Finite Group Databasebeta

← #90284

Subgroups — #90284

Order 448.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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×#120C211yesmaximal, characteristic, complemented
2#6516C211yesmaximal, characteristic, complemented
3C2×C22⋊Dic7C211yesmaximal, characteristic, complemented
4#6502C211yesmaximal, characteristic, complemented
5#6502C211yesmaximal, characteristic, complemented
6#6501C211yesmaximal, characteristic, complemented
7#6499C211yesmaximal, characteristic, complemented
8#6498C211yesmaximal, characteristic, complemented
9D4×Dic7C211yesmaximal, characteristic, complemented
10#6448C211yesmaximal, characteristic, complemented
11#6446C211yesmaximal, characteristic, complemented
12(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
13#6442C211yesmaximal, characteristic, complemented
14#6442C211yesmaximal, characteristic, complemented
15#6441C211yesmaximal, characteristic, complemented
16C24×C7C2211yescharacteristic
17C2×C7×D4C2211yescharacteristic
18C2×C7×D4C2211yescharacteristic
19C2×C7×D4C2211yescharacteristic
20C2×C22⋊D7C2211yescharacteristic
21C2×C22⋊D7C2211yescharacteristic
22C2×C22⋊D7C2211yescharacteristic
23C2×C22⋊D7C2211yescharacteristic, complemented
24C22×Dic7C2211yescharacteristic
25C22×Dic7C2211yescharacteristic
26C22×Dic7C2211yescharacteristic, complemented
27C2×C4×D7C2211yescharacteristic, complemented
28C2×Dic14C2211yescharacteristic, complemented
29C7×C22⋊C4C2211yescharacteristic
30C7×C22⋊C4C2211yescharacteristic
31C7×C22⋊C4C2211yescharacteristic
32C22⋊Dic7C2211yescharacteristic
33C22⋊Dic7C2211yescharacteristic
34C22⋊Dic7C2211yescharacteristic
35C22⋊Dic7C2211yescharacteristic
36C22⋊Dic7C2211yescharacteristic
37C22⋊Dic7C2211yescharacteristic
38C22⋊Dic7C2211yescharacteristic
39C22⋊Dic7C2211yescharacteristic, complemented
40C22⋊Dic7C2211yescharacteristic, complemented
41(C2×D7)⋊C4C2211yescharacteristic, complemented
42(C2×D7)⋊C4C2211yescharacteristic, complemented
43C4⋊Dic7C2211yescharacteristic, complemented
44C4⋊Dic7C2211yescharacteristic, complemented
45Dic7⋊C4C2211yescharacteristic, complemented
46Dic7⋊C4C2211yescharacteristic, complemented
47Dic7⋊C4C2211yescharacteristic, complemented
48Dic7⋊C4C2211yescharacteristic, complemented
49C4×Dic7C2211yescharacteristic, complemented
50C4×Dic7C2211yescharacteristic, complemented
51#538—71nomaximal, Sylow
52C23×C7—22no—
53C23×C7—41no—
54C23×C7—41no—
55C23×C7C2311yescharacteristic
56C23×C7C2311yescharacteristic
57C23×C7C2311yescharacteristic
58C23×C7C2311yescharacteristic
59C22×D7C2311yescharacteristic
60C7×D4—22no—
61C7×D4—41no—
62C7×D4—41no—
63C2×C4×C7C2311yescharacteristic
64C2×C4×C7C2311yescharacteristic
65C2×C4×C7C2311yescharacteristic
66C22⋊D7—22no—
67C22⋊D7—41no—
68C22⋊D7—41no—
69C22⋊D7—41no—
70C2×Dic7—41no—
71C2×Dic7—41no—
72C2×Dic7—41no—
73C2×Dic7C2311yescharacteristic
74C2×Dic7C2311yescharacteristic
75C2×Dic7C2311yescharacteristic
76C2×Dic7C2311yescharacteristic
77C2×Dic7C2311yescharacteristic
78C2×Dic7C2311yescharacteristic
79C2×Dic7C2311yescharacteristic
80C4×D7—41no—
81Dic14—41no—
82#126—71no—
83#126—71no—
84#124—71no—
85#124—71no—
86#123—71no—
87#123—71no—
88C22⋊Q8—71no—
89#121—71no—
90#121—71no—
91#121—71no—
92#120—71no—
93#120D711yescharacteristic, complemented
94C4×D4—71no—
95C4×D4—71no—
96C2×C22⋊C4—71no—
97C22×C7—21no—
98C22×C7—21no—
99C22×C7—21no—
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—44no—
109C22×C7#4112yes—
110C22×C7C2411yescharacteristic
111C2×D7—21no—
112C2×D7—21no—
113C2×D7—21no—
114C4×C7—21no—
115C4×C7—21no—
116C4×C7—21no—
117Dic7—21no—
118Dic7—21no—
119Dic7—21no—
120Dic7—21no—
121Dic7—21no—
122Dic7—21no—
123Dic7—21no—
124C24C2×D711yescharacteristic
125C2×Q8—71no—
126C2×D4—71no—
127C2×D4—71no—
128C2×D4—71no—
129C2×D4—71no—
130C2×D4C2×D711yescharacteristic
131C2×D4C2×D711yescharacteristic
132C2×D4C2×D711yescharacteristic
133C22×C4—71no—
134C22×C4—71no—
135C22×C4—71no—
136C22×C4—71no—
137C4⋊C4—71no—
138C4⋊C4—71no—
139C4⋊C4—71no—
140C4⋊C4—71no—
141C4⋊C4—71no—
142C4⋊C4—71no—
143C22⋊C4—71no—
144C22⋊C4—71no—
145C22⋊C4—71no—
146C22⋊C4—71no—
147C22⋊C4—71no—
148C22⋊C4—71no—
149C22⋊C4—71no—
150C22⋊C4—71no—
151C22⋊C4—71no—
152C22⋊C4—71no—
153C22⋊C4—71no—
154C22⋊C4C2×D711yescharacteristic
155C22⋊C4C2×D711yescharacteristic
156C22⋊C4C2×D711yescharacteristic
157C42—71no—
158C42—71no—
159C2×C7—22no—
160C2×C7—41no—
161C2×C7—41no—
162C2×C7—41no—
163C2×C7C2×#4111yescharacteristic
164C2×C7H(5,𝔽2)11yescharacteristic
165C2×C7H(5,𝔽2)11yescharacteristic
166D7—41no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—71no—
171C23C22×D711yescharacteristic
172C23C22×D711yescharacteristic
173C23C22×D711yescharacteristic
174C23C22×D711yescharacteristic
175Q8—281no—
176D4—22no—
177D4—41no—
178D4—41no—
179D4—142no—
180D4—281no—
181D4—281no—
182D4—281no—
183C2×C4—71no—
184C2×C4—71no—
185C2×C4—71no—
186C2×C4—71no—
187C2×C4—71no—
188C2×C4—71no—
189C2×C4—71no—
190C2×C4—281no—
191C2×C4—281no—
192C2×C4—281no—
193C2×C4—281no—
194C2×C4C22×D711yescharacteristic
195C2×C4C22×D711yescharacteristic
196C2×C4C22×D711yescharacteristic
197C7#53811yescharacteristic, complemented, Sylow
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—141no—
211C22—141no—
212C22—141no—
213C22#115912yes—
214C22C23×D711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—141no—
219C4—141no—
220C4—141no—
221C4—141no—
222C4—141no—
223C4—141no—
224C4—141no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—281no—
230C2C2×#115911yescharacteristic
231C2#654811yescharacteristic
232C2#654811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#120C2
hypercenter#120D7
hypocenterC7#538
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C7×#120C2semidirect
2. #6516C2semidirect
3. C2×C22⋊Dic7C2semidirect
4. #6502C2semidirect
5. #6502C2semidirect
6. #6501C2semidirect
7. #6499C2semidirect
8. #6498C2semidirect
9. D4×Dic7C2semidirect
10. #6448C2semidirect
11. #6446C2semidirect
12. (C22⋊D7)⋊C4C2semidirect
13. #6442C2semidirect
14. #6442C2semidirect
15. #6441C2semidirect
23. C2×C22⋊D7C22semidirect
26. C22×Dic7C22semidirect
27. C2×C4×D7C22semidirect
28. C2×Dic14C22semidirect
39. C22⋊Dic7C22semidirect
40. C22⋊Dic7C22semidirect
41. (C2×D7)⋊C4C22semidirect
42. (C2×D7)⋊C4C22semidirect
43. C4⋊Dic7C22semidirect
44. C4⋊Dic7C22semidirect
45. Dic7⋊C4C22semidirect
46. Dic7⋊C4C22semidirect
47. Dic7⋊C4C22semidirect
48. Dic7⋊C4C22semidirect
49. C4×Dic7C22semidirect
50. C4×Dic7C22semidirect
93. #120D7semidirect
197. C7#538semidirect

Maximal subgroups

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

subgroupindex
C7×#1202
#65162
C2×C22⋊Dic72
#65022
#65012
#64992
#64982
D4×Dic72
#64482
#64462
(C22⋊D7)⋊C42
#64422
#64412
#5387