Finite Group Databasebeta

← #72246661

Subgroups — #72246661

Order 4704.

818 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 144 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#72246657C212yesmaximal, complemented
2#72246656C214yesmaximal, complemented
3C7×#66672C212yesmaximal, complemented
4C3×#11716101C211yesmaximal, characteristic, complemented
5C4×D21⋊D7C211yesmaximal, characteristic, complemented
6D12×C7⋊D7C211yesmaximal, characteristic, complemented
7#72222113C212yesmaximal, complemented
8D3×#87639C212yesmaximal, complemented
9#13096694—31nomaximal, Hall
10C2×D21⋊D7C2211yescharacteristic
11C2×D3×C7⋊D7C2212yes—
12C4⋊(C3⋊(C7⋊D7))C2211yescharacteristic, complemented
13C4×C7×D21C2212yescomplemented
14C72×D12C2211yescharacteristic, complemented
15C3×C4×C7⋊D7C2211yescharacteristic
16C3×C7×Dic14C2212yescomplemented
17#11924252C2211yescharacteristic
18D21⋊Dic7C2212yescomplemented
19#11924248C2214yescomplemented
20#11924246C2214yescomplemented
21#11924242C2212yescomplemented
22C7×#4247C2214yescomplemented
23D3×C7×Dic7C2214yescomplemented
24C3×#87642C2212yescomplemented
25C3×#87639C2212yes—
26D4×C7⋊D7—31no—
27C7×#1159—32no—
28#11716115—32no—
29C2×#87639—32no—
30#11716108—34no—
31C4×D72—31no—
32#11716103—32no—
33#11716101D311yescharacteristic, complemented
34D7×#239—72nomaximal
35C2×C3⋊(C7⋊D7)C2312yes—
36C2×C7×D21C2312yes—
37C2×D3×C72C2312yes—
38C2×C3×C7⋊D7C2311yescharacteristic
39D21⋊D7—22no—
40D3×C7⋊D7—24no—
41C3×C4×C72C2311yescharacteristic
42C7×Dic21C2312yes—
43C3×C7⋊Dic7C2311yescharacteristic
44C3×C7×Dic7C2314yes—
45C22×C7⋊D7—32no—
46C2×D72—31no—
47C72×D4—31no—
48C22⋊(C7⋊D7)—32no—
49C4⋊(C7⋊D7)—31no—
50C4×C7⋊D7C2×D311yescharacteristic
51C7×C22⋊D7—34no—
52C2×C7×Dic7—34no—
53C4×C7×D7—32no—
54C7×Dic14C2×D312yescomplemented
55#87642C2×D312yescomplemented
56#87641—34no—
57#87639—31no—
58#87639—34no—
59#87639C2×D312yes—
60D7×Dic7—32no—
61#66732—72no—
62C7×#239—72no—
63C3×Q8×D7—72no—
64D7×D12—72no—
65D7×D12—74no—
66D7×D12—143no—
67C4×D3×D7—72no—
68C4×D3×D7—74no—
69#66673—72no—
70#66673—74no—
71#66672—74no—
72#66672D712yescomplemented
73C2×C3×C72C2411yescharacteristic
74C3⋊(C7⋊D7)—22no—
75C7×D21—22no—
76D3×C72—22no—
77C3×C7⋊D7#4112yescomplemented
78D7×#41—212no—
79C22×C72—32no—
80C2×C7⋊D7—32no—
81C2×C7⋊D7—64no—
82C2×C7⋊D7C22×D311yescharacteristic
83C2×C7×D7—32no—
84D72—62no—
85C4×C72C22×D311yescharacteristic
86C7⋊Dic7C22×D311yescharacteristic
87C7×Dic7—32no—
88C7×Dic7C22×D314yes—
89C2×D3×D7—72no—
90C2×D3×D7—74no—
91C2×D3×D7—146no—
92C3×C7×Q8—72no—
93D84—74no—
94D84—72no—
95D84—143no—
96C4×D21—74no—
97C4×D21C2×D712yescomplemented
98C7×D12—23no—
99C7×D12—74no—
100C7×D12C2×D712yescomplemented
101C4×D3×C7—74no—
102C4×D3×C7—72no—
103C3×C4×D7—72no—
104C3×C4×D7—74no—
105C3×C4×D7—143no—
106C3×Dic14—74no—
107C3×Dic14C2×D712yescomplemented
108#4247—74no—
109#4247—74no—
110#4247—146no—
111#4247C2×D714yescomplemented
112D21⋊C4—72no—
113D21⋊C4—74no—
114D3×Dic7—72no—
115D3×Dic7C2×D714yescomplemented
116Dic3×D7—72no—
117C3×C72C2×#4111yescharacteristic, complemented, Hall
118C7×#41—212no—
119#1161—212no—
120Q8×D7—72no—
121#1159—32no—
122#1159—214no—
123D4×D7—214no—
124D4×D7—212no—
125D4×D7—423no—
126#1157—214no—
127#1157—212no—
128C2×C4×D7—212no—
129C2×C4×D7—214no—
130C2×C72—62no—
131C2×C72C23×D311yescharacteristic
132C7⋊D7—62no—
133C7⋊D7#23912yescomplemented
134C7×D7—62no—
135C2×#239—491noHall
136C2×D21—74no—
137C2×D21—146no—
138C2×D21C22×D712yes—
139C2×D3×C7—26no—
140C2×D3×C7—72no—
141C2×D3×C7C22×D714yes—
142C2×C3×D7—72no—
143C2×C3×D7—143no—
144D3×D7—148no—
145D3×D7—144no—
146D3×D7—2812no—
147C3×C4×C7—23no—
148C3×C4×C7—74no—
149C3×C4×C7C22×D712yes—
150Dic21C22×D712yes—
151C3×Dic7—72no—
152C3×Dic7—143no—
153C3×Dic7C22×D714yes—
154C7×Dic3—72no—
155C22×D7—212no—
156C22×D7—214no—
157C22×D7—426no—
158C7×Q8—72no—
159C7×D4—32no—
160C7×D4—63no—
161C7×D4—214no—
162C2×C4×C7—214no—
163C2×C4×C7—212no—
164C22⋊D7—34no—
165C22⋊D7—214no—
166C22⋊D7—214no—
167C22⋊D7—426no—
168C2×Dic7—34no—
169C2×Dic7—212no—
170D28—214no—
171D28—212no—
172D28—423no—
173C4×D7—32no—
174C4×D7—72no—
175C4×D7—74no—
176C4×D7—143no—
177C4×D7—214no—
178C4×D7—212no—
179C4×D7—212no—
180C4×D7—214no—
181Dic14—74no—
182Dic14D3×D712yescomplemented
183C72C2×#23911yescharacteristic, complemented, Sylow
184C2×C3×Q8—491no—
185#239—72no—
186#239—494no—
187#239—492no—
188C2×D12—492no—
189C2×D12—491no—
190C2×C4×D3—492no—
191C2×C4×D3—491no—
192C2×C3×C7—23no—
193C2×C3×C7C23×D712yes—
194D21—22no—
195D21—144no—
196D21—286no—
197C3×D7—74no—
198C3×D7—146no—
199D3×C7—24no—
200D3×C7—46no—
201D3×C7—142no—
202C2×#41—1471noSylow
203C22×C7—34no—
204C22×C7—66no—
205C22×C7—212no—
206C2×D7—32no—
207C2×D7—72no—
208C2×D7—143no—
209C2×D7—214no—
210C2×D7—424no—
211C2×D7—428no—
212C2×D7—426no—
213C2×D7—8412no—
214C4×C7—23no—
215C4×C7—74no—
216C4×C7—212no—
217C4×C7C2×D3×D712yes—
218Dic7—32no—
219Dic7—72no—
220Dic7—143no—
221Dic7C2×D3×D714yes—
222C22×D3—491no—
223C22×D3—492no—
224C3×Q8—72no—
225C3×Q8—492no—
226C2×C3×C4—492no—
227C2×C3×C4—491no—
228C2×Dic3—491no—
229D12—74no—
230D12—494no—
231D12—492no—
232D12—491no—
233D12D7211yescharacteristic, complemented
234C4×D3—74no—
235C4×D3—72no—
236C4×D3—494no—
237C4×D3—492no—
238C3×C7—23no—
239C3×C7D7×#4112yescomplemented
240#41—212no—
241#41—1474no—
242#41—1472no—
243C2×Q8—491no—
244C2×D4—1471no—
245C2×D4—1472no—
246C22×C4—1472no—
247C22×C4—1471no—
248C2×C7—23no—
249C2×C7—64no—
250C2×C7—126no—
251C2×C7—422no—
252C2×C7C22×D3×D712yes—
253D7—62no—
254D7—74no—
255D7—146no—
256D7—424no—
257D7—846no—
258C22×C3—491no—
259C2×D3—72no—
260C2×D3—492no—
261C2×D3—984no—
262C2×D3—982no—
263C2×D3C2×D7212yes—
264C3×C4—74no—
265C3×C4—491no—
266C3×C4C2×D7211yescharacteristic
267Dic3—72no—
268C23—1472no—
269C23—1471no—
270Q8—72no—
271Q8—492no—
272D4—31no—
273D4—214no—
274D4—1474no—
275D4—1472no—
276D4—1471no—
277C2×C4—214no—
278C2×C4—212no—
279C2×C4—492no—
280C2×C4—491no—
281C2×C4—1474no—
282C2×C4—1471no—
283C2×C4—1472no—
284C7—23no—
285C7D7×#23912yescomplemented
286C2×C3—492no—
287C2×C3C22×D7211yescharacteristic
288D3—22no—
289D3—142no—
290D3—982no—
291C22—32no—
292C22—212no—
293C22—2944no—
294C22—2942no—
295C22—491no—
296C22—1472no—
297C4—74no—
298C4—212no—
299C4—491no—
300C4D3×D7211yescharacteristic
301C3#1309669411yescharacteristic, complemented, Sylow
302C2—62no—
303C2—2942no—
304C2—422no—
305C2—492no—
306C2C2×D3×D7211yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×D72
commutator, socle, cosocleC2×C3×C72C24
FittingC3×C4×C72C23
hypercenterC4D3×D72
hypocenterC3×C72C2×#41
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #72246657C2semidirect
2. #72246656C2semidirect
3. C7×#66672C2semidirect
4. C3×#11716101C2semidirect
5. C4×D21⋊D7C2semidirect
6. D12×C7⋊D7C2semidirect
7. #72222113C2semidirect
8. D3×#87639C2semidirect
12. C4⋊(C3⋊(C7⋊D7))C22semidirect
13. C4×C7×D21C22semidirect
14. C72×D12C22semidirect
16. C3×C7×Dic14C22semidirect
18. D21⋊Dic7C22semidirect
19. #11924248C22semidirect
20. #11924246C22semidirect
21. #11924242C22semidirect
22. C7×#4247C22semidirect
23. D3×C7×Dic7C22semidirect
24. C3×#87642C22semidirect
33. #11716101D3semidirect
54. C7×Dic14C2×D3semidirect
55. #87642C2×D3semidirect
72. #66672D7semidirect
77. C3×C7⋊D7#41semidirect
97. C4×D21C2×D7semidirect
100. C7×D12C2×D7semidirect
107. C3×Dic14C2×D7semidirect
111. #4247C2×D7semidirect
115. D3×Dic7C2×D7semidirect
117. C3×C72C2×#41semidirect
133. C7⋊D7#239semidirect
182. Dic14D3×D7semidirect
183. C72C2×#239semidirect
233. D12D72semidirect
239. C3×C7D7×#41semidirect
285. C7D7×#239semidirect
301. C3#13096694semidirect

Maximal subgroups

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

subgroupindex
#722466572
#722466562
C7×#666722
C3×#117161012
C4×D21⋊D72
D12×C7⋊D72
#722221132
D3×#876392
#130966943
D7×#2397