Finite Group Databasebeta

← #90393

Subgroups — #90393

Order 448.

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

normal subgroups only

This group has 212 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×#126C211yesmaximal, characteristic, complemented
2#6461C211yesmaximal, characteristic, complemented
3#6461C211yesmaximal, characteristic, complemented
4#6459C211yesmaximal, characteristic, complemented
5#6451C211yesmaximal, characteristic, complemented
6#6450C211yesmaximal, characteristic, complemented
7#6450C211yesmaximal, characteristic, complemented
8#6448C211yesmaximal, characteristic
9#6448C211yesmaximal, characteristic
10(C22⋊D7)⋊C4C211yesmaximal, characteristic
11#6442C211yesmaximal, characteristic, complemented
12#6441C211yesmaximal, characteristic, complemented
13#6440C211yesmaximal, characteristic, complemented
14#6435C211yesmaximal, characteristic, complemented
15C4×Dic14C211yesmaximal, 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
38Dic7⋊C4C2211yescharacteristic
39Dic7⋊C4C2211yescharacteristic
40Dic7⋊C4C2211yescharacteristic
41Dic7⋊C4C2211yescharacteristic
42Dic7⋊C4C2211yescharacteristic
43Dic7⋊C4C2211yescharacteristic
44Dic7⋊C4C2211yescharacteristic
45C4×Dic7C2211yescharacteristic
46C4×Dic7C2211yescharacteristic
47C4×Dic7C2211yescharacteristic
48C4×Dic7C2211yescharacteristic
49C4×Dic7C2211yescharacteristic
50C4×Dic7C2211yescharacteristic
51#556—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—22no—
61C2×Dic7—22no—
62C2×Dic7C2311yescharacteristic
63C2×Dic7C2311yescharacteristic
64C2×Dic7C2311yescharacteristic
65C2×Dic7C2311yescharacteristic
66C2×Dic7C2311yescharacteristic
67C2×Dic7C2311yescharacteristic
68C2×Dic7C2311yescharacteristic
69C4×D7—22no—
70Dic14—22no—
71Dic14—22no—
72Dic14—22no—
73C4⋊Q8—71no—
74#126—71no—
75#126—71no—
76#126—71no—
77#126D711yescharacteristic, complemented
78#124—71no—
79#124—71no—
80C22⋊Q8—71no—
81C22⋊Q8—71no—
82C4×Q8—71no—
83C4×Q8—71no—
84C4×Q8—71no—
85C4×D4—71no—
86#117—71no—
87#117—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#4112yes—
107Dic7#4112yes—
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⋊C4C2×D711yescharacteristic
124C4⋊C4C2×D711yescharacteristic
125C4⋊C4C2×D711yescharacteristic
126C22⋊C4—71no—
127C22⋊C4—71no—
128C22⋊C4—71no—
129C22⋊C4—71no—
130C22⋊C4—71no—
131C22⋊C4—71no—
132C22⋊C4—71no—
133C22⋊C4C2×D711yescharacteristic
134C22⋊C4C2×D711yescharacteristic
135C22⋊C4C2×D711yescharacteristic
136C42—71no—
137C42—71no—
138C42—71no—
139C42—71no—
140C42—71no—
141C42—71no—
142C42C2×D711yescharacteristic
143C2×C7—41no—
144C2×C7C2×#4111yescharacteristic
145C2×C7C2×#4111yescharacteristic
146C2×C72−1+411yescharacteristic
147D7—41no—
148C23—71no—
149C23C22×D711yescharacteristic
150Q8—142no—
151Q8—142no—
152Q8—142no—
153D4—142no—
154C2×C4—71no—
155C2×C4—71no—
156C2×C4—71no—
157C2×C4—71no—
158C2×C4—71no—
159C2×C4—71no—
160C2×C4—71no—
161C2×C4—142no—
162C2×C4—142no—
163C2×C4C22×D711yescharacteristic
164C2×C4C22×D711yescharacteristic
165C2×C4C22×D711yescharacteristic
166C2×C4C22×D711yescharacteristic
167C2×C4C22×D711yescharacteristic
168C2×C4C22×D711yescharacteristic
169C7#55611yescharacteristic, 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—72no—
184C4—72no—
185C4—141no—
186C4—141no—
187C4—141no—
188C4—141no—
189C4—141no—
190C2—41no—
191C2—281no—
192C2D7×#4111yescharacteristic
193C2D7×#4111yescharacteristic
194C2#655411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×#126C2
prime radicalC2×C22⋊D7C22
hypercenter#126D7
hypocenterC7#556
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. #6461C2semidirect
4. #6459C2semidirect
5. #6451C2semidirect
6. #6450C2semidirect
7. #6450C2semidirect
11. #6442C2semidirect
12. #6441C2semidirect
13. #6440C2semidirect
14. #6435C2semidirect
15. C4×Dic14C2semidirect
77. #126D7semidirect
169. C7#556semidirect

Maximal subgroups

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

subgroupindex
C7×#1262
#64612
#64592
#64512
#64502
#64482
(C22⋊D7)⋊C42
#64422
#64412
#64402
#64352
C4×Dic142
#5567