Finite Group Databasebeta

← #90311

Subgroups — #90311

Order 448.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C7×#121C211yesmaximal, characteristic, complemented
2#6503C211yesmaximal, characteristic, complemented
3#6502C211yesmaximal, characteristic, complemented
4#6502C211yesmaximal, characteristic, complemented
5#6501C211yesmaximal, characteristic, complemented
6#6500C211yesmaximal, characteristic, complemented
7#6500C211yesmaximal, characteristic, complemented
8#6493C211yesmaximal, characteristic, complemented
9#6492C211yesmaximal, characteristic, complemented
10#6461C211yesmaximal, characteristic, complemented
11#6457C211yesmaximal, characteristic, complemented
12#6449C211yesmaximal, characteristic, complemented
13#6447C211yesmaximal, characteristic, complemented
14#6445C211yesmaximal, characteristic, complemented
15#6442C211yesmaximal, characteristic, complemented
16C23×D7C2211yescharacteristic
17C2×C7×D4C2211yescharacteristic
18C2×C7×D4C2211yescharacteristic
19C2×C7×D4C2211yescharacteristic, complemented
20C22×C4×C7C2211yescharacteristic, complemented
21C2×C22⋊D7C2211yescharacteristic
22C2×C22⋊D7C2211yescharacteristic
23C2×C22⋊D7C2211yescharacteristic
24C2×C22⋊D7C2211yescharacteristic
25C2×C22⋊D7C2211yescharacteristic
26C2×C22⋊D7C2211yescharacteristic, complemented
27C2×C22⋊D7C2211yescharacteristic, complemented
28C22×Dic7C2211yescharacteristic, complemented
29C2×D28C2211yescharacteristic
30C2×D28C2211yescharacteristic
31C2×C4×D7C2211yescharacteristic, complemented
32C7×C4⋊C4C2211yescharacteristic, complemented
33C7×C22⋊C4C2211yescharacteristic
34C7×C22⋊C4C2211yescharacteristic, complemented
35C22⋊Dic7C2211yescharacteristic
36C22⋊Dic7C2211yescharacteristic
37C22⋊Dic7C2211yescharacteristic, complemented
38(C2×D7)⋊C4C2211yescharacteristic
39(C2×D7)⋊C4C2211yescharacteristic
40(C2×D7)⋊C4C2211yescharacteristic
41(C2×D7)⋊C4C2211yescharacteristic
42(C2×D7)⋊C4C2211yescharacteristic
43(C2×D7)⋊C4C2211yescharacteristic
44(C2×D7)⋊C4C2211yescharacteristic, complemented
45C4⋊Dic7C2211yescharacteristic, complemented
46C4⋊Dic7C2211yescharacteristic, complemented
47Dic7⋊C4C2211yescharacteristic, complemented
48Dic7⋊C4C2211yescharacteristic, complemented
49Dic7⋊C4C2211yescharacteristic, complemented
50C4×Dic7C2211yescharacteristic, complemented
51#560—71nomaximal, Sylow
52C23×C7C2311yescharacteristic
53C23×C7C2311yescharacteristic
54C23×C7C2311yescharacteristic
55C22×D7—41no—
56C22×D7—41no—
57C22×D7—41no—
58C22×D7C2311yescharacteristic
59C22×D7C2311yescharacteristic
60C22×D7C2311yescharacteristic
61C7×D4—41no—
62C7×D4—41no—
63C7×D4—41no—
64C2×C4×C7—41no—
65C2×C4×C7C2311yescharacteristic
66C2×C4×C7C2311yescharacteristic
67C2×C4×C7C2311yescharacteristic
68C2×C4×C7C2311yescharacteristic
69C22⋊D7—41no—
70C22⋊D7—41no—
71C22⋊D7—41no—
72C22⋊D7—41no—
73C22⋊D7—41no—
74C22⋊D7—41no—
75C22⋊D7—41no—
76C2×Dic7—41no—
77C2×Dic7C2311yescharacteristic
78C2×Dic7C2311yescharacteristic
79C2×Dic7C2311yescharacteristic
80C2×Dic7C2311yescharacteristic
81C2×Dic7C2311yescharacteristic
82D28—41no—
83D28—41no—
84C4×D7—41no—
85#127—71no—
86#126—71no—
87#126—71no—
88#123—71no—
89#123—71no—
90#123—71no—
91#121—71no—
92#121—71no—
93#121—71no—
94#121—71no—
95#121—71no—
96#121D711yescharacteristic, complemented
97#120—71no—
98#120—71no—
99#120—71no—
100C22×C7—21no—
101C22×C7—21no—
102C22×C7—21no—
103C22×C7—21no—
104C22×C7—21no—
105C22×C7—21no—
106C22×C7—21no—
107C22×C7—21no—
108C22×C7—21no—
109C22×C7C2411yescharacteristic
110C2×D7—21no—
111C2×D7—21no—
112C2×D7—21no—
113C2×D7—21no—
114C2×D7—21no—
115C2×D7—21no—
116C2×D7—21no—
117C2×D7—21no—
118C2×D7—21no—
119C2×D7—44no—
120C4×C7—21no—
121C4×C7—21no—
122C4×C7—21no—
123C4×C7—21no—
124Dic7—21no—
125Dic7—21no—
126Dic7—21no—
127Dic7—21no—
128Dic7—21no—
129C24—71no—
130C2×D4—71no—
131C2×D4—71no—
132C2×D4—71no—
133C2×D4—71no—
134C2×D4—71no—
135C2×D4—71no—
136C2×D4—71no—
137C2×D4—71no—
138C2×D4—71no—
139C2×D4C2×D711yescharacteristic
140C2×D4C2×D711yescharacteristic
141C2×D4C2×D711yescharacteristic, complemented
142C22×C4—71no—
143C22×C4—71no—
144C22×C4C2×D711yescharacteristic, complemented
145C4⋊C4—71no—
146C4⋊C4—71no—
147C4⋊C4—71no—
148C4⋊C4—71no—
149C4⋊C4—71no—
150C4⋊C4C2×D711yescharacteristic, complemented
151C22⋊C4—71no—
152C22⋊C4—71no—
153C22⋊C4—71no—
154C22⋊C4—71no—
155C22⋊C4—71no—
156C22⋊C4—71no—
157C22⋊C4—71no—
158C22⋊C4—71no—
159C22⋊C4—71no—
160C22⋊C4—71no—
161C22⋊C4C2×D711yescharacteristic
162C22⋊C4C2×D711yescharacteristic, complemented
163C42—71no—
164C2×C7—41no—
165C2×C7—41no—
166C2×C7—41no—
167C2×C7H(5,𝔽2)11yescharacteristic
168C2×C7H(5,𝔽2)11yescharacteristic
169C2×C7H(5,𝔽2)11yescharacteristic
170D7—41no—
171D7—41no—
172D7—41no—
173C23—71no—
174C23—71no—
175C23—71no—
176C23—281no—
177C23—281no—
178C23—281no—
179C23C22×D711yescharacteristic
180C23C22×D711yescharacteristic
181C23C22×D711yescharacteristic
182D4—41no—
183D4—41no—
184D4—41no—
185D4—281no—
186D4—281no—
187D4—281no—
188D4—281no—
189D4—281no—
190D4—281no—
191D4—281no—
192D4—281no—
193D4—281no—
194C2×C4—41no—
195C2×C4—71no—
196C2×C4—71no—
197C2×C4—71no—
198C2×C4—71no—
199C2×C4—71no—
200C2×C4—281no—
201C2×C4—281no—
202C2×C4C22×D711yescharacteristic
203C2×C4C22×D711yescharacteristic
204C2×C4C22×D711yescharacteristic
205C2×C4C22×D711yescharacteristic
206C7#56011yescharacteristic, complemented, Sylow
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—141no—
217C22—141no—
218C22—141no—
219C22—141no—
220C22—141no—
221C22—141no—
222C22—141no—
223C22—141no—
224C22—141no—
225C22—284no—
226C22C23×D711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—141no—
232C4—141no—
233C4—141no—
234C4—141no—
235C4—141no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—281no—
240C2—281no—
241C2—281no—
242C2#654811yescharacteristic
243C2#654811yescharacteristic
244C2#655311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#121C2
hypercenter#121D7
hypocenterC7#560
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C7×#121C2semidirect
2. #6503C2semidirect
3. #6502C2semidirect
4. #6502C2semidirect
5. #6501C2semidirect
6. #6500C2semidirect
7. #6500C2semidirect
8. #6493C2semidirect
9. #6492C2semidirect
10. #6461C2semidirect
11. #6457C2semidirect
12. #6449C2semidirect
13. #6447C2semidirect
14. #6445C2semidirect
15. #6442C2semidirect
19. C2×C7×D4C22semidirect
20. C22×C4×C7C22semidirect
26. C2×C22⋊D7C22semidirect
27. C2×C22⋊D7C22semidirect
28. C22×Dic7C22semidirect
31. C2×C4×D7C22semidirect
32. C7×C4⋊C4C22semidirect
34. C7×C22⋊C4C22semidirect
37. C22⋊Dic7C22semidirect
44. (C2×D7)⋊C4C22semidirect
45. C4⋊Dic7C22semidirect
46. C4⋊Dic7C22semidirect
47. Dic7⋊C4C22semidirect
48. Dic7⋊C4C22semidirect
49. Dic7⋊C4C22semidirect
50. C4×Dic7C22semidirect
96. #121D7semidirect
141. C2×D4C2×D7semidirect
144. C22×C4C2×D7semidirect
150. C4⋊C4C2×D7semidirect
162. C22⋊C4C2×D7semidirect
206. C7#560semidirect

Maximal subgroups

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

subgroupindex
C7×#1212
#65032
#65022
#65012
#65002
#64932
#64922
#64612
#64572
#64492
#64472
#64452
#64422
#5607