Finite Group Databasebeta

← #90304

Subgroups — #90304

Order 448.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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#6502C211yesmaximal, characteristic, complemented
3#6501C211yesmaximal, characteristic, complemented
4#6500C211yesmaximal, characteristic, complemented
5#6500C211yesmaximal, characteristic, complemented
6#6499C211yesmaximal, characteristic, complemented
7#6498C211yesmaximal, characteristic, complemented
8#6492C211yesmaximal, characteristic, complemented
9C4×C22⋊D7C211yesmaximal, characteristic, complemented
10#6461C211yesmaximal, characteristic, complemented
11#6459C211yesmaximal, characteristic, complemented
12#6448C211yesmaximal, characteristic, complemented
13(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
14D7×C22⋊C4C211yesmaximal, 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
25C22×Dic7C2211yescharacteristic, complemented
26C2×C4×D7C2211yescharacteristic
27C2×C4×D7C2211yescharacteristic
28C2×Dic14C2211yescharacteristic, complemented
29C7×C4⋊C4C2211yescharacteristic, complemented
30C7×C22⋊C4C2211yescharacteristic
31C7×C22⋊C4C2211yescharacteristic, complemented
32C22⋊Dic7C2211yescharacteristic
33C22⋊Dic7C2211yescharacteristic
34C22⋊Dic7C2211yescharacteristic
35C22⋊Dic7C2211yescharacteristic, complemented
36C22⋊Dic7C2211yescharacteristic, complemented
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
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×C7C2311yescharacteristic
53C23×C7C2311yescharacteristic
54C23×C7C2311yescharacteristic
55C22×D7—22no—
56C22×D7—41no—
57C22×D7—41no—
58C22×D7C2311yescharacteristic
59C22×D7C2311yescharacteristic
60C7×D4—41no—
61C7×D4—41no—
62C7×D4—41no—
63C2×C4×C7—41no—
64C2×C4×C7C2311yescharacteristic
65C2×C4×C7C2311yescharacteristic
66C2×C4×C7C2311yescharacteristic
67C2×C4×C7C2311yescharacteristic
68C22⋊D7—22no—
69C22⋊D7—22no—
70C22⋊D7—41no—
71C22⋊D7—41no—
72C2×Dic7—41no—
73C2×Dic7C2311yescharacteristic
74C2×Dic7C2311yescharacteristic
75C2×Dic7C2311yescharacteristic
76C2×Dic7C2311yescharacteristic
77C2×Dic7C2311yescharacteristic
78C2×Dic7C2311yescharacteristic
79C4×D7—41no—
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#121D711yescharacteristic, complemented
92#120—71no—
93#120—71no—
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×C7C2411yescharacteristic
107C2×D7—21no—
108C2×D7—21no—
109C2×D7—21no—
110C2×D7—21no—
111C2×D7—21no—
112C2×D7—44no—
113C2×D7#4112yes—
114C4×C7—21no—
115C4×C7—21no—
116C4×C7—21no—
117C4×C7—21no—
118Dic7—21no—
119Dic7—21no—
120Dic7—21no—
121Dic7—21no—
122Dic7—21no—
123Dic7—21no—
124C24—71no—
125C2×Q8—71no—
126C2×D4—71no—
127C2×D4—71no—
128C2×D4—71no—
129C2×D4—71no—
130C2×D4C2×D711yescharacteristic
131C2×D4C2×D711yescharacteristic
132C2×D4C2×D711yescharacteristic, complemented
133C22×C4—71no—
134C22×C4—71no—
135C22×C4—71no—
136C22×C4C2×D711yescharacteristic, complemented
137C4⋊C4—71no—
138C4⋊C4—71no—
139C4⋊C4—71no—
140C4⋊C4—71no—
141C4⋊C4—71no—
142C4⋊C4C2×D711yescharacteristic, complemented
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⋊C4—71no—
155C22⋊C4C2×D711yescharacteristic
156C22⋊C4C2×D711yescharacteristic, complemented
157C42—71no—
158C42—71no—
159C2×C7—41no—
160C2×C7—41no—
161C2×C7—41no—
162C2×C7C2×#4111yescharacteristic
163C2×C7H(5,𝔽2)11yescharacteristic
164C2×C7H(5,𝔽2)11yescharacteristic
165D7—22no—
166D7—41no—
167C23—71no—
168C23—71no—
169C23—142no—
170C23—281no—
171C23—281no—
172C23C22×D711yescharacteristic
173C23C22×D711yescharacteristic
174C23C22×D711yescharacteristic
175Q8—281no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—142no—
180D4—142no—
181D4—281no—
182D4—281no—
183C2×C4—41no—
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×C4C22×D711yescharacteristic
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—72no—
208C22—141no—
209C22—141no—
210C22—141no—
211C22—141no—
212C22—141no—
213C22—284no—
214C22C23×D711yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—141no—
220C4—141no—
221C4—141no—
222C4—141no—
223C4—141no—
224C4—141no—
225C2—41no—
226C2—41no—
227C2—41no—
228C2—142no—
229C2—281no—
230C2#654811yescharacteristic
231C2#654811yescharacteristic
232C2D7×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#121C2
hypercenter#121D7
hypocenterC7#538
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. #6502C2semidirect
3. #6501C2semidirect
4. #6500C2semidirect
5. #6500C2semidirect
6. #6499C2semidirect
7. #6498C2semidirect
8. #6492C2semidirect
9. C4×C22⋊D7C2semidirect
10. #6461C2semidirect
11. #6459C2semidirect
12. #6448C2semidirect
13. (C22⋊D7)⋊C4C2semidirect
14. D7×C22⋊C4C2semidirect
15. #6442C2semidirect
19. C2×C7×D4C22semidirect
20. C22×C4×C7C22semidirect
25. C22×Dic7C22semidirect
28. C2×Dic14C22semidirect
29. C7×C4⋊C4C22semidirect
31. C7×C22⋊C4C22semidirect
35. C22⋊Dic7C22semidirect
36. C22⋊Dic7C22semidirect
37. C22⋊Dic7C22semidirect
44. C4⋊Dic7C22semidirect
45. Dic7⋊C4C22semidirect
46. Dic7⋊C4C22semidirect
47. Dic7⋊C4C22semidirect
48. Dic7⋊C4C22semidirect
49. C4×Dic7C22semidirect
50. C4×Dic7C22semidirect
91. #121D7semidirect
132. C2×D4C2×D7semidirect
136. C22×C4C2×D7semidirect
142. C4⋊C4C2×D7semidirect
156. C22⋊C4C2×D7semidirect
197. C7#538semidirect

Maximal subgroups

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

subgroupindex
C7×#1212
#65022
#65012
#65002
#64992
#64982
#64922
C4×C22⋊D72
#64612
#64592
#64482
(C22⋊D7)⋊C42
D7×C22⋊C42
#64422
#5387