Finite Group Databasebeta

← #90403

Subgroups — #90403

Order 448.

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

normal subgroups only

This group has 196 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×#126C211yesmaximal, characteristic, complemented
2#6461C211yesmaximal, characteristic, complemented
3#6460C211yesmaximal, characteristic, complemented
4#6459C211yesmaximal, characteristic, complemented
5#6453C211yesmaximal, characteristic, complemented
6#6452C211yesmaximal, characteristic, complemented
7#6451C211yesmaximal, characteristic, complemented
8#6449C211yesmaximal, characteristic
9#6448C211yesmaximal, characteristic
10#6446C211yesmaximal, characteristic
11#6442C211yesmaximal, characteristic, complemented
12#6441C211yesmaximal, characteristic, complemented
13#6441C211yesmaximal, characteristic, complemented
14#6439C211yesmaximal, characteristic, complemented
15#6432C211yesmaximal, characteristic, complemented
16C2×C22⋊D7C2211yescharacteristic
17C22×Dic7C2211yescharacteristic
18C2×C4×D7C2211yescharacteristic
19C2×Dic14C2211yescharacteristic
20C2×Dic14C2211yescharacteristic
21C2×Dic14C2211yescharacteristic
22C7×C4⋊C4C2211yescharacteristic
23C7×C4⋊C4C2211yescharacteristic
24C7×C4⋊C4C2211yescharacteristic
25C7×C22⋊C4C2211yescharacteristic
26C7×C22⋊C4C2211yescharacteristic
27C7×C22⋊C4C2211yescharacteristic
28C42×C7C2211yescharacteristic
29C22⋊Dic7C2211yescharacteristic
30C22⋊Dic7C2211yescharacteristic
31(C2×D7)⋊C4C2211yescharacteristic
32(C2×D7)⋊C4C2211yescharacteristic
33(C2×D7)⋊C4C2211yescharacteristic
34(C2×D7)⋊C4C2211yescharacteristic
35(C2×D7)⋊C4C2211yescharacteristic
36C4⋊Dic7C2211yescharacteristic
37C4⋊Dic7C2211yescharacteristic
38C4⋊Dic7C2211yescharacteristic
39C4⋊Dic7C2211yescharacteristic
40C4⋊Dic7C2211yescharacteristic
41C4⋊Dic7C2211yescharacteristic
42Dic7⋊C4C2211yescharacteristic
43Dic7⋊C4C2211yescharacteristic
44Dic7⋊C4C2211yescharacteristic
45Dic7⋊C4C2211yescharacteristic
46Dic7⋊C4C2211yescharacteristic
47Dic7⋊C4C2211yescharacteristic
48Dic7⋊C4C2211yescharacteristic
49C4×Dic7C2211yescharacteristic
50C4×Dic7C2211yescharacteristic
51#563—71nomaximal, Sylow
52C23×C7C2311yescharacteristic
53C22×D7C2311yescharacteristic
54C2×C4×C7C2311yescharacteristic
55C2×C4×C7C2311yescharacteristic
56C2×C4×C7C2311yescharacteristic
57C2×C4×C7C2311yescharacteristic
58C2×C4×C7C2311yescharacteristic
59C2×C4×C7C2311yescharacteristic
60C22⋊D7—41no—
61C2×Dic7—41no—
62C2×Dic7C2311yescharacteristic
63C2×Dic7C2311yescharacteristic
64C2×Dic7C2311yescharacteristic
65C2×Dic7C2311yescharacteristic
66C2×Dic7C2311yescharacteristic
67C2×Dic7C2311yescharacteristic
68C2×Dic7C2311yescharacteristic
69C4×D7—41no—
70Dic14—41no—
71Dic14—41no—
72Dic14—41no—
73C4⋊Q8—71no—
74C4⋊Q8—71no—
75#126—71no—
76#126—71no—
77#126—71no—
78#126D711yescharacteristic, complemented
79#125—71no—
80#125—71no—
81#124—71no—
82#123—71no—
83#123—71no—
84C22⋊Q8—71no—
85C22⋊Q8—71no—
86C22⋊Q8—71no—
87C22⋊Q8—71no—
88C22×C7—21no—
89C22×C7—21no—
90C22×C7—21no—
91C22×C7C2411yescharacteristic
92C2×D7—21no—
93C2×D7—21no—
94C2×D7—21no—
95C4×C7—21no—
96C4×C7—21no—
97C4×C7—21no—
98C4×C7—21no—
99C4×C7—21no—
100C4×C7—21no—
101Dic7—21no—
102Dic7—21no—
103Dic7—21no—
104Dic7—21no—
105Dic7—21no—
106Dic7—21no—
107Dic7—21no—
108C2×Q8—71no—
109C2×Q8—71no—
110C2×Q8—71no—
111C2×D4—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⋊C4—71no—
125C4⋊C4—71no—
126C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
138C22⋊C4C2×D711yescharacteristic
139C22⋊C4C2×D711yescharacteristic
140C42—71no—
141C42—71no—
142C42C2×D711yescharacteristic
143C2×C7—41no—
144C2×C7H(5,𝔽2)11yescharacteristic
145C2×C72−1+411yescharacteristic
146C2×C72−1+411yescharacteristic
147D7—41no—
148C23—71no—
149C23C22×D711yescharacteristic
150Q8—281no—
151Q8—281no—
152Q8—281no—
153D4—281no—
154C2×C4—71no—
155C2×C4—71no—
156C2×C4—71no—
157C2×C4—71no—
158C2×C4—71no—
159C2×C4—71no—
160C2×C4—71no—
161C2×C4—281no—
162C2×C4—281no—
163C2×C4C22×D711yescharacteristic
164C2×C4C22×D711yescharacteristic
165C2×C4C22×D711yescharacteristic
166C2×C4C22×D711yescharacteristic
167C2×C4C22×D711yescharacteristic
168C2×C4C22×D711yescharacteristic
169C7#56311yescharacteristic, complemented, Sylow
170C22—21no—
171C22—21no—
172C22—21no—
173C22—141no—
174C22—141no—
175C22—141no—
176C22C23×D711yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—141no—
184C4—141no—
185C4—141no—
186C4—141no—
187C4—141no—
188C4—141no—
189C4—141no—
190C2—41no—
191C2—281no—
192C2#655311yescharacteristic
193C2#655411yescharacteristic
194C2#655411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#126C2
prime radicalC2×C22⋊D7C22
hypercenter#126D7
hypocenterC7#563
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. #6461C2semidirect
3. #6460C2semidirect
4. #6459C2semidirect
5. #6453C2semidirect
6. #6452C2semidirect
7. #6451C2semidirect
11. #6442C2semidirect
12. #6441C2semidirect
13. #6441C2semidirect
14. #6439C2semidirect
15. #6432C2semidirect
78. #126D7semidirect
169. C7#563semidirect

Maximal subgroups

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

subgroupindex
C7×#1262
#64612
#64602
#64592
#64532
#64522
#64512
#64492
#64482
#64462
#64422
#64412
#64392
#64322
#5637