Finite Group Databasebeta

← #90397

Subgroups — #90397

Order 448.

234 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 97 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×#126C211yesmaximal, characteristic, complemented
2#6459C211yesmaximal, characteristic, complemented
3#6457C211yesmaximal, characteristic, complemented
4#6456C211yesmaximal, characteristic, complemented
5#6455C211yesmaximal, characteristic, complemented
6#6452C211yesmaximal, characteristic, complemented
7#6450C211yesmaximal, characteristic, complemented
8#6448C211yesmaximal, characteristic, complemented
9#6447C211yesmaximal, characteristic
10#6446C211yesmaximal, characteristic, complemented
11(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
12(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
13#6440C211yesmaximal, characteristic, complemented
14#6439C211yesmaximal, characteristic, complemented
15C42×D7C211yesmaximal, characteristic, complemented
16C2×C22⋊D7C2211yescharacteristic
17C2×C22⋊D7C2211yescharacteristic
18C22×Dic7C2211yescharacteristic
19C2×D28C2211yescharacteristic
20C2×C4×D7C2211yescharacteristic
21C2×C4×D7C2211yescharacteristic
22C2×C4×D7C2211yescharacteristic
23C2×C4×D7C2211yescharacteristic
24C2×Dic14C2211yescharacteristic
25C7×C4⋊C4C2211yescharacteristic
26C7×C4⋊C4C2211yescharacteristic
27C7×C4⋊C4C2211yescharacteristic
28C7×C22⋊C4C2211yescharacteristic
29C7×C22⋊C4C2211yescharacteristic
30C7×C22⋊C4C2211yescharacteristic
31C42×C7C2211yescharacteristic
32C22⋊Dic7C2211yescharacteristic
33(C2×D7)⋊C4C2211yescharacteristic
34(C2×D7)⋊C4C2211yescharacteristic
35(C2×D7)⋊C4C2211yescharacteristic
36(C2×D7)⋊C4C2211yescharacteristic
37(C2×D7)⋊C4C2211yescharacteristic
38(C2×D7)⋊C4C2211yescharacteristic
39C4⋊Dic7C2211yescharacteristic
40Dic7⋊C4C2211yescharacteristic
41Dic7⋊C4C2211yescharacteristic
42Dic7⋊C4C2211yescharacteristic
43Dic7⋊C4C2211yescharacteristic
44Dic7⋊C4C2211yescharacteristic
45Dic7⋊C4C2211yescharacteristic
46C4×Dic7C2211yescharacteristic
47C4×Dic7C2211yescharacteristic
48C4×Dic7C2211yescharacteristic
49C4×Dic7C2211yescharacteristic
50C4×Dic7C2211yescharacteristic
51#529—71nomaximal, Sylow
52C23×C7C2311yescharacteristic
53C22×D7C2311yescharacteristic
54C22×D7C2311yescharacteristic
55C2×C4×C7C2311yescharacteristic
56C2×C4×C7C2311yescharacteristic
57C2×C4×C7C2311yescharacteristic
58C2×C4×C7C2311yescharacteristic
59C2×C4×C7C2311yescharacteristic
60C2×C4×C7C2311yescharacteristic
61C22⋊D7—22no—
62C22⋊D7—22no—
63C2×Dic7—22no—
64C2×Dic7C2311yescharacteristic
65C2×Dic7C2311yescharacteristic
66C2×Dic7C2311yescharacteristic
67C2×Dic7C2311yescharacteristic
68C2×Dic7C2311yescharacteristic
69C2×Dic7C2311yescharacteristic
70D28—22no—
71C4×D7—22no—
72C4×D7—22no—
73C4×D7—22no—
74C4×D7—22no—
75Dic14—22no—
76#126—71no—
77#126D711yescharacteristic, complemented
78#125—71no—
79#124—71no—
80#123—71no—
81#123—71no—
82C22⋊Q8—71no—
83#121—71no—
84C4×Q8—71no—
85C4×D4—71no—
86C4×D4—71no—
87C4×D4—71no—
88#117—71no—
89#117—71no—
90C2×C42—71no—
91C22×C7—21no—
92C22×C7—21no—
93C22×C7—21no—
94C22×C7C2411yescharacteristic
95C2×D7—21no—
96C2×D7—21no—
97C2×D7—21no—
98C2×D7—21no—
99C2×D7—21no—
100C2×D7#4112yes—
101C4×C7—21no—
102C4×C7—21no—
103C4×C7—21no—
104C4×C7—21no—
105C4×C7—21no—
106C4×C7—21no—
107Dic7—21no—
108Dic7—21no—
109Dic7—21no—
110Dic7—21no—
111Dic7#4112yes—
112Dic7#4112yes—
113C2×Q8—71no—
114C2×D4—71no—
115C2×D4—71no—
116C2×D4—71no—
117C22×C4—71no—
118C22×C4—71no—
119C22×C4—71no—
120C22×C4—71no—
121C22×C4—71no—
122C4⋊C4—71no—
123C4⋊C4—71no—
124C4⋊C4—71no—
125C4⋊C4—71no—
126C4⋊C4—71no—
127C4⋊C4—71no—
128C4⋊C4—71no—
129C4⋊C4C2×D711yescharacteristic
130C4⋊C4C2×D711yescharacteristic
131C4⋊C4C2×D711yescharacteristic
132C22⋊C4—71no—
133C22⋊C4—71no—
134C22⋊C4—71no—
135C22⋊C4—71no—
136C22⋊C4—71no—
137C22⋊C4—71no—
138C22⋊C4—71no—
139C22⋊C4C2×D711yescharacteristic
140C22⋊C4C2×D711yescharacteristic
141C22⋊C4C2×D711yescharacteristic
142C42—71no—
143C42—71no—
144C42—71no—
145C42—71no—
146C42—71no—
147C42C2×D711yescharacteristic
148C2×C7—41no—
149C2×C7C2×#4111yescharacteristic
150C2×C7C2×#4111yescharacteristic
151C2×C7C2×#4111yescharacteristic
152D7—22no—
153D7—41no—
154C23—71no—
155C23—71no—
156C23C22×D711yescharacteristic
157Q8—142no—
158D4—142no—
159D4—142no—
160D4—142no—
161C2×C4—71no—
162C2×C4—71no—
163C2×C4—71no—
164C2×C4—71no—
165C2×C4—71no—
166C2×C4—71no—
167C2×C4—142no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4C22×D711yescharacteristic
173C2×C4C22×D711yescharacteristic
174C2×C4C22×D711yescharacteristic
175C2×C4C22×D711yescharacteristic
176C2×C4C22×D711yescharacteristic
177C2×C4C22×D711yescharacteristic
178C7#52911yescharacteristic, complemented, Sylow
179C22—21no—
180C22—21no—
181C22—21no—
182C22—72no—
183C22—141no—
184C22—141no—
185C22—141no—
186C22—141no—
187C22—141no—
188C22C23×D711yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—72no—
196C4—72no—
197C4—141no—
198C4—141no—
199C4—141no—
200C4—141no—
201C2—41no—
202C2—142no—
203C2—281no—
204C2D7×#4111yescharacteristic
205C2D7×#4111yescharacteristic
206C2D7×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#126C2
prime radical#6447C2
hypercenter#126D7
hypocenterC7#529
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C7×#126C2semidirect
2. #6459C2semidirect
3. #6457C2semidirect
4. #6456C2semidirect
5. #6455C2semidirect
6. #6452C2semidirect
7. #6450C2semidirect
8. #6448C2semidirect
10. #6446C2semidirect
11. (C22⋊D7)⋊C4C2semidirect
12. (C22⋊D7)⋊C4C2semidirect
13. #6440C2semidirect
14. #6439C2semidirect
15. C42×D7C2semidirect
77. #126D7semidirect
178. C7#529semidirect

Maximal subgroups

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

subgroupindex
C7×#1262
#64592
#64572
#64562
#64552
#64522
#64502
#64482
#64472
#64462
(C22⋊D7)⋊C42
#64402
#64392
C42×D72
#5297