Finite Group Databasebeta

← #72254255

Subgroups — #72254255

Order 4704.

818 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 147 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#72254254C212yesmaximal, complemented
2#72254253C212yesmaximal, complemented
3#72254252C212yesmaximal, complemented
4C7×#66728C212yesmaximal, complemented
5#72246605C211yesmaximal, characteristic, complemented
6C4×D21⋊D7C212yesmaximal, complemented
7#72246460C211yesmaximal, characteristic, complemented
8C2×C3×C4×C7⋊D7C211yesmaximal, characteristic, complemented
9#72222120C212yesmaximal, complemented
10#13096688—31nomaximal, Hall
11C22×C3×C7⋊D7C2211yescharacteristic
12C2×D21⋊D7C2212yes—
13C2×C3×C4×C72C2211yescharacteristic, complemented
14C7×#4268C2214yescomplemented
15C7×D84C2212yescomplemented
16C4×C7×D21C2214yescomplemented
17C7×Dic42C2212yescomplemented
18C2×C3×C7⋊Dic7C2211yescharacteristic, complemented
19C3×C4×C7⋊D7C2211yescharacteristic
20C3×C4×C7⋊D7C2211yescharacteristic
21C3×C4×C7⋊D7C2211yescharacteristic, complemented
22C3×C4×C7⋊D7C2211yescharacteristic, complemented
23#11924255C2212yescomplemented
24#11924254C2212yescomplemented
25#11924253C2214yescomplemented
26#11924252C2212yes—
27D21⋊Dic7C2214yescomplemented
28C2×C4×C7⋊D7D311yescharacteristic, complemented
29C7×#1157—32no—
30#11716115—32no—
31#11716109—32no—
32#11716106—31no—
33C4×D72—32no—
34#11716102—32no—
35#11716101—31no—
36#11716100—32no—
37D7×#235—72nomaximal
38C22×C3×C72C2311yescharacteristic
39C2×C7×D21C2314yes—
40C2×C3×C7⋊D7—22no—
41C2×C3×C7⋊D7C2311yescharacteristic
42C2×C3×C7⋊D7C2311yescharacteristic
43D21⋊D7—24no—
44C3×C4×C72C2311yescharacteristic
45C3×C4×C72C2311yescharacteristic
46C7×Dic21C2314yes—
47C3×C7⋊Dic7C2311yescharacteristic
48C3×C7⋊Dic7C2311yescharacteristic
49C22×C7⋊D7C2×D311yescharacteristic
50C2×D72—32no—
51C2×C4×C72C2×D311yescharacteristic, complemented
52C2×C7⋊Dic7C2×D311yescharacteristic, complemented
53C4×C7⋊D7C2×D311yescharacteristic
54C4×C7⋊D7C2×D311yescharacteristic
55C4×C7⋊D7C2×D311yescharacteristic, complemented
56C4×C7⋊D7C2×D311yescharacteristic, complemented
57C7×C22⋊D7—34no—
58C7×D28—32no—
59C4×C7×D7—34no—
60C7×Dic14—32no—
61#87642—32no—
62#87641—34no—
63#87640—32no—
64#87639—32no—
65D7×Dic7—34no—
66#66728D712yescomplemented
67C7×#235—72no—
68C2×C3×C4×D7—72no—
69C2×C3×C4×D7—143no—
70D7×#67—74no—
71#66686—74no—
72D7×D12—72no—
73C4×D3×D7—74no—
74#66677—72no—
75#66676—72no—
76#66675—74no—
77D7×Dic6—72no—
78C2×C3×C72—21no—
79C2×C3×C72C2411yescharacteristic
80C7×D21—24no—
81C3×C7⋊D7—21no—
82C3×C7⋊D7#4112yescomplemented
83D7×#41—212no—
84C22×C72C22×D311yescharacteristic
85C2×C7⋊D7—22no—
86C2×C7⋊D7C22×D311yescharacteristic
87C2×C7⋊D7C22×D311yescharacteristic
88C2×C7×D7—34no—
89D72—64no—
90C4×C72C22×D311yescharacteristic
91C4×C72C22×D311yescharacteristic
92C7⋊Dic7C22×D311yescharacteristic
93C7⋊Dic7C22×D311yescharacteristic
94C7×Dic7—34no—
95C22×C3×D7—72no—
96C22×C3×D7—143no—
97C2×D3×D7—74no—
98C2×C3×C4×C7—23no—
99C2×C3×C4×C7C2×D712yescomplemented
100#4268C2×D714yescomplemented
101D84C2×D712yescomplemented
102C4×D21C2×D714yescomplemented
103Dic42C2×D712yescomplemented
104C7×#67—74no—
105C7×D12—72no—
106C4×D3×C7—74no—
107C7×Dic6—72no—
108C2×C3×Dic7—72no—
109C2×C3×Dic7—143no—
110C3×C4×D7—72no—
111C3×C4×D7—72no—
112C3×C4×D7—72no—
113C3×C4×D7—72no—
114C3×C4×D7—143no—
115C3×C4×D7—143no—
116C3×C4×D7—143no—
117C3×C4×D7—143no—
118#4248—74no—
119#4247—74no—
120#4246—74no—
121#4245—74no—
122D21⋊C4—74no—
123D3×Dic7—74no—
124Dic3×D7—74no—
125C3×C72C2×#4111yescharacteristic, complemented, Hall
126C7×#41—212no—
127#1161—212no—
128Q8×D7—212no—
129#1159—212no—
130#1159—214no—
131D4×D7—212no—
132D4×D7—214no—
133#1157—32no—
134#1157—214no—
135C2×C4×D7—72no—
136C2×C4×D7—143no—
137C2×C4×D7—214no—
138C2×C72—21no—
139C2×C72C23×D311yescharacteristic
140C7⋊D7—21no—
141C7⋊D7#23512yescomplemented
142C7×D7—64no—
143C2×#235—491noHall
144C22×C3×C7—23no—
145C22×C3×C7C22×D712yes—
146C2×D21C22×D714yes—
147C2×D3×C7—74no—
148C2×C3×D7—72no—
149C2×C3×D7—72no—
150C2×C3×D7—143no—
151C2×C3×D7—143no—
152C2×C3×D7—144no—
153C2×C3×D7—1412no—
154D3×D7—148no—
155C3×C4×C7—23no—
156C3×C4×C7—23no—
157C3×C4×C7C22×D712yes—
158C3×C4×C7C22×D712yes—
159Dic21C22×D714yes—
160C3×Dic7—72no—
161C3×Dic7—72no—
162C3×Dic7—143no—
163C3×Dic7—143no—
164C7×Dic3—74no—
165C22×D7—72no—
166C22×D7—143no—
167C22×D7—214no—
168C7×Q8—212no—
169C7×D4—214no—
170C7×D4—212no—
171C2×C4×C7—23no—
172C2×C4×C7—214no—
173C2×C4×C7D3×D712yescomplemented
174C22⋊D7—34no—
175C22⋊D7—214no—
176C22⋊D7—214no—
177C2×Dic7—72no—
178C2×Dic7—143no—
179C2×Dic7—214no—
180D28—32no—
181D28—214no—
182C4×D7—34no—
183C4×D7—72no—
184C4×D7—72no—
185C4×D7—72no—
186C4×D7—72no—
187C4×D7—143no—
188C4×D7—143no—
189C4×D7—143no—
190C4×D7—143no—
191C4×D7—214no—
192C4×D7—214no—
193Dic14—32no—
194Dic14—214no—
195C72C2×#23511yescharacteristic, complemented, Sylow
196C22×C3×C4—491no—
197C2×#67—492no—
198#235—72no—
199#235—492no—
200#235—492no—
201#235—492no—
202C2×D12—491no—
203C2×C4×D3—492no—
204C2×Dic6—491no—
205C2×C3×C7—26no—
206C2×C3×C7—22no—
207C2×C3×C7—23no—
208C2×C3×C7C23×D712yes—
209D21—24no—
210C3×D7—74no—
211C3×D7—146no—
212C3×D7—146no—
213C3×D7—142no—
214D3×C7—144no—
215C2×#41—1471noSylow
216C22×C7—23no—
217C22×C7—214no—
218C22×C7C2×D3×D712yes—
219C2×D7—34no—
220C2×D7—72no—
221C2×D7—72no—
222C2×D7—143no—
223C2×D7—143no—
224C2×D7—144no—
225C2×D7—1412no—
226C2×D7—428no—
227C4×C7—23no—
228C4×C7—23no—
229C4×C7—214no—
230C4×C7C2×D3×D712yes—
231C4×C7C2×D3×D712yes—
232Dic7—34no—
233Dic7—72no—
234Dic7—72no—
235Dic7—143no—
236Dic7—143no—
237C23×C3—491no—
238C22×D3—492no—
239C2×C3×C4—491no—
240C2×C3×C4—491no—
241C2×C3×C4—491no—
242C2×C3×C4—491no—
243C2×C3×C4—491no—
244C2×C3×C4D7211yescharacteristic, complemented
245#67—74no—
246#67—494no—
247C2×Dic3—492no—
248D12—72no—
249D12—492no—
250C4×D3—74no—
251C4×D3—494no—
252Dic6—72no—
253Dic6—492no—
254C3×C7—23no—
255C3×C7D7×#4112yescomplemented
256#41—212no—
257#41—1472no—
258#41—1472no—
259#41—1472no—
260C2×Q8—1471no—
261C2×D4—1471no—
262C2×D4—1472no—
263C22×C4—491no—
264C22×C4—1472no—
265C2×C7—26no—
266C2×C7—22no—
267C2×C7—23no—
268C2×C7—424no—
269C2×C7C22×D3×D712yes—
270D7—64no—
271D7—74no—
272D7—146no—
273D7—146no—
274D7—142no—
275C22×C3—491no—
276C22×C3—491no—
277C22×C3—982no—
278C22×C3C2×D7211yescharacteristic
279C2×D3—74no—
280C2×D3—984no—
281C3×C4—491no—
282C3×C4—491no—
283C3×C4C2×D7211yescharacteristic
284C3×C4C2×D7211yescharacteristic
285Dic3—74no—
286C23—491no—
287C23—1472no—
288Q8—212no—
289Q8—1472no—
290D4—214no—
291D4—212no—
292D4—1474no—
293D4—1472no—
294C2×C4—214no—
295C2×C4—491no—
296C2×C4—491no—
297C2×C4—491no—
298C2×C4—491no—
299C2×C4—491no—
300C2×C4—1472no—
301C2×C4—1474no—
302C2×C4D21⋊D711yescharacteristic, complemented
303C7—23no—
304C7D7×#23512yescomplemented
305C2×C3—21no—
306C2×C3—492no—
307C2×C3—981no—
308C2×C3C22×D7211yescharacteristic
309D3—144no—
310C22—214no—
311C22—2944no—
312C22—491no—
313C22—491no—
314C22—982no—
315C22C2×D21⋊D711yescharacteristic
316C4—214no—
317C4—491no—
318C4—491no—
319C4C2×D21⋊D711yescharacteristic
320C4C2×D21⋊D711yescharacteristic
321C3#1309668811yescharacteristic, complemented, Sylow
322C2—21no—
323C2—424no—
324C2—492no—
325C2—981no—
326C2C22×D21⋊D711yescharacteristic
subgroupG/H
centerC4C2×D21⋊D7
commutator, socle, cosocleC2×C3×C72C24
FrattiniC2C22×D21⋊D7
FittingC2×C3×C4×C72C22
hypercenterC2×C4D21⋊D7
hypocenterC3×C72C2×#41
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #72254254C2semidirect
2. #72254253C2semidirect
3. #72254252C2semidirect
4. C7×#66728C2semidirect
5. #72246605C2semidirect
6. C4×D21⋊D7C2semidirect
7. #72246460C2semidirect
8. C2×C3×C4×C7⋊D7C2semidirect
9. #72222120C2semidirect
13. C2×C3×C4×C72C22semidirect
14. C7×#4268C22semidirect
15. C7×D84C22semidirect
16. C4×C7×D21C22semidirect
17. C7×Dic42C22semidirect
18. C2×C3×C7⋊Dic7C22semidirect
21. C3×C4×C7⋊D7C22semidirect
22. C3×C4×C7⋊D7C22semidirect
23. #11924255C22semidirect
24. #11924254C22semidirect
25. #11924253C22semidirect
27. D21⋊Dic7C22semidirect
28. C2×C4×C7⋊D7D3semidirect
51. C2×C4×C72C2×D3semidirect
52. C2×C7⋊Dic7C2×D3semidirect
55. C4×C7⋊D7C2×D3semidirect
56. C4×C7⋊D7C2×D3semidirect
66. #66728D7semidirect
82. C3×C7⋊D7#41semidirect
99. C2×C3×C4×C7C2×D7semidirect
100. #4268C2×D7semidirect
101. D84C2×D7semidirect
102. C4×D21C2×D7semidirect
103. Dic42C2×D7semidirect
125. C3×C72C2×#41semidirect
141. C7⋊D7#235semidirect
173. C2×C4×C7D3×D7semidirect
195. C72C2×#235semidirect
244. C2×C3×C4D72semidirect
255. C3×C7D7×#41semidirect
302. C2×C4D21⋊D7semidirect
304. C7D7×#235semidirect
321. C3#13096688semidirect

Maximal subgroups

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

subgroupindex
#722542542
#722542532
#722542522
C7×#667282
#722466052
C4×D21⋊D72
#722464602
C2×C3×C4×C7⋊D72
#722221202
#130966883
D7×#2357