Finite Group Databasebeta

← #11726710

Subgroups — #11726710

Order 896.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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#90174C211yesmaximal, characteristic
2#89715C211yesmaximal, characteristic
3#89715C212yesmaximal
4C7×#342C211yesmaximal, characteristic
5#89322C212yesmaximal
6C23×Dic7C2211yescharacteristic
7C2×C7×C22⋊C4C2212yes—
8C2×C7×C22⋊C4C2211yescharacteristic
9C2×C22⋊Dic7C2211yescharacteristic
10C2×C22⋊Dic7C2212yes—
11#6487—22no—
12#6441—24no—
13C7⋊#95—22no—
14C7⋊#95—24no—
15#2030—71nomaximal, Sylow
16C24×C7C2311yescharacteristic
17C22×C4×C7—22no—
18C22×C4×C7D412yes—
19C22×Dic7—21no—
20C22×Dic7—22no—
21C22×Dic7—22no—
22C22×Dic7—21no—
23C22×Dic7—21no—
24C22×Dic7D414yes—
25C22×Dic7Q812yescomplemented
26C2×Dic14—22no—
27C7×C22⋊C4—22no—
28C7×C22⋊C4—22no—
29C7×C22⋊C4—22no—
30C22⋊Dic7—22no—
31C22⋊Dic7—22no—
32C22⋊Dic7—22no—
33C4⋊Dic7—22no—
34Dic7⋊C4—24no—
35#543—71no—
36#381—72no—
37#381—71no—
38#342—72no—
39#342D711yescharacteristic
40C23×C7—42no—
41C23×C7C2×D411yescharacteristic
42C23×C7C2×D412yes—
43C23×C7C2×Q811yescharacteristic
44C23×C7#4112yes—
45C23×C7#4111yescharacteristic
46C2×C4×C7—22no—
47C2×C4×C7—42no—
48C2×C4×C7—44no—
49C2×C4×C7—42no—
50C2×Dic7—24no—
51C2×Dic7—24no—
52C2×Dic7—22no—
53C2×Dic7—41no—
54C2×Dic7—41no—
55C2×Dic7—42no—
56C2×Dic7—44no—
57C2×Dic7—42no—
58C2×Dic7—41no—
59Dic14—82no—
60C23×C4—71no—
61C22⋊Q8—144no—
62C22⋊Q8—142no—
63C2×C22⋊C4—72no—
64C2×C22⋊C4—71no—
65C2×C22⋊C4C2×D712yes—
66C2×C22⋊C4C2×D711yescharacteristic
67#95—144no—
68#95—142no—
69C22×C7—22no—
70C22×C7—24no—
71C22×C7—41no—
72C22×C7—42no—
73C22×C7—41no—
74C22×C7#12011yescharacteristic
75C22×C7C22⋊Q812yes—
76C22×C7C22⋊Q811yescharacteristic
77C22×C7#12411yescharacteristic
78C22×C7#12412yes—
79C4×C7—42no—
80C4×C7—82no—
81Dic7—42no—
82Dic7—44no—
83Dic7—81no—
84C24C22×D711yescharacteristic
85C2×Q8—142no—
86C22×C4—22no—
87C22×C4—72no—
88C22×C4—74no—
89C22×C4—141no—
90C22×C4—142no—
91C22×C4—142no—
92C22×C4—141no—
93C22×C4—141no—
94C22×C4D2812yes—
95C4⋊C4—144no—
96C4⋊C4—142no—
97C22⋊C4—22no—
98C22⋊C4—22no—
99C22⋊C4—22no—
100C22⋊C4—142no—
101C22⋊C4—142no—
102C22⋊C4—142no—
103C2×C7—21no—
104C2×C7—22no—
105C2×C7—22no—
106C2×C7—21no—
107C2×C7#39311yescharacteristic
108C2×C7#45812yes—
109C23—42no—
110C23C2×D2811yescharacteristic
111C23#115712yes—
112C23D4×D712yes—
113C23#115911yescharacteristic
114C23Q8×D711yescharacteristic
115Q8—562no—
116C2×C4—22no—
117C2×C4—42no—
118C2×C4—44no—
119C2×C4—42no—
120C2×C4—144no—
121C2×C4—144no—
122C2×C4—142no—
123C2×C4—281no—
124C2×C4—281no—
125C2×C4—282no—
126C2×C4—284no—
127C2×C4—282no—
128C2×C4—281no—
129C7#203011yescharacteristic, complemented, Sylow
130C22—22no—
131C22—24no—
132C22—42no—
133C22—41no—
134C22—41no—
135C22#643811yescharacteristic
136C22#644511yescharacteristic
137C22#644812yes—
138C22#645912yes—
139C22#646011yescharacteristic
140C4—42no—
141C4—82no—
142C4—282no—
143C4—284no—
144C4—561no—
145C2—22no—
146C2—21no—
147C2—22no—
148C2—21no—
149C2#8944511yescharacteristic
150C2#8951412yes—
subgroupG/H
centerC22#6445
commutator, prime radical, cosocleC24×C7C23
FrattiniC24C22×D7
FittingC7×#342C2
socleC22×C7#120
hypercenter#342D7
hypocenterC7#2030
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
25. C22×Dic7Q8semidirect
129. C7#2030semidirect

Maximal subgroups

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

subgroupindex
#901742
#897152
C7×#3422
#893222
#20307