Finite Group Databasebeta

← #11729005

Subgroups — #11729005

Order 896.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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
1#90391C211yesmaximal, characteristic
2C7×#491C211yesmaximal, characteristic, complemented
3#89861C212yesmaximal, complemented
4#89338C211yesmaximal, characteristic, complemented
5#89259C212yesmaximal, complemented
6C7×C4⋊Q8C2211yescharacteristic
7C7×#125C2211yescharacteristic
8#6459—22no—
9#6458—22no—
10#6457—22no—
11#6438C2212yes—
12C7×#103—22no—
13C7×#97C2211yescharacteristic
14#6384—22no—
15#6383—22no—
16#6377C2212yes—
17#1662—71nomaximal, Sylow
18C2×C7×Q8—21no—
19C2×D28—22no—
20C2×D28D412yes—
21C2×C4×D7—22no—
22C2×Dic14D412yes—
23C2×C7×C8—21no—
24C7×C4⋊C4—22no—
25C7×C4⋊C4—22no—
26C7×C4⋊C4D412yes—
27C42×C7C2311yescharacteristic
28(C2×D7)⋊C4—22no—
29(C2×D7)⋊C4—22no—
30(C2×D7)⋊C4—22no—
31Dic7⋊C4—22no—
32C2×C7⋊C8—22no—
33#562—71no—
34#491D711yescharacteristic, complemented
35#489—72no—
36#330—71no—
37#329—72no—
38C22×D7—22no—
39C7×Q8—42no—
40C2×C4×C7—22no—
41C2×C4×C7—41no—
42C2×C4×C7C2×D412yes—
43C2×C4×C7C2×D411yescharacteristic
44C2×Dic7—22no—
45D28—42no—
46D28—82no—
47C4×D7—82no—
48Dic14—42no—
49C7×C8—41no—
50C7⋊C8—42no—
51C4⋊Q8C2×D711yescharacteristic
52#125C2×D711yescharacteristic
53#124—72no—
54#123—142no—
55C22⋊Q8—142no—
56#121—142no—
57#103—22no—
58#103—142no—
59#102—142no—
60#97—72no—
61#97C2×D711yescharacteristic
62C22×C7#12011yescharacteristic
63C2×D7—42no—
64C2×D7—42no—
65C2×D7—42no—
66C4×C7—21no—
67C4×C7—22no—
68C4×C7—42no—
69C4×C7—81no—
70Dic7—42no—
71C2×Q8—21no—
72C2×Q8—72no—
73C2×D4—72no—
74C2×D4—142no—
75C22×C4—142no—
76C2×C8—21no—
77C2×C8—142no—
78C4⋊C4—22no—
79C4⋊C4—22no—
80C4⋊C4—142no—
81C4⋊C4C22⋊D712yes—
82C22⋊C4—142no—
83C22⋊C4—142no—
84C22⋊C4—142no—
85C42C22×D711yescharacteristic
86C2×C7#45411yescharacteristic
87C2×C7#45512yes—
88D7—82no—
89C23—142no—
90Q8—42no—
91Q8—282no—
92D4—282no—
93D4—562no—
94C2×C4—22no—
95C2×C4—41no—
96C2×C4—142no—
97C2×C4—562no—
98C2×C4D4×D712yes—
99C2×C4C2×C22⋊D711yescharacteristic
100C8—41no—
101C8—282no—
102C7#166211yescharacteristic, complemented, Sylow
103C22—282no—
104C22—282no—
105C22—282no—
106C22#650011yescharacteristic
107C4—22no—
108C4—21no—
109C4—42no—
110C4—81no—
111C4—282no—
112C2—562no—
113C2#8983411yescharacteristic
114C2#8997412yes—
subgroupG/H
centerC22#6500
commutator, cosocleC42×C7C23
FrattiniC42C22×D7
FittingC7×#491C2
socleC22×C7#120
prime radical#90391C2
hypercenter#491D7
hypocenterC7#1662
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. C7×#491C2semidirect
3. #89861C2semidirect
4. #89338C2semidirect
5. #89259C2semidirect
34. #491D7semidirect
102. C7#1662semidirect

Maximal subgroups

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

subgroupindex
#903912
C7×#4912
#898612
#893382
#892592
#16627