Finite Group Databasebeta

← D7×C22⋊Q8 #90317

Subgroups — D7×C22⋊Q8 #90317

Order 448.

322 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×Q8×D7C211yesmaximal, characteristic, complemented
2C22×C4×D7C211yesmaximal, characteristic
3C7×C22⋊Q8C211yesmaximal, characteristic, complemented
4#6509C211yesmaximal, characteristic, complemented
5#6487C211yesmaximal, characteristic, complemented
6#6460C211yesmaximal, characteristic, complemented
7#6459C212yesmaximal, complemented
8D7×C4⋊C4C211yesmaximal, characteristic, complemented
9D7×C4⋊C4C212yesmaximal, complemented
10D7×C22⋊C4C212yesmaximal
11#6441C212yesmaximal, complemented
12C23×D7C2211yescharacteristic
13C2×C7×Q8C2211yescharacteristic, complemented
14C22×C4×C7C2211yescharacteristic
15C22×Dic7C2211yescharacteristic
16Q8×D7—24no—
17C2×C4×D7—24no—
18C2×C4×D7C2212yes—
19C2×C4×D7C2212yes—
20C2×C4×D7C2211yescharacteristic
21C2×C4×D7C2211yescharacteristic
22C2×C4×D7C2211yescharacteristic
23C2×C4×D7C2211yescharacteristic
24C2×Dic14C2212yescomplemented
25C2×Dic14C2211yescharacteristic, complemented
26C7×C4⋊C4C2212yescomplemented
27C7×C4⋊C4C2211yescharacteristic, complemented
28C7×C22⋊C4C2212yes—
29C22⋊Dic7C2212yes—
30(C2×D7)⋊C4C2212yes—
31(C2×D7)⋊C4C2212yes—
32C4⋊Dic7C2212yescomplemented
33C4⋊Dic7C2211yescharacteristic, complemented
34Dic7⋊C4C2212yescomplemented
35Dic7⋊C4C2212yescomplemented
36Dic7⋊C4C2212yescomplemented
37C2×C22⋊Q8—71nomaximal, Sylow
38C23×C7C2311yescharacteristic
39C22×D7—22no—
40C22×D7—22no—
41C22×D7Q814yescomplemented
42C22×D7C2311yescharacteristic
43C22×D7C2311yescharacteristic
44C7×Q8—22no—
45C2×C4×C7—22no—
46C2×C4×C7C2311yescharacteristic
47C2×C4×C7C2312yes—
48C2×C4×C7C2312yes—
49C2×C4×C7C2311yescharacteristic
50C2×Dic7—22no—
51C2×Dic7C2312yes—
52C2×Dic7C2311yescharacteristic
53C2×Dic7C2312yes—
54C2×Dic7C2311yescharacteristic
55C4×D7—22no—
56C4×D7—24no—
57C4×D7—24no—
58C4×D7—22no—
59C4×D7—22no—
60C4×D7D414yes—
61Dic14—22no—
62Dic14—24no—
63C22×Q8—71no—
64C23×C4—71no—
65C22⋊Q8—71no—
66C22⋊Q8—71no—
67C22⋊Q8—72no—
68C22⋊Q8—71no—
69C22⋊Q8—72no—
70C22⋊Q8D711yescharacteristic, direct product factor
71C2×C4⋊C4—71no—
72C2×C4⋊C4—72no—
73C2×C22⋊C4—72no—
74C22×C7—21no—
75C22×C7—21no—
76C22×C7C2×Q812yescomplemented
77C22×C7C2411yescharacteristic
78C2×D7—21no—
79C2×D7—21no—
80C2×D7—28no—
81C2×D7C2×D412yes—
82C2×D7C2×Q812yes—
83C2×D7C2×Q812yescomplemented
84C2×D7#4112yes—
85C4×C7—22no—
86C4×C7—21no—
87C4×C7—22no—
88C4×C7C2×D412yes—
89Dic7—22no—
90Dic7—21no—
91Dic7—22no—
92Dic7C2×D412yes—
93C24—71no—
94C2×Q8—72no—
95C2×Q8—71no—
96C2×Q8—144no—
97C2×Q8C2×D711yescharacteristic, complemented
98C22×C4—71no—
99C22×C4—71no—
100C22×C4—72no—
101C22×C4—72no—
102C22×C4—71no—
103C22×C4—71no—
104C22×C4—71no—
105C22×C4—144no—
106C22×C4C2×D711yescharacteristic
107C4⋊C4—72no—
108C4⋊C4—72no—
109C4⋊C4—72no—
110C4⋊C4—71no—
111C4⋊C4—72no—
112C4⋊C4C2×D711yescharacteristic, complemented
113C4⋊C4C2×D712yescomplemented
114C22⋊C4—72no—
115C22⋊C4—72no—
116C22⋊C4—72no—
117C22⋊C4C2×D712yes—
118C2×C7—22no—
119C2×C7C22×D411yescharacteristic
120C2×C7C22×Q811yescharacteristic
121C2×C7C2×#4111yescharacteristic
122D7—22no—
123D7C22⋊Q814yesdirect product factor
124C23—71no—
125C23—74no—
126C23—71no—
127C23—142no—
128C23—142no—
129C23C22×D711yescharacteristic
130Q8—22no—
131Q8—144no—
132Q8—142no—
133C2×C4—22no—
134C2×C4—71no—
135C2×C4—74no—
136C2×C4—72no—
137C2×C4—72no—
138C2×C4—71no—
139C2×C4—142no—
140C2×C4—144no—
141C2×C4—142no—
142C2×C4—144no—
143C2×C4—142no—
144C2×C4—142no—
145C2×C4C22×D712yes—
146C2×C4C22×D712yes—
147C2×C4C22×D711yescharacteristic
148C2×C4C22×D711yescharacteristic
149C7C2×C22⋊Q811yescharacteristic, complemented, Sylow
150C22—21no—
151C22—21no—
152C22—72no—
153C22—72no—
154C22—72no—
155C22—72no—
156C22—148no—
157C22—141no—
158C22—141no—
159C22Q8×D712yescomplemented
160C22C23×D711yescharacteristic
161C4—21no—
162C4—22no—
163C4—22no—
164C4—72no—
165C4—141no—
166C4—142no—
167C4—142no—
168C4D4×D712yes—
169C2—22no—
170C2—74no—
171C2—142no—
172C2C2×D4×D711yescharacteristic
173C2C2×Q8×D711yescharacteristic
174C2D7×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×C22⋊Q8C2
prime radicalC23×D7C22
hypercenterC22⋊Q8D7
hypocenterC7C2×C22⋊Q8
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D7C2semidirect
3. C7×C22⋊Q8C2semidirect
4. #6509C2semidirect
5. #6487C2semidirect
6. #6460C2semidirect
7. #6459C2semidirect
8. D7×C4⋊C4C2semidirect
9. D7×C4⋊C4C2semidirect
11. #6441C2semidirect
13. C2×C7×Q8C22semidirect
24. C2×Dic14C22semidirect
25. C2×Dic14C22semidirect
26. C7×C4⋊C4C22semidirect
27. C7×C4⋊C4C22semidirect
32. C4⋊Dic7C22semidirect
33. C4⋊Dic7C22semidirect
34. Dic7⋊C4C22semidirect
35. Dic7⋊C4C22semidirect
36. Dic7⋊C4C22semidirect
41. C22×D7Q8semidirect
70. C22⋊Q8D7direct (N × H)
76. C22×C7C2×Q8semidirect
83. C2×D7C2×Q8semidirect
97. C2×Q8C2×D7semidirect
112. C4⋊C4C2×D7semidirect
113. C4⋊C4C2×D7semidirect
123. D7C22⋊Q8direct (N × H)
149. C7C2×C22⋊Q8semidirect
159. C22Q8×D7semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D72
C22×C4×D72
C7×C22⋊Q82
#65092
#64872
#64602
#64592
D7×C4⋊C42
D7×C22⋊C42
#64412
C2×C22⋊Q87