Finite Group Databasebeta

← #90384

Subgroups — #90384

Order 448.

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

normal subgroups only

This group has 214 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×#125C211yesmaximal, characteristic, complemented
2#6461C211yesmaximal, characteristic, complemented
3#6461C211yesmaximal, characteristic, complemented
4#6459C211yesmaximal, characteristic, complemented
5#6459C211yesmaximal, characteristic, complemented
6#6457C211yesmaximal, characteristic
7#6457C211yesmaximal, characteristic
8#6456C211yesmaximal, characteristic
9#6455C211yesmaximal, characteristic, complemented
10D7×C4⋊C4C211yesmaximal, characteristic, complemented
11#6452C211yesmaximal, characteristic, complemented
12#6452C211yesmaximal, characteristic, complemented
13#6450C211yesmaximal, characteristic, complemented
14#6435C211yesmaximal, characteristic, complemented
15#6435C211yesmaximal, characteristic, complemented
16C2×D28C2211yescharacteristic
17C2×C4×D7C2211yescharacteristic
18C2×C4×D7C2211yescharacteristic
19C2×C4×D7C2211yescharacteristic
20C2×C4×D7C2211yescharacteristic
21C2×Dic14C2211yescharacteristic
22C7×C4⋊C4C2211yescharacteristic
23C7×C4⋊C4C2211yescharacteristic
24C7×C4⋊C4C2211yescharacteristic
25C7×C4⋊C4C2211yescharacteristic
26C7×C4⋊C4C2211yescharacteristic
27C7×C4⋊C4C2211yescharacteristic
28C42×C7C2211yescharacteristic
29(C2×D7)⋊C4C2211yescharacteristic
30(C2×D7)⋊C4C2211yescharacteristic
31(C2×D7)⋊C4C2211yescharacteristic
32(C2×D7)⋊C4C2211yescharacteristic
33(C2×D7)⋊C4C2211yescharacteristic
34(C2×D7)⋊C4C2211yescharacteristic
35(C2×D7)⋊C4C2211yescharacteristic
36(C2×D7)⋊C4C2211yescharacteristic
37C4⋊Dic7C2211yescharacteristic
38C4⋊Dic7C2211yescharacteristic
39Dic7⋊C4C2211yescharacteristic
40Dic7⋊C4C2211yescharacteristic
41Dic7⋊C4C2211yescharacteristic
42Dic7⋊C4C2211yescharacteristic
43Dic7⋊C4C2211yescharacteristic
44Dic7⋊C4C2211yescharacteristic
45Dic7⋊C4C2211yescharacteristic
46Dic7⋊C4C2211yescharacteristic
47C4×Dic7C2211yescharacteristic
48C4×Dic7C2211yescharacteristic
49C4×Dic7C2211yescharacteristic
50C4×Dic7C2211yescharacteristic
51#552—71nomaximal, Sylow
52C22×D7C2311yescharacteristic
53C22×D7C2311yescharacteristic
54C2×C4×C7C2311yescharacteristic
55C2×C4×C7C2311yescharacteristic
56C2×C4×C7C2311yescharacteristic
57C2×C4×C7C2311yescharacteristic
58C2×C4×C7C2311yescharacteristic
59C2×C4×C7C2311yescharacteristic
60C2×C4×C7C2311yescharacteristic
61C2×Dic7C2311yescharacteristic
62C2×Dic7C2311yescharacteristic
63C2×Dic7C2311yescharacteristic
64C2×Dic7C2311yescharacteristic
65C2×Dic7C2311yescharacteristic
66C2×Dic7C2311yescharacteristic
67D28—22no—
68C4×D7—22no—
69C4×D7—22no—
70C4×D7—22no—
71C4×D7—22no—
72Dic14—22no—
73#126—71no—
74#126—71no—
75#125—71no—
76#125—71no—
77#125D711yescharacteristic, complemented
78#123—71no—
79#123—71no—
80C22⋊Q8—71no—
81C22⋊Q8—71no—
82C4×Q8—71no—
83C4×D4—71no—
84#117—71no—
85#117—71no—
86#117—71no—
87C2×C4⋊C4—71no—
88C22×C7C2411yescharacteristic
89C2×D7—21no—
90C2×D7—21no—
91C2×D7—21no—
92C2×D7—21no—
93C2×D7—21no—
94C2×D7#4112yes—
95C4×C7—21no—
96C4×C7—21no—
97C4×C7—21no—
98C4×C7—21no—
99C4×C7—21no—
100C4×C7—21no—
101C4×C7—21no—
102Dic7—21no—
103Dic7—21no—
104Dic7—21no—
105Dic7—21no—
106Dic7—21no—
107Dic7#4112yes—
108C2×Q8—71no—
109C2×D4—71no—
110C22×C4—71no—
111C22×C4—71no—
112C22×C4—71no—
113C22×C4—71no—
114C4⋊C4—71no—
115C4⋊C4—71no—
116C4⋊C4—71no—
117C4⋊C4—71no—
118C4⋊C4—71no—
119C4⋊C4—71no—
120C4⋊C4—71no—
121C4⋊C4—71no—
122C4⋊C4—71no—
123C4⋊C4—71no—
124C4⋊C4C2×D711yescharacteristic
125C4⋊C4C2×D711yescharacteristic
126C4⋊C4C2×D711yescharacteristic
127C4⋊C4C2×D711yescharacteristic
128C4⋊C4C2×D711yescharacteristic
129C4⋊C4C2×D711yescharacteristic
130C22⋊C4—71no—
131C22⋊C4—71no—
132C22⋊C4—71no—
133C22⋊C4—71no—
134C22⋊C4—71no—
135C22⋊C4—71no—
136C22⋊C4—71no—
137C22⋊C4—71no—
138C42—71no—
139C42—71no—
140C42—71no—
141C42—71no—
142C42C2×D711yescharacteristic
143C2×C7C2×#4111yescharacteristic
144C2×C7C2×#4111yescharacteristic
145C2×C72−1+411yescharacteristic
146D7—22no—
147D7—41no—
148C23—71no—
149C23—71no—
150Q8—142no—
151D4—142no—
152C2×C4—71no—
153C2×C4—71no—
154C2×C4—71no—
155C2×C4—71no—
156C2×C4—71no—
157C2×C4—71no—
158C2×C4—142no—
159C2×C4—142no—
160C2×C4—142no—
161C2×C4—142no—
162C2×C4C22×D711yescharacteristic
163C2×C4C22×D711yescharacteristic
164C2×C4C22×D711yescharacteristic
165C2×C4C22×D711yescharacteristic
166C2×C4C22×D711yescharacteristic
167C2×C4C22×D711yescharacteristic
168C2×C4C22×D711yescharacteristic
169C7#55211yescharacteristic, complemented, Sylow
170C22—72no—
171C22—141no—
172C22—141no—
173C22—141no—
174C22—141no—
175C22—141no—
176C22C23×D711yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—72no—
185C4—141no—
186C4—141no—
187C4—141no—
188C4—141no—
189C4—141no—
190C2—142no—
191C2—281no—
192C2#655111yescharacteristic
193C2D7×#4111yescharacteristic
194C2D7×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#125C2
prime radicalC2×D28C22
hypercenter#125D7
hypocenterC7#552
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C7×#125C2semidirect
2. #6461C2semidirect
3. #6461C2semidirect
4. #6459C2semidirect
5. #6459C2semidirect
9. #6455C2semidirect
10. D7×C4⋊C4C2semidirect
11. #6452C2semidirect
12. #6452C2semidirect
13. #6450C2semidirect
14. #6435C2semidirect
15. #6435C2semidirect
77. #125D7semidirect
169. C7#552semidirect

Maximal subgroups

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

subgroupindex
C7×#1252
#64612
#64592
#64572
#64562
#64552
D7×C4⋊C42
#64522
#64502
#64352
#5527