C22
hypercenterD4D21⋊D7
hypocenterC3×C72C22×D4
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #72246553C2semidirect
2. #72246551C2semidirect
3. C7×D4×D21C2semidirect
4. C4×D21⋊D7C2semidirect
5. #72246460C2semidirect
6. C22×D21⋊D7C2semidirect
7. C3×D4×C7⋊D7C2semidirect
8. #72222120C2semidirect
10. C22×C7×D21C22semidirect
11. C22×C3×C7⋊D7C22semidirect
14. C2×D21⋊D7C22semidirect
15. C2×D21⋊D7C22semidirect
16. C3×C72×D4C22semidirect
17. C7×#4268C22semidirect
18. C7×D84C22semidirect
19. C4×C7×D21C22semidirect
20. C3×C22⋊(C7⋊D7)C22semidirect
21. C3×C4⋊(C7⋊D7)C22semidirect
22. C3×C4×C7⋊D7C22semidirect
23. #11924254C22semidirect
24. #11924253C22semidirect
25. #11924252C22semidirect
26. D21⋊Dic7C22semidirect
28. D4×C7⋊D7D3semidirect
36. C22×C3×C72C23semidirect
39. C2×C7×D21C23semidirect
42. C2×C3×C7⋊D7C23semidirect
45. D21⋊D7D4direct (N × H)
46. C3×C4×C72C23semidirect
47. C7×Dic21C23semidirect
48. C3×C7⋊Dic7C23semidirect
49. C22×C7⋊D7C2×D3semidirect
55. C72×D4C2×D3semidirect
56. C22⋊(C7⋊D7)C2×D3semidirect
57. C4⋊(C7⋊D7)C2×D3semidirect
58. C4×C7⋊D7C2×D3semidirect
67. D4×D21D7semidirect
81. C7×D21C2×D4semidirect
83. C3×C7⋊D7C2×D4semidirect
85. C22×C72C22×D3semidirect
88. C2×C7⋊D7C22×D3semidirect
95. C4×C72C22×D3semidirect
96. C7⋊Dic7C22×D3semidirect
98. C22×D21C2×D7semidirect
108. C3×C7×D4C2×D7semidirect
109. #4268C2×D7semidirect
110. D84C2×D7semidirect
111. C4×D21C2×D7semidirect
127. C3×C72C22×D4semidirect
143. C7⋊D7D3×D4semidirect
148. C22×C3×C7C22×D7semidirect
151. C2×D21C22×D7semidirect
166. C3×C4×C7C22×D7semidirect
167. Dic21C22×D7semidirect
183. C7×D4D3×D7semidirect
200. C72C2×D3×D4semidirect
214. D21D4×D7semidirect
226. C22×C7C2×D3×D7semidirect
242. C4×C7C2×D3×D7semidirect
254. C3×D4D72semidirect
264. C3×C7C2×D4×D7semidirect
288. C22×C3C2×D72semidirect
296. C3×C4C2×D72semidirect
310. D4D21⋊D7direct (N × H)
316. C7D3×D4×D7semidirect
332. C22C2×D21⋊D7semidirect
335. C4C2×D21⋊D7semidirect
336. C3D4×D72semidirect

Maximal subgroups

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

subgroupindex
#722465532
#722465512
C7×D4×D212
C4×D21⋊D72
#722464602
C22×D21⋊D72
C3×D4×C7⋊D72
#722221202
D4×D723
D3×D4×D77